KILLED proof of input_WROFAnYAwk.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). (0) CpxTRS (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxTRS (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (4) CpxTRS (5) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxWeightedTrs (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CpxTypedWeightedTrs (9) CompletionProof [UPPER BOUND(ID), 0 ms] (10) CpxTypedWeightedCompleteTrs (11) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (12) CpxTypedWeightedCompleteTrs (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (14) CpxRNTS (15) InliningProof [UPPER BOUND(ID), 391 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), 0 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), 0 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), 0 ms] (48) CdtProblem (49) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (50) CdtProblem (51) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (52) CdtProblem (53) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (54) CdtProblem (55) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (56) CdtProblem (57) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (58) CdtProblem (59) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (60) CdtProblem (61) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (62) CdtProblem (63) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (64) CdtProblem (65) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (66) CdtProblem (67) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (68) CdtProblem (69) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (70) CdtProblem (71) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (72) CdtProblem (73) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (74) CdtProblem (75) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (76) CdtProblem (77) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (78) CdtProblem (79) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (80) CdtProblem (81) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (82) CdtProblem (83) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (84) CdtProblem (85) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (86) CdtProblem (87) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (88) CdtProblem (89) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (90) CdtProblem (91) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (92) CdtProblem (93) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (94) CdtProblem (95) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (96) CdtProblem (97) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (98) CdtProblem (99) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (100) CdtProblem (101) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (102) CdtProblem (103) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (104) CdtProblem (105) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (106) CdtProblem (107) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 4 ms] (108) CdtProblem (109) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (110) CdtProblem (111) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (112) CdtProblem (113) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (114) CdtProblem (115) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (116) CdtProblem (117) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (118) CdtProblem (119) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (120) CdtProblem (121) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (122) CdtProblem (123) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 37 ms] (124) CdtProblem (125) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (126) CdtProblem (127) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 110 ms] (128) CdtProblem (129) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (130) CdtProblem (131) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 129 ms] (132) CdtProblem (133) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (134) CdtProblem (135) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 199 ms] (136) CdtProblem (137) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (138) CdtProblem (139) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 199 ms] (140) CdtProblem (141) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (142) CdtProblem (143) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 274 ms] (144) CdtProblem (145) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (146) CdtProblem (147) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 254 ms] (148) CdtProblem (149) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (150) CdtProblem (151) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 357 ms] (152) CdtProblem (153) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (154) CdtProblem (155) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 342 ms] (156) CdtProblem (157) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (158) CdtProblem (159) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 5 ms] (160) CdtProblem (161) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (162) CdtProblem (163) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (164) CdtProblem (165) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (166) CdtProblem (167) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (168) CdtProblem (169) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms] (170) CdtProblem (171) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (172) CdtProblem (173) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (174) CdtProblem (175) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (176) CdtProblem (177) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (178) CdtProblem (179) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 91 ms] (180) CdtProblem (181) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (182) CdtProblem (183) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (184) CdtProblem (185) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 73 ms] (186) CdtProblem (187) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (188) CdtProblem (189) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 6 ms] (190) CdtProblem (191) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 443 ms] (192) CdtProblem (193) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 298 ms] (194) CdtProblem (195) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 271 ms] (196) CdtProblem (197) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 298 ms] (198) CdtProblem (199) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 829 ms] (200) 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__nats -> cons(0, incr(nats)) a__pairs -> cons(0, incr(odds)) a__odds -> a__incr(a__pairs) a__incr(cons(X, XS)) -> cons(s(mark(X)), incr(XS)) a__head(cons(X, XS)) -> mark(X) a__tail(cons(X, XS)) -> mark(XS) mark(nats) -> a__nats mark(incr(X)) -> a__incr(mark(X)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(head(X)) -> a__head(mark(X)) mark(tail(X)) -> a__tail(mark(X)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0) -> 0 mark(s(X)) -> s(mark(X)) a__nats -> nats a__incr(X) -> incr(X) a__pairs -> pairs a__odds -> odds 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__nats -> cons(0', incr(nats)) a__pairs -> cons(0', incr(odds)) a__odds -> a__incr(a__pairs) a__incr(cons(X, XS)) -> cons(s(mark(X)), incr(XS)) a__head(cons(X, XS)) -> mark(X) a__tail(cons(X, XS)) -> mark(XS) mark(nats) -> a__nats mark(incr(X)) -> a__incr(mark(X)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(head(X)) -> a__head(mark(X)) mark(tail(X)) -> a__tail(mark(X)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0') -> 0' mark(s(X)) -> s(mark(X)) a__nats -> nats a__incr(X) -> incr(X) a__pairs -> pairs a__odds -> odds 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__nats -> cons(0, incr(nats)) a__pairs -> cons(0, incr(odds)) a__odds -> a__incr(a__pairs) a__incr(cons(X, XS)) -> cons(s(mark(X)), incr(XS)) a__head(cons(X, XS)) -> mark(X) a__tail(cons(X, XS)) -> mark(XS) mark(nats) -> a__nats mark(incr(X)) -> a__incr(mark(X)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(head(X)) -> a__head(mark(X)) mark(tail(X)) -> a__tail(mark(X)) mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0) -> 0 mark(s(X)) -> s(mark(X)) a__nats -> nats a__incr(X) -> incr(X) a__pairs -> pairs a__odds -> odds 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__nats -> cons(0, incr(nats)) [1] a__pairs -> cons(0, incr(odds)) [1] a__odds -> a__incr(a__pairs) [1] a__incr(cons(X, XS)) -> cons(s(mark(X)), incr(XS)) [1] a__head(cons(X, XS)) -> mark(X) [1] a__tail(cons(X, XS)) -> mark(XS) [1] mark(nats) -> a__nats [1] mark(incr(X)) -> a__incr(mark(X)) [1] mark(pairs) -> a__pairs [1] mark(odds) -> a__odds [1] mark(head(X)) -> a__head(mark(X)) [1] mark(tail(X)) -> a__tail(mark(X)) [1] mark(cons(X1, X2)) -> cons(mark(X1), X2) [1] mark(0) -> 0 [1] mark(s(X)) -> s(mark(X)) [1] a__nats -> nats [1] a__incr(X) -> incr(X) [1] a__pairs -> pairs [1] a__odds -> odds [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__nats -> cons(0, incr(nats)) [1] a__pairs -> cons(0, incr(odds)) [1] a__odds -> a__incr(a__pairs) [1] a__incr(cons(X, XS)) -> cons(s(mark(X)), incr(XS)) [1] a__head(cons(X, XS)) -> mark(X) [1] a__tail(cons(X, XS)) -> mark(XS) [1] mark(nats) -> a__nats [1] mark(incr(X)) -> a__incr(mark(X)) [1] mark(pairs) -> a__pairs [1] mark(odds) -> a__odds [1] mark(head(X)) -> a__head(mark(X)) [1] mark(tail(X)) -> a__tail(mark(X)) [1] mark(cons(X1, X2)) -> cons(mark(X1), X2) [1] mark(0) -> 0 [1] mark(s(X)) -> s(mark(X)) [1] a__nats -> nats [1] a__incr(X) -> incr(X) [1] a__pairs -> pairs [1] a__odds -> odds [1] a__head(X) -> head(X) [1] a__tail(X) -> tail(X) [1] The TRS has the following type information: a__nats :: 0:nats:incr:cons:odds:s:pairs:head:tail cons :: 0:nats:incr:cons:odds:s:pairs:head:tail -> 0:nats:incr:cons:odds:s:pairs:head:tail -> 0:nats:incr:cons:odds:s:pairs:head:tail 0 :: 0:nats:incr:cons:odds:s:pairs:head:tail incr :: 0:nats:incr:cons:odds:s:pairs:head:tail -> 0:nats:incr:cons:odds:s:pairs:head:tail nats :: 0:nats:incr:cons:odds:s:pairs:head:tail a__pairs :: 0:nats:incr:cons:odds:s:pairs:head:tail odds :: 0:nats:incr:cons:odds:s:pairs:head:tail a__odds :: 0:nats:incr:cons:odds:s:pairs:head:tail a__incr :: 0:nats:incr:cons:odds:s:pairs:head:tail -> 0:nats:incr:cons:odds:s:pairs:head:tail s :: 0:nats:incr:cons:odds:s:pairs:head:tail -> 0:nats:incr:cons:odds:s:pairs:head:tail mark :: 0:nats:incr:cons:odds:s:pairs:head:tail -> 0:nats:incr:cons:odds:s:pairs:head:tail a__head :: 0:nats:incr:cons:odds:s:pairs:head:tail -> 0:nats:incr:cons:odds:s:pairs:head:tail a__tail :: 0:nats:incr:cons:odds:s:pairs:head:tail -> 0:nats:incr:cons:odds:s:pairs:head:tail pairs :: 0:nats:incr:cons:odds:s:pairs:head:tail head :: 0:nats:incr:cons:odds:s:pairs:head:tail -> 0:nats:incr:cons:odds:s:pairs:head:tail tail :: 0:nats:incr:cons:odds:s:pairs:head:tail -> 0:nats:incr:cons:odds:s:pairs: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__pairs a__tail_1 a__odds a__incr_1 a__nats a__head_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__nats -> cons(0, incr(nats)) [1] a__pairs -> cons(0, incr(odds)) [1] a__odds -> a__incr(a__pairs) [1] a__incr(cons(X, XS)) -> cons(s(mark(X)), incr(XS)) [1] a__head(cons(X, XS)) -> mark(X) [1] a__tail(cons(X, XS)) -> mark(XS) [1] mark(nats) -> a__nats [1] mark(incr(X)) -> a__incr(mark(X)) [1] mark(pairs) -> a__pairs [1] mark(odds) -> a__odds [1] mark(head(X)) -> a__head(mark(X)) [1] mark(tail(X)) -> a__tail(mark(X)) [1] mark(cons(X1, X2)) -> cons(mark(X1), X2) [1] mark(0) -> 0 [1] mark(s(X)) -> s(mark(X)) [1] a__nats -> nats [1] a__incr(X) -> incr(X) [1] a__pairs -> pairs [1] a__odds -> odds [1] a__head(X) -> head(X) [1] a__tail(X) -> tail(X) [1] The TRS has the following type information: a__nats :: 0:nats:incr:cons:odds:s:pairs:head:tail cons :: 0:nats:incr:cons:odds:s:pairs:head:tail -> 0:nats:incr:cons:odds:s:pairs:head:tail -> 0:nats:incr:cons:odds:s:pairs:head:tail 0 :: 0:nats:incr:cons:odds:s:pairs:head:tail incr :: 0:nats:incr:cons:odds:s:pairs:head:tail -> 0:nats:incr:cons:odds:s:pairs:head:tail nats :: 0:nats:incr:cons:odds:s:pairs:head:tail a__pairs :: 0:nats:incr:cons:odds:s:pairs:head:tail odds :: 0:nats:incr:cons:odds:s:pairs:head:tail a__odds :: 0:nats:incr:cons:odds:s:pairs:head:tail a__incr :: 0:nats:incr:cons:odds:s:pairs:head:tail -> 0:nats:incr:cons:odds:s:pairs:head:tail s :: 0:nats:incr:cons:odds:s:pairs:head:tail -> 0:nats:incr:cons:odds:s:pairs:head:tail mark :: 0:nats:incr:cons:odds:s:pairs:head:tail -> 0:nats:incr:cons:odds:s:pairs:head:tail a__head :: 0:nats:incr:cons:odds:s:pairs:head:tail -> 0:nats:incr:cons:odds:s:pairs:head:tail a__tail :: 0:nats:incr:cons:odds:s:pairs:head:tail -> 0:nats:incr:cons:odds:s:pairs:head:tail pairs :: 0:nats:incr:cons:odds:s:pairs:head:tail head :: 0:nats:incr:cons:odds:s:pairs:head:tail -> 0:nats:incr:cons:odds:s:pairs:head:tail tail :: 0:nats:incr:cons:odds:s:pairs:head:tail -> 0:nats:incr:cons:odds:s:pairs: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__nats -> cons(0, incr(nats)) [1] a__pairs -> cons(0, incr(odds)) [1] a__odds -> a__incr(cons(0, incr(odds))) [2] a__odds -> a__incr(pairs) [2] a__incr(cons(X, XS)) -> cons(s(mark(X)), incr(XS)) [1] a__head(cons(X, XS)) -> mark(X) [1] a__tail(cons(X, XS)) -> mark(XS) [1] mark(nats) -> a__nats [1] mark(incr(nats)) -> a__incr(a__nats) [2] mark(incr(incr(X'))) -> a__incr(a__incr(mark(X'))) [2] mark(incr(pairs)) -> a__incr(a__pairs) [2] mark(incr(odds)) -> a__incr(a__odds) [2] mark(incr(head(X''))) -> a__incr(a__head(mark(X''))) [2] mark(incr(tail(X3))) -> a__incr(a__tail(mark(X3))) [2] mark(incr(cons(X1', X2'))) -> a__incr(cons(mark(X1'), X2')) [2] mark(incr(0)) -> a__incr(0) [2] mark(incr(s(X4))) -> a__incr(s(mark(X4))) [2] mark(pairs) -> a__pairs [1] mark(odds) -> a__odds [1] mark(head(nats)) -> a__head(a__nats) [2] mark(head(incr(X5))) -> a__head(a__incr(mark(X5))) [2] mark(head(pairs)) -> a__head(a__pairs) [2] mark(head(odds)) -> a__head(a__odds) [2] mark(head(head(X6))) -> a__head(a__head(mark(X6))) [2] mark(head(tail(X7))) -> a__head(a__tail(mark(X7))) [2] mark(head(cons(X1'', X2''))) -> a__head(cons(mark(X1''), X2'')) [2] mark(head(0)) -> a__head(0) [2] mark(head(s(X8))) -> a__head(s(mark(X8))) [2] mark(tail(nats)) -> a__tail(a__nats) [2] mark(tail(incr(X9))) -> a__tail(a__incr(mark(X9))) [2] mark(tail(pairs)) -> a__tail(a__pairs) [2] mark(tail(odds)) -> a__tail(a__odds) [2] mark(tail(head(X10))) -> a__tail(a__head(mark(X10))) [2] mark(tail(tail(X11))) -> a__tail(a__tail(mark(X11))) [2] mark(tail(cons(X12, X21))) -> a__tail(cons(mark(X12), X21)) [2] mark(tail(0)) -> a__tail(0) [2] mark(tail(s(X13))) -> a__tail(s(mark(X13))) [2] mark(cons(X1, X2)) -> cons(mark(X1), X2) [1] mark(0) -> 0 [1] mark(s(X)) -> s(mark(X)) [1] a__nats -> nats [1] a__incr(X) -> incr(X) [1] a__pairs -> pairs [1] a__odds -> odds [1] a__head(X) -> head(X) [1] a__tail(X) -> tail(X) [1] The TRS has the following type information: a__nats :: 0:nats:incr:cons:odds:s:pairs:head:tail cons :: 0:nats:incr:cons:odds:s:pairs:head:tail -> 0:nats:incr:cons:odds:s:pairs:head:tail -> 0:nats:incr:cons:odds:s:pairs:head:tail 0 :: 0:nats:incr:cons:odds:s:pairs:head:tail incr :: 0:nats:incr:cons:odds:s:pairs:head:tail -> 0:nats:incr:cons:odds:s:pairs:head:tail nats :: 0:nats:incr:cons:odds:s:pairs:head:tail a__pairs :: 0:nats:incr:cons:odds:s:pairs:head:tail odds :: 0:nats:incr:cons:odds:s:pairs:head:tail a__odds :: 0:nats:incr:cons:odds:s:pairs:head:tail a__incr :: 0:nats:incr:cons:odds:s:pairs:head:tail -> 0:nats:incr:cons:odds:s:pairs:head:tail s :: 0:nats:incr:cons:odds:s:pairs:head:tail -> 0:nats:incr:cons:odds:s:pairs:head:tail mark :: 0:nats:incr:cons:odds:s:pairs:head:tail -> 0:nats:incr:cons:odds:s:pairs:head:tail a__head :: 0:nats:incr:cons:odds:s:pairs:head:tail -> 0:nats:incr:cons:odds:s:pairs:head:tail a__tail :: 0:nats:incr:cons:odds:s:pairs:head:tail -> 0:nats:incr:cons:odds:s:pairs:head:tail pairs :: 0:nats:incr:cons:odds:s:pairs:head:tail head :: 0:nats:incr:cons:odds:s:pairs:head:tail -> 0:nats:incr:cons:odds:s:pairs:head:tail tail :: 0:nats:incr:cons:odds:s:pairs:head:tail -> 0:nats:incr:cons:odds:s:pairs: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: 0 => 0 nats => 1 odds => 2 pairs => 3 ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: a__head(z) -{ 1 }-> mark(X) :|: z = 1 + X + XS, X >= 0, XS >= 0 a__head(z) -{ 1 }-> 1 + X :|: X >= 0, z = X a__incr(z) -{ 1 }-> 1 + X :|: X >= 0, z = X a__incr(z) -{ 1 }-> 1 + (1 + mark(X)) + (1 + XS) :|: z = 1 + X + XS, X >= 0, XS >= 0 a__nats -{ 1 }-> 1 :|: a__nats -{ 1 }-> 1 + 0 + (1 + 1) :|: a__odds -{ 2 }-> a__incr(3) :|: a__odds -{ 2 }-> a__incr(1 + 0 + (1 + 2)) :|: a__odds -{ 1 }-> 2 :|: a__pairs -{ 1 }-> 3 :|: a__pairs -{ 1 }-> 1 + 0 + (1 + 2) :|: a__tail(z) -{ 1 }-> mark(XS) :|: z = 1 + X + XS, X >= 0, XS >= 0 a__tail(z) -{ 1 }-> 1 + X :|: X >= 0, z = X mark(z) -{ 2 }-> a__tail(a__tail(mark(X11))) :|: X11 >= 0, z = 1 + (1 + X11) mark(z) -{ 2 }-> a__tail(a__pairs) :|: z = 1 + 3 mark(z) -{ 2 }-> a__tail(a__odds) :|: z = 1 + 2 mark(z) -{ 2 }-> a__tail(a__nats) :|: z = 1 + 1 mark(z) -{ 2 }-> a__tail(a__incr(mark(X9))) :|: X9 >= 0, z = 1 + (1 + X9) mark(z) -{ 2 }-> a__tail(a__head(mark(X10))) :|: z = 1 + (1 + X10), X10 >= 0 mark(z) -{ 2 }-> a__tail(0) :|: z = 1 + 0 mark(z) -{ 2 }-> a__tail(1 + mark(X13)) :|: z = 1 + (1 + X13), X13 >= 0 mark(z) -{ 2 }-> a__tail(1 + mark(X12) + X21) :|: X12 >= 0, z = 1 + (1 + X12 + X21), X21 >= 0 mark(z) -{ 1 }-> a__pairs :|: z = 3 mark(z) -{ 1 }-> a__odds :|: z = 2 mark(z) -{ 1 }-> a__nats :|: z = 1 mark(z) -{ 2 }-> a__incr(a__tail(mark(X3))) :|: z = 1 + (1 + X3), X3 >= 0 mark(z) -{ 2 }-> a__incr(a__pairs) :|: z = 1 + 3 mark(z) -{ 2 }-> a__incr(a__odds) :|: z = 1 + 2 mark(z) -{ 2 }-> a__incr(a__nats) :|: z = 1 + 1 mark(z) -{ 2 }-> a__incr(a__incr(mark(X'))) :|: X' >= 0, z = 1 + (1 + X') mark(z) -{ 2 }-> a__incr(a__head(mark(X''))) :|: z = 1 + (1 + X''), X'' >= 0 mark(z) -{ 2 }-> a__incr(0) :|: z = 1 + 0 mark(z) -{ 2 }-> a__incr(1 + mark(X4)) :|: z = 1 + (1 + X4), X4 >= 0 mark(z) -{ 2 }-> a__incr(1 + mark(X1') + X2') :|: z = 1 + (1 + X1' + X2'), X2' >= 0, X1' >= 0 mark(z) -{ 2 }-> a__head(a__tail(mark(X7))) :|: X7 >= 0, z = 1 + (1 + X7) mark(z) -{ 2 }-> a__head(a__pairs) :|: z = 1 + 3 mark(z) -{ 2 }-> a__head(a__odds) :|: z = 1 + 2 mark(z) -{ 2 }-> a__head(a__nats) :|: z = 1 + 1 mark(z) -{ 2 }-> a__head(a__incr(mark(X5))) :|: X5 >= 0, z = 1 + (1 + X5) mark(z) -{ 2 }-> a__head(a__head(mark(X6))) :|: X6 >= 0, z = 1 + (1 + X6) mark(z) -{ 2 }-> a__head(0) :|: z = 1 + 0 mark(z) -{ 2 }-> a__head(1 + mark(X8)) :|: X8 >= 0, z = 1 + (1 + X8) mark(z) -{ 2 }-> a__head(1 + mark(X1'') + X2'') :|: X1'' >= 0, z = 1 + (1 + X1'' + X2''), X2'' >= 0 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__nats -{ 1 }-> 1 + 0 + (1 + 1) :|: a__nats -{ 1 }-> 1 :|: a__pairs -{ 1 }-> 3 :|: a__pairs -{ 1 }-> 1 + 0 + (1 + 2) :|: ---------------------------------------- (16) Obligation: Complexity RNTS consisting of the following rules: a__head(z) -{ 1 }-> mark(X) :|: z = 1 + X + XS, X >= 0, XS >= 0 a__head(z) -{ 1 }-> 1 + X :|: X >= 0, z = X a__incr(z) -{ 1 }-> 1 + X :|: X >= 0, z = X a__incr(z) -{ 1 }-> 1 + (1 + mark(X)) + (1 + XS) :|: z = 1 + X + XS, X >= 0, XS >= 0 a__nats -{ 1 }-> 1 :|: a__nats -{ 1 }-> 1 + 0 + (1 + 1) :|: a__odds -{ 2 }-> a__incr(3) :|: a__odds -{ 2 }-> a__incr(1 + 0 + (1 + 2)) :|: a__odds -{ 1 }-> 2 :|: a__pairs -{ 1 }-> 3 :|: a__pairs -{ 1 }-> 1 + 0 + (1 + 2) :|: a__tail(z) -{ 1 }-> mark(XS) :|: z = 1 + X + XS, X >= 0, XS >= 0 a__tail(z) -{ 1 }-> 1 + X :|: X >= 0, z = X mark(z) -{ 2 }-> a__tail(a__tail(mark(X11))) :|: X11 >= 0, z = 1 + (1 + X11) mark(z) -{ 2 }-> a__tail(a__odds) :|: z = 1 + 2 mark(z) -{ 2 }-> a__tail(a__incr(mark(X9))) :|: X9 >= 0, z = 1 + (1 + X9) mark(z) -{ 2 }-> a__tail(a__head(mark(X10))) :|: z = 1 + (1 + X10), X10 >= 0 mark(z) -{ 3 }-> a__tail(3) :|: z = 1 + 3 mark(z) -{ 3 }-> a__tail(1) :|: z = 1 + 1 mark(z) -{ 2 }-> a__tail(0) :|: z = 1 + 0 mark(z) -{ 2 }-> a__tail(1 + mark(X13)) :|: z = 1 + (1 + X13), X13 >= 0 mark(z) -{ 2 }-> a__tail(1 + mark(X12) + X21) :|: X12 >= 0, z = 1 + (1 + X12 + X21), X21 >= 0 mark(z) -{ 3 }-> a__tail(1 + 0 + (1 + 2)) :|: z = 1 + 3 mark(z) -{ 3 }-> a__tail(1 + 0 + (1 + 1)) :|: z = 1 + 1 mark(z) -{ 1 }-> a__odds :|: z = 2 mark(z) -{ 2 }-> a__incr(a__tail(mark(X3))) :|: z = 1 + (1 + X3), X3 >= 0 mark(z) -{ 2 }-> a__incr(a__odds) :|: z = 1 + 2 mark(z) -{ 2 }-> a__incr(a__incr(mark(X'))) :|: X' >= 0, z = 1 + (1 + X') mark(z) -{ 2 }-> a__incr(a__head(mark(X''))) :|: z = 1 + (1 + X''), X'' >= 0 mark(z) -{ 3 }-> a__incr(3) :|: z = 1 + 3 mark(z) -{ 3 }-> a__incr(1) :|: z = 1 + 1 mark(z) -{ 2 }-> a__incr(0) :|: z = 1 + 0 mark(z) -{ 2 }-> a__incr(1 + mark(X4)) :|: z = 1 + (1 + X4), X4 >= 0 mark(z) -{ 2 }-> a__incr(1 + mark(X1') + X2') :|: z = 1 + (1 + X1' + X2'), X2' >= 0, X1' >= 0 mark(z) -{ 3 }-> a__incr(1 + 0 + (1 + 2)) :|: z = 1 + 3 mark(z) -{ 3 }-> a__incr(1 + 0 + (1 + 1)) :|: z = 1 + 1 mark(z) -{ 2 }-> a__head(a__tail(mark(X7))) :|: X7 >= 0, z = 1 + (1 + X7) mark(z) -{ 2 }-> a__head(a__odds) :|: z = 1 + 2 mark(z) -{ 2 }-> a__head(a__incr(mark(X5))) :|: X5 >= 0, z = 1 + (1 + X5) mark(z) -{ 2 }-> a__head(a__head(mark(X6))) :|: X6 >= 0, z = 1 + (1 + X6) mark(z) -{ 3 }-> a__head(3) :|: z = 1 + 3 mark(z) -{ 3 }-> a__head(1) :|: z = 1 + 1 mark(z) -{ 2 }-> a__head(0) :|: z = 1 + 0 mark(z) -{ 2 }-> a__head(1 + mark(X8)) :|: X8 >= 0, z = 1 + (1 + X8) mark(z) -{ 2 }-> a__head(1 + mark(X1'') + X2'') :|: X1'' >= 0, z = 1 + (1 + X1'' + X2''), X2'' >= 0 mark(z) -{ 3 }-> a__head(1 + 0 + (1 + 2)) :|: z = 1 + 3 mark(z) -{ 3 }-> a__head(1 + 0 + (1 + 1)) :|: z = 1 + 1 mark(z) -{ 2 }-> 3 :|: z = 3 mark(z) -{ 2 }-> 1 :|: z = 1 mark(z) -{ 1 }-> 0 :|: z = 0 mark(z) -{ 1 }-> 1 + mark(X) :|: z = 1 + X, X >= 0 mark(z) -{ 1 }-> 1 + mark(X1) + X2 :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2 mark(z) -{ 2 }-> 1 + 0 + (1 + 2) :|: z = 3 mark(z) -{ 2 }-> 1 + 0 + (1 + 1) :|: z = 1 ---------------------------------------- (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__head(z) -{ 1 }-> mark(X) :|: z = 1 + X + XS, X >= 0, XS >= 0 a__head(z) -{ 1 }-> 1 + z :|: z >= 0 a__incr(z) -{ 1 }-> 1 + z :|: z >= 0 a__incr(z) -{ 1 }-> 1 + (1 + mark(X)) + (1 + XS) :|: z = 1 + X + XS, X >= 0, XS >= 0 a__nats -{ 1 }-> 1 :|: a__nats -{ 1 }-> 1 + 0 + (1 + 1) :|: a__odds -{ 2 }-> a__incr(3) :|: a__odds -{ 2 }-> a__incr(1 + 0 + (1 + 2)) :|: a__odds -{ 1 }-> 2 :|: a__pairs -{ 1 }-> 3 :|: a__pairs -{ 1 }-> 1 + 0 + (1 + 2) :|: a__tail(z) -{ 1 }-> mark(XS) :|: z = 1 + X + XS, X >= 0, XS >= 0 a__tail(z) -{ 1 }-> 1 + z :|: z >= 0 mark(z) -{ 2 }-> a__tail(a__tail(mark(z - 2))) :|: z - 2 >= 0 mark(z) -{ 2 }-> a__tail(a__odds) :|: z = 1 + 2 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) -{ 3 }-> a__tail(3) :|: z = 1 + 3 mark(z) -{ 3 }-> a__tail(1) :|: z = 1 + 1 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(X12) + X21) :|: X12 >= 0, z = 1 + (1 + X12 + X21), X21 >= 0 mark(z) -{ 3 }-> a__tail(1 + 0 + (1 + 2)) :|: z = 1 + 3 mark(z) -{ 3 }-> a__tail(1 + 0 + (1 + 1)) :|: z = 1 + 1 mark(z) -{ 1 }-> a__odds :|: z = 2 mark(z) -{ 2 }-> a__incr(a__tail(mark(z - 2))) :|: z - 2 >= 0 mark(z) -{ 2 }-> a__incr(a__odds) :|: z = 1 + 2 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) -{ 3 }-> a__incr(3) :|: z = 1 + 3 mark(z) -{ 3 }-> a__incr(1) :|: z = 1 + 1 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') :|: z = 1 + (1 + X1' + X2'), X2' >= 0, X1' >= 0 mark(z) -{ 3 }-> a__incr(1 + 0 + (1 + 2)) :|: z = 1 + 3 mark(z) -{ 3 }-> a__incr(1 + 0 + (1 + 1)) :|: z = 1 + 1 mark(z) -{ 2 }-> a__head(a__tail(mark(z - 2))) :|: z - 2 >= 0 mark(z) -{ 2 }-> a__head(a__odds) :|: z = 1 + 2 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) -{ 3 }-> a__head(3) :|: z = 1 + 3 mark(z) -{ 3 }-> a__head(1) :|: z = 1 + 1 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(X1'') + X2'') :|: X1'' >= 0, z = 1 + (1 + X1'' + X2''), X2'' >= 0 mark(z) -{ 3 }-> a__head(1 + 0 + (1 + 2)) :|: z = 1 + 3 mark(z) -{ 3 }-> a__head(1 + 0 + (1 + 1)) :|: z = 1 + 1 mark(z) -{ 2 }-> 3 :|: z = 3 mark(z) -{ 2 }-> 1 :|: z = 1 mark(z) -{ 1 }-> 0 :|: z = 0 mark(z) -{ 1 }-> 1 + mark(z - 1) :|: z - 1 >= 0 mark(z) -{ 1 }-> 1 + mark(X1) + X2 :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2 mark(z) -{ 2 }-> 1 + 0 + (1 + 2) :|: z = 3 mark(z) -{ 2 }-> 1 + 0 + (1 + 1) :|: z = 1 ---------------------------------------- (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__nats -> cons(0, incr(nats)) [1] a__pairs -> cons(0, incr(odds)) [1] a__odds -> a__incr(a__pairs) [1] a__incr(cons(X, XS)) -> cons(s(mark(X)), incr(XS)) [1] a__head(cons(X, XS)) -> mark(X) [1] a__tail(cons(X, XS)) -> mark(XS) [1] mark(nats) -> a__nats [1] mark(incr(X)) -> a__incr(mark(X)) [1] mark(pairs) -> a__pairs [1] mark(odds) -> a__odds [1] mark(head(X)) -> a__head(mark(X)) [1] mark(tail(X)) -> a__tail(mark(X)) [1] mark(cons(X1, X2)) -> cons(mark(X1), X2) [1] mark(0) -> 0 [1] mark(s(X)) -> s(mark(X)) [1] a__nats -> nats [1] a__incr(X) -> incr(X) [1] a__pairs -> pairs [1] a__odds -> odds [1] a__head(X) -> head(X) [1] a__tail(X) -> tail(X) [1] The TRS has the following type information: a__nats :: 0:nats:incr:cons:odds:s:pairs:head:tail cons :: 0:nats:incr:cons:odds:s:pairs:head:tail -> 0:nats:incr:cons:odds:s:pairs:head:tail -> 0:nats:incr:cons:odds:s:pairs:head:tail 0 :: 0:nats:incr:cons:odds:s:pairs:head:tail incr :: 0:nats:incr:cons:odds:s:pairs:head:tail -> 0:nats:incr:cons:odds:s:pairs:head:tail nats :: 0:nats:incr:cons:odds:s:pairs:head:tail a__pairs :: 0:nats:incr:cons:odds:s:pairs:head:tail odds :: 0:nats:incr:cons:odds:s:pairs:head:tail a__odds :: 0:nats:incr:cons:odds:s:pairs:head:tail a__incr :: 0:nats:incr:cons:odds:s:pairs:head:tail -> 0:nats:incr:cons:odds:s:pairs:head:tail s :: 0:nats:incr:cons:odds:s:pairs:head:tail -> 0:nats:incr:cons:odds:s:pairs:head:tail mark :: 0:nats:incr:cons:odds:s:pairs:head:tail -> 0:nats:incr:cons:odds:s:pairs:head:tail a__head :: 0:nats:incr:cons:odds:s:pairs:head:tail -> 0:nats:incr:cons:odds:s:pairs:head:tail a__tail :: 0:nats:incr:cons:odds:s:pairs:head:tail -> 0:nats:incr:cons:odds:s:pairs:head:tail pairs :: 0:nats:incr:cons:odds:s:pairs:head:tail head :: 0:nats:incr:cons:odds:s:pairs:head:tail -> 0:nats:incr:cons:odds:s:pairs:head:tail tail :: 0:nats:incr:cons:odds:s:pairs:head:tail -> 0:nats:incr:cons:odds:s:pairs: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: 0 => 0 nats => 1 odds => 2 pairs => 3 ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: a__head(z) -{ 1 }-> mark(X) :|: z = 1 + X + XS, X >= 0, XS >= 0 a__head(z) -{ 1 }-> 1 + X :|: X >= 0, z = X a__incr(z) -{ 1 }-> 1 + X :|: X >= 0, z = X a__incr(z) -{ 1 }-> 1 + (1 + mark(X)) + (1 + XS) :|: z = 1 + X + XS, X >= 0, XS >= 0 a__nats -{ 1 }-> 1 :|: a__nats -{ 1 }-> 1 + 0 + (1 + 1) :|: a__odds -{ 1 }-> a__incr(a__pairs) :|: a__odds -{ 1 }-> 2 :|: a__pairs -{ 1 }-> 3 :|: a__pairs -{ 1 }-> 1 + 0 + (1 + 2) :|: a__tail(z) -{ 1 }-> mark(XS) :|: z = 1 + X + XS, X >= 0, XS >= 0 a__tail(z) -{ 1 }-> 1 + X :|: X >= 0, z = X mark(z) -{ 1 }-> a__tail(mark(X)) :|: z = 1 + X, X >= 0 mark(z) -{ 1 }-> a__pairs :|: z = 3 mark(z) -{ 1 }-> a__odds :|: z = 2 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 }-> 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__nats -> cons(0, incr(nats)) a__nats -> nats a__pairs -> cons(0, incr(odds)) a__pairs -> pairs a__odds -> a__incr(a__pairs) a__odds -> odds a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) 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(nats) -> a__nats mark(incr(z0)) -> a__incr(mark(z0)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__NATS -> c A__NATS -> c1 A__PAIRS -> c2 A__PAIRS -> c3 A__ODDS -> c4(A__INCR(a__pairs), A__PAIRS) A__ODDS -> c5 A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__INCR(z0) -> c7 A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__HEAD(z0) -> c9 A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) A__TAIL(z0) -> c11 MARK(nats) -> c12(A__NATS) MARK(incr(z0)) -> c13(A__INCR(mark(z0)), MARK(z0)) MARK(pairs) -> c14(A__PAIRS) MARK(odds) -> c15(A__ODDS) MARK(head(z0)) -> c16(A__HEAD(mark(z0)), MARK(z0)) MARK(tail(z0)) -> c17(A__TAIL(mark(z0)), MARK(z0)) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(0) -> c19 MARK(s(z0)) -> c20(MARK(z0)) S tuples: A__NATS -> c A__NATS -> c1 A__PAIRS -> c2 A__PAIRS -> c3 A__ODDS -> c4(A__INCR(a__pairs), A__PAIRS) A__ODDS -> c5 A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__INCR(z0) -> c7 A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__HEAD(z0) -> c9 A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) A__TAIL(z0) -> c11 MARK(nats) -> c12(A__NATS) MARK(incr(z0)) -> c13(A__INCR(mark(z0)), MARK(z0)) MARK(pairs) -> c14(A__PAIRS) MARK(odds) -> c15(A__ODDS) MARK(head(z0)) -> c16(A__HEAD(mark(z0)), MARK(z0)) MARK(tail(z0)) -> c17(A__TAIL(mark(z0)), MARK(z0)) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(0) -> c19 MARK(s(z0)) -> c20(MARK(z0)) K tuples:none Defined Rule Symbols: a__nats, a__pairs, a__odds, a__incr_1, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__NATS, A__PAIRS, A__ODDS, A__INCR_1, A__HEAD_1, A__TAIL_1, MARK_1 Compound Symbols: c, c1, c2, c3, c4_2, c5, c6_1, c7, c8_1, c9, c10_1, c11, c12_1, c13_2, c14_1, c15_1, c16_2, c17_2, c18_1, c19, c20_1 ---------------------------------------- (25) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 11 trailing nodes: A__ODDS -> c5 A__PAIRS -> c3 A__NATS -> c1 A__TAIL(z0) -> c11 A__NATS -> c MARK(pairs) -> c14(A__PAIRS) A__INCR(z0) -> c7 MARK(0) -> c19 A__HEAD(z0) -> c9 MARK(nats) -> c12(A__NATS) A__PAIRS -> c2 ---------------------------------------- (26) Obligation: Complexity Dependency Tuples Problem Rules: a__nats -> cons(0, incr(nats)) a__nats -> nats a__pairs -> cons(0, incr(odds)) a__pairs -> pairs a__odds -> a__incr(a__pairs) a__odds -> odds a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) 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(nats) -> a__nats mark(incr(z0)) -> a__incr(mark(z0)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__ODDS -> c4(A__INCR(a__pairs), A__PAIRS) A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(incr(z0)) -> c13(A__INCR(mark(z0)), MARK(z0)) MARK(odds) -> c15(A__ODDS) MARK(head(z0)) -> c16(A__HEAD(mark(z0)), MARK(z0)) MARK(tail(z0)) -> c17(A__TAIL(mark(z0)), MARK(z0)) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) S tuples: A__ODDS -> c4(A__INCR(a__pairs), A__PAIRS) A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(incr(z0)) -> c13(A__INCR(mark(z0)), MARK(z0)) MARK(odds) -> c15(A__ODDS) MARK(head(z0)) -> c16(A__HEAD(mark(z0)), MARK(z0)) MARK(tail(z0)) -> c17(A__TAIL(mark(z0)), MARK(z0)) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) K tuples:none Defined Rule Symbols: a__nats, a__pairs, a__odds, a__incr_1, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__ODDS, A__INCR_1, A__HEAD_1, A__TAIL_1, MARK_1 Compound Symbols: c4_2, c6_1, c8_1, c10_1, c13_2, c15_1, c16_2, c17_2, c18_1, c20_1 ---------------------------------------- (27) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (28) Obligation: Complexity Dependency Tuples Problem Rules: a__nats -> cons(0, incr(nats)) a__nats -> nats a__pairs -> cons(0, incr(odds)) a__pairs -> pairs a__odds -> a__incr(a__pairs) a__odds -> odds a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) 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(nats) -> a__nats mark(incr(z0)) -> a__incr(mark(z0)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(incr(z0)) -> c13(A__INCR(mark(z0)), MARK(z0)) MARK(odds) -> c15(A__ODDS) MARK(head(z0)) -> c16(A__HEAD(mark(z0)), MARK(z0)) MARK(tail(z0)) -> c17(A__TAIL(mark(z0)), MARK(z0)) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) A__ODDS -> c4(A__INCR(a__pairs)) S tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(incr(z0)) -> c13(A__INCR(mark(z0)), MARK(z0)) MARK(odds) -> c15(A__ODDS) MARK(head(z0)) -> c16(A__HEAD(mark(z0)), MARK(z0)) MARK(tail(z0)) -> c17(A__TAIL(mark(z0)), MARK(z0)) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) A__ODDS -> c4(A__INCR(a__pairs)) K tuples:none Defined Rule Symbols: a__nats, a__pairs, a__odds, a__incr_1, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__HEAD_1, A__TAIL_1, MARK_1, A__ODDS Compound Symbols: c6_1, c8_1, c10_1, c13_2, c15_1, c16_2, c17_2, c18_1, c20_1, c4_1 ---------------------------------------- (29) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(incr(z0)) -> c13(A__INCR(mark(z0)), MARK(z0)) by MARK(incr(nats)) -> c13(A__INCR(a__nats), MARK(nats)) MARK(incr(incr(z0))) -> c13(A__INCR(a__incr(mark(z0))), MARK(incr(z0))) MARK(incr(pairs)) -> c13(A__INCR(a__pairs), MARK(pairs)) MARK(incr(odds)) -> c13(A__INCR(a__odds), MARK(odds)) MARK(incr(head(z0))) -> c13(A__INCR(a__head(mark(z0))), MARK(head(z0))) MARK(incr(tail(z0))) -> c13(A__INCR(a__tail(mark(z0))), MARK(tail(z0))) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(0)) -> c13(A__INCR(0), MARK(0)) MARK(incr(s(z0))) -> c13(A__INCR(s(mark(z0))), MARK(s(z0))) ---------------------------------------- (30) Obligation: Complexity Dependency Tuples Problem Rules: a__nats -> cons(0, incr(nats)) a__nats -> nats a__pairs -> cons(0, incr(odds)) a__pairs -> pairs a__odds -> a__incr(a__pairs) a__odds -> odds a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) 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(nats) -> a__nats mark(incr(z0)) -> a__incr(mark(z0)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(head(z0)) -> c16(A__HEAD(mark(z0)), MARK(z0)) MARK(tail(z0)) -> c17(A__TAIL(mark(z0)), MARK(z0)) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) A__ODDS -> c4(A__INCR(a__pairs)) MARK(incr(nats)) -> c13(A__INCR(a__nats), MARK(nats)) MARK(incr(incr(z0))) -> c13(A__INCR(a__incr(mark(z0))), MARK(incr(z0))) MARK(incr(pairs)) -> c13(A__INCR(a__pairs), MARK(pairs)) MARK(incr(odds)) -> c13(A__INCR(a__odds), MARK(odds)) MARK(incr(head(z0))) -> c13(A__INCR(a__head(mark(z0))), MARK(head(z0))) MARK(incr(tail(z0))) -> c13(A__INCR(a__tail(mark(z0))), MARK(tail(z0))) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(0)) -> c13(A__INCR(0), MARK(0)) MARK(incr(s(z0))) -> c13(A__INCR(s(mark(z0))), MARK(s(z0))) S tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(head(z0)) -> c16(A__HEAD(mark(z0)), MARK(z0)) MARK(tail(z0)) -> c17(A__TAIL(mark(z0)), MARK(z0)) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) A__ODDS -> c4(A__INCR(a__pairs)) MARK(incr(nats)) -> c13(A__INCR(a__nats), MARK(nats)) MARK(incr(incr(z0))) -> c13(A__INCR(a__incr(mark(z0))), MARK(incr(z0))) MARK(incr(pairs)) -> c13(A__INCR(a__pairs), MARK(pairs)) MARK(incr(odds)) -> c13(A__INCR(a__odds), MARK(odds)) MARK(incr(head(z0))) -> c13(A__INCR(a__head(mark(z0))), MARK(head(z0))) MARK(incr(tail(z0))) -> c13(A__INCR(a__tail(mark(z0))), MARK(tail(z0))) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(0)) -> c13(A__INCR(0), MARK(0)) MARK(incr(s(z0))) -> c13(A__INCR(s(mark(z0))), MARK(s(z0))) K tuples:none Defined Rule Symbols: a__nats, a__pairs, a__odds, a__incr_1, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__HEAD_1, A__TAIL_1, MARK_1, A__ODDS Compound Symbols: c6_1, c8_1, c10_1, c15_1, c16_2, c17_2, c18_1, c20_1, c4_1, c13_2 ---------------------------------------- (31) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(incr(0)) -> c13(A__INCR(0), MARK(0)) ---------------------------------------- (32) Obligation: Complexity Dependency Tuples Problem Rules: a__nats -> cons(0, incr(nats)) a__nats -> nats a__pairs -> cons(0, incr(odds)) a__pairs -> pairs a__odds -> a__incr(a__pairs) a__odds -> odds a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) 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(nats) -> a__nats mark(incr(z0)) -> a__incr(mark(z0)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(head(z0)) -> c16(A__HEAD(mark(z0)), MARK(z0)) MARK(tail(z0)) -> c17(A__TAIL(mark(z0)), MARK(z0)) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) A__ODDS -> c4(A__INCR(a__pairs)) MARK(incr(nats)) -> c13(A__INCR(a__nats), MARK(nats)) MARK(incr(incr(z0))) -> c13(A__INCR(a__incr(mark(z0))), MARK(incr(z0))) MARK(incr(pairs)) -> c13(A__INCR(a__pairs), MARK(pairs)) MARK(incr(odds)) -> c13(A__INCR(a__odds), MARK(odds)) MARK(incr(head(z0))) -> c13(A__INCR(a__head(mark(z0))), MARK(head(z0))) MARK(incr(tail(z0))) -> c13(A__INCR(a__tail(mark(z0))), MARK(tail(z0))) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c13(A__INCR(s(mark(z0))), MARK(s(z0))) S tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(head(z0)) -> c16(A__HEAD(mark(z0)), MARK(z0)) MARK(tail(z0)) -> c17(A__TAIL(mark(z0)), MARK(z0)) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) A__ODDS -> c4(A__INCR(a__pairs)) MARK(incr(nats)) -> c13(A__INCR(a__nats), MARK(nats)) MARK(incr(incr(z0))) -> c13(A__INCR(a__incr(mark(z0))), MARK(incr(z0))) MARK(incr(pairs)) -> c13(A__INCR(a__pairs), MARK(pairs)) MARK(incr(odds)) -> c13(A__INCR(a__odds), MARK(odds)) MARK(incr(head(z0))) -> c13(A__INCR(a__head(mark(z0))), MARK(head(z0))) MARK(incr(tail(z0))) -> c13(A__INCR(a__tail(mark(z0))), MARK(tail(z0))) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c13(A__INCR(s(mark(z0))), MARK(s(z0))) K tuples:none Defined Rule Symbols: a__nats, a__pairs, a__odds, a__incr_1, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__HEAD_1, A__TAIL_1, MARK_1, A__ODDS Compound Symbols: c6_1, c8_1, c10_1, c15_1, c16_2, c17_2, c18_1, c20_1, c4_1, c13_2 ---------------------------------------- (33) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (34) Obligation: Complexity Dependency Tuples Problem Rules: a__nats -> cons(0, incr(nats)) a__nats -> nats a__pairs -> cons(0, incr(odds)) a__pairs -> pairs a__odds -> a__incr(a__pairs) a__odds -> odds a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) 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(nats) -> a__nats mark(incr(z0)) -> a__incr(mark(z0)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(head(z0)) -> c16(A__HEAD(mark(z0)), MARK(z0)) MARK(tail(z0)) -> c17(A__TAIL(mark(z0)), MARK(z0)) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) A__ODDS -> c4(A__INCR(a__pairs)) MARK(incr(incr(z0))) -> c13(A__INCR(a__incr(mark(z0))), MARK(incr(z0))) MARK(incr(odds)) -> c13(A__INCR(a__odds), MARK(odds)) MARK(incr(head(z0))) -> c13(A__INCR(a__head(mark(z0))), MARK(head(z0))) MARK(incr(tail(z0))) -> c13(A__INCR(a__tail(mark(z0))), MARK(tail(z0))) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(nats)) -> c13(A__INCR(a__nats)) MARK(incr(pairs)) -> c13(A__INCR(a__pairs)) MARK(incr(s(z0))) -> c13(MARK(s(z0))) S tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(head(z0)) -> c16(A__HEAD(mark(z0)), MARK(z0)) MARK(tail(z0)) -> c17(A__TAIL(mark(z0)), MARK(z0)) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) A__ODDS -> c4(A__INCR(a__pairs)) MARK(incr(incr(z0))) -> c13(A__INCR(a__incr(mark(z0))), MARK(incr(z0))) MARK(incr(odds)) -> c13(A__INCR(a__odds), MARK(odds)) MARK(incr(head(z0))) -> c13(A__INCR(a__head(mark(z0))), MARK(head(z0))) MARK(incr(tail(z0))) -> c13(A__INCR(a__tail(mark(z0))), MARK(tail(z0))) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(nats)) -> c13(A__INCR(a__nats)) MARK(incr(pairs)) -> c13(A__INCR(a__pairs)) MARK(incr(s(z0))) -> c13(MARK(s(z0))) K tuples:none Defined Rule Symbols: a__nats, a__pairs, a__odds, a__incr_1, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__HEAD_1, A__TAIL_1, MARK_1, A__ODDS Compound Symbols: c6_1, c8_1, c10_1, c15_1, c16_2, c17_2, c18_1, c20_1, c4_1, c13_2, c13_1 ---------------------------------------- (35) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(head(z0)) -> c16(A__HEAD(mark(z0)), MARK(z0)) by MARK(head(nats)) -> c16(A__HEAD(a__nats), MARK(nats)) MARK(head(incr(z0))) -> c16(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(pairs)) -> c16(A__HEAD(a__pairs), MARK(pairs)) MARK(head(odds)) -> c16(A__HEAD(a__odds), MARK(odds)) MARK(head(head(z0))) -> c16(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c16(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(0)) -> c16(A__HEAD(0), MARK(0)) MARK(head(s(z0))) -> c16(A__HEAD(s(mark(z0))), MARK(s(z0))) ---------------------------------------- (36) Obligation: Complexity Dependency Tuples Problem Rules: a__nats -> cons(0, incr(nats)) a__nats -> nats a__pairs -> cons(0, incr(odds)) a__pairs -> pairs a__odds -> a__incr(a__pairs) a__odds -> odds a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) 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(nats) -> a__nats mark(incr(z0)) -> a__incr(mark(z0)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(tail(z0)) -> c17(A__TAIL(mark(z0)), MARK(z0)) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) A__ODDS -> c4(A__INCR(a__pairs)) MARK(incr(incr(z0))) -> c13(A__INCR(a__incr(mark(z0))), MARK(incr(z0))) MARK(incr(odds)) -> c13(A__INCR(a__odds), MARK(odds)) MARK(incr(head(z0))) -> c13(A__INCR(a__head(mark(z0))), MARK(head(z0))) MARK(incr(tail(z0))) -> c13(A__INCR(a__tail(mark(z0))), MARK(tail(z0))) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(nats)) -> c13(A__INCR(a__nats)) MARK(incr(pairs)) -> c13(A__INCR(a__pairs)) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(nats)) -> c16(A__HEAD(a__nats), MARK(nats)) MARK(head(incr(z0))) -> c16(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(pairs)) -> c16(A__HEAD(a__pairs), MARK(pairs)) MARK(head(odds)) -> c16(A__HEAD(a__odds), MARK(odds)) MARK(head(head(z0))) -> c16(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c16(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(0)) -> c16(A__HEAD(0), MARK(0)) MARK(head(s(z0))) -> c16(A__HEAD(s(mark(z0))), MARK(s(z0))) S tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(tail(z0)) -> c17(A__TAIL(mark(z0)), MARK(z0)) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) A__ODDS -> c4(A__INCR(a__pairs)) MARK(incr(incr(z0))) -> c13(A__INCR(a__incr(mark(z0))), MARK(incr(z0))) MARK(incr(odds)) -> c13(A__INCR(a__odds), MARK(odds)) MARK(incr(head(z0))) -> c13(A__INCR(a__head(mark(z0))), MARK(head(z0))) MARK(incr(tail(z0))) -> c13(A__INCR(a__tail(mark(z0))), MARK(tail(z0))) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(nats)) -> c13(A__INCR(a__nats)) MARK(incr(pairs)) -> c13(A__INCR(a__pairs)) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(nats)) -> c16(A__HEAD(a__nats), MARK(nats)) MARK(head(incr(z0))) -> c16(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(pairs)) -> c16(A__HEAD(a__pairs), MARK(pairs)) MARK(head(odds)) -> c16(A__HEAD(a__odds), MARK(odds)) MARK(head(head(z0))) -> c16(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c16(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(0)) -> c16(A__HEAD(0), MARK(0)) MARK(head(s(z0))) -> c16(A__HEAD(s(mark(z0))), MARK(s(z0))) K tuples:none Defined Rule Symbols: a__nats, a__pairs, a__odds, a__incr_1, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__HEAD_1, A__TAIL_1, MARK_1, A__ODDS Compound Symbols: c6_1, c8_1, c10_1, c15_1, c17_2, c18_1, c20_1, c4_1, c13_2, c13_1, c16_2 ---------------------------------------- (37) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(head(0)) -> c16(A__HEAD(0), MARK(0)) ---------------------------------------- (38) Obligation: Complexity Dependency Tuples Problem Rules: a__nats -> cons(0, incr(nats)) a__nats -> nats a__pairs -> cons(0, incr(odds)) a__pairs -> pairs a__odds -> a__incr(a__pairs) a__odds -> odds a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) 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(nats) -> a__nats mark(incr(z0)) -> a__incr(mark(z0)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(tail(z0)) -> c17(A__TAIL(mark(z0)), MARK(z0)) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) A__ODDS -> c4(A__INCR(a__pairs)) MARK(incr(incr(z0))) -> c13(A__INCR(a__incr(mark(z0))), MARK(incr(z0))) MARK(incr(odds)) -> c13(A__INCR(a__odds), MARK(odds)) MARK(incr(head(z0))) -> c13(A__INCR(a__head(mark(z0))), MARK(head(z0))) MARK(incr(tail(z0))) -> c13(A__INCR(a__tail(mark(z0))), MARK(tail(z0))) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(nats)) -> c13(A__INCR(a__nats)) MARK(incr(pairs)) -> c13(A__INCR(a__pairs)) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(nats)) -> c16(A__HEAD(a__nats), MARK(nats)) MARK(head(incr(z0))) -> c16(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(pairs)) -> c16(A__HEAD(a__pairs), MARK(pairs)) MARK(head(odds)) -> c16(A__HEAD(a__odds), MARK(odds)) MARK(head(head(z0))) -> c16(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c16(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c16(A__HEAD(s(mark(z0))), MARK(s(z0))) S tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(tail(z0)) -> c17(A__TAIL(mark(z0)), MARK(z0)) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) A__ODDS -> c4(A__INCR(a__pairs)) MARK(incr(incr(z0))) -> c13(A__INCR(a__incr(mark(z0))), MARK(incr(z0))) MARK(incr(odds)) -> c13(A__INCR(a__odds), MARK(odds)) MARK(incr(head(z0))) -> c13(A__INCR(a__head(mark(z0))), MARK(head(z0))) MARK(incr(tail(z0))) -> c13(A__INCR(a__tail(mark(z0))), MARK(tail(z0))) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(nats)) -> c13(A__INCR(a__nats)) MARK(incr(pairs)) -> c13(A__INCR(a__pairs)) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(nats)) -> c16(A__HEAD(a__nats), MARK(nats)) MARK(head(incr(z0))) -> c16(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(pairs)) -> c16(A__HEAD(a__pairs), MARK(pairs)) MARK(head(odds)) -> c16(A__HEAD(a__odds), MARK(odds)) MARK(head(head(z0))) -> c16(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c16(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c16(A__HEAD(s(mark(z0))), MARK(s(z0))) K tuples:none Defined Rule Symbols: a__nats, a__pairs, a__odds, a__incr_1, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__HEAD_1, A__TAIL_1, MARK_1, A__ODDS Compound Symbols: c6_1, c8_1, c10_1, c15_1, c17_2, c18_1, c20_1, c4_1, c13_2, c13_1, c16_2 ---------------------------------------- (39) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (40) Obligation: Complexity Dependency Tuples Problem Rules: a__nats -> cons(0, incr(nats)) a__nats -> nats a__pairs -> cons(0, incr(odds)) a__pairs -> pairs a__odds -> a__incr(a__pairs) a__odds -> odds a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) 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(nats) -> a__nats mark(incr(z0)) -> a__incr(mark(z0)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(tail(z0)) -> c17(A__TAIL(mark(z0)), MARK(z0)) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) A__ODDS -> c4(A__INCR(a__pairs)) MARK(incr(incr(z0))) -> c13(A__INCR(a__incr(mark(z0))), MARK(incr(z0))) MARK(incr(odds)) -> c13(A__INCR(a__odds), MARK(odds)) MARK(incr(head(z0))) -> c13(A__INCR(a__head(mark(z0))), MARK(head(z0))) MARK(incr(tail(z0))) -> c13(A__INCR(a__tail(mark(z0))), MARK(tail(z0))) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(nats)) -> c13(A__INCR(a__nats)) MARK(incr(pairs)) -> c13(A__INCR(a__pairs)) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(incr(z0))) -> c16(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(odds)) -> c16(A__HEAD(a__odds), MARK(odds)) MARK(head(head(z0))) -> c16(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c16(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(nats)) -> c16(A__HEAD(a__nats)) MARK(head(pairs)) -> c16(A__HEAD(a__pairs)) MARK(head(s(z0))) -> c16(MARK(s(z0))) S tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(tail(z0)) -> c17(A__TAIL(mark(z0)), MARK(z0)) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) A__ODDS -> c4(A__INCR(a__pairs)) MARK(incr(incr(z0))) -> c13(A__INCR(a__incr(mark(z0))), MARK(incr(z0))) MARK(incr(odds)) -> c13(A__INCR(a__odds), MARK(odds)) MARK(incr(head(z0))) -> c13(A__INCR(a__head(mark(z0))), MARK(head(z0))) MARK(incr(tail(z0))) -> c13(A__INCR(a__tail(mark(z0))), MARK(tail(z0))) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(nats)) -> c13(A__INCR(a__nats)) MARK(incr(pairs)) -> c13(A__INCR(a__pairs)) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(incr(z0))) -> c16(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(odds)) -> c16(A__HEAD(a__odds), MARK(odds)) MARK(head(head(z0))) -> c16(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c16(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(nats)) -> c16(A__HEAD(a__nats)) MARK(head(pairs)) -> c16(A__HEAD(a__pairs)) MARK(head(s(z0))) -> c16(MARK(s(z0))) K tuples:none Defined Rule Symbols: a__nats, a__pairs, a__odds, a__incr_1, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__HEAD_1, A__TAIL_1, MARK_1, A__ODDS Compound Symbols: c6_1, c8_1, c10_1, c15_1, c17_2, c18_1, c20_1, c4_1, c13_2, c13_1, c16_2, c16_1 ---------------------------------------- (41) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(tail(z0)) -> c17(A__TAIL(mark(z0)), MARK(z0)) by MARK(tail(nats)) -> c17(A__TAIL(a__nats), MARK(nats)) MARK(tail(incr(z0))) -> c17(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(pairs)) -> c17(A__TAIL(a__pairs), MARK(pairs)) MARK(tail(odds)) -> c17(A__TAIL(a__odds), MARK(odds)) MARK(tail(head(z0))) -> c17(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c17(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(0)) -> c17(A__TAIL(0), MARK(0)) MARK(tail(s(z0))) -> c17(A__TAIL(s(mark(z0))), MARK(s(z0))) ---------------------------------------- (42) Obligation: Complexity Dependency Tuples Problem Rules: a__nats -> cons(0, incr(nats)) a__nats -> nats a__pairs -> cons(0, incr(odds)) a__pairs -> pairs a__odds -> a__incr(a__pairs) a__odds -> odds a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) 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(nats) -> a__nats mark(incr(z0)) -> a__incr(mark(z0)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) A__ODDS -> c4(A__INCR(a__pairs)) MARK(incr(incr(z0))) -> c13(A__INCR(a__incr(mark(z0))), MARK(incr(z0))) MARK(incr(odds)) -> c13(A__INCR(a__odds), MARK(odds)) MARK(incr(head(z0))) -> c13(A__INCR(a__head(mark(z0))), MARK(head(z0))) MARK(incr(tail(z0))) -> c13(A__INCR(a__tail(mark(z0))), MARK(tail(z0))) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(nats)) -> c13(A__INCR(a__nats)) MARK(incr(pairs)) -> c13(A__INCR(a__pairs)) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(incr(z0))) -> c16(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(odds)) -> c16(A__HEAD(a__odds), MARK(odds)) MARK(head(head(z0))) -> c16(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c16(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(nats)) -> c16(A__HEAD(a__nats)) MARK(head(pairs)) -> c16(A__HEAD(a__pairs)) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(nats)) -> c17(A__TAIL(a__nats), MARK(nats)) MARK(tail(incr(z0))) -> c17(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(pairs)) -> c17(A__TAIL(a__pairs), MARK(pairs)) MARK(tail(odds)) -> c17(A__TAIL(a__odds), MARK(odds)) MARK(tail(head(z0))) -> c17(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c17(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(0)) -> c17(A__TAIL(0), MARK(0)) MARK(tail(s(z0))) -> c17(A__TAIL(s(mark(z0))), MARK(s(z0))) S tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) A__ODDS -> c4(A__INCR(a__pairs)) MARK(incr(incr(z0))) -> c13(A__INCR(a__incr(mark(z0))), MARK(incr(z0))) MARK(incr(odds)) -> c13(A__INCR(a__odds), MARK(odds)) MARK(incr(head(z0))) -> c13(A__INCR(a__head(mark(z0))), MARK(head(z0))) MARK(incr(tail(z0))) -> c13(A__INCR(a__tail(mark(z0))), MARK(tail(z0))) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(nats)) -> c13(A__INCR(a__nats)) MARK(incr(pairs)) -> c13(A__INCR(a__pairs)) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(incr(z0))) -> c16(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(odds)) -> c16(A__HEAD(a__odds), MARK(odds)) MARK(head(head(z0))) -> c16(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c16(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(nats)) -> c16(A__HEAD(a__nats)) MARK(head(pairs)) -> c16(A__HEAD(a__pairs)) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(nats)) -> c17(A__TAIL(a__nats), MARK(nats)) MARK(tail(incr(z0))) -> c17(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(pairs)) -> c17(A__TAIL(a__pairs), MARK(pairs)) MARK(tail(odds)) -> c17(A__TAIL(a__odds), MARK(odds)) MARK(tail(head(z0))) -> c17(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c17(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(0)) -> c17(A__TAIL(0), MARK(0)) MARK(tail(s(z0))) -> c17(A__TAIL(s(mark(z0))), MARK(s(z0))) K tuples:none Defined Rule Symbols: a__nats, a__pairs, a__odds, a__incr_1, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__HEAD_1, A__TAIL_1, MARK_1, A__ODDS Compound Symbols: c6_1, c8_1, c10_1, c15_1, c18_1, c20_1, c4_1, c13_2, c13_1, c16_2, c16_1, c17_2 ---------------------------------------- (43) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(tail(0)) -> c17(A__TAIL(0), MARK(0)) ---------------------------------------- (44) Obligation: Complexity Dependency Tuples Problem Rules: a__nats -> cons(0, incr(nats)) a__nats -> nats a__pairs -> cons(0, incr(odds)) a__pairs -> pairs a__odds -> a__incr(a__pairs) a__odds -> odds a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) 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(nats) -> a__nats mark(incr(z0)) -> a__incr(mark(z0)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) A__ODDS -> c4(A__INCR(a__pairs)) MARK(incr(incr(z0))) -> c13(A__INCR(a__incr(mark(z0))), MARK(incr(z0))) MARK(incr(odds)) -> c13(A__INCR(a__odds), MARK(odds)) MARK(incr(head(z0))) -> c13(A__INCR(a__head(mark(z0))), MARK(head(z0))) MARK(incr(tail(z0))) -> c13(A__INCR(a__tail(mark(z0))), MARK(tail(z0))) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(nats)) -> c13(A__INCR(a__nats)) MARK(incr(pairs)) -> c13(A__INCR(a__pairs)) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(incr(z0))) -> c16(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(odds)) -> c16(A__HEAD(a__odds), MARK(odds)) MARK(head(head(z0))) -> c16(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c16(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(nats)) -> c16(A__HEAD(a__nats)) MARK(head(pairs)) -> c16(A__HEAD(a__pairs)) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(nats)) -> c17(A__TAIL(a__nats), MARK(nats)) MARK(tail(incr(z0))) -> c17(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(pairs)) -> c17(A__TAIL(a__pairs), MARK(pairs)) MARK(tail(odds)) -> c17(A__TAIL(a__odds), MARK(odds)) MARK(tail(head(z0))) -> c17(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c17(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c17(A__TAIL(s(mark(z0))), MARK(s(z0))) S tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) A__ODDS -> c4(A__INCR(a__pairs)) MARK(incr(incr(z0))) -> c13(A__INCR(a__incr(mark(z0))), MARK(incr(z0))) MARK(incr(odds)) -> c13(A__INCR(a__odds), MARK(odds)) MARK(incr(head(z0))) -> c13(A__INCR(a__head(mark(z0))), MARK(head(z0))) MARK(incr(tail(z0))) -> c13(A__INCR(a__tail(mark(z0))), MARK(tail(z0))) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(nats)) -> c13(A__INCR(a__nats)) MARK(incr(pairs)) -> c13(A__INCR(a__pairs)) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(incr(z0))) -> c16(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(odds)) -> c16(A__HEAD(a__odds), MARK(odds)) MARK(head(head(z0))) -> c16(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c16(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(nats)) -> c16(A__HEAD(a__nats)) MARK(head(pairs)) -> c16(A__HEAD(a__pairs)) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(nats)) -> c17(A__TAIL(a__nats), MARK(nats)) MARK(tail(incr(z0))) -> c17(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(pairs)) -> c17(A__TAIL(a__pairs), MARK(pairs)) MARK(tail(odds)) -> c17(A__TAIL(a__odds), MARK(odds)) MARK(tail(head(z0))) -> c17(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c17(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c17(A__TAIL(s(mark(z0))), MARK(s(z0))) K tuples:none Defined Rule Symbols: a__nats, a__pairs, a__odds, a__incr_1, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__HEAD_1, A__TAIL_1, MARK_1, A__ODDS Compound Symbols: c6_1, c8_1, c10_1, c15_1, c18_1, c20_1, c4_1, c13_2, c13_1, c16_2, c16_1, c17_2 ---------------------------------------- (45) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (46) Obligation: Complexity Dependency Tuples Problem Rules: a__nats -> cons(0, incr(nats)) a__nats -> nats a__pairs -> cons(0, incr(odds)) a__pairs -> pairs a__odds -> a__incr(a__pairs) a__odds -> odds a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) 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(nats) -> a__nats mark(incr(z0)) -> a__incr(mark(z0)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) A__ODDS -> c4(A__INCR(a__pairs)) MARK(incr(incr(z0))) -> c13(A__INCR(a__incr(mark(z0))), MARK(incr(z0))) MARK(incr(odds)) -> c13(A__INCR(a__odds), MARK(odds)) MARK(incr(head(z0))) -> c13(A__INCR(a__head(mark(z0))), MARK(head(z0))) MARK(incr(tail(z0))) -> c13(A__INCR(a__tail(mark(z0))), MARK(tail(z0))) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(nats)) -> c13(A__INCR(a__nats)) MARK(incr(pairs)) -> c13(A__INCR(a__pairs)) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(incr(z0))) -> c16(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(odds)) -> c16(A__HEAD(a__odds), MARK(odds)) MARK(head(head(z0))) -> c16(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c16(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(nats)) -> c16(A__HEAD(a__nats)) MARK(head(pairs)) -> c16(A__HEAD(a__pairs)) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(incr(z0))) -> c17(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(odds)) -> c17(A__TAIL(a__odds), MARK(odds)) MARK(tail(head(z0))) -> c17(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c17(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(nats)) -> c17(A__TAIL(a__nats)) MARK(tail(pairs)) -> c17(A__TAIL(a__pairs)) MARK(tail(s(z0))) -> c17(MARK(s(z0))) S tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) A__ODDS -> c4(A__INCR(a__pairs)) MARK(incr(incr(z0))) -> c13(A__INCR(a__incr(mark(z0))), MARK(incr(z0))) MARK(incr(odds)) -> c13(A__INCR(a__odds), MARK(odds)) MARK(incr(head(z0))) -> c13(A__INCR(a__head(mark(z0))), MARK(head(z0))) MARK(incr(tail(z0))) -> c13(A__INCR(a__tail(mark(z0))), MARK(tail(z0))) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(nats)) -> c13(A__INCR(a__nats)) MARK(incr(pairs)) -> c13(A__INCR(a__pairs)) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(incr(z0))) -> c16(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(odds)) -> c16(A__HEAD(a__odds), MARK(odds)) MARK(head(head(z0))) -> c16(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c16(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(nats)) -> c16(A__HEAD(a__nats)) MARK(head(pairs)) -> c16(A__HEAD(a__pairs)) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(incr(z0))) -> c17(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(odds)) -> c17(A__TAIL(a__odds), MARK(odds)) MARK(tail(head(z0))) -> c17(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c17(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(nats)) -> c17(A__TAIL(a__nats)) MARK(tail(pairs)) -> c17(A__TAIL(a__pairs)) MARK(tail(s(z0))) -> c17(MARK(s(z0))) K tuples:none Defined Rule Symbols: a__nats, a__pairs, a__odds, a__incr_1, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__HEAD_1, A__TAIL_1, MARK_1, A__ODDS Compound Symbols: c6_1, c8_1, c10_1, c15_1, c18_1, c20_1, c4_1, c13_2, c13_1, c16_2, c16_1, c17_2, c17_1 ---------------------------------------- (47) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace A__ODDS -> c4(A__INCR(a__pairs)) by A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) A__ODDS -> c4(A__INCR(pairs)) ---------------------------------------- (48) Obligation: Complexity Dependency Tuples Problem Rules: a__nats -> cons(0, incr(nats)) a__nats -> nats a__pairs -> cons(0, incr(odds)) a__pairs -> pairs a__odds -> a__incr(a__pairs) a__odds -> odds a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) 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(nats) -> a__nats mark(incr(z0)) -> a__incr(mark(z0)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(incr(z0))) -> c13(A__INCR(a__incr(mark(z0))), MARK(incr(z0))) MARK(incr(odds)) -> c13(A__INCR(a__odds), MARK(odds)) MARK(incr(head(z0))) -> c13(A__INCR(a__head(mark(z0))), MARK(head(z0))) MARK(incr(tail(z0))) -> c13(A__INCR(a__tail(mark(z0))), MARK(tail(z0))) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(nats)) -> c13(A__INCR(a__nats)) MARK(incr(pairs)) -> c13(A__INCR(a__pairs)) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(incr(z0))) -> c16(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(odds)) -> c16(A__HEAD(a__odds), MARK(odds)) MARK(head(head(z0))) -> c16(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c16(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(nats)) -> c16(A__HEAD(a__nats)) MARK(head(pairs)) -> c16(A__HEAD(a__pairs)) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(incr(z0))) -> c17(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(odds)) -> c17(A__TAIL(a__odds), MARK(odds)) MARK(tail(head(z0))) -> c17(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c17(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(nats)) -> c17(A__TAIL(a__nats)) MARK(tail(pairs)) -> c17(A__TAIL(a__pairs)) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) A__ODDS -> c4(A__INCR(pairs)) S tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(incr(z0))) -> c13(A__INCR(a__incr(mark(z0))), MARK(incr(z0))) MARK(incr(odds)) -> c13(A__INCR(a__odds), MARK(odds)) MARK(incr(head(z0))) -> c13(A__INCR(a__head(mark(z0))), MARK(head(z0))) MARK(incr(tail(z0))) -> c13(A__INCR(a__tail(mark(z0))), MARK(tail(z0))) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(nats)) -> c13(A__INCR(a__nats)) MARK(incr(pairs)) -> c13(A__INCR(a__pairs)) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(incr(z0))) -> c16(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(odds)) -> c16(A__HEAD(a__odds), MARK(odds)) MARK(head(head(z0))) -> c16(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c16(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(nats)) -> c16(A__HEAD(a__nats)) MARK(head(pairs)) -> c16(A__HEAD(a__pairs)) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(incr(z0))) -> c17(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(odds)) -> c17(A__TAIL(a__odds), MARK(odds)) MARK(tail(head(z0))) -> c17(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c17(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(nats)) -> c17(A__TAIL(a__nats)) MARK(tail(pairs)) -> c17(A__TAIL(a__pairs)) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) A__ODDS -> c4(A__INCR(pairs)) K tuples:none Defined Rule Symbols: a__nats, a__pairs, a__odds, a__incr_1, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__HEAD_1, A__TAIL_1, MARK_1, A__ODDS Compound Symbols: c6_1, c8_1, c10_1, c15_1, c18_1, c20_1, c13_2, c13_1, c16_2, c16_1, c17_2, c17_1, c4_1 ---------------------------------------- (49) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: A__ODDS -> c4(A__INCR(pairs)) ---------------------------------------- (50) Obligation: Complexity Dependency Tuples Problem Rules: a__nats -> cons(0, incr(nats)) a__nats -> nats a__pairs -> cons(0, incr(odds)) a__pairs -> pairs a__odds -> a__incr(a__pairs) a__odds -> odds a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) 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(nats) -> a__nats mark(incr(z0)) -> a__incr(mark(z0)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(incr(z0))) -> c13(A__INCR(a__incr(mark(z0))), MARK(incr(z0))) MARK(incr(odds)) -> c13(A__INCR(a__odds), MARK(odds)) MARK(incr(head(z0))) -> c13(A__INCR(a__head(mark(z0))), MARK(head(z0))) MARK(incr(tail(z0))) -> c13(A__INCR(a__tail(mark(z0))), MARK(tail(z0))) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(nats)) -> c13(A__INCR(a__nats)) MARK(incr(pairs)) -> c13(A__INCR(a__pairs)) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(incr(z0))) -> c16(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(odds)) -> c16(A__HEAD(a__odds), MARK(odds)) MARK(head(head(z0))) -> c16(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c16(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(nats)) -> c16(A__HEAD(a__nats)) MARK(head(pairs)) -> c16(A__HEAD(a__pairs)) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(incr(z0))) -> c17(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(odds)) -> c17(A__TAIL(a__odds), MARK(odds)) MARK(tail(head(z0))) -> c17(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c17(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(nats)) -> c17(A__TAIL(a__nats)) MARK(tail(pairs)) -> c17(A__TAIL(a__pairs)) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) S tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(incr(z0))) -> c13(A__INCR(a__incr(mark(z0))), MARK(incr(z0))) MARK(incr(odds)) -> c13(A__INCR(a__odds), MARK(odds)) MARK(incr(head(z0))) -> c13(A__INCR(a__head(mark(z0))), MARK(head(z0))) MARK(incr(tail(z0))) -> c13(A__INCR(a__tail(mark(z0))), MARK(tail(z0))) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(nats)) -> c13(A__INCR(a__nats)) MARK(incr(pairs)) -> c13(A__INCR(a__pairs)) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(incr(z0))) -> c16(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(odds)) -> c16(A__HEAD(a__odds), MARK(odds)) MARK(head(head(z0))) -> c16(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c16(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(nats)) -> c16(A__HEAD(a__nats)) MARK(head(pairs)) -> c16(A__HEAD(a__pairs)) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(incr(z0))) -> c17(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(odds)) -> c17(A__TAIL(a__odds), MARK(odds)) MARK(tail(head(z0))) -> c17(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c17(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(nats)) -> c17(A__TAIL(a__nats)) MARK(tail(pairs)) -> c17(A__TAIL(a__pairs)) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) K tuples:none Defined Rule Symbols: a__nats, a__pairs, a__odds, a__incr_1, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__HEAD_1, A__TAIL_1, MARK_1, A__ODDS Compound Symbols: c6_1, c8_1, c10_1, c15_1, c18_1, c20_1, c13_2, c13_1, c16_2, c16_1, c17_2, c17_1, c4_1 ---------------------------------------- (51) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(incr(incr(z0))) -> c13(A__INCR(a__incr(mark(z0))), MARK(incr(z0))) by MARK(incr(incr(x0))) -> c13(A__INCR(incr(mark(x0))), MARK(incr(x0))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(0))) -> c13(A__INCR(a__incr(0)), MARK(incr(0))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) ---------------------------------------- (52) Obligation: Complexity Dependency Tuples Problem Rules: a__nats -> cons(0, incr(nats)) a__nats -> nats a__pairs -> cons(0, incr(odds)) a__pairs -> pairs a__odds -> a__incr(a__pairs) a__odds -> odds a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) 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(nats) -> a__nats mark(incr(z0)) -> a__incr(mark(z0)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(odds)) -> c13(A__INCR(a__odds), MARK(odds)) MARK(incr(head(z0))) -> c13(A__INCR(a__head(mark(z0))), MARK(head(z0))) MARK(incr(tail(z0))) -> c13(A__INCR(a__tail(mark(z0))), MARK(tail(z0))) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(nats)) -> c13(A__INCR(a__nats)) MARK(incr(pairs)) -> c13(A__INCR(a__pairs)) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(incr(z0))) -> c16(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(odds)) -> c16(A__HEAD(a__odds), MARK(odds)) MARK(head(head(z0))) -> c16(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c16(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(nats)) -> c16(A__HEAD(a__nats)) MARK(head(pairs)) -> c16(A__HEAD(a__pairs)) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(incr(z0))) -> c17(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(odds)) -> c17(A__TAIL(a__odds), MARK(odds)) MARK(tail(head(z0))) -> c17(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c17(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(nats)) -> c17(A__TAIL(a__nats)) MARK(tail(pairs)) -> c17(A__TAIL(a__pairs)) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(x0))) -> c13(A__INCR(incr(mark(x0))), MARK(incr(x0))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(0))) -> c13(A__INCR(a__incr(0)), MARK(incr(0))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) S tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(odds)) -> c13(A__INCR(a__odds), MARK(odds)) MARK(incr(head(z0))) -> c13(A__INCR(a__head(mark(z0))), MARK(head(z0))) MARK(incr(tail(z0))) -> c13(A__INCR(a__tail(mark(z0))), MARK(tail(z0))) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(nats)) -> c13(A__INCR(a__nats)) MARK(incr(pairs)) -> c13(A__INCR(a__pairs)) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(incr(z0))) -> c16(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(odds)) -> c16(A__HEAD(a__odds), MARK(odds)) MARK(head(head(z0))) -> c16(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c16(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(nats)) -> c16(A__HEAD(a__nats)) MARK(head(pairs)) -> c16(A__HEAD(a__pairs)) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(incr(z0))) -> c17(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(odds)) -> c17(A__TAIL(a__odds), MARK(odds)) MARK(tail(head(z0))) -> c17(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c17(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(nats)) -> c17(A__TAIL(a__nats)) MARK(tail(pairs)) -> c17(A__TAIL(a__pairs)) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(x0))) -> c13(A__INCR(incr(mark(x0))), MARK(incr(x0))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(0))) -> c13(A__INCR(a__incr(0)), MARK(incr(0))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) K tuples:none Defined Rule Symbols: a__nats, a__pairs, a__odds, a__incr_1, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__HEAD_1, A__TAIL_1, MARK_1, A__ODDS Compound Symbols: c6_1, c8_1, c10_1, c15_1, c18_1, c20_1, c13_2, c13_1, c16_2, c16_1, c17_2, c17_1, c4_1 ---------------------------------------- (53) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (54) Obligation: Complexity Dependency Tuples Problem Rules: a__nats -> cons(0, incr(nats)) a__nats -> nats a__pairs -> cons(0, incr(odds)) a__pairs -> pairs a__odds -> a__incr(a__pairs) a__odds -> odds a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) 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(nats) -> a__nats mark(incr(z0)) -> a__incr(mark(z0)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(odds)) -> c13(A__INCR(a__odds), MARK(odds)) MARK(incr(head(z0))) -> c13(A__INCR(a__head(mark(z0))), MARK(head(z0))) MARK(incr(tail(z0))) -> c13(A__INCR(a__tail(mark(z0))), MARK(tail(z0))) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(nats)) -> c13(A__INCR(a__nats)) MARK(incr(pairs)) -> c13(A__INCR(a__pairs)) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(incr(z0))) -> c16(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(odds)) -> c16(A__HEAD(a__odds), MARK(odds)) MARK(head(head(z0))) -> c16(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c16(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(nats)) -> c16(A__HEAD(a__nats)) MARK(head(pairs)) -> c16(A__HEAD(a__pairs)) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(incr(z0))) -> c17(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(odds)) -> c17(A__TAIL(a__odds), MARK(odds)) MARK(tail(head(z0))) -> c17(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c17(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(nats)) -> c17(A__TAIL(a__nats)) MARK(tail(pairs)) -> c17(A__TAIL(a__pairs)) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(incr(0))) -> c13(A__INCR(a__incr(0))) S tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(odds)) -> c13(A__INCR(a__odds), MARK(odds)) MARK(incr(head(z0))) -> c13(A__INCR(a__head(mark(z0))), MARK(head(z0))) MARK(incr(tail(z0))) -> c13(A__INCR(a__tail(mark(z0))), MARK(tail(z0))) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(nats)) -> c13(A__INCR(a__nats)) MARK(incr(pairs)) -> c13(A__INCR(a__pairs)) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(incr(z0))) -> c16(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(odds)) -> c16(A__HEAD(a__odds), MARK(odds)) MARK(head(head(z0))) -> c16(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c16(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(nats)) -> c16(A__HEAD(a__nats)) MARK(head(pairs)) -> c16(A__HEAD(a__pairs)) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(incr(z0))) -> c17(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(odds)) -> c17(A__TAIL(a__odds), MARK(odds)) MARK(tail(head(z0))) -> c17(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c17(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(nats)) -> c17(A__TAIL(a__nats)) MARK(tail(pairs)) -> c17(A__TAIL(a__pairs)) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(incr(0))) -> c13(A__INCR(a__incr(0))) K tuples:none Defined Rule Symbols: a__nats, a__pairs, a__odds, a__incr_1, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__HEAD_1, A__TAIL_1, MARK_1, A__ODDS Compound Symbols: c6_1, c8_1, c10_1, c15_1, c18_1, c20_1, c13_2, c13_1, c16_2, c16_1, c17_2, c17_1, c4_1 ---------------------------------------- (55) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(incr(odds)) -> c13(A__INCR(a__odds), MARK(odds)) by MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(A__INCR(odds), MARK(odds)) ---------------------------------------- (56) Obligation: Complexity Dependency Tuples Problem Rules: a__nats -> cons(0, incr(nats)) a__nats -> nats a__pairs -> cons(0, incr(odds)) a__pairs -> pairs a__odds -> a__incr(a__pairs) a__odds -> odds a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) 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(nats) -> a__nats mark(incr(z0)) -> a__incr(mark(z0)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(head(z0))) -> c13(A__INCR(a__head(mark(z0))), MARK(head(z0))) MARK(incr(tail(z0))) -> c13(A__INCR(a__tail(mark(z0))), MARK(tail(z0))) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(nats)) -> c13(A__INCR(a__nats)) MARK(incr(pairs)) -> c13(A__INCR(a__pairs)) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(incr(z0))) -> c16(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(odds)) -> c16(A__HEAD(a__odds), MARK(odds)) MARK(head(head(z0))) -> c16(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c16(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(nats)) -> c16(A__HEAD(a__nats)) MARK(head(pairs)) -> c16(A__HEAD(a__pairs)) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(incr(z0))) -> c17(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(odds)) -> c17(A__TAIL(a__odds), MARK(odds)) MARK(tail(head(z0))) -> c17(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c17(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(nats)) -> c17(A__TAIL(a__nats)) MARK(tail(pairs)) -> c17(A__TAIL(a__pairs)) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(incr(0))) -> c13(A__INCR(a__incr(0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(A__INCR(odds), MARK(odds)) S tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(head(z0))) -> c13(A__INCR(a__head(mark(z0))), MARK(head(z0))) MARK(incr(tail(z0))) -> c13(A__INCR(a__tail(mark(z0))), MARK(tail(z0))) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(nats)) -> c13(A__INCR(a__nats)) MARK(incr(pairs)) -> c13(A__INCR(a__pairs)) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(incr(z0))) -> c16(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(odds)) -> c16(A__HEAD(a__odds), MARK(odds)) MARK(head(head(z0))) -> c16(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c16(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(nats)) -> c16(A__HEAD(a__nats)) MARK(head(pairs)) -> c16(A__HEAD(a__pairs)) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(incr(z0))) -> c17(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(odds)) -> c17(A__TAIL(a__odds), MARK(odds)) MARK(tail(head(z0))) -> c17(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c17(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(nats)) -> c17(A__TAIL(a__nats)) MARK(tail(pairs)) -> c17(A__TAIL(a__pairs)) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(incr(0))) -> c13(A__INCR(a__incr(0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(A__INCR(odds), MARK(odds)) K tuples:none Defined Rule Symbols: a__nats, a__pairs, a__odds, a__incr_1, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__HEAD_1, A__TAIL_1, MARK_1, A__ODDS Compound Symbols: c6_1, c8_1, c10_1, c15_1, c18_1, c20_1, c13_2, c13_1, c16_2, c16_1, c17_2, c17_1, c4_1 ---------------------------------------- (57) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (58) Obligation: Complexity Dependency Tuples Problem Rules: a__nats -> cons(0, incr(nats)) a__nats -> nats a__pairs -> cons(0, incr(odds)) a__pairs -> pairs a__odds -> a__incr(a__pairs) a__odds -> odds a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) 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(nats) -> a__nats mark(incr(z0)) -> a__incr(mark(z0)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(head(z0))) -> c13(A__INCR(a__head(mark(z0))), MARK(head(z0))) MARK(incr(tail(z0))) -> c13(A__INCR(a__tail(mark(z0))), MARK(tail(z0))) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(nats)) -> c13(A__INCR(a__nats)) MARK(incr(pairs)) -> c13(A__INCR(a__pairs)) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(incr(z0))) -> c16(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(odds)) -> c16(A__HEAD(a__odds), MARK(odds)) MARK(head(head(z0))) -> c16(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c16(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(nats)) -> c16(A__HEAD(a__nats)) MARK(head(pairs)) -> c16(A__HEAD(a__pairs)) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(incr(z0))) -> c17(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(odds)) -> c17(A__TAIL(a__odds), MARK(odds)) MARK(tail(head(z0))) -> c17(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c17(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(nats)) -> c17(A__TAIL(a__nats)) MARK(tail(pairs)) -> c17(A__TAIL(a__pairs)) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(incr(0))) -> c13(A__INCR(a__incr(0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) S tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(head(z0))) -> c13(A__INCR(a__head(mark(z0))), MARK(head(z0))) MARK(incr(tail(z0))) -> c13(A__INCR(a__tail(mark(z0))), MARK(tail(z0))) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(nats)) -> c13(A__INCR(a__nats)) MARK(incr(pairs)) -> c13(A__INCR(a__pairs)) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(incr(z0))) -> c16(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(odds)) -> c16(A__HEAD(a__odds), MARK(odds)) MARK(head(head(z0))) -> c16(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c16(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(nats)) -> c16(A__HEAD(a__nats)) MARK(head(pairs)) -> c16(A__HEAD(a__pairs)) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(incr(z0))) -> c17(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(odds)) -> c17(A__TAIL(a__odds), MARK(odds)) MARK(tail(head(z0))) -> c17(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c17(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(nats)) -> c17(A__TAIL(a__nats)) MARK(tail(pairs)) -> c17(A__TAIL(a__pairs)) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(incr(0))) -> c13(A__INCR(a__incr(0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) K tuples:none Defined Rule Symbols: a__nats, a__pairs, a__odds, a__incr_1, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__HEAD_1, A__TAIL_1, MARK_1, A__ODDS Compound Symbols: c6_1, c8_1, c10_1, c15_1, c18_1, c20_1, c13_2, c13_1, c16_2, c16_1, c17_2, c17_1, c4_1 ---------------------------------------- (59) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(incr(head(z0))) -> c13(A__INCR(a__head(mark(z0))), MARK(head(z0))) by MARK(incr(head(x0))) -> c13(A__INCR(head(mark(x0))), MARK(head(x0))) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(0))) -> c13(A__INCR(a__head(0)), MARK(head(0))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) ---------------------------------------- (60) Obligation: Complexity Dependency Tuples Problem Rules: a__nats -> cons(0, incr(nats)) a__nats -> nats a__pairs -> cons(0, incr(odds)) a__pairs -> pairs a__odds -> a__incr(a__pairs) a__odds -> odds a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) 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(nats) -> a__nats mark(incr(z0)) -> a__incr(mark(z0)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(tail(z0))) -> c13(A__INCR(a__tail(mark(z0))), MARK(tail(z0))) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(nats)) -> c13(A__INCR(a__nats)) MARK(incr(pairs)) -> c13(A__INCR(a__pairs)) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(incr(z0))) -> c16(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(odds)) -> c16(A__HEAD(a__odds), MARK(odds)) MARK(head(head(z0))) -> c16(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c16(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(nats)) -> c16(A__HEAD(a__nats)) MARK(head(pairs)) -> c16(A__HEAD(a__pairs)) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(incr(z0))) -> c17(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(odds)) -> c17(A__TAIL(a__odds), MARK(odds)) MARK(tail(head(z0))) -> c17(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c17(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(nats)) -> c17(A__TAIL(a__nats)) MARK(tail(pairs)) -> c17(A__TAIL(a__pairs)) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(incr(0))) -> c13(A__INCR(a__incr(0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(x0))) -> c13(A__INCR(head(mark(x0))), MARK(head(x0))) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(0))) -> c13(A__INCR(a__head(0)), MARK(head(0))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) S tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(tail(z0))) -> c13(A__INCR(a__tail(mark(z0))), MARK(tail(z0))) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(nats)) -> c13(A__INCR(a__nats)) MARK(incr(pairs)) -> c13(A__INCR(a__pairs)) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(incr(z0))) -> c16(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(odds)) -> c16(A__HEAD(a__odds), MARK(odds)) MARK(head(head(z0))) -> c16(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c16(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(nats)) -> c16(A__HEAD(a__nats)) MARK(head(pairs)) -> c16(A__HEAD(a__pairs)) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(incr(z0))) -> c17(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(odds)) -> c17(A__TAIL(a__odds), MARK(odds)) MARK(tail(head(z0))) -> c17(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c17(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(nats)) -> c17(A__TAIL(a__nats)) MARK(tail(pairs)) -> c17(A__TAIL(a__pairs)) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(incr(0))) -> c13(A__INCR(a__incr(0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(x0))) -> c13(A__INCR(head(mark(x0))), MARK(head(x0))) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(0))) -> c13(A__INCR(a__head(0)), MARK(head(0))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) K tuples:none Defined Rule Symbols: a__nats, a__pairs, a__odds, a__incr_1, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__HEAD_1, A__TAIL_1, MARK_1, A__ODDS Compound Symbols: c6_1, c8_1, c10_1, c15_1, c18_1, c20_1, c13_2, c13_1, c16_2, c16_1, c17_2, c17_1, c4_1 ---------------------------------------- (61) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: a__nats -> cons(0, incr(nats)) a__nats -> nats a__pairs -> cons(0, incr(odds)) a__pairs -> pairs a__odds -> a__incr(a__pairs) a__odds -> odds a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) 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(nats) -> a__nats mark(incr(z0)) -> a__incr(mark(z0)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(tail(z0))) -> c13(A__INCR(a__tail(mark(z0))), MARK(tail(z0))) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(nats)) -> c13(A__INCR(a__nats)) MARK(incr(pairs)) -> c13(A__INCR(a__pairs)) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(incr(z0))) -> c16(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(odds)) -> c16(A__HEAD(a__odds), MARK(odds)) MARK(head(head(z0))) -> c16(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c16(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(nats)) -> c16(A__HEAD(a__nats)) MARK(head(pairs)) -> c16(A__HEAD(a__pairs)) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(incr(z0))) -> c17(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(odds)) -> c17(A__TAIL(a__odds), MARK(odds)) MARK(tail(head(z0))) -> c17(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c17(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(nats)) -> c17(A__TAIL(a__nats)) MARK(tail(pairs)) -> c17(A__TAIL(a__pairs)) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(incr(0))) -> c13(A__INCR(a__incr(0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(head(0))) -> c13(A__INCR(a__head(0))) S tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(tail(z0))) -> c13(A__INCR(a__tail(mark(z0))), MARK(tail(z0))) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(nats)) -> c13(A__INCR(a__nats)) MARK(incr(pairs)) -> c13(A__INCR(a__pairs)) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(incr(z0))) -> c16(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(odds)) -> c16(A__HEAD(a__odds), MARK(odds)) MARK(head(head(z0))) -> c16(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c16(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(nats)) -> c16(A__HEAD(a__nats)) MARK(head(pairs)) -> c16(A__HEAD(a__pairs)) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(incr(z0))) -> c17(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(odds)) -> c17(A__TAIL(a__odds), MARK(odds)) MARK(tail(head(z0))) -> c17(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c17(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(nats)) -> c17(A__TAIL(a__nats)) MARK(tail(pairs)) -> c17(A__TAIL(a__pairs)) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(incr(0))) -> c13(A__INCR(a__incr(0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(head(0))) -> c13(A__INCR(a__head(0))) K tuples:none Defined Rule Symbols: a__nats, a__pairs, a__odds, a__incr_1, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__HEAD_1, A__TAIL_1, MARK_1, A__ODDS Compound Symbols: c6_1, c8_1, c10_1, c15_1, c18_1, c20_1, c13_2, c13_1, c16_2, c16_1, c17_2, c17_1, c4_1 ---------------------------------------- (63) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(incr(tail(z0))) -> c13(A__INCR(a__tail(mark(z0))), MARK(tail(z0))) by MARK(incr(tail(x0))) -> c13(A__INCR(tail(mark(x0))), MARK(tail(x0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(0))) -> c13(A__INCR(a__tail(0)), MARK(tail(0))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) ---------------------------------------- (64) Obligation: Complexity Dependency Tuples Problem Rules: a__nats -> cons(0, incr(nats)) a__nats -> nats a__pairs -> cons(0, incr(odds)) a__pairs -> pairs a__odds -> a__incr(a__pairs) a__odds -> odds a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) 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(nats) -> a__nats mark(incr(z0)) -> a__incr(mark(z0)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(nats)) -> c13(A__INCR(a__nats)) MARK(incr(pairs)) -> c13(A__INCR(a__pairs)) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(incr(z0))) -> c16(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(odds)) -> c16(A__HEAD(a__odds), MARK(odds)) MARK(head(head(z0))) -> c16(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c16(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(nats)) -> c16(A__HEAD(a__nats)) MARK(head(pairs)) -> c16(A__HEAD(a__pairs)) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(incr(z0))) -> c17(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(odds)) -> c17(A__TAIL(a__odds), MARK(odds)) MARK(tail(head(z0))) -> c17(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c17(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(nats)) -> c17(A__TAIL(a__nats)) MARK(tail(pairs)) -> c17(A__TAIL(a__pairs)) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(incr(0))) -> c13(A__INCR(a__incr(0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(head(0))) -> c13(A__INCR(a__head(0))) MARK(incr(tail(x0))) -> c13(A__INCR(tail(mark(x0))), MARK(tail(x0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(0))) -> c13(A__INCR(a__tail(0)), MARK(tail(0))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) S tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(nats)) -> c13(A__INCR(a__nats)) MARK(incr(pairs)) -> c13(A__INCR(a__pairs)) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(incr(z0))) -> c16(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(odds)) -> c16(A__HEAD(a__odds), MARK(odds)) MARK(head(head(z0))) -> c16(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c16(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(nats)) -> c16(A__HEAD(a__nats)) MARK(head(pairs)) -> c16(A__HEAD(a__pairs)) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(incr(z0))) -> c17(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(odds)) -> c17(A__TAIL(a__odds), MARK(odds)) MARK(tail(head(z0))) -> c17(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c17(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(nats)) -> c17(A__TAIL(a__nats)) MARK(tail(pairs)) -> c17(A__TAIL(a__pairs)) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(incr(0))) -> c13(A__INCR(a__incr(0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(head(0))) -> c13(A__INCR(a__head(0))) MARK(incr(tail(x0))) -> c13(A__INCR(tail(mark(x0))), MARK(tail(x0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(0))) -> c13(A__INCR(a__tail(0)), MARK(tail(0))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) K tuples:none Defined Rule Symbols: a__nats, a__pairs, a__odds, a__incr_1, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__HEAD_1, A__TAIL_1, MARK_1, A__ODDS Compound Symbols: c6_1, c8_1, c10_1, c15_1, c18_1, c20_1, c13_2, c13_1, c16_2, c16_1, c17_2, c17_1, c4_1 ---------------------------------------- (65) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (66) Obligation: Complexity Dependency Tuples Problem Rules: a__nats -> cons(0, incr(nats)) a__nats -> nats a__pairs -> cons(0, incr(odds)) a__pairs -> pairs a__odds -> a__incr(a__pairs) a__odds -> odds a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) 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(nats) -> a__nats mark(incr(z0)) -> a__incr(mark(z0)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(nats)) -> c13(A__INCR(a__nats)) MARK(incr(pairs)) -> c13(A__INCR(a__pairs)) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(incr(z0))) -> c16(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(odds)) -> c16(A__HEAD(a__odds), MARK(odds)) MARK(head(head(z0))) -> c16(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c16(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(nats)) -> c16(A__HEAD(a__nats)) MARK(head(pairs)) -> c16(A__HEAD(a__pairs)) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(incr(z0))) -> c17(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(odds)) -> c17(A__TAIL(a__odds), MARK(odds)) MARK(tail(head(z0))) -> c17(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c17(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(nats)) -> c17(A__TAIL(a__nats)) MARK(tail(pairs)) -> c17(A__TAIL(a__pairs)) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(incr(0))) -> c13(A__INCR(a__incr(0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(head(0))) -> c13(A__INCR(a__head(0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(incr(tail(0))) -> c13(A__INCR(a__tail(0))) S tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(nats)) -> c13(A__INCR(a__nats)) MARK(incr(pairs)) -> c13(A__INCR(a__pairs)) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(incr(z0))) -> c16(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(odds)) -> c16(A__HEAD(a__odds), MARK(odds)) MARK(head(head(z0))) -> c16(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c16(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(nats)) -> c16(A__HEAD(a__nats)) MARK(head(pairs)) -> c16(A__HEAD(a__pairs)) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(incr(z0))) -> c17(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(odds)) -> c17(A__TAIL(a__odds), MARK(odds)) MARK(tail(head(z0))) -> c17(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c17(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(nats)) -> c17(A__TAIL(a__nats)) MARK(tail(pairs)) -> c17(A__TAIL(a__pairs)) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(incr(0))) -> c13(A__INCR(a__incr(0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(head(0))) -> c13(A__INCR(a__head(0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(incr(tail(0))) -> c13(A__INCR(a__tail(0))) K tuples:none Defined Rule Symbols: a__nats, a__pairs, a__odds, a__incr_1, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__HEAD_1, A__TAIL_1, MARK_1, A__ODDS Compound Symbols: c6_1, c8_1, c10_1, c15_1, c18_1, c20_1, c13_2, c13_1, c16_2, c16_1, c17_2, c17_1, c4_1 ---------------------------------------- (67) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(incr(nats)) -> c13(A__INCR(a__nats)) by MARK(incr(nats)) -> c13(A__INCR(cons(0, incr(nats)))) MARK(incr(nats)) -> c13(A__INCR(nats)) ---------------------------------------- (68) Obligation: Complexity Dependency Tuples Problem Rules: a__nats -> cons(0, incr(nats)) a__nats -> nats a__pairs -> cons(0, incr(odds)) a__pairs -> pairs a__odds -> a__incr(a__pairs) a__odds -> odds a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) 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(nats) -> a__nats mark(incr(z0)) -> a__incr(mark(z0)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(pairs)) -> c13(A__INCR(a__pairs)) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(incr(z0))) -> c16(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(odds)) -> c16(A__HEAD(a__odds), MARK(odds)) MARK(head(head(z0))) -> c16(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c16(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(nats)) -> c16(A__HEAD(a__nats)) MARK(head(pairs)) -> c16(A__HEAD(a__pairs)) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(incr(z0))) -> c17(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(odds)) -> c17(A__TAIL(a__odds), MARK(odds)) MARK(tail(head(z0))) -> c17(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c17(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(nats)) -> c17(A__TAIL(a__nats)) MARK(tail(pairs)) -> c17(A__TAIL(a__pairs)) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(incr(0))) -> c13(A__INCR(a__incr(0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(head(0))) -> c13(A__INCR(a__head(0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(incr(tail(0))) -> c13(A__INCR(a__tail(0))) MARK(incr(nats)) -> c13(A__INCR(cons(0, incr(nats)))) MARK(incr(nats)) -> c13(A__INCR(nats)) S tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(pairs)) -> c13(A__INCR(a__pairs)) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(incr(z0))) -> c16(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(odds)) -> c16(A__HEAD(a__odds), MARK(odds)) MARK(head(head(z0))) -> c16(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c16(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(nats)) -> c16(A__HEAD(a__nats)) MARK(head(pairs)) -> c16(A__HEAD(a__pairs)) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(incr(z0))) -> c17(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(odds)) -> c17(A__TAIL(a__odds), MARK(odds)) MARK(tail(head(z0))) -> c17(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c17(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(nats)) -> c17(A__TAIL(a__nats)) MARK(tail(pairs)) -> c17(A__TAIL(a__pairs)) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(incr(0))) -> c13(A__INCR(a__incr(0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(head(0))) -> c13(A__INCR(a__head(0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(incr(tail(0))) -> c13(A__INCR(a__tail(0))) MARK(incr(nats)) -> c13(A__INCR(cons(0, incr(nats)))) MARK(incr(nats)) -> c13(A__INCR(nats)) K tuples:none Defined Rule Symbols: a__nats, a__pairs, a__odds, a__incr_1, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__HEAD_1, A__TAIL_1, MARK_1, A__ODDS Compound Symbols: c6_1, c8_1, c10_1, c15_1, c18_1, c20_1, c13_2, c13_1, c16_2, c16_1, c17_2, c17_1, c4_1 ---------------------------------------- (69) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(incr(nats)) -> c13(A__INCR(nats)) ---------------------------------------- (70) Obligation: Complexity Dependency Tuples Problem Rules: a__nats -> cons(0, incr(nats)) a__nats -> nats a__pairs -> cons(0, incr(odds)) a__pairs -> pairs a__odds -> a__incr(a__pairs) a__odds -> odds a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) 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(nats) -> a__nats mark(incr(z0)) -> a__incr(mark(z0)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(pairs)) -> c13(A__INCR(a__pairs)) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(incr(z0))) -> c16(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(odds)) -> c16(A__HEAD(a__odds), MARK(odds)) MARK(head(head(z0))) -> c16(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c16(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(nats)) -> c16(A__HEAD(a__nats)) MARK(head(pairs)) -> c16(A__HEAD(a__pairs)) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(incr(z0))) -> c17(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(odds)) -> c17(A__TAIL(a__odds), MARK(odds)) MARK(tail(head(z0))) -> c17(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c17(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(nats)) -> c17(A__TAIL(a__nats)) MARK(tail(pairs)) -> c17(A__TAIL(a__pairs)) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(incr(0))) -> c13(A__INCR(a__incr(0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(head(0))) -> c13(A__INCR(a__head(0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(incr(tail(0))) -> c13(A__INCR(a__tail(0))) MARK(incr(nats)) -> c13(A__INCR(cons(0, incr(nats)))) S tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(pairs)) -> c13(A__INCR(a__pairs)) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(incr(z0))) -> c16(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(odds)) -> c16(A__HEAD(a__odds), MARK(odds)) MARK(head(head(z0))) -> c16(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c16(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(nats)) -> c16(A__HEAD(a__nats)) MARK(head(pairs)) -> c16(A__HEAD(a__pairs)) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(incr(z0))) -> c17(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(odds)) -> c17(A__TAIL(a__odds), MARK(odds)) MARK(tail(head(z0))) -> c17(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c17(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(nats)) -> c17(A__TAIL(a__nats)) MARK(tail(pairs)) -> c17(A__TAIL(a__pairs)) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(incr(0))) -> c13(A__INCR(a__incr(0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(head(0))) -> c13(A__INCR(a__head(0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(incr(tail(0))) -> c13(A__INCR(a__tail(0))) MARK(incr(nats)) -> c13(A__INCR(cons(0, incr(nats)))) K tuples:none Defined Rule Symbols: a__nats, a__pairs, a__odds, a__incr_1, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__HEAD_1, A__TAIL_1, MARK_1, A__ODDS Compound Symbols: c6_1, c8_1, c10_1, c15_1, c18_1, c20_1, c13_2, c13_1, c16_2, c16_1, c17_2, c17_1, c4_1 ---------------------------------------- (71) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(incr(pairs)) -> c13(A__INCR(a__pairs)) by MARK(incr(pairs)) -> c13(A__INCR(cons(0, incr(odds)))) MARK(incr(pairs)) -> c13(A__INCR(pairs)) ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: a__nats -> cons(0, incr(nats)) a__nats -> nats a__pairs -> cons(0, incr(odds)) a__pairs -> pairs a__odds -> a__incr(a__pairs) a__odds -> odds a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) 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(nats) -> a__nats mark(incr(z0)) -> a__incr(mark(z0)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(incr(z0))) -> c16(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(odds)) -> c16(A__HEAD(a__odds), MARK(odds)) MARK(head(head(z0))) -> c16(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c16(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(nats)) -> c16(A__HEAD(a__nats)) MARK(head(pairs)) -> c16(A__HEAD(a__pairs)) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(incr(z0))) -> c17(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(odds)) -> c17(A__TAIL(a__odds), MARK(odds)) MARK(tail(head(z0))) -> c17(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c17(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(nats)) -> c17(A__TAIL(a__nats)) MARK(tail(pairs)) -> c17(A__TAIL(a__pairs)) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(incr(0))) -> c13(A__INCR(a__incr(0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(head(0))) -> c13(A__INCR(a__head(0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(incr(tail(0))) -> c13(A__INCR(a__tail(0))) MARK(incr(nats)) -> c13(A__INCR(cons(0, incr(nats)))) MARK(incr(pairs)) -> c13(A__INCR(cons(0, incr(odds)))) MARK(incr(pairs)) -> c13(A__INCR(pairs)) S tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(incr(z0))) -> c16(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(odds)) -> c16(A__HEAD(a__odds), MARK(odds)) MARK(head(head(z0))) -> c16(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c16(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(nats)) -> c16(A__HEAD(a__nats)) MARK(head(pairs)) -> c16(A__HEAD(a__pairs)) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(incr(z0))) -> c17(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(odds)) -> c17(A__TAIL(a__odds), MARK(odds)) MARK(tail(head(z0))) -> c17(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c17(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(nats)) -> c17(A__TAIL(a__nats)) MARK(tail(pairs)) -> c17(A__TAIL(a__pairs)) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(incr(0))) -> c13(A__INCR(a__incr(0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(head(0))) -> c13(A__INCR(a__head(0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(incr(tail(0))) -> c13(A__INCR(a__tail(0))) MARK(incr(nats)) -> c13(A__INCR(cons(0, incr(nats)))) MARK(incr(pairs)) -> c13(A__INCR(cons(0, incr(odds)))) MARK(incr(pairs)) -> c13(A__INCR(pairs)) K tuples:none Defined Rule Symbols: a__nats, a__pairs, a__odds, a__incr_1, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__HEAD_1, A__TAIL_1, MARK_1, A__ODDS Compound Symbols: c6_1, c8_1, c10_1, c15_1, c18_1, c20_1, c13_2, c13_1, c16_2, c16_1, c17_2, c17_1, c4_1 ---------------------------------------- (73) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(incr(pairs)) -> c13(A__INCR(pairs)) ---------------------------------------- (74) Obligation: Complexity Dependency Tuples Problem Rules: a__nats -> cons(0, incr(nats)) a__nats -> nats a__pairs -> cons(0, incr(odds)) a__pairs -> pairs a__odds -> a__incr(a__pairs) a__odds -> odds a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) 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(nats) -> a__nats mark(incr(z0)) -> a__incr(mark(z0)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(incr(z0))) -> c16(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(odds)) -> c16(A__HEAD(a__odds), MARK(odds)) MARK(head(head(z0))) -> c16(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c16(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(nats)) -> c16(A__HEAD(a__nats)) MARK(head(pairs)) -> c16(A__HEAD(a__pairs)) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(incr(z0))) -> c17(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(odds)) -> c17(A__TAIL(a__odds), MARK(odds)) MARK(tail(head(z0))) -> c17(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c17(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(nats)) -> c17(A__TAIL(a__nats)) MARK(tail(pairs)) -> c17(A__TAIL(a__pairs)) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(incr(0))) -> c13(A__INCR(a__incr(0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(head(0))) -> c13(A__INCR(a__head(0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(incr(tail(0))) -> c13(A__INCR(a__tail(0))) MARK(incr(nats)) -> c13(A__INCR(cons(0, incr(nats)))) MARK(incr(pairs)) -> c13(A__INCR(cons(0, incr(odds)))) S tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(incr(z0))) -> c16(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) MARK(head(odds)) -> c16(A__HEAD(a__odds), MARK(odds)) MARK(head(head(z0))) -> c16(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c16(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(nats)) -> c16(A__HEAD(a__nats)) MARK(head(pairs)) -> c16(A__HEAD(a__pairs)) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(incr(z0))) -> c17(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(odds)) -> c17(A__TAIL(a__odds), MARK(odds)) MARK(tail(head(z0))) -> c17(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c17(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(nats)) -> c17(A__TAIL(a__nats)) MARK(tail(pairs)) -> c17(A__TAIL(a__pairs)) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(incr(0))) -> c13(A__INCR(a__incr(0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(head(0))) -> c13(A__INCR(a__head(0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(incr(tail(0))) -> c13(A__INCR(a__tail(0))) MARK(incr(nats)) -> c13(A__INCR(cons(0, incr(nats)))) MARK(incr(pairs)) -> c13(A__INCR(cons(0, incr(odds)))) K tuples:none Defined Rule Symbols: a__nats, a__pairs, a__odds, a__incr_1, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__HEAD_1, A__TAIL_1, MARK_1, A__ODDS Compound Symbols: c6_1, c8_1, c10_1, c15_1, c18_1, c20_1, c13_2, c13_1, c16_2, c16_1, c17_2, c17_1, c4_1 ---------------------------------------- (75) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(head(incr(z0))) -> c16(A__HEAD(a__incr(mark(z0))), MARK(incr(z0))) by MARK(head(incr(x0))) -> c16(A__HEAD(incr(mark(x0))), MARK(incr(x0))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs)), MARK(incr(pairs))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(0))) -> c16(A__HEAD(a__incr(0)), MARK(incr(0))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) ---------------------------------------- (76) Obligation: Complexity Dependency Tuples Problem Rules: a__nats -> cons(0, incr(nats)) a__nats -> nats a__pairs -> cons(0, incr(odds)) a__pairs -> pairs a__odds -> a__incr(a__pairs) a__odds -> odds a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) 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(nats) -> a__nats mark(incr(z0)) -> a__incr(mark(z0)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(odds)) -> c16(A__HEAD(a__odds), MARK(odds)) MARK(head(head(z0))) -> c16(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c16(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(nats)) -> c16(A__HEAD(a__nats)) MARK(head(pairs)) -> c16(A__HEAD(a__pairs)) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(incr(z0))) -> c17(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(odds)) -> c17(A__TAIL(a__odds), MARK(odds)) MARK(tail(head(z0))) -> c17(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c17(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(nats)) -> c17(A__TAIL(a__nats)) MARK(tail(pairs)) -> c17(A__TAIL(a__pairs)) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(incr(0))) -> c13(A__INCR(a__incr(0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(head(0))) -> c13(A__INCR(a__head(0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(incr(tail(0))) -> c13(A__INCR(a__tail(0))) MARK(incr(nats)) -> c13(A__INCR(cons(0, incr(nats)))) MARK(incr(pairs)) -> c13(A__INCR(cons(0, incr(odds)))) MARK(head(incr(x0))) -> c16(A__HEAD(incr(mark(x0))), MARK(incr(x0))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs)), MARK(incr(pairs))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(0))) -> c16(A__HEAD(a__incr(0)), MARK(incr(0))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) S tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(odds)) -> c16(A__HEAD(a__odds), MARK(odds)) MARK(head(head(z0))) -> c16(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c16(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(nats)) -> c16(A__HEAD(a__nats)) MARK(head(pairs)) -> c16(A__HEAD(a__pairs)) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(incr(z0))) -> c17(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(odds)) -> c17(A__TAIL(a__odds), MARK(odds)) MARK(tail(head(z0))) -> c17(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c17(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(nats)) -> c17(A__TAIL(a__nats)) MARK(tail(pairs)) -> c17(A__TAIL(a__pairs)) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(incr(0))) -> c13(A__INCR(a__incr(0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(head(0))) -> c13(A__INCR(a__head(0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(incr(tail(0))) -> c13(A__INCR(a__tail(0))) MARK(incr(nats)) -> c13(A__INCR(cons(0, incr(nats)))) MARK(incr(pairs)) -> c13(A__INCR(cons(0, incr(odds)))) MARK(head(incr(x0))) -> c16(A__HEAD(incr(mark(x0))), MARK(incr(x0))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs)), MARK(incr(pairs))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(0))) -> c16(A__HEAD(a__incr(0)), MARK(incr(0))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) K tuples:none Defined Rule Symbols: a__nats, a__pairs, a__odds, a__incr_1, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__HEAD_1, A__TAIL_1, MARK_1, A__ODDS Compound Symbols: c6_1, c8_1, c10_1, c15_1, c18_1, c20_1, c13_2, c13_1, c16_2, c16_1, c17_2, c17_1, c4_1 ---------------------------------------- (77) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (78) Obligation: Complexity Dependency Tuples Problem Rules: a__nats -> cons(0, incr(nats)) a__nats -> nats a__pairs -> cons(0, incr(odds)) a__pairs -> pairs a__odds -> a__incr(a__pairs) a__odds -> odds a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) 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(nats) -> a__nats mark(incr(z0)) -> a__incr(mark(z0)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(odds)) -> c16(A__HEAD(a__odds), MARK(odds)) MARK(head(head(z0))) -> c16(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c16(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(nats)) -> c16(A__HEAD(a__nats)) MARK(head(pairs)) -> c16(A__HEAD(a__pairs)) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(incr(z0))) -> c17(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(odds)) -> c17(A__TAIL(a__odds), MARK(odds)) MARK(tail(head(z0))) -> c17(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c17(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(nats)) -> c17(A__TAIL(a__nats)) MARK(tail(pairs)) -> c17(A__TAIL(a__pairs)) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(incr(0))) -> c13(A__INCR(a__incr(0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(head(0))) -> c13(A__INCR(a__head(0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(incr(tail(0))) -> c13(A__INCR(a__tail(0))) MARK(incr(nats)) -> c13(A__INCR(cons(0, incr(nats)))) MARK(incr(pairs)) -> c13(A__INCR(cons(0, incr(odds)))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs)), MARK(incr(pairs))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(incr(0))) -> c16(A__HEAD(a__incr(0))) S tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(odds)) -> c16(A__HEAD(a__odds), MARK(odds)) MARK(head(head(z0))) -> c16(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c16(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(nats)) -> c16(A__HEAD(a__nats)) MARK(head(pairs)) -> c16(A__HEAD(a__pairs)) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(incr(z0))) -> c17(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(odds)) -> c17(A__TAIL(a__odds), MARK(odds)) MARK(tail(head(z0))) -> c17(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c17(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(nats)) -> c17(A__TAIL(a__nats)) MARK(tail(pairs)) -> c17(A__TAIL(a__pairs)) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(incr(0))) -> c13(A__INCR(a__incr(0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(head(0))) -> c13(A__INCR(a__head(0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(incr(tail(0))) -> c13(A__INCR(a__tail(0))) MARK(incr(nats)) -> c13(A__INCR(cons(0, incr(nats)))) MARK(incr(pairs)) -> c13(A__INCR(cons(0, incr(odds)))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs)), MARK(incr(pairs))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(incr(0))) -> c16(A__HEAD(a__incr(0))) K tuples:none Defined Rule Symbols: a__nats, a__pairs, a__odds, a__incr_1, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__HEAD_1, A__TAIL_1, MARK_1, A__ODDS Compound Symbols: c6_1, c8_1, c10_1, c15_1, c18_1, c20_1, c13_2, c13_1, c16_2, c16_1, c17_2, c17_1, c4_1 ---------------------------------------- (79) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(head(odds)) -> c16(A__HEAD(a__odds), MARK(odds)) by MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs)), MARK(odds)) MARK(head(odds)) -> c16(A__HEAD(odds), MARK(odds)) ---------------------------------------- (80) Obligation: Complexity Dependency Tuples Problem Rules: a__nats -> cons(0, incr(nats)) a__nats -> nats a__pairs -> cons(0, incr(odds)) a__pairs -> pairs a__odds -> a__incr(a__pairs) a__odds -> odds a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) 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(nats) -> a__nats mark(incr(z0)) -> a__incr(mark(z0)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(head(z0))) -> c16(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c16(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(nats)) -> c16(A__HEAD(a__nats)) MARK(head(pairs)) -> c16(A__HEAD(a__pairs)) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(incr(z0))) -> c17(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(odds)) -> c17(A__TAIL(a__odds), MARK(odds)) MARK(tail(head(z0))) -> c17(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c17(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(nats)) -> c17(A__TAIL(a__nats)) MARK(tail(pairs)) -> c17(A__TAIL(a__pairs)) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(incr(0))) -> c13(A__INCR(a__incr(0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(head(0))) -> c13(A__INCR(a__head(0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(incr(tail(0))) -> c13(A__INCR(a__tail(0))) MARK(incr(nats)) -> c13(A__INCR(cons(0, incr(nats)))) MARK(incr(pairs)) -> c13(A__INCR(cons(0, incr(odds)))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs)), MARK(incr(pairs))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(incr(0))) -> c16(A__HEAD(a__incr(0))) MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs)), MARK(odds)) MARK(head(odds)) -> c16(A__HEAD(odds), MARK(odds)) S tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(head(z0))) -> c16(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c16(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(nats)) -> c16(A__HEAD(a__nats)) MARK(head(pairs)) -> c16(A__HEAD(a__pairs)) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(incr(z0))) -> c17(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(odds)) -> c17(A__TAIL(a__odds), MARK(odds)) MARK(tail(head(z0))) -> c17(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c17(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(nats)) -> c17(A__TAIL(a__nats)) MARK(tail(pairs)) -> c17(A__TAIL(a__pairs)) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(incr(0))) -> c13(A__INCR(a__incr(0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(head(0))) -> c13(A__INCR(a__head(0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(incr(tail(0))) -> c13(A__INCR(a__tail(0))) MARK(incr(nats)) -> c13(A__INCR(cons(0, incr(nats)))) MARK(incr(pairs)) -> c13(A__INCR(cons(0, incr(odds)))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs)), MARK(incr(pairs))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(incr(0))) -> c16(A__HEAD(a__incr(0))) MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs)), MARK(odds)) MARK(head(odds)) -> c16(A__HEAD(odds), MARK(odds)) K tuples:none Defined Rule Symbols: a__nats, a__pairs, a__odds, a__incr_1, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__HEAD_1, A__TAIL_1, MARK_1, A__ODDS Compound Symbols: c6_1, c8_1, c10_1, c15_1, c18_1, c20_1, c13_2, c13_1, c16_2, c16_1, c17_2, c17_1, c4_1 ---------------------------------------- (81) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (82) Obligation: Complexity Dependency Tuples Problem Rules: a__nats -> cons(0, incr(nats)) a__nats -> nats a__pairs -> cons(0, incr(odds)) a__pairs -> pairs a__odds -> a__incr(a__pairs) a__odds -> odds a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) 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(nats) -> a__nats mark(incr(z0)) -> a__incr(mark(z0)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(head(z0))) -> c16(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c16(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(nats)) -> c16(A__HEAD(a__nats)) MARK(head(pairs)) -> c16(A__HEAD(a__pairs)) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(incr(z0))) -> c17(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(odds)) -> c17(A__TAIL(a__odds), MARK(odds)) MARK(tail(head(z0))) -> c17(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c17(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(nats)) -> c17(A__TAIL(a__nats)) MARK(tail(pairs)) -> c17(A__TAIL(a__pairs)) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(incr(0))) -> c13(A__INCR(a__incr(0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(head(0))) -> c13(A__INCR(a__head(0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(incr(tail(0))) -> c13(A__INCR(a__tail(0))) MARK(incr(nats)) -> c13(A__INCR(cons(0, incr(nats)))) MARK(incr(pairs)) -> c13(A__INCR(cons(0, incr(odds)))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs)), MARK(incr(pairs))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(incr(0))) -> c16(A__HEAD(a__incr(0))) MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs)), MARK(odds)) MARK(head(odds)) -> c16(MARK(odds)) S tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(head(z0))) -> c16(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c16(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(nats)) -> c16(A__HEAD(a__nats)) MARK(head(pairs)) -> c16(A__HEAD(a__pairs)) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(incr(z0))) -> c17(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(odds)) -> c17(A__TAIL(a__odds), MARK(odds)) MARK(tail(head(z0))) -> c17(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c17(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(nats)) -> c17(A__TAIL(a__nats)) MARK(tail(pairs)) -> c17(A__TAIL(a__pairs)) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(incr(0))) -> c13(A__INCR(a__incr(0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(head(0))) -> c13(A__INCR(a__head(0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(incr(tail(0))) -> c13(A__INCR(a__tail(0))) MARK(incr(nats)) -> c13(A__INCR(cons(0, incr(nats)))) MARK(incr(pairs)) -> c13(A__INCR(cons(0, incr(odds)))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs)), MARK(incr(pairs))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(incr(0))) -> c16(A__HEAD(a__incr(0))) MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs)), MARK(odds)) MARK(head(odds)) -> c16(MARK(odds)) K tuples:none Defined Rule Symbols: a__nats, a__pairs, a__odds, a__incr_1, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__HEAD_1, A__TAIL_1, MARK_1, A__ODDS Compound Symbols: c6_1, c8_1, c10_1, c15_1, c18_1, c20_1, c13_2, c13_1, c16_2, c16_1, c17_2, c17_1, c4_1 ---------------------------------------- (83) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(head(head(z0))) -> c16(A__HEAD(a__head(mark(z0))), MARK(head(z0))) by MARK(head(head(x0))) -> c16(A__HEAD(head(mark(x0))), MARK(head(x0))) MARK(head(head(nats))) -> c16(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(pairs))) -> c16(A__HEAD(a__head(a__pairs)), MARK(head(pairs))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(0))) -> c16(A__HEAD(a__head(0)), MARK(head(0))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) ---------------------------------------- (84) Obligation: Complexity Dependency Tuples Problem Rules: a__nats -> cons(0, incr(nats)) a__nats -> nats a__pairs -> cons(0, incr(odds)) a__pairs -> pairs a__odds -> a__incr(a__pairs) a__odds -> odds a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) 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(nats) -> a__nats mark(incr(z0)) -> a__incr(mark(z0)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(tail(z0))) -> c16(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(nats)) -> c16(A__HEAD(a__nats)) MARK(head(pairs)) -> c16(A__HEAD(a__pairs)) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(incr(z0))) -> c17(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(odds)) -> c17(A__TAIL(a__odds), MARK(odds)) MARK(tail(head(z0))) -> c17(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c17(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(nats)) -> c17(A__TAIL(a__nats)) MARK(tail(pairs)) -> c17(A__TAIL(a__pairs)) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(incr(0))) -> c13(A__INCR(a__incr(0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(head(0))) -> c13(A__INCR(a__head(0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(incr(tail(0))) -> c13(A__INCR(a__tail(0))) MARK(incr(nats)) -> c13(A__INCR(cons(0, incr(nats)))) MARK(incr(pairs)) -> c13(A__INCR(cons(0, incr(odds)))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs)), MARK(incr(pairs))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(incr(0))) -> c16(A__HEAD(a__incr(0))) MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs)), MARK(odds)) MARK(head(odds)) -> c16(MARK(odds)) MARK(head(head(x0))) -> c16(A__HEAD(head(mark(x0))), MARK(head(x0))) MARK(head(head(nats))) -> c16(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(pairs))) -> c16(A__HEAD(a__head(a__pairs)), MARK(head(pairs))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(0))) -> c16(A__HEAD(a__head(0)), MARK(head(0))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) S tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(tail(z0))) -> c16(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(nats)) -> c16(A__HEAD(a__nats)) MARK(head(pairs)) -> c16(A__HEAD(a__pairs)) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(incr(z0))) -> c17(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(odds)) -> c17(A__TAIL(a__odds), MARK(odds)) MARK(tail(head(z0))) -> c17(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c17(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(nats)) -> c17(A__TAIL(a__nats)) MARK(tail(pairs)) -> c17(A__TAIL(a__pairs)) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(incr(0))) -> c13(A__INCR(a__incr(0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(head(0))) -> c13(A__INCR(a__head(0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(incr(tail(0))) -> c13(A__INCR(a__tail(0))) MARK(incr(nats)) -> c13(A__INCR(cons(0, incr(nats)))) MARK(incr(pairs)) -> c13(A__INCR(cons(0, incr(odds)))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs)), MARK(incr(pairs))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(incr(0))) -> c16(A__HEAD(a__incr(0))) MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs)), MARK(odds)) MARK(head(odds)) -> c16(MARK(odds)) MARK(head(head(x0))) -> c16(A__HEAD(head(mark(x0))), MARK(head(x0))) MARK(head(head(nats))) -> c16(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(pairs))) -> c16(A__HEAD(a__head(a__pairs)), MARK(head(pairs))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(0))) -> c16(A__HEAD(a__head(0)), MARK(head(0))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) K tuples:none Defined Rule Symbols: a__nats, a__pairs, a__odds, a__incr_1, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__HEAD_1, A__TAIL_1, MARK_1, A__ODDS Compound Symbols: c6_1, c8_1, c10_1, c15_1, c18_1, c20_1, c13_2, c13_1, c16_2, c16_1, c17_2, c17_1, c4_1 ---------------------------------------- (85) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (86) Obligation: Complexity Dependency Tuples Problem Rules: a__nats -> cons(0, incr(nats)) a__nats -> nats a__pairs -> cons(0, incr(odds)) a__pairs -> pairs a__odds -> a__incr(a__pairs) a__odds -> odds a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) 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(nats) -> a__nats mark(incr(z0)) -> a__incr(mark(z0)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(tail(z0))) -> c16(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(nats)) -> c16(A__HEAD(a__nats)) MARK(head(pairs)) -> c16(A__HEAD(a__pairs)) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(incr(z0))) -> c17(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(odds)) -> c17(A__TAIL(a__odds), MARK(odds)) MARK(tail(head(z0))) -> c17(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c17(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(nats)) -> c17(A__TAIL(a__nats)) MARK(tail(pairs)) -> c17(A__TAIL(a__pairs)) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(incr(0))) -> c13(A__INCR(a__incr(0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(head(0))) -> c13(A__INCR(a__head(0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(incr(tail(0))) -> c13(A__INCR(a__tail(0))) MARK(incr(nats)) -> c13(A__INCR(cons(0, incr(nats)))) MARK(incr(pairs)) -> c13(A__INCR(cons(0, incr(odds)))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs)), MARK(incr(pairs))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(incr(0))) -> c16(A__HEAD(a__incr(0))) MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs)), MARK(odds)) MARK(head(odds)) -> c16(MARK(odds)) MARK(head(head(nats))) -> c16(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(pairs))) -> c16(A__HEAD(a__head(a__pairs)), MARK(head(pairs))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(head(0))) -> c16(A__HEAD(a__head(0))) S tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(tail(z0))) -> c16(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(nats)) -> c16(A__HEAD(a__nats)) MARK(head(pairs)) -> c16(A__HEAD(a__pairs)) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(incr(z0))) -> c17(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(odds)) -> c17(A__TAIL(a__odds), MARK(odds)) MARK(tail(head(z0))) -> c17(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c17(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(nats)) -> c17(A__TAIL(a__nats)) MARK(tail(pairs)) -> c17(A__TAIL(a__pairs)) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(incr(0))) -> c13(A__INCR(a__incr(0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(head(0))) -> c13(A__INCR(a__head(0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(incr(tail(0))) -> c13(A__INCR(a__tail(0))) MARK(incr(nats)) -> c13(A__INCR(cons(0, incr(nats)))) MARK(incr(pairs)) -> c13(A__INCR(cons(0, incr(odds)))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs)), MARK(incr(pairs))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(incr(0))) -> c16(A__HEAD(a__incr(0))) MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs)), MARK(odds)) MARK(head(odds)) -> c16(MARK(odds)) MARK(head(head(nats))) -> c16(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(pairs))) -> c16(A__HEAD(a__head(a__pairs)), MARK(head(pairs))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(head(0))) -> c16(A__HEAD(a__head(0))) K tuples:none Defined Rule Symbols: a__nats, a__pairs, a__odds, a__incr_1, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__HEAD_1, A__TAIL_1, MARK_1, A__ODDS Compound Symbols: c6_1, c8_1, c10_1, c15_1, c18_1, c20_1, c13_2, c13_1, c16_2, c16_1, c17_2, c17_1, c4_1 ---------------------------------------- (87) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(head(tail(z0))) -> c16(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) by MARK(head(tail(x0))) -> c16(A__HEAD(tail(mark(x0))), MARK(tail(x0))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(0))) -> c16(A__HEAD(a__tail(0)), MARK(tail(0))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) ---------------------------------------- (88) Obligation: Complexity Dependency Tuples Problem Rules: a__nats -> cons(0, incr(nats)) a__nats -> nats a__pairs -> cons(0, incr(odds)) a__pairs -> pairs a__odds -> a__incr(a__pairs) a__odds -> odds a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) 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(nats) -> a__nats mark(incr(z0)) -> a__incr(mark(z0)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(nats)) -> c16(A__HEAD(a__nats)) MARK(head(pairs)) -> c16(A__HEAD(a__pairs)) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(incr(z0))) -> c17(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(odds)) -> c17(A__TAIL(a__odds), MARK(odds)) MARK(tail(head(z0))) -> c17(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c17(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(nats)) -> c17(A__TAIL(a__nats)) MARK(tail(pairs)) -> c17(A__TAIL(a__pairs)) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(incr(0))) -> c13(A__INCR(a__incr(0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(head(0))) -> c13(A__INCR(a__head(0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(incr(tail(0))) -> c13(A__INCR(a__tail(0))) MARK(incr(nats)) -> c13(A__INCR(cons(0, incr(nats)))) MARK(incr(pairs)) -> c13(A__INCR(cons(0, incr(odds)))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs)), MARK(incr(pairs))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(incr(0))) -> c16(A__HEAD(a__incr(0))) MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs)), MARK(odds)) MARK(head(odds)) -> c16(MARK(odds)) MARK(head(head(nats))) -> c16(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(pairs))) -> c16(A__HEAD(a__head(a__pairs)), MARK(head(pairs))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(head(0))) -> c16(A__HEAD(a__head(0))) MARK(head(tail(x0))) -> c16(A__HEAD(tail(mark(x0))), MARK(tail(x0))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(0))) -> c16(A__HEAD(a__tail(0)), MARK(tail(0))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) S tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(nats)) -> c16(A__HEAD(a__nats)) MARK(head(pairs)) -> c16(A__HEAD(a__pairs)) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(incr(z0))) -> c17(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(odds)) -> c17(A__TAIL(a__odds), MARK(odds)) MARK(tail(head(z0))) -> c17(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c17(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(nats)) -> c17(A__TAIL(a__nats)) MARK(tail(pairs)) -> c17(A__TAIL(a__pairs)) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(incr(0))) -> c13(A__INCR(a__incr(0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(head(0))) -> c13(A__INCR(a__head(0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(incr(tail(0))) -> c13(A__INCR(a__tail(0))) MARK(incr(nats)) -> c13(A__INCR(cons(0, incr(nats)))) MARK(incr(pairs)) -> c13(A__INCR(cons(0, incr(odds)))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs)), MARK(incr(pairs))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(incr(0))) -> c16(A__HEAD(a__incr(0))) MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs)), MARK(odds)) MARK(head(odds)) -> c16(MARK(odds)) MARK(head(head(nats))) -> c16(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(pairs))) -> c16(A__HEAD(a__head(a__pairs)), MARK(head(pairs))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(head(0))) -> c16(A__HEAD(a__head(0))) MARK(head(tail(x0))) -> c16(A__HEAD(tail(mark(x0))), MARK(tail(x0))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(0))) -> c16(A__HEAD(a__tail(0)), MARK(tail(0))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) K tuples:none Defined Rule Symbols: a__nats, a__pairs, a__odds, a__incr_1, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__HEAD_1, A__TAIL_1, MARK_1, A__ODDS Compound Symbols: c6_1, c8_1, c10_1, c15_1, c18_1, c20_1, c13_2, c13_1, c16_2, c16_1, c17_2, c17_1, c4_1 ---------------------------------------- (89) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (90) Obligation: Complexity Dependency Tuples Problem Rules: a__nats -> cons(0, incr(nats)) a__nats -> nats a__pairs -> cons(0, incr(odds)) a__pairs -> pairs a__odds -> a__incr(a__pairs) a__odds -> odds a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) 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(nats) -> a__nats mark(incr(z0)) -> a__incr(mark(z0)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(nats)) -> c16(A__HEAD(a__nats)) MARK(head(pairs)) -> c16(A__HEAD(a__pairs)) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(incr(z0))) -> c17(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(odds)) -> c17(A__TAIL(a__odds), MARK(odds)) MARK(tail(head(z0))) -> c17(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c17(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(nats)) -> c17(A__TAIL(a__nats)) MARK(tail(pairs)) -> c17(A__TAIL(a__pairs)) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(incr(0))) -> c13(A__INCR(a__incr(0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(head(0))) -> c13(A__INCR(a__head(0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(incr(tail(0))) -> c13(A__INCR(a__tail(0))) MARK(incr(nats)) -> c13(A__INCR(cons(0, incr(nats)))) MARK(incr(pairs)) -> c13(A__INCR(cons(0, incr(odds)))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs)), MARK(incr(pairs))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(incr(0))) -> c16(A__HEAD(a__incr(0))) MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs)), MARK(odds)) MARK(head(odds)) -> c16(MARK(odds)) MARK(head(head(nats))) -> c16(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(pairs))) -> c16(A__HEAD(a__head(a__pairs)), MARK(head(pairs))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(head(0))) -> c16(A__HEAD(a__head(0))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(head(tail(0))) -> c16(A__HEAD(a__tail(0))) S tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(nats)) -> c16(A__HEAD(a__nats)) MARK(head(pairs)) -> c16(A__HEAD(a__pairs)) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(incr(z0))) -> c17(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(odds)) -> c17(A__TAIL(a__odds), MARK(odds)) MARK(tail(head(z0))) -> c17(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c17(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(nats)) -> c17(A__TAIL(a__nats)) MARK(tail(pairs)) -> c17(A__TAIL(a__pairs)) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(incr(0))) -> c13(A__INCR(a__incr(0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(head(0))) -> c13(A__INCR(a__head(0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(incr(tail(0))) -> c13(A__INCR(a__tail(0))) MARK(incr(nats)) -> c13(A__INCR(cons(0, incr(nats)))) MARK(incr(pairs)) -> c13(A__INCR(cons(0, incr(odds)))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs)), MARK(incr(pairs))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(incr(0))) -> c16(A__HEAD(a__incr(0))) MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs)), MARK(odds)) MARK(head(odds)) -> c16(MARK(odds)) MARK(head(head(nats))) -> c16(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(pairs))) -> c16(A__HEAD(a__head(a__pairs)), MARK(head(pairs))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(head(0))) -> c16(A__HEAD(a__head(0))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(head(tail(0))) -> c16(A__HEAD(a__tail(0))) K tuples:none Defined Rule Symbols: a__nats, a__pairs, a__odds, a__incr_1, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__HEAD_1, A__TAIL_1, MARK_1, A__ODDS Compound Symbols: c6_1, c8_1, c10_1, c15_1, c18_1, c20_1, c13_2, c13_1, c16_2, c16_1, c17_2, c17_1, c4_1 ---------------------------------------- (91) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(head(nats)) -> c16(A__HEAD(a__nats)) by MARK(head(nats)) -> c16(A__HEAD(cons(0, incr(nats)))) MARK(head(nats)) -> c16(A__HEAD(nats)) ---------------------------------------- (92) Obligation: Complexity Dependency Tuples Problem Rules: a__nats -> cons(0, incr(nats)) a__nats -> nats a__pairs -> cons(0, incr(odds)) a__pairs -> pairs a__odds -> a__incr(a__pairs) a__odds -> odds a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) 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(nats) -> a__nats mark(incr(z0)) -> a__incr(mark(z0)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(pairs)) -> c16(A__HEAD(a__pairs)) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(incr(z0))) -> c17(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(odds)) -> c17(A__TAIL(a__odds), MARK(odds)) MARK(tail(head(z0))) -> c17(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c17(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(nats)) -> c17(A__TAIL(a__nats)) MARK(tail(pairs)) -> c17(A__TAIL(a__pairs)) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(incr(0))) -> c13(A__INCR(a__incr(0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(head(0))) -> c13(A__INCR(a__head(0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(incr(tail(0))) -> c13(A__INCR(a__tail(0))) MARK(incr(nats)) -> c13(A__INCR(cons(0, incr(nats)))) MARK(incr(pairs)) -> c13(A__INCR(cons(0, incr(odds)))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs)), MARK(incr(pairs))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(incr(0))) -> c16(A__HEAD(a__incr(0))) MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs)), MARK(odds)) MARK(head(odds)) -> c16(MARK(odds)) MARK(head(head(nats))) -> c16(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(pairs))) -> c16(A__HEAD(a__head(a__pairs)), MARK(head(pairs))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(head(0))) -> c16(A__HEAD(a__head(0))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(head(tail(0))) -> c16(A__HEAD(a__tail(0))) MARK(head(nats)) -> c16(A__HEAD(cons(0, incr(nats)))) MARK(head(nats)) -> c16(A__HEAD(nats)) S tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(pairs)) -> c16(A__HEAD(a__pairs)) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(incr(z0))) -> c17(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(odds)) -> c17(A__TAIL(a__odds), MARK(odds)) MARK(tail(head(z0))) -> c17(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c17(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(nats)) -> c17(A__TAIL(a__nats)) MARK(tail(pairs)) -> c17(A__TAIL(a__pairs)) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(incr(0))) -> c13(A__INCR(a__incr(0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(head(0))) -> c13(A__INCR(a__head(0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(incr(tail(0))) -> c13(A__INCR(a__tail(0))) MARK(incr(nats)) -> c13(A__INCR(cons(0, incr(nats)))) MARK(incr(pairs)) -> c13(A__INCR(cons(0, incr(odds)))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs)), MARK(incr(pairs))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(incr(0))) -> c16(A__HEAD(a__incr(0))) MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs)), MARK(odds)) MARK(head(odds)) -> c16(MARK(odds)) MARK(head(head(nats))) -> c16(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(pairs))) -> c16(A__HEAD(a__head(a__pairs)), MARK(head(pairs))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(head(0))) -> c16(A__HEAD(a__head(0))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(head(tail(0))) -> c16(A__HEAD(a__tail(0))) MARK(head(nats)) -> c16(A__HEAD(cons(0, incr(nats)))) MARK(head(nats)) -> c16(A__HEAD(nats)) K tuples:none Defined Rule Symbols: a__nats, a__pairs, a__odds, a__incr_1, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__HEAD_1, A__TAIL_1, MARK_1, A__ODDS Compound Symbols: c6_1, c8_1, c10_1, c15_1, c18_1, c20_1, c13_2, c13_1, c16_2, c16_1, c17_2, c17_1, c4_1 ---------------------------------------- (93) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(head(nats)) -> c16(A__HEAD(nats)) ---------------------------------------- (94) Obligation: Complexity Dependency Tuples Problem Rules: a__nats -> cons(0, incr(nats)) a__nats -> nats a__pairs -> cons(0, incr(odds)) a__pairs -> pairs a__odds -> a__incr(a__pairs) a__odds -> odds a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) 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(nats) -> a__nats mark(incr(z0)) -> a__incr(mark(z0)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(pairs)) -> c16(A__HEAD(a__pairs)) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(incr(z0))) -> c17(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(odds)) -> c17(A__TAIL(a__odds), MARK(odds)) MARK(tail(head(z0))) -> c17(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c17(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(nats)) -> c17(A__TAIL(a__nats)) MARK(tail(pairs)) -> c17(A__TAIL(a__pairs)) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(incr(0))) -> c13(A__INCR(a__incr(0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(head(0))) -> c13(A__INCR(a__head(0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(incr(tail(0))) -> c13(A__INCR(a__tail(0))) MARK(incr(nats)) -> c13(A__INCR(cons(0, incr(nats)))) MARK(incr(pairs)) -> c13(A__INCR(cons(0, incr(odds)))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs)), MARK(incr(pairs))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(incr(0))) -> c16(A__HEAD(a__incr(0))) MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs)), MARK(odds)) MARK(head(odds)) -> c16(MARK(odds)) MARK(head(head(nats))) -> c16(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(pairs))) -> c16(A__HEAD(a__head(a__pairs)), MARK(head(pairs))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(head(0))) -> c16(A__HEAD(a__head(0))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(head(tail(0))) -> c16(A__HEAD(a__tail(0))) MARK(head(nats)) -> c16(A__HEAD(cons(0, incr(nats)))) S tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(pairs)) -> c16(A__HEAD(a__pairs)) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(incr(z0))) -> c17(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(odds)) -> c17(A__TAIL(a__odds), MARK(odds)) MARK(tail(head(z0))) -> c17(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c17(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(nats)) -> c17(A__TAIL(a__nats)) MARK(tail(pairs)) -> c17(A__TAIL(a__pairs)) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(incr(0))) -> c13(A__INCR(a__incr(0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(head(0))) -> c13(A__INCR(a__head(0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(incr(tail(0))) -> c13(A__INCR(a__tail(0))) MARK(incr(nats)) -> c13(A__INCR(cons(0, incr(nats)))) MARK(incr(pairs)) -> c13(A__INCR(cons(0, incr(odds)))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs)), MARK(incr(pairs))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(incr(0))) -> c16(A__HEAD(a__incr(0))) MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs)), MARK(odds)) MARK(head(odds)) -> c16(MARK(odds)) MARK(head(head(nats))) -> c16(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(pairs))) -> c16(A__HEAD(a__head(a__pairs)), MARK(head(pairs))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(head(0))) -> c16(A__HEAD(a__head(0))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(head(tail(0))) -> c16(A__HEAD(a__tail(0))) MARK(head(nats)) -> c16(A__HEAD(cons(0, incr(nats)))) K tuples:none Defined Rule Symbols: a__nats, a__pairs, a__odds, a__incr_1, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__HEAD_1, A__TAIL_1, MARK_1, A__ODDS Compound Symbols: c6_1, c8_1, c10_1, c15_1, c18_1, c20_1, c13_2, c13_1, c16_2, c16_1, c17_2, c17_1, c4_1 ---------------------------------------- (95) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(head(pairs)) -> c16(A__HEAD(a__pairs)) by MARK(head(pairs)) -> c16(A__HEAD(cons(0, incr(odds)))) MARK(head(pairs)) -> c16(A__HEAD(pairs)) ---------------------------------------- (96) Obligation: Complexity Dependency Tuples Problem Rules: a__nats -> cons(0, incr(nats)) a__nats -> nats a__pairs -> cons(0, incr(odds)) a__pairs -> pairs a__odds -> a__incr(a__pairs) a__odds -> odds a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) 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(nats) -> a__nats mark(incr(z0)) -> a__incr(mark(z0)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(incr(z0))) -> c17(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(odds)) -> c17(A__TAIL(a__odds), MARK(odds)) MARK(tail(head(z0))) -> c17(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c17(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(nats)) -> c17(A__TAIL(a__nats)) MARK(tail(pairs)) -> c17(A__TAIL(a__pairs)) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(incr(0))) -> c13(A__INCR(a__incr(0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(head(0))) -> c13(A__INCR(a__head(0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(incr(tail(0))) -> c13(A__INCR(a__tail(0))) MARK(incr(nats)) -> c13(A__INCR(cons(0, incr(nats)))) MARK(incr(pairs)) -> c13(A__INCR(cons(0, incr(odds)))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs)), MARK(incr(pairs))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(incr(0))) -> c16(A__HEAD(a__incr(0))) MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs)), MARK(odds)) MARK(head(odds)) -> c16(MARK(odds)) MARK(head(head(nats))) -> c16(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(pairs))) -> c16(A__HEAD(a__head(a__pairs)), MARK(head(pairs))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(head(0))) -> c16(A__HEAD(a__head(0))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(head(tail(0))) -> c16(A__HEAD(a__tail(0))) MARK(head(nats)) -> c16(A__HEAD(cons(0, incr(nats)))) MARK(head(pairs)) -> c16(A__HEAD(cons(0, incr(odds)))) MARK(head(pairs)) -> c16(A__HEAD(pairs)) S tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(incr(z0))) -> c17(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(odds)) -> c17(A__TAIL(a__odds), MARK(odds)) MARK(tail(head(z0))) -> c17(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c17(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(nats)) -> c17(A__TAIL(a__nats)) MARK(tail(pairs)) -> c17(A__TAIL(a__pairs)) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(incr(0))) -> c13(A__INCR(a__incr(0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(head(0))) -> c13(A__INCR(a__head(0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(incr(tail(0))) -> c13(A__INCR(a__tail(0))) MARK(incr(nats)) -> c13(A__INCR(cons(0, incr(nats)))) MARK(incr(pairs)) -> c13(A__INCR(cons(0, incr(odds)))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs)), MARK(incr(pairs))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(incr(0))) -> c16(A__HEAD(a__incr(0))) MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs)), MARK(odds)) MARK(head(odds)) -> c16(MARK(odds)) MARK(head(head(nats))) -> c16(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(pairs))) -> c16(A__HEAD(a__head(a__pairs)), MARK(head(pairs))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(head(0))) -> c16(A__HEAD(a__head(0))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(head(tail(0))) -> c16(A__HEAD(a__tail(0))) MARK(head(nats)) -> c16(A__HEAD(cons(0, incr(nats)))) MARK(head(pairs)) -> c16(A__HEAD(cons(0, incr(odds)))) MARK(head(pairs)) -> c16(A__HEAD(pairs)) K tuples:none Defined Rule Symbols: a__nats, a__pairs, a__odds, a__incr_1, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__HEAD_1, A__TAIL_1, MARK_1, A__ODDS Compound Symbols: c6_1, c8_1, c10_1, c15_1, c18_1, c20_1, c13_2, c13_1, c16_2, c16_1, c17_2, c17_1, c4_1 ---------------------------------------- (97) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(head(pairs)) -> c16(A__HEAD(pairs)) ---------------------------------------- (98) Obligation: Complexity Dependency Tuples Problem Rules: a__nats -> cons(0, incr(nats)) a__nats -> nats a__pairs -> cons(0, incr(odds)) a__pairs -> pairs a__odds -> a__incr(a__pairs) a__odds -> odds a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) 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(nats) -> a__nats mark(incr(z0)) -> a__incr(mark(z0)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(incr(z0))) -> c17(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(odds)) -> c17(A__TAIL(a__odds), MARK(odds)) MARK(tail(head(z0))) -> c17(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c17(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(nats)) -> c17(A__TAIL(a__nats)) MARK(tail(pairs)) -> c17(A__TAIL(a__pairs)) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(incr(0))) -> c13(A__INCR(a__incr(0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(head(0))) -> c13(A__INCR(a__head(0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(incr(tail(0))) -> c13(A__INCR(a__tail(0))) MARK(incr(nats)) -> c13(A__INCR(cons(0, incr(nats)))) MARK(incr(pairs)) -> c13(A__INCR(cons(0, incr(odds)))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs)), MARK(incr(pairs))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(incr(0))) -> c16(A__HEAD(a__incr(0))) MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs)), MARK(odds)) MARK(head(odds)) -> c16(MARK(odds)) MARK(head(head(nats))) -> c16(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(pairs))) -> c16(A__HEAD(a__head(a__pairs)), MARK(head(pairs))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(head(0))) -> c16(A__HEAD(a__head(0))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(head(tail(0))) -> c16(A__HEAD(a__tail(0))) MARK(head(nats)) -> c16(A__HEAD(cons(0, incr(nats)))) MARK(head(pairs)) -> c16(A__HEAD(cons(0, incr(odds)))) S tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(incr(z0))) -> c17(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) MARK(tail(odds)) -> c17(A__TAIL(a__odds), MARK(odds)) MARK(tail(head(z0))) -> c17(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c17(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(nats)) -> c17(A__TAIL(a__nats)) MARK(tail(pairs)) -> c17(A__TAIL(a__pairs)) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(incr(0))) -> c13(A__INCR(a__incr(0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(head(0))) -> c13(A__INCR(a__head(0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(incr(tail(0))) -> c13(A__INCR(a__tail(0))) MARK(incr(nats)) -> c13(A__INCR(cons(0, incr(nats)))) MARK(incr(pairs)) -> c13(A__INCR(cons(0, incr(odds)))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs)), MARK(incr(pairs))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(incr(0))) -> c16(A__HEAD(a__incr(0))) MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs)), MARK(odds)) MARK(head(odds)) -> c16(MARK(odds)) MARK(head(head(nats))) -> c16(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(pairs))) -> c16(A__HEAD(a__head(a__pairs)), MARK(head(pairs))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(head(0))) -> c16(A__HEAD(a__head(0))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(head(tail(0))) -> c16(A__HEAD(a__tail(0))) MARK(head(nats)) -> c16(A__HEAD(cons(0, incr(nats)))) MARK(head(pairs)) -> c16(A__HEAD(cons(0, incr(odds)))) K tuples:none Defined Rule Symbols: a__nats, a__pairs, a__odds, a__incr_1, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__HEAD_1, A__TAIL_1, MARK_1, A__ODDS Compound Symbols: c6_1, c8_1, c10_1, c15_1, c18_1, c20_1, c13_2, c13_1, c16_2, c16_1, c17_2, c17_1, c4_1 ---------------------------------------- (99) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(tail(incr(z0))) -> c17(A__TAIL(a__incr(mark(z0))), MARK(incr(z0))) by MARK(tail(incr(x0))) -> c17(A__TAIL(incr(mark(x0))), MARK(incr(x0))) MARK(tail(incr(nats))) -> c17(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(incr(z0)))) -> c17(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(pairs))) -> c17(A__TAIL(a__incr(a__pairs)), MARK(incr(pairs))) MARK(tail(incr(odds))) -> c17(A__TAIL(a__incr(a__odds)), MARK(incr(odds))) MARK(tail(incr(head(z0)))) -> c17(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c17(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c17(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(0))) -> c17(A__TAIL(a__incr(0)), MARK(incr(0))) MARK(tail(incr(s(z0)))) -> c17(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) ---------------------------------------- (100) Obligation: Complexity Dependency Tuples Problem Rules: a__nats -> cons(0, incr(nats)) a__nats -> nats a__pairs -> cons(0, incr(odds)) a__pairs -> pairs a__odds -> a__incr(a__pairs) a__odds -> odds a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) 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(nats) -> a__nats mark(incr(z0)) -> a__incr(mark(z0)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(odds)) -> c17(A__TAIL(a__odds), MARK(odds)) MARK(tail(head(z0))) -> c17(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c17(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(nats)) -> c17(A__TAIL(a__nats)) MARK(tail(pairs)) -> c17(A__TAIL(a__pairs)) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(incr(0))) -> c13(A__INCR(a__incr(0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(head(0))) -> c13(A__INCR(a__head(0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(incr(tail(0))) -> c13(A__INCR(a__tail(0))) MARK(incr(nats)) -> c13(A__INCR(cons(0, incr(nats)))) MARK(incr(pairs)) -> c13(A__INCR(cons(0, incr(odds)))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs)), MARK(incr(pairs))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(incr(0))) -> c16(A__HEAD(a__incr(0))) MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs)), MARK(odds)) MARK(head(odds)) -> c16(MARK(odds)) MARK(head(head(nats))) -> c16(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(pairs))) -> c16(A__HEAD(a__head(a__pairs)), MARK(head(pairs))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(head(0))) -> c16(A__HEAD(a__head(0))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(head(tail(0))) -> c16(A__HEAD(a__tail(0))) MARK(head(nats)) -> c16(A__HEAD(cons(0, incr(nats)))) MARK(head(pairs)) -> c16(A__HEAD(cons(0, incr(odds)))) MARK(tail(incr(x0))) -> c17(A__TAIL(incr(mark(x0))), MARK(incr(x0))) MARK(tail(incr(nats))) -> c17(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(incr(z0)))) -> c17(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(pairs))) -> c17(A__TAIL(a__incr(a__pairs)), MARK(incr(pairs))) MARK(tail(incr(odds))) -> c17(A__TAIL(a__incr(a__odds)), MARK(incr(odds))) MARK(tail(incr(head(z0)))) -> c17(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c17(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c17(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(0))) -> c17(A__TAIL(a__incr(0)), MARK(incr(0))) MARK(tail(incr(s(z0)))) -> c17(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) S tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(odds)) -> c17(A__TAIL(a__odds), MARK(odds)) MARK(tail(head(z0))) -> c17(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c17(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(nats)) -> c17(A__TAIL(a__nats)) MARK(tail(pairs)) -> c17(A__TAIL(a__pairs)) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(incr(0))) -> c13(A__INCR(a__incr(0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(head(0))) -> c13(A__INCR(a__head(0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(incr(tail(0))) -> c13(A__INCR(a__tail(0))) MARK(incr(nats)) -> c13(A__INCR(cons(0, incr(nats)))) MARK(incr(pairs)) -> c13(A__INCR(cons(0, incr(odds)))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs)), MARK(incr(pairs))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(incr(0))) -> c16(A__HEAD(a__incr(0))) MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs)), MARK(odds)) MARK(head(odds)) -> c16(MARK(odds)) MARK(head(head(nats))) -> c16(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(pairs))) -> c16(A__HEAD(a__head(a__pairs)), MARK(head(pairs))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(head(0))) -> c16(A__HEAD(a__head(0))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(head(tail(0))) -> c16(A__HEAD(a__tail(0))) MARK(head(nats)) -> c16(A__HEAD(cons(0, incr(nats)))) MARK(head(pairs)) -> c16(A__HEAD(cons(0, incr(odds)))) MARK(tail(incr(x0))) -> c17(A__TAIL(incr(mark(x0))), MARK(incr(x0))) MARK(tail(incr(nats))) -> c17(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(incr(z0)))) -> c17(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(pairs))) -> c17(A__TAIL(a__incr(a__pairs)), MARK(incr(pairs))) MARK(tail(incr(odds))) -> c17(A__TAIL(a__incr(a__odds)), MARK(incr(odds))) MARK(tail(incr(head(z0)))) -> c17(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c17(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c17(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(0))) -> c17(A__TAIL(a__incr(0)), MARK(incr(0))) MARK(tail(incr(s(z0)))) -> c17(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) K tuples:none Defined Rule Symbols: a__nats, a__pairs, a__odds, a__incr_1, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__HEAD_1, A__TAIL_1, MARK_1, A__ODDS Compound Symbols: c6_1, c8_1, c10_1, c15_1, c18_1, c20_1, c13_2, c13_1, c16_2, c16_1, c17_2, c17_1, c4_1 ---------------------------------------- (101) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (102) Obligation: Complexity Dependency Tuples Problem Rules: a__nats -> cons(0, incr(nats)) a__nats -> nats a__pairs -> cons(0, incr(odds)) a__pairs -> pairs a__odds -> a__incr(a__pairs) a__odds -> odds a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) 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(nats) -> a__nats mark(incr(z0)) -> a__incr(mark(z0)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(odds)) -> c17(A__TAIL(a__odds), MARK(odds)) MARK(tail(head(z0))) -> c17(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c17(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(nats)) -> c17(A__TAIL(a__nats)) MARK(tail(pairs)) -> c17(A__TAIL(a__pairs)) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(incr(0))) -> c13(A__INCR(a__incr(0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(head(0))) -> c13(A__INCR(a__head(0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(incr(tail(0))) -> c13(A__INCR(a__tail(0))) MARK(incr(nats)) -> c13(A__INCR(cons(0, incr(nats)))) MARK(incr(pairs)) -> c13(A__INCR(cons(0, incr(odds)))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs)), MARK(incr(pairs))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(incr(0))) -> c16(A__HEAD(a__incr(0))) MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs)), MARK(odds)) MARK(head(odds)) -> c16(MARK(odds)) MARK(head(head(nats))) -> c16(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(pairs))) -> c16(A__HEAD(a__head(a__pairs)), MARK(head(pairs))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(head(0))) -> c16(A__HEAD(a__head(0))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(head(tail(0))) -> c16(A__HEAD(a__tail(0))) MARK(head(nats)) -> c16(A__HEAD(cons(0, incr(nats)))) MARK(head(pairs)) -> c16(A__HEAD(cons(0, incr(odds)))) MARK(tail(incr(nats))) -> c17(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(incr(z0)))) -> c17(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(pairs))) -> c17(A__TAIL(a__incr(a__pairs)), MARK(incr(pairs))) MARK(tail(incr(odds))) -> c17(A__TAIL(a__incr(a__odds)), MARK(incr(odds))) MARK(tail(incr(head(z0)))) -> c17(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c17(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c17(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c17(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c17(MARK(incr(x0))) MARK(tail(incr(0))) -> c17(A__TAIL(a__incr(0))) S tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(odds)) -> c17(A__TAIL(a__odds), MARK(odds)) MARK(tail(head(z0))) -> c17(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c17(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(nats)) -> c17(A__TAIL(a__nats)) MARK(tail(pairs)) -> c17(A__TAIL(a__pairs)) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(incr(0))) -> c13(A__INCR(a__incr(0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(head(0))) -> c13(A__INCR(a__head(0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(incr(tail(0))) -> c13(A__INCR(a__tail(0))) MARK(incr(nats)) -> c13(A__INCR(cons(0, incr(nats)))) MARK(incr(pairs)) -> c13(A__INCR(cons(0, incr(odds)))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs)), MARK(incr(pairs))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(incr(0))) -> c16(A__HEAD(a__incr(0))) MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs)), MARK(odds)) MARK(head(odds)) -> c16(MARK(odds)) MARK(head(head(nats))) -> c16(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(pairs))) -> c16(A__HEAD(a__head(a__pairs)), MARK(head(pairs))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(head(0))) -> c16(A__HEAD(a__head(0))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(head(tail(0))) -> c16(A__HEAD(a__tail(0))) MARK(head(nats)) -> c16(A__HEAD(cons(0, incr(nats)))) MARK(head(pairs)) -> c16(A__HEAD(cons(0, incr(odds)))) MARK(tail(incr(nats))) -> c17(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(incr(z0)))) -> c17(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(pairs))) -> c17(A__TAIL(a__incr(a__pairs)), MARK(incr(pairs))) MARK(tail(incr(odds))) -> c17(A__TAIL(a__incr(a__odds)), MARK(incr(odds))) MARK(tail(incr(head(z0)))) -> c17(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c17(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c17(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c17(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c17(MARK(incr(x0))) MARK(tail(incr(0))) -> c17(A__TAIL(a__incr(0))) K tuples:none Defined Rule Symbols: a__nats, a__pairs, a__odds, a__incr_1, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__HEAD_1, A__TAIL_1, MARK_1, A__ODDS Compound Symbols: c6_1, c8_1, c10_1, c15_1, c18_1, c20_1, c13_2, c13_1, c16_2, c16_1, c17_2, c17_1, c4_1 ---------------------------------------- (103) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(tail(odds)) -> c17(A__TAIL(a__odds), MARK(odds)) by MARK(tail(odds)) -> c17(A__TAIL(a__incr(a__pairs)), MARK(odds)) MARK(tail(odds)) -> c17(A__TAIL(odds), MARK(odds)) ---------------------------------------- (104) Obligation: Complexity Dependency Tuples Problem Rules: a__nats -> cons(0, incr(nats)) a__nats -> nats a__pairs -> cons(0, incr(odds)) a__pairs -> pairs a__odds -> a__incr(a__pairs) a__odds -> odds a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) 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(nats) -> a__nats mark(incr(z0)) -> a__incr(mark(z0)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(head(z0))) -> c17(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c17(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(nats)) -> c17(A__TAIL(a__nats)) MARK(tail(pairs)) -> c17(A__TAIL(a__pairs)) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(incr(0))) -> c13(A__INCR(a__incr(0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(head(0))) -> c13(A__INCR(a__head(0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(incr(tail(0))) -> c13(A__INCR(a__tail(0))) MARK(incr(nats)) -> c13(A__INCR(cons(0, incr(nats)))) MARK(incr(pairs)) -> c13(A__INCR(cons(0, incr(odds)))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs)), MARK(incr(pairs))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(incr(0))) -> c16(A__HEAD(a__incr(0))) MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs)), MARK(odds)) MARK(head(odds)) -> c16(MARK(odds)) MARK(head(head(nats))) -> c16(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(pairs))) -> c16(A__HEAD(a__head(a__pairs)), MARK(head(pairs))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(head(0))) -> c16(A__HEAD(a__head(0))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(head(tail(0))) -> c16(A__HEAD(a__tail(0))) MARK(head(nats)) -> c16(A__HEAD(cons(0, incr(nats)))) MARK(head(pairs)) -> c16(A__HEAD(cons(0, incr(odds)))) MARK(tail(incr(nats))) -> c17(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(incr(z0)))) -> c17(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(pairs))) -> c17(A__TAIL(a__incr(a__pairs)), MARK(incr(pairs))) MARK(tail(incr(odds))) -> c17(A__TAIL(a__incr(a__odds)), MARK(incr(odds))) MARK(tail(incr(head(z0)))) -> c17(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c17(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c17(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c17(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c17(MARK(incr(x0))) MARK(tail(incr(0))) -> c17(A__TAIL(a__incr(0))) MARK(tail(odds)) -> c17(A__TAIL(a__incr(a__pairs)), MARK(odds)) MARK(tail(odds)) -> c17(A__TAIL(odds), MARK(odds)) S tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(head(z0))) -> c17(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c17(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(nats)) -> c17(A__TAIL(a__nats)) MARK(tail(pairs)) -> c17(A__TAIL(a__pairs)) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(incr(0))) -> c13(A__INCR(a__incr(0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(head(0))) -> c13(A__INCR(a__head(0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(incr(tail(0))) -> c13(A__INCR(a__tail(0))) MARK(incr(nats)) -> c13(A__INCR(cons(0, incr(nats)))) MARK(incr(pairs)) -> c13(A__INCR(cons(0, incr(odds)))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs)), MARK(incr(pairs))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(incr(0))) -> c16(A__HEAD(a__incr(0))) MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs)), MARK(odds)) MARK(head(odds)) -> c16(MARK(odds)) MARK(head(head(nats))) -> c16(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(pairs))) -> c16(A__HEAD(a__head(a__pairs)), MARK(head(pairs))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(head(0))) -> c16(A__HEAD(a__head(0))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(head(tail(0))) -> c16(A__HEAD(a__tail(0))) MARK(head(nats)) -> c16(A__HEAD(cons(0, incr(nats)))) MARK(head(pairs)) -> c16(A__HEAD(cons(0, incr(odds)))) MARK(tail(incr(nats))) -> c17(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(incr(z0)))) -> c17(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(pairs))) -> c17(A__TAIL(a__incr(a__pairs)), MARK(incr(pairs))) MARK(tail(incr(odds))) -> c17(A__TAIL(a__incr(a__odds)), MARK(incr(odds))) MARK(tail(incr(head(z0)))) -> c17(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c17(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c17(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c17(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c17(MARK(incr(x0))) MARK(tail(incr(0))) -> c17(A__TAIL(a__incr(0))) MARK(tail(odds)) -> c17(A__TAIL(a__incr(a__pairs)), MARK(odds)) MARK(tail(odds)) -> c17(A__TAIL(odds), MARK(odds)) K tuples:none Defined Rule Symbols: a__nats, a__pairs, a__odds, a__incr_1, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__HEAD_1, A__TAIL_1, MARK_1, A__ODDS Compound Symbols: c6_1, c8_1, c10_1, c15_1, c18_1, c20_1, c13_2, c13_1, c16_2, c16_1, c17_2, c17_1, c4_1 ---------------------------------------- (105) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (106) Obligation: Complexity Dependency Tuples Problem Rules: a__nats -> cons(0, incr(nats)) a__nats -> nats a__pairs -> cons(0, incr(odds)) a__pairs -> pairs a__odds -> a__incr(a__pairs) a__odds -> odds a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) 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(nats) -> a__nats mark(incr(z0)) -> a__incr(mark(z0)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(head(z0))) -> c17(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c17(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(nats)) -> c17(A__TAIL(a__nats)) MARK(tail(pairs)) -> c17(A__TAIL(a__pairs)) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(incr(0))) -> c13(A__INCR(a__incr(0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(head(0))) -> c13(A__INCR(a__head(0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(incr(tail(0))) -> c13(A__INCR(a__tail(0))) MARK(incr(nats)) -> c13(A__INCR(cons(0, incr(nats)))) MARK(incr(pairs)) -> c13(A__INCR(cons(0, incr(odds)))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs)), MARK(incr(pairs))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(incr(0))) -> c16(A__HEAD(a__incr(0))) MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs)), MARK(odds)) MARK(head(odds)) -> c16(MARK(odds)) MARK(head(head(nats))) -> c16(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(pairs))) -> c16(A__HEAD(a__head(a__pairs)), MARK(head(pairs))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(head(0))) -> c16(A__HEAD(a__head(0))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(head(tail(0))) -> c16(A__HEAD(a__tail(0))) MARK(head(nats)) -> c16(A__HEAD(cons(0, incr(nats)))) MARK(head(pairs)) -> c16(A__HEAD(cons(0, incr(odds)))) MARK(tail(incr(nats))) -> c17(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(incr(z0)))) -> c17(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(pairs))) -> c17(A__TAIL(a__incr(a__pairs)), MARK(incr(pairs))) MARK(tail(incr(odds))) -> c17(A__TAIL(a__incr(a__odds)), MARK(incr(odds))) MARK(tail(incr(head(z0)))) -> c17(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c17(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c17(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c17(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c17(MARK(incr(x0))) MARK(tail(incr(0))) -> c17(A__TAIL(a__incr(0))) MARK(tail(odds)) -> c17(A__TAIL(a__incr(a__pairs)), MARK(odds)) MARK(tail(odds)) -> c17(MARK(odds)) S tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(head(z0))) -> c17(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c17(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(nats)) -> c17(A__TAIL(a__nats)) MARK(tail(pairs)) -> c17(A__TAIL(a__pairs)) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(incr(0))) -> c13(A__INCR(a__incr(0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(head(0))) -> c13(A__INCR(a__head(0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(incr(tail(0))) -> c13(A__INCR(a__tail(0))) MARK(incr(nats)) -> c13(A__INCR(cons(0, incr(nats)))) MARK(incr(pairs)) -> c13(A__INCR(cons(0, incr(odds)))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs)), MARK(incr(pairs))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(incr(0))) -> c16(A__HEAD(a__incr(0))) MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs)), MARK(odds)) MARK(head(odds)) -> c16(MARK(odds)) MARK(head(head(nats))) -> c16(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(pairs))) -> c16(A__HEAD(a__head(a__pairs)), MARK(head(pairs))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(head(0))) -> c16(A__HEAD(a__head(0))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(head(tail(0))) -> c16(A__HEAD(a__tail(0))) MARK(head(nats)) -> c16(A__HEAD(cons(0, incr(nats)))) MARK(head(pairs)) -> c16(A__HEAD(cons(0, incr(odds)))) MARK(tail(incr(nats))) -> c17(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(incr(z0)))) -> c17(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(pairs))) -> c17(A__TAIL(a__incr(a__pairs)), MARK(incr(pairs))) MARK(tail(incr(odds))) -> c17(A__TAIL(a__incr(a__odds)), MARK(incr(odds))) MARK(tail(incr(head(z0)))) -> c17(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c17(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c17(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c17(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c17(MARK(incr(x0))) MARK(tail(incr(0))) -> c17(A__TAIL(a__incr(0))) MARK(tail(odds)) -> c17(A__TAIL(a__incr(a__pairs)), MARK(odds)) MARK(tail(odds)) -> c17(MARK(odds)) K tuples:none Defined Rule Symbols: a__nats, a__pairs, a__odds, a__incr_1, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__HEAD_1, A__TAIL_1, MARK_1, A__ODDS Compound Symbols: c6_1, c8_1, c10_1, c15_1, c18_1, c20_1, c13_2, c13_1, c16_2, c16_1, c17_2, c17_1, c4_1 ---------------------------------------- (107) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(tail(head(z0))) -> c17(A__TAIL(a__head(mark(z0))), MARK(head(z0))) by MARK(tail(head(x0))) -> c17(A__TAIL(head(mark(x0))), MARK(head(x0))) MARK(tail(head(nats))) -> c17(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(incr(z0)))) -> c17(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(pairs))) -> c17(A__TAIL(a__head(a__pairs)), MARK(head(pairs))) MARK(tail(head(odds))) -> c17(A__TAIL(a__head(a__odds)), MARK(head(odds))) MARK(tail(head(head(z0)))) -> c17(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c17(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c17(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(0))) -> c17(A__TAIL(a__head(0)), MARK(head(0))) MARK(tail(head(s(z0)))) -> c17(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) ---------------------------------------- (108) Obligation: Complexity Dependency Tuples Problem Rules: a__nats -> cons(0, incr(nats)) a__nats -> nats a__pairs -> cons(0, incr(odds)) a__pairs -> pairs a__odds -> a__incr(a__pairs) a__odds -> odds a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) 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(nats) -> a__nats mark(incr(z0)) -> a__incr(mark(z0)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(tail(z0))) -> c17(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(nats)) -> c17(A__TAIL(a__nats)) MARK(tail(pairs)) -> c17(A__TAIL(a__pairs)) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(incr(0))) -> c13(A__INCR(a__incr(0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(head(0))) -> c13(A__INCR(a__head(0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(incr(tail(0))) -> c13(A__INCR(a__tail(0))) MARK(incr(nats)) -> c13(A__INCR(cons(0, incr(nats)))) MARK(incr(pairs)) -> c13(A__INCR(cons(0, incr(odds)))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs)), MARK(incr(pairs))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(incr(0))) -> c16(A__HEAD(a__incr(0))) MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs)), MARK(odds)) MARK(head(odds)) -> c16(MARK(odds)) MARK(head(head(nats))) -> c16(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(pairs))) -> c16(A__HEAD(a__head(a__pairs)), MARK(head(pairs))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(head(0))) -> c16(A__HEAD(a__head(0))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(head(tail(0))) -> c16(A__HEAD(a__tail(0))) MARK(head(nats)) -> c16(A__HEAD(cons(0, incr(nats)))) MARK(head(pairs)) -> c16(A__HEAD(cons(0, incr(odds)))) MARK(tail(incr(nats))) -> c17(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(incr(z0)))) -> c17(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(pairs))) -> c17(A__TAIL(a__incr(a__pairs)), MARK(incr(pairs))) MARK(tail(incr(odds))) -> c17(A__TAIL(a__incr(a__odds)), MARK(incr(odds))) MARK(tail(incr(head(z0)))) -> c17(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c17(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c17(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c17(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c17(MARK(incr(x0))) MARK(tail(incr(0))) -> c17(A__TAIL(a__incr(0))) MARK(tail(odds)) -> c17(A__TAIL(a__incr(a__pairs)), MARK(odds)) MARK(tail(odds)) -> c17(MARK(odds)) MARK(tail(head(x0))) -> c17(A__TAIL(head(mark(x0))), MARK(head(x0))) MARK(tail(head(nats))) -> c17(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(incr(z0)))) -> c17(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(pairs))) -> c17(A__TAIL(a__head(a__pairs)), MARK(head(pairs))) MARK(tail(head(odds))) -> c17(A__TAIL(a__head(a__odds)), MARK(head(odds))) MARK(tail(head(head(z0)))) -> c17(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c17(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c17(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(0))) -> c17(A__TAIL(a__head(0)), MARK(head(0))) MARK(tail(head(s(z0)))) -> c17(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) S tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(tail(z0))) -> c17(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(nats)) -> c17(A__TAIL(a__nats)) MARK(tail(pairs)) -> c17(A__TAIL(a__pairs)) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(incr(0))) -> c13(A__INCR(a__incr(0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(head(0))) -> c13(A__INCR(a__head(0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(incr(tail(0))) -> c13(A__INCR(a__tail(0))) MARK(incr(nats)) -> c13(A__INCR(cons(0, incr(nats)))) MARK(incr(pairs)) -> c13(A__INCR(cons(0, incr(odds)))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs)), MARK(incr(pairs))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(incr(0))) -> c16(A__HEAD(a__incr(0))) MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs)), MARK(odds)) MARK(head(odds)) -> c16(MARK(odds)) MARK(head(head(nats))) -> c16(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(pairs))) -> c16(A__HEAD(a__head(a__pairs)), MARK(head(pairs))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(head(0))) -> c16(A__HEAD(a__head(0))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(head(tail(0))) -> c16(A__HEAD(a__tail(0))) MARK(head(nats)) -> c16(A__HEAD(cons(0, incr(nats)))) MARK(head(pairs)) -> c16(A__HEAD(cons(0, incr(odds)))) MARK(tail(incr(nats))) -> c17(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(incr(z0)))) -> c17(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(pairs))) -> c17(A__TAIL(a__incr(a__pairs)), MARK(incr(pairs))) MARK(tail(incr(odds))) -> c17(A__TAIL(a__incr(a__odds)), MARK(incr(odds))) MARK(tail(incr(head(z0)))) -> c17(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c17(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c17(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c17(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c17(MARK(incr(x0))) MARK(tail(incr(0))) -> c17(A__TAIL(a__incr(0))) MARK(tail(odds)) -> c17(A__TAIL(a__incr(a__pairs)), MARK(odds)) MARK(tail(odds)) -> c17(MARK(odds)) MARK(tail(head(x0))) -> c17(A__TAIL(head(mark(x0))), MARK(head(x0))) MARK(tail(head(nats))) -> c17(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(incr(z0)))) -> c17(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(pairs))) -> c17(A__TAIL(a__head(a__pairs)), MARK(head(pairs))) MARK(tail(head(odds))) -> c17(A__TAIL(a__head(a__odds)), MARK(head(odds))) MARK(tail(head(head(z0)))) -> c17(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c17(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c17(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(0))) -> c17(A__TAIL(a__head(0)), MARK(head(0))) MARK(tail(head(s(z0)))) -> c17(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) K tuples:none Defined Rule Symbols: a__nats, a__pairs, a__odds, a__incr_1, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__HEAD_1, A__TAIL_1, MARK_1, A__ODDS Compound Symbols: c6_1, c8_1, c10_1, c15_1, c18_1, c20_1, c13_2, c13_1, c16_2, c16_1, c17_2, c17_1, c4_1 ---------------------------------------- (109) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (110) Obligation: Complexity Dependency Tuples Problem Rules: a__nats -> cons(0, incr(nats)) a__nats -> nats a__pairs -> cons(0, incr(odds)) a__pairs -> pairs a__odds -> a__incr(a__pairs) a__odds -> odds a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) 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(nats) -> a__nats mark(incr(z0)) -> a__incr(mark(z0)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(tail(z0))) -> c17(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(nats)) -> c17(A__TAIL(a__nats)) MARK(tail(pairs)) -> c17(A__TAIL(a__pairs)) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(incr(0))) -> c13(A__INCR(a__incr(0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(head(0))) -> c13(A__INCR(a__head(0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(incr(tail(0))) -> c13(A__INCR(a__tail(0))) MARK(incr(nats)) -> c13(A__INCR(cons(0, incr(nats)))) MARK(incr(pairs)) -> c13(A__INCR(cons(0, incr(odds)))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs)), MARK(incr(pairs))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(incr(0))) -> c16(A__HEAD(a__incr(0))) MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs)), MARK(odds)) MARK(head(odds)) -> c16(MARK(odds)) MARK(head(head(nats))) -> c16(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(pairs))) -> c16(A__HEAD(a__head(a__pairs)), MARK(head(pairs))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(head(0))) -> c16(A__HEAD(a__head(0))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(head(tail(0))) -> c16(A__HEAD(a__tail(0))) MARK(head(nats)) -> c16(A__HEAD(cons(0, incr(nats)))) MARK(head(pairs)) -> c16(A__HEAD(cons(0, incr(odds)))) MARK(tail(incr(nats))) -> c17(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(incr(z0)))) -> c17(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(pairs))) -> c17(A__TAIL(a__incr(a__pairs)), MARK(incr(pairs))) MARK(tail(incr(odds))) -> c17(A__TAIL(a__incr(a__odds)), MARK(incr(odds))) MARK(tail(incr(head(z0)))) -> c17(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c17(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c17(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c17(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c17(MARK(incr(x0))) MARK(tail(incr(0))) -> c17(A__TAIL(a__incr(0))) MARK(tail(odds)) -> c17(A__TAIL(a__incr(a__pairs)), MARK(odds)) MARK(tail(odds)) -> c17(MARK(odds)) MARK(tail(head(nats))) -> c17(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(incr(z0)))) -> c17(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(pairs))) -> c17(A__TAIL(a__head(a__pairs)), MARK(head(pairs))) MARK(tail(head(odds))) -> c17(A__TAIL(a__head(a__odds)), MARK(head(odds))) MARK(tail(head(head(z0)))) -> c17(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c17(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c17(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c17(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c17(MARK(head(x0))) MARK(tail(head(0))) -> c17(A__TAIL(a__head(0))) S tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(tail(z0))) -> c17(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(nats)) -> c17(A__TAIL(a__nats)) MARK(tail(pairs)) -> c17(A__TAIL(a__pairs)) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(incr(0))) -> c13(A__INCR(a__incr(0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(head(0))) -> c13(A__INCR(a__head(0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(incr(tail(0))) -> c13(A__INCR(a__tail(0))) MARK(incr(nats)) -> c13(A__INCR(cons(0, incr(nats)))) MARK(incr(pairs)) -> c13(A__INCR(cons(0, incr(odds)))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs)), MARK(incr(pairs))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(incr(0))) -> c16(A__HEAD(a__incr(0))) MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs)), MARK(odds)) MARK(head(odds)) -> c16(MARK(odds)) MARK(head(head(nats))) -> c16(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(pairs))) -> c16(A__HEAD(a__head(a__pairs)), MARK(head(pairs))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(head(0))) -> c16(A__HEAD(a__head(0))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(head(tail(0))) -> c16(A__HEAD(a__tail(0))) MARK(head(nats)) -> c16(A__HEAD(cons(0, incr(nats)))) MARK(head(pairs)) -> c16(A__HEAD(cons(0, incr(odds)))) MARK(tail(incr(nats))) -> c17(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(incr(z0)))) -> c17(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(pairs))) -> c17(A__TAIL(a__incr(a__pairs)), MARK(incr(pairs))) MARK(tail(incr(odds))) -> c17(A__TAIL(a__incr(a__odds)), MARK(incr(odds))) MARK(tail(incr(head(z0)))) -> c17(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c17(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c17(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c17(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c17(MARK(incr(x0))) MARK(tail(incr(0))) -> c17(A__TAIL(a__incr(0))) MARK(tail(odds)) -> c17(A__TAIL(a__incr(a__pairs)), MARK(odds)) MARK(tail(odds)) -> c17(MARK(odds)) MARK(tail(head(nats))) -> c17(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(incr(z0)))) -> c17(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(pairs))) -> c17(A__TAIL(a__head(a__pairs)), MARK(head(pairs))) MARK(tail(head(odds))) -> c17(A__TAIL(a__head(a__odds)), MARK(head(odds))) MARK(tail(head(head(z0)))) -> c17(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c17(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c17(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c17(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c17(MARK(head(x0))) MARK(tail(head(0))) -> c17(A__TAIL(a__head(0))) K tuples:none Defined Rule Symbols: a__nats, a__pairs, a__odds, a__incr_1, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__HEAD_1, A__TAIL_1, MARK_1, A__ODDS Compound Symbols: c6_1, c8_1, c10_1, c15_1, c18_1, c20_1, c13_2, c13_1, c16_2, c16_1, c17_2, c17_1, c4_1 ---------------------------------------- (111) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(tail(tail(z0))) -> c17(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) by MARK(tail(tail(x0))) -> c17(A__TAIL(tail(mark(x0))), MARK(tail(x0))) MARK(tail(tail(nats))) -> c17(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(incr(z0)))) -> c17(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(pairs))) -> c17(A__TAIL(a__tail(a__pairs)), MARK(tail(pairs))) MARK(tail(tail(odds))) -> c17(A__TAIL(a__tail(a__odds)), MARK(tail(odds))) MARK(tail(tail(head(z0)))) -> c17(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c17(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c17(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(0))) -> c17(A__TAIL(a__tail(0)), MARK(tail(0))) MARK(tail(tail(s(z0)))) -> c17(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) ---------------------------------------- (112) Obligation: Complexity Dependency Tuples Problem Rules: a__nats -> cons(0, incr(nats)) a__nats -> nats a__pairs -> cons(0, incr(odds)) a__pairs -> pairs a__odds -> a__incr(a__pairs) a__odds -> odds a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) 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(nats) -> a__nats mark(incr(z0)) -> a__incr(mark(z0)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(nats)) -> c17(A__TAIL(a__nats)) MARK(tail(pairs)) -> c17(A__TAIL(a__pairs)) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(incr(0))) -> c13(A__INCR(a__incr(0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(head(0))) -> c13(A__INCR(a__head(0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(incr(tail(0))) -> c13(A__INCR(a__tail(0))) MARK(incr(nats)) -> c13(A__INCR(cons(0, incr(nats)))) MARK(incr(pairs)) -> c13(A__INCR(cons(0, incr(odds)))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs)), MARK(incr(pairs))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(incr(0))) -> c16(A__HEAD(a__incr(0))) MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs)), MARK(odds)) MARK(head(odds)) -> c16(MARK(odds)) MARK(head(head(nats))) -> c16(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(pairs))) -> c16(A__HEAD(a__head(a__pairs)), MARK(head(pairs))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(head(0))) -> c16(A__HEAD(a__head(0))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(head(tail(0))) -> c16(A__HEAD(a__tail(0))) MARK(head(nats)) -> c16(A__HEAD(cons(0, incr(nats)))) MARK(head(pairs)) -> c16(A__HEAD(cons(0, incr(odds)))) MARK(tail(incr(nats))) -> c17(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(incr(z0)))) -> c17(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(pairs))) -> c17(A__TAIL(a__incr(a__pairs)), MARK(incr(pairs))) MARK(tail(incr(odds))) -> c17(A__TAIL(a__incr(a__odds)), MARK(incr(odds))) MARK(tail(incr(head(z0)))) -> c17(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c17(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c17(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c17(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c17(MARK(incr(x0))) MARK(tail(incr(0))) -> c17(A__TAIL(a__incr(0))) MARK(tail(odds)) -> c17(A__TAIL(a__incr(a__pairs)), MARK(odds)) MARK(tail(odds)) -> c17(MARK(odds)) MARK(tail(head(nats))) -> c17(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(incr(z0)))) -> c17(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(pairs))) -> c17(A__TAIL(a__head(a__pairs)), MARK(head(pairs))) MARK(tail(head(odds))) -> c17(A__TAIL(a__head(a__odds)), MARK(head(odds))) MARK(tail(head(head(z0)))) -> c17(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c17(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c17(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c17(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c17(MARK(head(x0))) MARK(tail(head(0))) -> c17(A__TAIL(a__head(0))) MARK(tail(tail(x0))) -> c17(A__TAIL(tail(mark(x0))), MARK(tail(x0))) MARK(tail(tail(nats))) -> c17(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(incr(z0)))) -> c17(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(pairs))) -> c17(A__TAIL(a__tail(a__pairs)), MARK(tail(pairs))) MARK(tail(tail(odds))) -> c17(A__TAIL(a__tail(a__odds)), MARK(tail(odds))) MARK(tail(tail(head(z0)))) -> c17(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c17(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c17(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(0))) -> c17(A__TAIL(a__tail(0)), MARK(tail(0))) MARK(tail(tail(s(z0)))) -> c17(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) S tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(nats)) -> c17(A__TAIL(a__nats)) MARK(tail(pairs)) -> c17(A__TAIL(a__pairs)) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(incr(0))) -> c13(A__INCR(a__incr(0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(head(0))) -> c13(A__INCR(a__head(0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(incr(tail(0))) -> c13(A__INCR(a__tail(0))) MARK(incr(nats)) -> c13(A__INCR(cons(0, incr(nats)))) MARK(incr(pairs)) -> c13(A__INCR(cons(0, incr(odds)))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs)), MARK(incr(pairs))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(incr(0))) -> c16(A__HEAD(a__incr(0))) MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs)), MARK(odds)) MARK(head(odds)) -> c16(MARK(odds)) MARK(head(head(nats))) -> c16(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(pairs))) -> c16(A__HEAD(a__head(a__pairs)), MARK(head(pairs))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(head(0))) -> c16(A__HEAD(a__head(0))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(head(tail(0))) -> c16(A__HEAD(a__tail(0))) MARK(head(nats)) -> c16(A__HEAD(cons(0, incr(nats)))) MARK(head(pairs)) -> c16(A__HEAD(cons(0, incr(odds)))) MARK(tail(incr(nats))) -> c17(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(incr(z0)))) -> c17(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(pairs))) -> c17(A__TAIL(a__incr(a__pairs)), MARK(incr(pairs))) MARK(tail(incr(odds))) -> c17(A__TAIL(a__incr(a__odds)), MARK(incr(odds))) MARK(tail(incr(head(z0)))) -> c17(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c17(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c17(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c17(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c17(MARK(incr(x0))) MARK(tail(incr(0))) -> c17(A__TAIL(a__incr(0))) MARK(tail(odds)) -> c17(A__TAIL(a__incr(a__pairs)), MARK(odds)) MARK(tail(odds)) -> c17(MARK(odds)) MARK(tail(head(nats))) -> c17(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(incr(z0)))) -> c17(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(pairs))) -> c17(A__TAIL(a__head(a__pairs)), MARK(head(pairs))) MARK(tail(head(odds))) -> c17(A__TAIL(a__head(a__odds)), MARK(head(odds))) MARK(tail(head(head(z0)))) -> c17(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c17(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c17(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c17(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c17(MARK(head(x0))) MARK(tail(head(0))) -> c17(A__TAIL(a__head(0))) MARK(tail(tail(x0))) -> c17(A__TAIL(tail(mark(x0))), MARK(tail(x0))) MARK(tail(tail(nats))) -> c17(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(incr(z0)))) -> c17(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(pairs))) -> c17(A__TAIL(a__tail(a__pairs)), MARK(tail(pairs))) MARK(tail(tail(odds))) -> c17(A__TAIL(a__tail(a__odds)), MARK(tail(odds))) MARK(tail(tail(head(z0)))) -> c17(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c17(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c17(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(0))) -> c17(A__TAIL(a__tail(0)), MARK(tail(0))) MARK(tail(tail(s(z0)))) -> c17(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) K tuples:none Defined Rule Symbols: a__nats, a__pairs, a__odds, a__incr_1, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__HEAD_1, A__TAIL_1, MARK_1, A__ODDS Compound Symbols: c6_1, c8_1, c10_1, c15_1, c18_1, c20_1, c13_2, c13_1, c16_2, c16_1, c17_2, c17_1, c4_1 ---------------------------------------- (113) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (114) Obligation: Complexity Dependency Tuples Problem Rules: a__nats -> cons(0, incr(nats)) a__nats -> nats a__pairs -> cons(0, incr(odds)) a__pairs -> pairs a__odds -> a__incr(a__pairs) a__odds -> odds a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) 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(nats) -> a__nats mark(incr(z0)) -> a__incr(mark(z0)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(nats)) -> c17(A__TAIL(a__nats)) MARK(tail(pairs)) -> c17(A__TAIL(a__pairs)) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(incr(0))) -> c13(A__INCR(a__incr(0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(head(0))) -> c13(A__INCR(a__head(0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(incr(tail(0))) -> c13(A__INCR(a__tail(0))) MARK(incr(nats)) -> c13(A__INCR(cons(0, incr(nats)))) MARK(incr(pairs)) -> c13(A__INCR(cons(0, incr(odds)))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs)), MARK(incr(pairs))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(incr(0))) -> c16(A__HEAD(a__incr(0))) MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs)), MARK(odds)) MARK(head(odds)) -> c16(MARK(odds)) MARK(head(head(nats))) -> c16(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(pairs))) -> c16(A__HEAD(a__head(a__pairs)), MARK(head(pairs))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(head(0))) -> c16(A__HEAD(a__head(0))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(head(tail(0))) -> c16(A__HEAD(a__tail(0))) MARK(head(nats)) -> c16(A__HEAD(cons(0, incr(nats)))) MARK(head(pairs)) -> c16(A__HEAD(cons(0, incr(odds)))) MARK(tail(incr(nats))) -> c17(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(incr(z0)))) -> c17(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(pairs))) -> c17(A__TAIL(a__incr(a__pairs)), MARK(incr(pairs))) MARK(tail(incr(odds))) -> c17(A__TAIL(a__incr(a__odds)), MARK(incr(odds))) MARK(tail(incr(head(z0)))) -> c17(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c17(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c17(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c17(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c17(MARK(incr(x0))) MARK(tail(incr(0))) -> c17(A__TAIL(a__incr(0))) MARK(tail(odds)) -> c17(A__TAIL(a__incr(a__pairs)), MARK(odds)) MARK(tail(odds)) -> c17(MARK(odds)) MARK(tail(head(nats))) -> c17(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(incr(z0)))) -> c17(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(pairs))) -> c17(A__TAIL(a__head(a__pairs)), MARK(head(pairs))) MARK(tail(head(odds))) -> c17(A__TAIL(a__head(a__odds)), MARK(head(odds))) MARK(tail(head(head(z0)))) -> c17(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c17(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c17(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c17(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c17(MARK(head(x0))) MARK(tail(head(0))) -> c17(A__TAIL(a__head(0))) MARK(tail(tail(nats))) -> c17(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(incr(z0)))) -> c17(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(pairs))) -> c17(A__TAIL(a__tail(a__pairs)), MARK(tail(pairs))) MARK(tail(tail(odds))) -> c17(A__TAIL(a__tail(a__odds)), MARK(tail(odds))) MARK(tail(tail(head(z0)))) -> c17(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c17(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c17(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c17(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c17(MARK(tail(x0))) MARK(tail(tail(0))) -> c17(A__TAIL(a__tail(0))) S tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(nats)) -> c17(A__TAIL(a__nats)) MARK(tail(pairs)) -> c17(A__TAIL(a__pairs)) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(incr(0))) -> c13(A__INCR(a__incr(0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(head(0))) -> c13(A__INCR(a__head(0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(incr(tail(0))) -> c13(A__INCR(a__tail(0))) MARK(incr(nats)) -> c13(A__INCR(cons(0, incr(nats)))) MARK(incr(pairs)) -> c13(A__INCR(cons(0, incr(odds)))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs)), MARK(incr(pairs))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(incr(0))) -> c16(A__HEAD(a__incr(0))) MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs)), MARK(odds)) MARK(head(odds)) -> c16(MARK(odds)) MARK(head(head(nats))) -> c16(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(pairs))) -> c16(A__HEAD(a__head(a__pairs)), MARK(head(pairs))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(head(0))) -> c16(A__HEAD(a__head(0))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(head(tail(0))) -> c16(A__HEAD(a__tail(0))) MARK(head(nats)) -> c16(A__HEAD(cons(0, incr(nats)))) MARK(head(pairs)) -> c16(A__HEAD(cons(0, incr(odds)))) MARK(tail(incr(nats))) -> c17(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(incr(z0)))) -> c17(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(pairs))) -> c17(A__TAIL(a__incr(a__pairs)), MARK(incr(pairs))) MARK(tail(incr(odds))) -> c17(A__TAIL(a__incr(a__odds)), MARK(incr(odds))) MARK(tail(incr(head(z0)))) -> c17(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c17(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c17(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c17(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c17(MARK(incr(x0))) MARK(tail(incr(0))) -> c17(A__TAIL(a__incr(0))) MARK(tail(odds)) -> c17(A__TAIL(a__incr(a__pairs)), MARK(odds)) MARK(tail(odds)) -> c17(MARK(odds)) MARK(tail(head(nats))) -> c17(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(incr(z0)))) -> c17(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(pairs))) -> c17(A__TAIL(a__head(a__pairs)), MARK(head(pairs))) MARK(tail(head(odds))) -> c17(A__TAIL(a__head(a__odds)), MARK(head(odds))) MARK(tail(head(head(z0)))) -> c17(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c17(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c17(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c17(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c17(MARK(head(x0))) MARK(tail(head(0))) -> c17(A__TAIL(a__head(0))) MARK(tail(tail(nats))) -> c17(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(incr(z0)))) -> c17(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(pairs))) -> c17(A__TAIL(a__tail(a__pairs)), MARK(tail(pairs))) MARK(tail(tail(odds))) -> c17(A__TAIL(a__tail(a__odds)), MARK(tail(odds))) MARK(tail(tail(head(z0)))) -> c17(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c17(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c17(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c17(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c17(MARK(tail(x0))) MARK(tail(tail(0))) -> c17(A__TAIL(a__tail(0))) K tuples:none Defined Rule Symbols: a__nats, a__pairs, a__odds, a__incr_1, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__HEAD_1, A__TAIL_1, MARK_1, A__ODDS Compound Symbols: c6_1, c8_1, c10_1, c15_1, c18_1, c20_1, c13_2, c13_1, c16_2, c16_1, c17_2, c17_1, c4_1 ---------------------------------------- (115) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(tail(nats)) -> c17(A__TAIL(a__nats)) by MARK(tail(nats)) -> c17(A__TAIL(cons(0, incr(nats)))) MARK(tail(nats)) -> c17(A__TAIL(nats)) ---------------------------------------- (116) Obligation: Complexity Dependency Tuples Problem Rules: a__nats -> cons(0, incr(nats)) a__nats -> nats a__pairs -> cons(0, incr(odds)) a__pairs -> pairs a__odds -> a__incr(a__pairs) a__odds -> odds a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) 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(nats) -> a__nats mark(incr(z0)) -> a__incr(mark(z0)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(pairs)) -> c17(A__TAIL(a__pairs)) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(incr(0))) -> c13(A__INCR(a__incr(0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(head(0))) -> c13(A__INCR(a__head(0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(incr(tail(0))) -> c13(A__INCR(a__tail(0))) MARK(incr(nats)) -> c13(A__INCR(cons(0, incr(nats)))) MARK(incr(pairs)) -> c13(A__INCR(cons(0, incr(odds)))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs)), MARK(incr(pairs))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(incr(0))) -> c16(A__HEAD(a__incr(0))) MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs)), MARK(odds)) MARK(head(odds)) -> c16(MARK(odds)) MARK(head(head(nats))) -> c16(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(pairs))) -> c16(A__HEAD(a__head(a__pairs)), MARK(head(pairs))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(head(0))) -> c16(A__HEAD(a__head(0))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(head(tail(0))) -> c16(A__HEAD(a__tail(0))) MARK(head(nats)) -> c16(A__HEAD(cons(0, incr(nats)))) MARK(head(pairs)) -> c16(A__HEAD(cons(0, incr(odds)))) MARK(tail(incr(nats))) -> c17(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(incr(z0)))) -> c17(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(pairs))) -> c17(A__TAIL(a__incr(a__pairs)), MARK(incr(pairs))) MARK(tail(incr(odds))) -> c17(A__TAIL(a__incr(a__odds)), MARK(incr(odds))) MARK(tail(incr(head(z0)))) -> c17(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c17(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c17(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c17(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c17(MARK(incr(x0))) MARK(tail(incr(0))) -> c17(A__TAIL(a__incr(0))) MARK(tail(odds)) -> c17(A__TAIL(a__incr(a__pairs)), MARK(odds)) MARK(tail(odds)) -> c17(MARK(odds)) MARK(tail(head(nats))) -> c17(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(incr(z0)))) -> c17(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(pairs))) -> c17(A__TAIL(a__head(a__pairs)), MARK(head(pairs))) MARK(tail(head(odds))) -> c17(A__TAIL(a__head(a__odds)), MARK(head(odds))) MARK(tail(head(head(z0)))) -> c17(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c17(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c17(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c17(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c17(MARK(head(x0))) MARK(tail(head(0))) -> c17(A__TAIL(a__head(0))) MARK(tail(tail(nats))) -> c17(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(incr(z0)))) -> c17(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(pairs))) -> c17(A__TAIL(a__tail(a__pairs)), MARK(tail(pairs))) MARK(tail(tail(odds))) -> c17(A__TAIL(a__tail(a__odds)), MARK(tail(odds))) MARK(tail(tail(head(z0)))) -> c17(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c17(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c17(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c17(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c17(MARK(tail(x0))) MARK(tail(tail(0))) -> c17(A__TAIL(a__tail(0))) MARK(tail(nats)) -> c17(A__TAIL(cons(0, incr(nats)))) MARK(tail(nats)) -> c17(A__TAIL(nats)) S tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(pairs)) -> c17(A__TAIL(a__pairs)) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(incr(0))) -> c13(A__INCR(a__incr(0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(head(0))) -> c13(A__INCR(a__head(0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(incr(tail(0))) -> c13(A__INCR(a__tail(0))) MARK(incr(nats)) -> c13(A__INCR(cons(0, incr(nats)))) MARK(incr(pairs)) -> c13(A__INCR(cons(0, incr(odds)))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs)), MARK(incr(pairs))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(incr(0))) -> c16(A__HEAD(a__incr(0))) MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs)), MARK(odds)) MARK(head(odds)) -> c16(MARK(odds)) MARK(head(head(nats))) -> c16(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(pairs))) -> c16(A__HEAD(a__head(a__pairs)), MARK(head(pairs))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(head(0))) -> c16(A__HEAD(a__head(0))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(head(tail(0))) -> c16(A__HEAD(a__tail(0))) MARK(head(nats)) -> c16(A__HEAD(cons(0, incr(nats)))) MARK(head(pairs)) -> c16(A__HEAD(cons(0, incr(odds)))) MARK(tail(incr(nats))) -> c17(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(incr(z0)))) -> c17(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(pairs))) -> c17(A__TAIL(a__incr(a__pairs)), MARK(incr(pairs))) MARK(tail(incr(odds))) -> c17(A__TAIL(a__incr(a__odds)), MARK(incr(odds))) MARK(tail(incr(head(z0)))) -> c17(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c17(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c17(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c17(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c17(MARK(incr(x0))) MARK(tail(incr(0))) -> c17(A__TAIL(a__incr(0))) MARK(tail(odds)) -> c17(A__TAIL(a__incr(a__pairs)), MARK(odds)) MARK(tail(odds)) -> c17(MARK(odds)) MARK(tail(head(nats))) -> c17(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(incr(z0)))) -> c17(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(pairs))) -> c17(A__TAIL(a__head(a__pairs)), MARK(head(pairs))) MARK(tail(head(odds))) -> c17(A__TAIL(a__head(a__odds)), MARK(head(odds))) MARK(tail(head(head(z0)))) -> c17(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c17(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c17(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c17(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c17(MARK(head(x0))) MARK(tail(head(0))) -> c17(A__TAIL(a__head(0))) MARK(tail(tail(nats))) -> c17(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(incr(z0)))) -> c17(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(pairs))) -> c17(A__TAIL(a__tail(a__pairs)), MARK(tail(pairs))) MARK(tail(tail(odds))) -> c17(A__TAIL(a__tail(a__odds)), MARK(tail(odds))) MARK(tail(tail(head(z0)))) -> c17(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c17(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c17(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c17(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c17(MARK(tail(x0))) MARK(tail(tail(0))) -> c17(A__TAIL(a__tail(0))) MARK(tail(nats)) -> c17(A__TAIL(cons(0, incr(nats)))) MARK(tail(nats)) -> c17(A__TAIL(nats)) K tuples:none Defined Rule Symbols: a__nats, a__pairs, a__odds, a__incr_1, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__HEAD_1, A__TAIL_1, MARK_1, A__ODDS Compound Symbols: c6_1, c8_1, c10_1, c15_1, c18_1, c20_1, c13_2, c13_1, c16_2, c16_1, c17_2, c17_1, c4_1 ---------------------------------------- (117) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(tail(nats)) -> c17(A__TAIL(nats)) ---------------------------------------- (118) Obligation: Complexity Dependency Tuples Problem Rules: a__nats -> cons(0, incr(nats)) a__nats -> nats a__pairs -> cons(0, incr(odds)) a__pairs -> pairs a__odds -> a__incr(a__pairs) a__odds -> odds a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) 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(nats) -> a__nats mark(incr(z0)) -> a__incr(mark(z0)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(pairs)) -> c17(A__TAIL(a__pairs)) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(incr(0))) -> c13(A__INCR(a__incr(0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(head(0))) -> c13(A__INCR(a__head(0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(incr(tail(0))) -> c13(A__INCR(a__tail(0))) MARK(incr(nats)) -> c13(A__INCR(cons(0, incr(nats)))) MARK(incr(pairs)) -> c13(A__INCR(cons(0, incr(odds)))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs)), MARK(incr(pairs))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(incr(0))) -> c16(A__HEAD(a__incr(0))) MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs)), MARK(odds)) MARK(head(odds)) -> c16(MARK(odds)) MARK(head(head(nats))) -> c16(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(pairs))) -> c16(A__HEAD(a__head(a__pairs)), MARK(head(pairs))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(head(0))) -> c16(A__HEAD(a__head(0))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(head(tail(0))) -> c16(A__HEAD(a__tail(0))) MARK(head(nats)) -> c16(A__HEAD(cons(0, incr(nats)))) MARK(head(pairs)) -> c16(A__HEAD(cons(0, incr(odds)))) MARK(tail(incr(nats))) -> c17(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(incr(z0)))) -> c17(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(pairs))) -> c17(A__TAIL(a__incr(a__pairs)), MARK(incr(pairs))) MARK(tail(incr(odds))) -> c17(A__TAIL(a__incr(a__odds)), MARK(incr(odds))) MARK(tail(incr(head(z0)))) -> c17(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c17(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c17(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c17(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c17(MARK(incr(x0))) MARK(tail(incr(0))) -> c17(A__TAIL(a__incr(0))) MARK(tail(odds)) -> c17(A__TAIL(a__incr(a__pairs)), MARK(odds)) MARK(tail(odds)) -> c17(MARK(odds)) MARK(tail(head(nats))) -> c17(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(incr(z0)))) -> c17(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(pairs))) -> c17(A__TAIL(a__head(a__pairs)), MARK(head(pairs))) MARK(tail(head(odds))) -> c17(A__TAIL(a__head(a__odds)), MARK(head(odds))) MARK(tail(head(head(z0)))) -> c17(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c17(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c17(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c17(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c17(MARK(head(x0))) MARK(tail(head(0))) -> c17(A__TAIL(a__head(0))) MARK(tail(tail(nats))) -> c17(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(incr(z0)))) -> c17(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(pairs))) -> c17(A__TAIL(a__tail(a__pairs)), MARK(tail(pairs))) MARK(tail(tail(odds))) -> c17(A__TAIL(a__tail(a__odds)), MARK(tail(odds))) MARK(tail(tail(head(z0)))) -> c17(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c17(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c17(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c17(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c17(MARK(tail(x0))) MARK(tail(tail(0))) -> c17(A__TAIL(a__tail(0))) MARK(tail(nats)) -> c17(A__TAIL(cons(0, incr(nats)))) S tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(pairs)) -> c17(A__TAIL(a__pairs)) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(incr(0))) -> c13(A__INCR(a__incr(0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(head(0))) -> c13(A__INCR(a__head(0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(incr(tail(0))) -> c13(A__INCR(a__tail(0))) MARK(incr(nats)) -> c13(A__INCR(cons(0, incr(nats)))) MARK(incr(pairs)) -> c13(A__INCR(cons(0, incr(odds)))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs)), MARK(incr(pairs))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(incr(0))) -> c16(A__HEAD(a__incr(0))) MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs)), MARK(odds)) MARK(head(odds)) -> c16(MARK(odds)) MARK(head(head(nats))) -> c16(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(pairs))) -> c16(A__HEAD(a__head(a__pairs)), MARK(head(pairs))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(head(0))) -> c16(A__HEAD(a__head(0))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(head(tail(0))) -> c16(A__HEAD(a__tail(0))) MARK(head(nats)) -> c16(A__HEAD(cons(0, incr(nats)))) MARK(head(pairs)) -> c16(A__HEAD(cons(0, incr(odds)))) MARK(tail(incr(nats))) -> c17(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(incr(z0)))) -> c17(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(pairs))) -> c17(A__TAIL(a__incr(a__pairs)), MARK(incr(pairs))) MARK(tail(incr(odds))) -> c17(A__TAIL(a__incr(a__odds)), MARK(incr(odds))) MARK(tail(incr(head(z0)))) -> c17(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c17(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c17(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c17(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c17(MARK(incr(x0))) MARK(tail(incr(0))) -> c17(A__TAIL(a__incr(0))) MARK(tail(odds)) -> c17(A__TAIL(a__incr(a__pairs)), MARK(odds)) MARK(tail(odds)) -> c17(MARK(odds)) MARK(tail(head(nats))) -> c17(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(incr(z0)))) -> c17(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(pairs))) -> c17(A__TAIL(a__head(a__pairs)), MARK(head(pairs))) MARK(tail(head(odds))) -> c17(A__TAIL(a__head(a__odds)), MARK(head(odds))) MARK(tail(head(head(z0)))) -> c17(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c17(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c17(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c17(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c17(MARK(head(x0))) MARK(tail(head(0))) -> c17(A__TAIL(a__head(0))) MARK(tail(tail(nats))) -> c17(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(incr(z0)))) -> c17(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(pairs))) -> c17(A__TAIL(a__tail(a__pairs)), MARK(tail(pairs))) MARK(tail(tail(odds))) -> c17(A__TAIL(a__tail(a__odds)), MARK(tail(odds))) MARK(tail(tail(head(z0)))) -> c17(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c17(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c17(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c17(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c17(MARK(tail(x0))) MARK(tail(tail(0))) -> c17(A__TAIL(a__tail(0))) MARK(tail(nats)) -> c17(A__TAIL(cons(0, incr(nats)))) K tuples:none Defined Rule Symbols: a__nats, a__pairs, a__odds, a__incr_1, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__HEAD_1, A__TAIL_1, MARK_1, A__ODDS Compound Symbols: c6_1, c8_1, c10_1, c15_1, c18_1, c20_1, c13_2, c13_1, c16_2, c16_1, c17_2, c17_1, c4_1 ---------------------------------------- (119) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(tail(pairs)) -> c17(A__TAIL(a__pairs)) by MARK(tail(pairs)) -> c17(A__TAIL(cons(0, incr(odds)))) MARK(tail(pairs)) -> c17(A__TAIL(pairs)) ---------------------------------------- (120) Obligation: Complexity Dependency Tuples Problem Rules: a__nats -> cons(0, incr(nats)) a__nats -> nats a__pairs -> cons(0, incr(odds)) a__pairs -> pairs a__odds -> a__incr(a__pairs) a__odds -> odds a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) 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(nats) -> a__nats mark(incr(z0)) -> a__incr(mark(z0)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(incr(0))) -> c13(A__INCR(a__incr(0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(head(0))) -> c13(A__INCR(a__head(0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(incr(tail(0))) -> c13(A__INCR(a__tail(0))) MARK(incr(nats)) -> c13(A__INCR(cons(0, incr(nats)))) MARK(incr(pairs)) -> c13(A__INCR(cons(0, incr(odds)))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs)), MARK(incr(pairs))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(incr(0))) -> c16(A__HEAD(a__incr(0))) MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs)), MARK(odds)) MARK(head(odds)) -> c16(MARK(odds)) MARK(head(head(nats))) -> c16(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(pairs))) -> c16(A__HEAD(a__head(a__pairs)), MARK(head(pairs))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(head(0))) -> c16(A__HEAD(a__head(0))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(head(tail(0))) -> c16(A__HEAD(a__tail(0))) MARK(head(nats)) -> c16(A__HEAD(cons(0, incr(nats)))) MARK(head(pairs)) -> c16(A__HEAD(cons(0, incr(odds)))) MARK(tail(incr(nats))) -> c17(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(incr(z0)))) -> c17(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(pairs))) -> c17(A__TAIL(a__incr(a__pairs)), MARK(incr(pairs))) MARK(tail(incr(odds))) -> c17(A__TAIL(a__incr(a__odds)), MARK(incr(odds))) MARK(tail(incr(head(z0)))) -> c17(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c17(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c17(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c17(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c17(MARK(incr(x0))) MARK(tail(incr(0))) -> c17(A__TAIL(a__incr(0))) MARK(tail(odds)) -> c17(A__TAIL(a__incr(a__pairs)), MARK(odds)) MARK(tail(odds)) -> c17(MARK(odds)) MARK(tail(head(nats))) -> c17(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(incr(z0)))) -> c17(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(pairs))) -> c17(A__TAIL(a__head(a__pairs)), MARK(head(pairs))) MARK(tail(head(odds))) -> c17(A__TAIL(a__head(a__odds)), MARK(head(odds))) MARK(tail(head(head(z0)))) -> c17(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c17(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c17(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c17(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c17(MARK(head(x0))) MARK(tail(head(0))) -> c17(A__TAIL(a__head(0))) MARK(tail(tail(nats))) -> c17(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(incr(z0)))) -> c17(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(pairs))) -> c17(A__TAIL(a__tail(a__pairs)), MARK(tail(pairs))) MARK(tail(tail(odds))) -> c17(A__TAIL(a__tail(a__odds)), MARK(tail(odds))) MARK(tail(tail(head(z0)))) -> c17(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c17(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c17(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c17(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c17(MARK(tail(x0))) MARK(tail(tail(0))) -> c17(A__TAIL(a__tail(0))) MARK(tail(nats)) -> c17(A__TAIL(cons(0, incr(nats)))) MARK(tail(pairs)) -> c17(A__TAIL(cons(0, incr(odds)))) MARK(tail(pairs)) -> c17(A__TAIL(pairs)) S tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(incr(0))) -> c13(A__INCR(a__incr(0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(head(0))) -> c13(A__INCR(a__head(0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(incr(tail(0))) -> c13(A__INCR(a__tail(0))) MARK(incr(nats)) -> c13(A__INCR(cons(0, incr(nats)))) MARK(incr(pairs)) -> c13(A__INCR(cons(0, incr(odds)))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs)), MARK(incr(pairs))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(incr(0))) -> c16(A__HEAD(a__incr(0))) MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs)), MARK(odds)) MARK(head(odds)) -> c16(MARK(odds)) MARK(head(head(nats))) -> c16(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(pairs))) -> c16(A__HEAD(a__head(a__pairs)), MARK(head(pairs))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(head(0))) -> c16(A__HEAD(a__head(0))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(head(tail(0))) -> c16(A__HEAD(a__tail(0))) MARK(head(nats)) -> c16(A__HEAD(cons(0, incr(nats)))) MARK(head(pairs)) -> c16(A__HEAD(cons(0, incr(odds)))) MARK(tail(incr(nats))) -> c17(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(incr(z0)))) -> c17(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(pairs))) -> c17(A__TAIL(a__incr(a__pairs)), MARK(incr(pairs))) MARK(tail(incr(odds))) -> c17(A__TAIL(a__incr(a__odds)), MARK(incr(odds))) MARK(tail(incr(head(z0)))) -> c17(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c17(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c17(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c17(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c17(MARK(incr(x0))) MARK(tail(incr(0))) -> c17(A__TAIL(a__incr(0))) MARK(tail(odds)) -> c17(A__TAIL(a__incr(a__pairs)), MARK(odds)) MARK(tail(odds)) -> c17(MARK(odds)) MARK(tail(head(nats))) -> c17(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(incr(z0)))) -> c17(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(pairs))) -> c17(A__TAIL(a__head(a__pairs)), MARK(head(pairs))) MARK(tail(head(odds))) -> c17(A__TAIL(a__head(a__odds)), MARK(head(odds))) MARK(tail(head(head(z0)))) -> c17(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c17(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c17(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c17(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c17(MARK(head(x0))) MARK(tail(head(0))) -> c17(A__TAIL(a__head(0))) MARK(tail(tail(nats))) -> c17(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(incr(z0)))) -> c17(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(pairs))) -> c17(A__TAIL(a__tail(a__pairs)), MARK(tail(pairs))) MARK(tail(tail(odds))) -> c17(A__TAIL(a__tail(a__odds)), MARK(tail(odds))) MARK(tail(tail(head(z0)))) -> c17(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c17(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c17(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c17(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c17(MARK(tail(x0))) MARK(tail(tail(0))) -> c17(A__TAIL(a__tail(0))) MARK(tail(nats)) -> c17(A__TAIL(cons(0, incr(nats)))) MARK(tail(pairs)) -> c17(A__TAIL(cons(0, incr(odds)))) MARK(tail(pairs)) -> c17(A__TAIL(pairs)) K tuples:none Defined Rule Symbols: a__nats, a__pairs, a__odds, a__incr_1, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__HEAD_1, A__TAIL_1, MARK_1, A__ODDS Compound Symbols: c6_1, c8_1, c10_1, c15_1, c18_1, c20_1, c13_2, c13_1, c16_2, c16_1, c17_2, c17_1, c4_1 ---------------------------------------- (121) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(tail(pairs)) -> c17(A__TAIL(pairs)) ---------------------------------------- (122) Obligation: Complexity Dependency Tuples Problem Rules: a__nats -> cons(0, incr(nats)) a__nats -> nats a__pairs -> cons(0, incr(odds)) a__pairs -> pairs a__odds -> a__incr(a__pairs) a__odds -> odds a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) 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(nats) -> a__nats mark(incr(z0)) -> a__incr(mark(z0)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(incr(0))) -> c13(A__INCR(a__incr(0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(head(0))) -> c13(A__INCR(a__head(0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(incr(tail(0))) -> c13(A__INCR(a__tail(0))) MARK(incr(nats)) -> c13(A__INCR(cons(0, incr(nats)))) MARK(incr(pairs)) -> c13(A__INCR(cons(0, incr(odds)))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs)), MARK(incr(pairs))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(incr(0))) -> c16(A__HEAD(a__incr(0))) MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs)), MARK(odds)) MARK(head(odds)) -> c16(MARK(odds)) MARK(head(head(nats))) -> c16(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(pairs))) -> c16(A__HEAD(a__head(a__pairs)), MARK(head(pairs))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(head(0))) -> c16(A__HEAD(a__head(0))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(head(tail(0))) -> c16(A__HEAD(a__tail(0))) MARK(head(nats)) -> c16(A__HEAD(cons(0, incr(nats)))) MARK(head(pairs)) -> c16(A__HEAD(cons(0, incr(odds)))) MARK(tail(incr(nats))) -> c17(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(incr(z0)))) -> c17(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(pairs))) -> c17(A__TAIL(a__incr(a__pairs)), MARK(incr(pairs))) MARK(tail(incr(odds))) -> c17(A__TAIL(a__incr(a__odds)), MARK(incr(odds))) MARK(tail(incr(head(z0)))) -> c17(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c17(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c17(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c17(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c17(MARK(incr(x0))) MARK(tail(incr(0))) -> c17(A__TAIL(a__incr(0))) MARK(tail(odds)) -> c17(A__TAIL(a__incr(a__pairs)), MARK(odds)) MARK(tail(odds)) -> c17(MARK(odds)) MARK(tail(head(nats))) -> c17(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(incr(z0)))) -> c17(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(pairs))) -> c17(A__TAIL(a__head(a__pairs)), MARK(head(pairs))) MARK(tail(head(odds))) -> c17(A__TAIL(a__head(a__odds)), MARK(head(odds))) MARK(tail(head(head(z0)))) -> c17(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c17(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c17(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c17(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c17(MARK(head(x0))) MARK(tail(head(0))) -> c17(A__TAIL(a__head(0))) MARK(tail(tail(nats))) -> c17(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(incr(z0)))) -> c17(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(pairs))) -> c17(A__TAIL(a__tail(a__pairs)), MARK(tail(pairs))) MARK(tail(tail(odds))) -> c17(A__TAIL(a__tail(a__odds)), MARK(tail(odds))) MARK(tail(tail(head(z0)))) -> c17(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c17(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c17(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c17(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c17(MARK(tail(x0))) MARK(tail(tail(0))) -> c17(A__TAIL(a__tail(0))) MARK(tail(nats)) -> c17(A__TAIL(cons(0, incr(nats)))) MARK(tail(pairs)) -> c17(A__TAIL(cons(0, incr(odds)))) S tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(incr(0))) -> c13(A__INCR(a__incr(0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(head(0))) -> c13(A__INCR(a__head(0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(incr(tail(0))) -> c13(A__INCR(a__tail(0))) MARK(incr(nats)) -> c13(A__INCR(cons(0, incr(nats)))) MARK(incr(pairs)) -> c13(A__INCR(cons(0, incr(odds)))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs)), MARK(incr(pairs))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(incr(0))) -> c16(A__HEAD(a__incr(0))) MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs)), MARK(odds)) MARK(head(odds)) -> c16(MARK(odds)) MARK(head(head(nats))) -> c16(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(pairs))) -> c16(A__HEAD(a__head(a__pairs)), MARK(head(pairs))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(head(0))) -> c16(A__HEAD(a__head(0))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(head(tail(0))) -> c16(A__HEAD(a__tail(0))) MARK(head(nats)) -> c16(A__HEAD(cons(0, incr(nats)))) MARK(head(pairs)) -> c16(A__HEAD(cons(0, incr(odds)))) MARK(tail(incr(nats))) -> c17(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(incr(z0)))) -> c17(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(pairs))) -> c17(A__TAIL(a__incr(a__pairs)), MARK(incr(pairs))) MARK(tail(incr(odds))) -> c17(A__TAIL(a__incr(a__odds)), MARK(incr(odds))) MARK(tail(incr(head(z0)))) -> c17(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c17(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c17(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c17(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c17(MARK(incr(x0))) MARK(tail(incr(0))) -> c17(A__TAIL(a__incr(0))) MARK(tail(odds)) -> c17(A__TAIL(a__incr(a__pairs)), MARK(odds)) MARK(tail(odds)) -> c17(MARK(odds)) MARK(tail(head(nats))) -> c17(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(incr(z0)))) -> c17(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(pairs))) -> c17(A__TAIL(a__head(a__pairs)), MARK(head(pairs))) MARK(tail(head(odds))) -> c17(A__TAIL(a__head(a__odds)), MARK(head(odds))) MARK(tail(head(head(z0)))) -> c17(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c17(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c17(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c17(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c17(MARK(head(x0))) MARK(tail(head(0))) -> c17(A__TAIL(a__head(0))) MARK(tail(tail(nats))) -> c17(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(incr(z0)))) -> c17(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(pairs))) -> c17(A__TAIL(a__tail(a__pairs)), MARK(tail(pairs))) MARK(tail(tail(odds))) -> c17(A__TAIL(a__tail(a__odds)), MARK(tail(odds))) MARK(tail(tail(head(z0)))) -> c17(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c17(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c17(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c17(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c17(MARK(tail(x0))) MARK(tail(tail(0))) -> c17(A__TAIL(a__tail(0))) MARK(tail(nats)) -> c17(A__TAIL(cons(0, incr(nats)))) MARK(tail(pairs)) -> c17(A__TAIL(cons(0, incr(odds)))) K tuples:none Defined Rule Symbols: a__nats, a__pairs, a__odds, a__incr_1, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__HEAD_1, A__TAIL_1, MARK_1, A__ODDS Compound Symbols: c6_1, c8_1, c10_1, c15_1, c18_1, c20_1, c13_2, c13_1, c16_2, c16_1, c17_2, c17_1, c4_1 ---------------------------------------- (123) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(incr(incr(0))) -> c13(A__INCR(a__incr(0))) by MARK(incr(incr(0))) -> c13(A__INCR(incr(0))) ---------------------------------------- (124) Obligation: Complexity Dependency Tuples Problem Rules: a__nats -> cons(0, incr(nats)) a__nats -> nats a__pairs -> cons(0, incr(odds)) a__pairs -> pairs a__odds -> a__incr(a__pairs) a__odds -> odds a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) 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(nats) -> a__nats mark(incr(z0)) -> a__incr(mark(z0)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(head(0))) -> c13(A__INCR(a__head(0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(incr(tail(0))) -> c13(A__INCR(a__tail(0))) MARK(incr(nats)) -> c13(A__INCR(cons(0, incr(nats)))) MARK(incr(pairs)) -> c13(A__INCR(cons(0, incr(odds)))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs)), MARK(incr(pairs))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(incr(0))) -> c16(A__HEAD(a__incr(0))) MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs)), MARK(odds)) MARK(head(odds)) -> c16(MARK(odds)) MARK(head(head(nats))) -> c16(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(pairs))) -> c16(A__HEAD(a__head(a__pairs)), MARK(head(pairs))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(head(0))) -> c16(A__HEAD(a__head(0))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(head(tail(0))) -> c16(A__HEAD(a__tail(0))) MARK(head(nats)) -> c16(A__HEAD(cons(0, incr(nats)))) MARK(head(pairs)) -> c16(A__HEAD(cons(0, incr(odds)))) MARK(tail(incr(nats))) -> c17(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(incr(z0)))) -> c17(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(pairs))) -> c17(A__TAIL(a__incr(a__pairs)), MARK(incr(pairs))) MARK(tail(incr(odds))) -> c17(A__TAIL(a__incr(a__odds)), MARK(incr(odds))) MARK(tail(incr(head(z0)))) -> c17(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c17(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c17(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c17(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c17(MARK(incr(x0))) MARK(tail(incr(0))) -> c17(A__TAIL(a__incr(0))) MARK(tail(odds)) -> c17(A__TAIL(a__incr(a__pairs)), MARK(odds)) MARK(tail(odds)) -> c17(MARK(odds)) MARK(tail(head(nats))) -> c17(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(incr(z0)))) -> c17(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(pairs))) -> c17(A__TAIL(a__head(a__pairs)), MARK(head(pairs))) MARK(tail(head(odds))) -> c17(A__TAIL(a__head(a__odds)), MARK(head(odds))) MARK(tail(head(head(z0)))) -> c17(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c17(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c17(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c17(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c17(MARK(head(x0))) MARK(tail(head(0))) -> c17(A__TAIL(a__head(0))) MARK(tail(tail(nats))) -> c17(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(incr(z0)))) -> c17(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(pairs))) -> c17(A__TAIL(a__tail(a__pairs)), MARK(tail(pairs))) MARK(tail(tail(odds))) -> c17(A__TAIL(a__tail(a__odds)), MARK(tail(odds))) MARK(tail(tail(head(z0)))) -> c17(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c17(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c17(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c17(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c17(MARK(tail(x0))) MARK(tail(tail(0))) -> c17(A__TAIL(a__tail(0))) MARK(tail(nats)) -> c17(A__TAIL(cons(0, incr(nats)))) MARK(tail(pairs)) -> c17(A__TAIL(cons(0, incr(odds)))) MARK(incr(incr(0))) -> c13(A__INCR(incr(0))) S tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(head(0))) -> c13(A__INCR(a__head(0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(incr(tail(0))) -> c13(A__INCR(a__tail(0))) MARK(incr(nats)) -> c13(A__INCR(cons(0, incr(nats)))) MARK(incr(pairs)) -> c13(A__INCR(cons(0, incr(odds)))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs)), MARK(incr(pairs))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(incr(0))) -> c16(A__HEAD(a__incr(0))) MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs)), MARK(odds)) MARK(head(odds)) -> c16(MARK(odds)) MARK(head(head(nats))) -> c16(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(pairs))) -> c16(A__HEAD(a__head(a__pairs)), MARK(head(pairs))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(head(0))) -> c16(A__HEAD(a__head(0))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(head(tail(0))) -> c16(A__HEAD(a__tail(0))) MARK(head(nats)) -> c16(A__HEAD(cons(0, incr(nats)))) MARK(head(pairs)) -> c16(A__HEAD(cons(0, incr(odds)))) MARK(tail(incr(nats))) -> c17(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(incr(z0)))) -> c17(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(pairs))) -> c17(A__TAIL(a__incr(a__pairs)), MARK(incr(pairs))) MARK(tail(incr(odds))) -> c17(A__TAIL(a__incr(a__odds)), MARK(incr(odds))) MARK(tail(incr(head(z0)))) -> c17(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c17(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c17(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c17(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c17(MARK(incr(x0))) MARK(tail(incr(0))) -> c17(A__TAIL(a__incr(0))) MARK(tail(odds)) -> c17(A__TAIL(a__incr(a__pairs)), MARK(odds)) MARK(tail(odds)) -> c17(MARK(odds)) MARK(tail(head(nats))) -> c17(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(incr(z0)))) -> c17(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(pairs))) -> c17(A__TAIL(a__head(a__pairs)), MARK(head(pairs))) MARK(tail(head(odds))) -> c17(A__TAIL(a__head(a__odds)), MARK(head(odds))) MARK(tail(head(head(z0)))) -> c17(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c17(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c17(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c17(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c17(MARK(head(x0))) MARK(tail(head(0))) -> c17(A__TAIL(a__head(0))) MARK(tail(tail(nats))) -> c17(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(incr(z0)))) -> c17(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(pairs))) -> c17(A__TAIL(a__tail(a__pairs)), MARK(tail(pairs))) MARK(tail(tail(odds))) -> c17(A__TAIL(a__tail(a__odds)), MARK(tail(odds))) MARK(tail(tail(head(z0)))) -> c17(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c17(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c17(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c17(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c17(MARK(tail(x0))) MARK(tail(tail(0))) -> c17(A__TAIL(a__tail(0))) MARK(tail(nats)) -> c17(A__TAIL(cons(0, incr(nats)))) MARK(tail(pairs)) -> c17(A__TAIL(cons(0, incr(odds)))) MARK(incr(incr(0))) -> c13(A__INCR(incr(0))) K tuples:none Defined Rule Symbols: a__nats, a__pairs, a__odds, a__incr_1, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__HEAD_1, A__TAIL_1, MARK_1, A__ODDS Compound Symbols: c6_1, c8_1, c10_1, c15_1, c18_1, c20_1, c13_2, c13_1, c16_2, c16_1, c17_2, c17_1, c4_1 ---------------------------------------- (125) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(incr(incr(0))) -> c13(A__INCR(incr(0))) ---------------------------------------- (126) Obligation: Complexity Dependency Tuples Problem Rules: a__nats -> cons(0, incr(nats)) a__nats -> nats a__pairs -> cons(0, incr(odds)) a__pairs -> pairs a__odds -> a__incr(a__pairs) a__odds -> odds a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) 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(nats) -> a__nats mark(incr(z0)) -> a__incr(mark(z0)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(head(0))) -> c13(A__INCR(a__head(0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(incr(tail(0))) -> c13(A__INCR(a__tail(0))) MARK(incr(nats)) -> c13(A__INCR(cons(0, incr(nats)))) MARK(incr(pairs)) -> c13(A__INCR(cons(0, incr(odds)))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs)), MARK(incr(pairs))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(incr(0))) -> c16(A__HEAD(a__incr(0))) MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs)), MARK(odds)) MARK(head(odds)) -> c16(MARK(odds)) MARK(head(head(nats))) -> c16(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(pairs))) -> c16(A__HEAD(a__head(a__pairs)), MARK(head(pairs))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(head(0))) -> c16(A__HEAD(a__head(0))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(head(tail(0))) -> c16(A__HEAD(a__tail(0))) MARK(head(nats)) -> c16(A__HEAD(cons(0, incr(nats)))) MARK(head(pairs)) -> c16(A__HEAD(cons(0, incr(odds)))) MARK(tail(incr(nats))) -> c17(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(incr(z0)))) -> c17(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(pairs))) -> c17(A__TAIL(a__incr(a__pairs)), MARK(incr(pairs))) MARK(tail(incr(odds))) -> c17(A__TAIL(a__incr(a__odds)), MARK(incr(odds))) MARK(tail(incr(head(z0)))) -> c17(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c17(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c17(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c17(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c17(MARK(incr(x0))) MARK(tail(incr(0))) -> c17(A__TAIL(a__incr(0))) MARK(tail(odds)) -> c17(A__TAIL(a__incr(a__pairs)), MARK(odds)) MARK(tail(odds)) -> c17(MARK(odds)) MARK(tail(head(nats))) -> c17(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(incr(z0)))) -> c17(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(pairs))) -> c17(A__TAIL(a__head(a__pairs)), MARK(head(pairs))) MARK(tail(head(odds))) -> c17(A__TAIL(a__head(a__odds)), MARK(head(odds))) MARK(tail(head(head(z0)))) -> c17(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c17(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c17(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c17(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c17(MARK(head(x0))) MARK(tail(head(0))) -> c17(A__TAIL(a__head(0))) MARK(tail(tail(nats))) -> c17(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(incr(z0)))) -> c17(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(pairs))) -> c17(A__TAIL(a__tail(a__pairs)), MARK(tail(pairs))) MARK(tail(tail(odds))) -> c17(A__TAIL(a__tail(a__odds)), MARK(tail(odds))) MARK(tail(tail(head(z0)))) -> c17(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c17(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c17(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c17(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c17(MARK(tail(x0))) MARK(tail(tail(0))) -> c17(A__TAIL(a__tail(0))) MARK(tail(nats)) -> c17(A__TAIL(cons(0, incr(nats)))) MARK(tail(pairs)) -> c17(A__TAIL(cons(0, incr(odds)))) S tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(head(0))) -> c13(A__INCR(a__head(0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(incr(tail(0))) -> c13(A__INCR(a__tail(0))) MARK(incr(nats)) -> c13(A__INCR(cons(0, incr(nats)))) MARK(incr(pairs)) -> c13(A__INCR(cons(0, incr(odds)))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs)), MARK(incr(pairs))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(incr(0))) -> c16(A__HEAD(a__incr(0))) MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs)), MARK(odds)) MARK(head(odds)) -> c16(MARK(odds)) MARK(head(head(nats))) -> c16(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(pairs))) -> c16(A__HEAD(a__head(a__pairs)), MARK(head(pairs))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(head(0))) -> c16(A__HEAD(a__head(0))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(head(tail(0))) -> c16(A__HEAD(a__tail(0))) MARK(head(nats)) -> c16(A__HEAD(cons(0, incr(nats)))) MARK(head(pairs)) -> c16(A__HEAD(cons(0, incr(odds)))) MARK(tail(incr(nats))) -> c17(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(incr(z0)))) -> c17(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(pairs))) -> c17(A__TAIL(a__incr(a__pairs)), MARK(incr(pairs))) MARK(tail(incr(odds))) -> c17(A__TAIL(a__incr(a__odds)), MARK(incr(odds))) MARK(tail(incr(head(z0)))) -> c17(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c17(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c17(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c17(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c17(MARK(incr(x0))) MARK(tail(incr(0))) -> c17(A__TAIL(a__incr(0))) MARK(tail(odds)) -> c17(A__TAIL(a__incr(a__pairs)), MARK(odds)) MARK(tail(odds)) -> c17(MARK(odds)) MARK(tail(head(nats))) -> c17(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(incr(z0)))) -> c17(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(pairs))) -> c17(A__TAIL(a__head(a__pairs)), MARK(head(pairs))) MARK(tail(head(odds))) -> c17(A__TAIL(a__head(a__odds)), MARK(head(odds))) MARK(tail(head(head(z0)))) -> c17(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c17(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c17(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c17(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c17(MARK(head(x0))) MARK(tail(head(0))) -> c17(A__TAIL(a__head(0))) MARK(tail(tail(nats))) -> c17(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(incr(z0)))) -> c17(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(pairs))) -> c17(A__TAIL(a__tail(a__pairs)), MARK(tail(pairs))) MARK(tail(tail(odds))) -> c17(A__TAIL(a__tail(a__odds)), MARK(tail(odds))) MARK(tail(tail(head(z0)))) -> c17(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c17(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c17(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c17(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c17(MARK(tail(x0))) MARK(tail(tail(0))) -> c17(A__TAIL(a__tail(0))) MARK(tail(nats)) -> c17(A__TAIL(cons(0, incr(nats)))) MARK(tail(pairs)) -> c17(A__TAIL(cons(0, incr(odds)))) K tuples:none Defined Rule Symbols: a__nats, a__pairs, a__odds, a__incr_1, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__HEAD_1, A__TAIL_1, MARK_1, A__ODDS Compound Symbols: c6_1, c8_1, c10_1, c15_1, c18_1, c20_1, c13_2, c13_1, c16_2, c16_1, c17_2, c17_1, c4_1 ---------------------------------------- (127) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(incr(head(0))) -> c13(A__INCR(a__head(0))) by MARK(incr(head(0))) -> c13(A__INCR(head(0))) ---------------------------------------- (128) Obligation: Complexity Dependency Tuples Problem Rules: a__nats -> cons(0, incr(nats)) a__nats -> nats a__pairs -> cons(0, incr(odds)) a__pairs -> pairs a__odds -> a__incr(a__pairs) a__odds -> odds a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) 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(nats) -> a__nats mark(incr(z0)) -> a__incr(mark(z0)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(incr(tail(0))) -> c13(A__INCR(a__tail(0))) MARK(incr(nats)) -> c13(A__INCR(cons(0, incr(nats)))) MARK(incr(pairs)) -> c13(A__INCR(cons(0, incr(odds)))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs)), MARK(incr(pairs))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(incr(0))) -> c16(A__HEAD(a__incr(0))) MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs)), MARK(odds)) MARK(head(odds)) -> c16(MARK(odds)) MARK(head(head(nats))) -> c16(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(pairs))) -> c16(A__HEAD(a__head(a__pairs)), MARK(head(pairs))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(head(0))) -> c16(A__HEAD(a__head(0))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(head(tail(0))) -> c16(A__HEAD(a__tail(0))) MARK(head(nats)) -> c16(A__HEAD(cons(0, incr(nats)))) MARK(head(pairs)) -> c16(A__HEAD(cons(0, incr(odds)))) MARK(tail(incr(nats))) -> c17(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(incr(z0)))) -> c17(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(pairs))) -> c17(A__TAIL(a__incr(a__pairs)), MARK(incr(pairs))) MARK(tail(incr(odds))) -> c17(A__TAIL(a__incr(a__odds)), MARK(incr(odds))) MARK(tail(incr(head(z0)))) -> c17(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c17(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c17(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c17(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c17(MARK(incr(x0))) MARK(tail(incr(0))) -> c17(A__TAIL(a__incr(0))) MARK(tail(odds)) -> c17(A__TAIL(a__incr(a__pairs)), MARK(odds)) MARK(tail(odds)) -> c17(MARK(odds)) MARK(tail(head(nats))) -> c17(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(incr(z0)))) -> c17(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(pairs))) -> c17(A__TAIL(a__head(a__pairs)), MARK(head(pairs))) MARK(tail(head(odds))) -> c17(A__TAIL(a__head(a__odds)), MARK(head(odds))) MARK(tail(head(head(z0)))) -> c17(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c17(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c17(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c17(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c17(MARK(head(x0))) MARK(tail(head(0))) -> c17(A__TAIL(a__head(0))) MARK(tail(tail(nats))) -> c17(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(incr(z0)))) -> c17(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(pairs))) -> c17(A__TAIL(a__tail(a__pairs)), MARK(tail(pairs))) MARK(tail(tail(odds))) -> c17(A__TAIL(a__tail(a__odds)), MARK(tail(odds))) MARK(tail(tail(head(z0)))) -> c17(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c17(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c17(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c17(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c17(MARK(tail(x0))) MARK(tail(tail(0))) -> c17(A__TAIL(a__tail(0))) MARK(tail(nats)) -> c17(A__TAIL(cons(0, incr(nats)))) MARK(tail(pairs)) -> c17(A__TAIL(cons(0, incr(odds)))) MARK(incr(head(0))) -> c13(A__INCR(head(0))) S tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(incr(tail(0))) -> c13(A__INCR(a__tail(0))) MARK(incr(nats)) -> c13(A__INCR(cons(0, incr(nats)))) MARK(incr(pairs)) -> c13(A__INCR(cons(0, incr(odds)))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs)), MARK(incr(pairs))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(incr(0))) -> c16(A__HEAD(a__incr(0))) MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs)), MARK(odds)) MARK(head(odds)) -> c16(MARK(odds)) MARK(head(head(nats))) -> c16(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(pairs))) -> c16(A__HEAD(a__head(a__pairs)), MARK(head(pairs))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(head(0))) -> c16(A__HEAD(a__head(0))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(head(tail(0))) -> c16(A__HEAD(a__tail(0))) MARK(head(nats)) -> c16(A__HEAD(cons(0, incr(nats)))) MARK(head(pairs)) -> c16(A__HEAD(cons(0, incr(odds)))) MARK(tail(incr(nats))) -> c17(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(incr(z0)))) -> c17(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(pairs))) -> c17(A__TAIL(a__incr(a__pairs)), MARK(incr(pairs))) MARK(tail(incr(odds))) -> c17(A__TAIL(a__incr(a__odds)), MARK(incr(odds))) MARK(tail(incr(head(z0)))) -> c17(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c17(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c17(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c17(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c17(MARK(incr(x0))) MARK(tail(incr(0))) -> c17(A__TAIL(a__incr(0))) MARK(tail(odds)) -> c17(A__TAIL(a__incr(a__pairs)), MARK(odds)) MARK(tail(odds)) -> c17(MARK(odds)) MARK(tail(head(nats))) -> c17(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(incr(z0)))) -> c17(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(pairs))) -> c17(A__TAIL(a__head(a__pairs)), MARK(head(pairs))) MARK(tail(head(odds))) -> c17(A__TAIL(a__head(a__odds)), MARK(head(odds))) MARK(tail(head(head(z0)))) -> c17(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c17(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c17(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c17(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c17(MARK(head(x0))) MARK(tail(head(0))) -> c17(A__TAIL(a__head(0))) MARK(tail(tail(nats))) -> c17(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(incr(z0)))) -> c17(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(pairs))) -> c17(A__TAIL(a__tail(a__pairs)), MARK(tail(pairs))) MARK(tail(tail(odds))) -> c17(A__TAIL(a__tail(a__odds)), MARK(tail(odds))) MARK(tail(tail(head(z0)))) -> c17(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c17(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c17(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c17(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c17(MARK(tail(x0))) MARK(tail(tail(0))) -> c17(A__TAIL(a__tail(0))) MARK(tail(nats)) -> c17(A__TAIL(cons(0, incr(nats)))) MARK(tail(pairs)) -> c17(A__TAIL(cons(0, incr(odds)))) MARK(incr(head(0))) -> c13(A__INCR(head(0))) K tuples:none Defined Rule Symbols: a__nats, a__pairs, a__odds, a__incr_1, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__HEAD_1, A__TAIL_1, MARK_1, A__ODDS Compound Symbols: c6_1, c8_1, c10_1, c15_1, c18_1, c20_1, c13_2, c13_1, c16_2, c16_1, c17_2, c17_1, c4_1 ---------------------------------------- (129) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(incr(head(0))) -> c13(A__INCR(head(0))) ---------------------------------------- (130) Obligation: Complexity Dependency Tuples Problem Rules: a__nats -> cons(0, incr(nats)) a__nats -> nats a__pairs -> cons(0, incr(odds)) a__pairs -> pairs a__odds -> a__incr(a__pairs) a__odds -> odds a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) 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(nats) -> a__nats mark(incr(z0)) -> a__incr(mark(z0)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(incr(tail(0))) -> c13(A__INCR(a__tail(0))) MARK(incr(nats)) -> c13(A__INCR(cons(0, incr(nats)))) MARK(incr(pairs)) -> c13(A__INCR(cons(0, incr(odds)))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs)), MARK(incr(pairs))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(incr(0))) -> c16(A__HEAD(a__incr(0))) MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs)), MARK(odds)) MARK(head(odds)) -> c16(MARK(odds)) MARK(head(head(nats))) -> c16(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(pairs))) -> c16(A__HEAD(a__head(a__pairs)), MARK(head(pairs))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(head(0))) -> c16(A__HEAD(a__head(0))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(head(tail(0))) -> c16(A__HEAD(a__tail(0))) MARK(head(nats)) -> c16(A__HEAD(cons(0, incr(nats)))) MARK(head(pairs)) -> c16(A__HEAD(cons(0, incr(odds)))) MARK(tail(incr(nats))) -> c17(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(incr(z0)))) -> c17(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(pairs))) -> c17(A__TAIL(a__incr(a__pairs)), MARK(incr(pairs))) MARK(tail(incr(odds))) -> c17(A__TAIL(a__incr(a__odds)), MARK(incr(odds))) MARK(tail(incr(head(z0)))) -> c17(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c17(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c17(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c17(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c17(MARK(incr(x0))) MARK(tail(incr(0))) -> c17(A__TAIL(a__incr(0))) MARK(tail(odds)) -> c17(A__TAIL(a__incr(a__pairs)), MARK(odds)) MARK(tail(odds)) -> c17(MARK(odds)) MARK(tail(head(nats))) -> c17(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(incr(z0)))) -> c17(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(pairs))) -> c17(A__TAIL(a__head(a__pairs)), MARK(head(pairs))) MARK(tail(head(odds))) -> c17(A__TAIL(a__head(a__odds)), MARK(head(odds))) MARK(tail(head(head(z0)))) -> c17(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c17(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c17(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c17(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c17(MARK(head(x0))) MARK(tail(head(0))) -> c17(A__TAIL(a__head(0))) MARK(tail(tail(nats))) -> c17(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(incr(z0)))) -> c17(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(pairs))) -> c17(A__TAIL(a__tail(a__pairs)), MARK(tail(pairs))) MARK(tail(tail(odds))) -> c17(A__TAIL(a__tail(a__odds)), MARK(tail(odds))) MARK(tail(tail(head(z0)))) -> c17(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c17(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c17(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c17(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c17(MARK(tail(x0))) MARK(tail(tail(0))) -> c17(A__TAIL(a__tail(0))) MARK(tail(nats)) -> c17(A__TAIL(cons(0, incr(nats)))) MARK(tail(pairs)) -> c17(A__TAIL(cons(0, incr(odds)))) S tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(incr(tail(0))) -> c13(A__INCR(a__tail(0))) MARK(incr(nats)) -> c13(A__INCR(cons(0, incr(nats)))) MARK(incr(pairs)) -> c13(A__INCR(cons(0, incr(odds)))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs)), MARK(incr(pairs))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(incr(0))) -> c16(A__HEAD(a__incr(0))) MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs)), MARK(odds)) MARK(head(odds)) -> c16(MARK(odds)) MARK(head(head(nats))) -> c16(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(pairs))) -> c16(A__HEAD(a__head(a__pairs)), MARK(head(pairs))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(head(0))) -> c16(A__HEAD(a__head(0))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(head(tail(0))) -> c16(A__HEAD(a__tail(0))) MARK(head(nats)) -> c16(A__HEAD(cons(0, incr(nats)))) MARK(head(pairs)) -> c16(A__HEAD(cons(0, incr(odds)))) MARK(tail(incr(nats))) -> c17(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(incr(z0)))) -> c17(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(pairs))) -> c17(A__TAIL(a__incr(a__pairs)), MARK(incr(pairs))) MARK(tail(incr(odds))) -> c17(A__TAIL(a__incr(a__odds)), MARK(incr(odds))) MARK(tail(incr(head(z0)))) -> c17(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c17(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c17(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c17(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c17(MARK(incr(x0))) MARK(tail(incr(0))) -> c17(A__TAIL(a__incr(0))) MARK(tail(odds)) -> c17(A__TAIL(a__incr(a__pairs)), MARK(odds)) MARK(tail(odds)) -> c17(MARK(odds)) MARK(tail(head(nats))) -> c17(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(incr(z0)))) -> c17(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(pairs))) -> c17(A__TAIL(a__head(a__pairs)), MARK(head(pairs))) MARK(tail(head(odds))) -> c17(A__TAIL(a__head(a__odds)), MARK(head(odds))) MARK(tail(head(head(z0)))) -> c17(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c17(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c17(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c17(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c17(MARK(head(x0))) MARK(tail(head(0))) -> c17(A__TAIL(a__head(0))) MARK(tail(tail(nats))) -> c17(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(incr(z0)))) -> c17(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(pairs))) -> c17(A__TAIL(a__tail(a__pairs)), MARK(tail(pairs))) MARK(tail(tail(odds))) -> c17(A__TAIL(a__tail(a__odds)), MARK(tail(odds))) MARK(tail(tail(head(z0)))) -> c17(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c17(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c17(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c17(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c17(MARK(tail(x0))) MARK(tail(tail(0))) -> c17(A__TAIL(a__tail(0))) MARK(tail(nats)) -> c17(A__TAIL(cons(0, incr(nats)))) MARK(tail(pairs)) -> c17(A__TAIL(cons(0, incr(odds)))) K tuples:none Defined Rule Symbols: a__nats, a__pairs, a__odds, a__incr_1, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__HEAD_1, A__TAIL_1, MARK_1, A__ODDS Compound Symbols: c6_1, c8_1, c10_1, c15_1, c18_1, c20_1, c13_2, c13_1, c16_2, c16_1, c17_2, c17_1, c4_1 ---------------------------------------- (131) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(incr(tail(0))) -> c13(A__INCR(a__tail(0))) by MARK(incr(tail(0))) -> c13(A__INCR(tail(0))) ---------------------------------------- (132) Obligation: Complexity Dependency Tuples Problem Rules: a__nats -> cons(0, incr(nats)) a__nats -> nats a__pairs -> cons(0, incr(odds)) a__pairs -> pairs a__odds -> a__incr(a__pairs) a__odds -> odds a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) 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(nats) -> a__nats mark(incr(z0)) -> a__incr(mark(z0)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(incr(nats)) -> c13(A__INCR(cons(0, incr(nats)))) MARK(incr(pairs)) -> c13(A__INCR(cons(0, incr(odds)))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs)), MARK(incr(pairs))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(incr(0))) -> c16(A__HEAD(a__incr(0))) MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs)), MARK(odds)) MARK(head(odds)) -> c16(MARK(odds)) MARK(head(head(nats))) -> c16(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(pairs))) -> c16(A__HEAD(a__head(a__pairs)), MARK(head(pairs))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(head(0))) -> c16(A__HEAD(a__head(0))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(head(tail(0))) -> c16(A__HEAD(a__tail(0))) MARK(head(nats)) -> c16(A__HEAD(cons(0, incr(nats)))) MARK(head(pairs)) -> c16(A__HEAD(cons(0, incr(odds)))) MARK(tail(incr(nats))) -> c17(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(incr(z0)))) -> c17(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(pairs))) -> c17(A__TAIL(a__incr(a__pairs)), MARK(incr(pairs))) MARK(tail(incr(odds))) -> c17(A__TAIL(a__incr(a__odds)), MARK(incr(odds))) MARK(tail(incr(head(z0)))) -> c17(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c17(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c17(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c17(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c17(MARK(incr(x0))) MARK(tail(incr(0))) -> c17(A__TAIL(a__incr(0))) MARK(tail(odds)) -> c17(A__TAIL(a__incr(a__pairs)), MARK(odds)) MARK(tail(odds)) -> c17(MARK(odds)) MARK(tail(head(nats))) -> c17(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(incr(z0)))) -> c17(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(pairs))) -> c17(A__TAIL(a__head(a__pairs)), MARK(head(pairs))) MARK(tail(head(odds))) -> c17(A__TAIL(a__head(a__odds)), MARK(head(odds))) MARK(tail(head(head(z0)))) -> c17(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c17(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c17(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c17(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c17(MARK(head(x0))) MARK(tail(head(0))) -> c17(A__TAIL(a__head(0))) MARK(tail(tail(nats))) -> c17(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(incr(z0)))) -> c17(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(pairs))) -> c17(A__TAIL(a__tail(a__pairs)), MARK(tail(pairs))) MARK(tail(tail(odds))) -> c17(A__TAIL(a__tail(a__odds)), MARK(tail(odds))) MARK(tail(tail(head(z0)))) -> c17(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c17(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c17(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c17(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c17(MARK(tail(x0))) MARK(tail(tail(0))) -> c17(A__TAIL(a__tail(0))) MARK(tail(nats)) -> c17(A__TAIL(cons(0, incr(nats)))) MARK(tail(pairs)) -> c17(A__TAIL(cons(0, incr(odds)))) MARK(incr(tail(0))) -> c13(A__INCR(tail(0))) S tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(incr(nats)) -> c13(A__INCR(cons(0, incr(nats)))) MARK(incr(pairs)) -> c13(A__INCR(cons(0, incr(odds)))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs)), MARK(incr(pairs))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(incr(0))) -> c16(A__HEAD(a__incr(0))) MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs)), MARK(odds)) MARK(head(odds)) -> c16(MARK(odds)) MARK(head(head(nats))) -> c16(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(pairs))) -> c16(A__HEAD(a__head(a__pairs)), MARK(head(pairs))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(head(0))) -> c16(A__HEAD(a__head(0))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(head(tail(0))) -> c16(A__HEAD(a__tail(0))) MARK(head(nats)) -> c16(A__HEAD(cons(0, incr(nats)))) MARK(head(pairs)) -> c16(A__HEAD(cons(0, incr(odds)))) MARK(tail(incr(nats))) -> c17(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(incr(z0)))) -> c17(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(pairs))) -> c17(A__TAIL(a__incr(a__pairs)), MARK(incr(pairs))) MARK(tail(incr(odds))) -> c17(A__TAIL(a__incr(a__odds)), MARK(incr(odds))) MARK(tail(incr(head(z0)))) -> c17(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c17(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c17(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c17(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c17(MARK(incr(x0))) MARK(tail(incr(0))) -> c17(A__TAIL(a__incr(0))) MARK(tail(odds)) -> c17(A__TAIL(a__incr(a__pairs)), MARK(odds)) MARK(tail(odds)) -> c17(MARK(odds)) MARK(tail(head(nats))) -> c17(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(incr(z0)))) -> c17(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(pairs))) -> c17(A__TAIL(a__head(a__pairs)), MARK(head(pairs))) MARK(tail(head(odds))) -> c17(A__TAIL(a__head(a__odds)), MARK(head(odds))) MARK(tail(head(head(z0)))) -> c17(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c17(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c17(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c17(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c17(MARK(head(x0))) MARK(tail(head(0))) -> c17(A__TAIL(a__head(0))) MARK(tail(tail(nats))) -> c17(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(incr(z0)))) -> c17(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(pairs))) -> c17(A__TAIL(a__tail(a__pairs)), MARK(tail(pairs))) MARK(tail(tail(odds))) -> c17(A__TAIL(a__tail(a__odds)), MARK(tail(odds))) MARK(tail(tail(head(z0)))) -> c17(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c17(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c17(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c17(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c17(MARK(tail(x0))) MARK(tail(tail(0))) -> c17(A__TAIL(a__tail(0))) MARK(tail(nats)) -> c17(A__TAIL(cons(0, incr(nats)))) MARK(tail(pairs)) -> c17(A__TAIL(cons(0, incr(odds)))) MARK(incr(tail(0))) -> c13(A__INCR(tail(0))) K tuples:none Defined Rule Symbols: a__nats, a__pairs, a__odds, a__incr_1, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__HEAD_1, A__TAIL_1, MARK_1, A__ODDS Compound Symbols: c6_1, c8_1, c10_1, c15_1, c18_1, c20_1, c13_2, c13_1, c16_2, c16_1, c17_2, c17_1, c4_1 ---------------------------------------- (133) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(incr(tail(0))) -> c13(A__INCR(tail(0))) ---------------------------------------- (134) Obligation: Complexity Dependency Tuples Problem Rules: a__nats -> cons(0, incr(nats)) a__nats -> nats a__pairs -> cons(0, incr(odds)) a__pairs -> pairs a__odds -> a__incr(a__pairs) a__odds -> odds a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) 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(nats) -> a__nats mark(incr(z0)) -> a__incr(mark(z0)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(incr(nats)) -> c13(A__INCR(cons(0, incr(nats)))) MARK(incr(pairs)) -> c13(A__INCR(cons(0, incr(odds)))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs)), MARK(incr(pairs))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(incr(0))) -> c16(A__HEAD(a__incr(0))) MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs)), MARK(odds)) MARK(head(odds)) -> c16(MARK(odds)) MARK(head(head(nats))) -> c16(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(pairs))) -> c16(A__HEAD(a__head(a__pairs)), MARK(head(pairs))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(head(0))) -> c16(A__HEAD(a__head(0))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(head(tail(0))) -> c16(A__HEAD(a__tail(0))) MARK(head(nats)) -> c16(A__HEAD(cons(0, incr(nats)))) MARK(head(pairs)) -> c16(A__HEAD(cons(0, incr(odds)))) MARK(tail(incr(nats))) -> c17(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(incr(z0)))) -> c17(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(pairs))) -> c17(A__TAIL(a__incr(a__pairs)), MARK(incr(pairs))) MARK(tail(incr(odds))) -> c17(A__TAIL(a__incr(a__odds)), MARK(incr(odds))) MARK(tail(incr(head(z0)))) -> c17(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c17(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c17(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c17(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c17(MARK(incr(x0))) MARK(tail(incr(0))) -> c17(A__TAIL(a__incr(0))) MARK(tail(odds)) -> c17(A__TAIL(a__incr(a__pairs)), MARK(odds)) MARK(tail(odds)) -> c17(MARK(odds)) MARK(tail(head(nats))) -> c17(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(incr(z0)))) -> c17(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(pairs))) -> c17(A__TAIL(a__head(a__pairs)), MARK(head(pairs))) MARK(tail(head(odds))) -> c17(A__TAIL(a__head(a__odds)), MARK(head(odds))) MARK(tail(head(head(z0)))) -> c17(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c17(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c17(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c17(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c17(MARK(head(x0))) MARK(tail(head(0))) -> c17(A__TAIL(a__head(0))) MARK(tail(tail(nats))) -> c17(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(incr(z0)))) -> c17(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(pairs))) -> c17(A__TAIL(a__tail(a__pairs)), MARK(tail(pairs))) MARK(tail(tail(odds))) -> c17(A__TAIL(a__tail(a__odds)), MARK(tail(odds))) MARK(tail(tail(head(z0)))) -> c17(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c17(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c17(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c17(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c17(MARK(tail(x0))) MARK(tail(tail(0))) -> c17(A__TAIL(a__tail(0))) MARK(tail(nats)) -> c17(A__TAIL(cons(0, incr(nats)))) MARK(tail(pairs)) -> c17(A__TAIL(cons(0, incr(odds)))) S tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(incr(nats)) -> c13(A__INCR(cons(0, incr(nats)))) MARK(incr(pairs)) -> c13(A__INCR(cons(0, incr(odds)))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs)), MARK(incr(pairs))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(incr(0))) -> c16(A__HEAD(a__incr(0))) MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs)), MARK(odds)) MARK(head(odds)) -> c16(MARK(odds)) MARK(head(head(nats))) -> c16(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(pairs))) -> c16(A__HEAD(a__head(a__pairs)), MARK(head(pairs))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(head(0))) -> c16(A__HEAD(a__head(0))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(head(tail(0))) -> c16(A__HEAD(a__tail(0))) MARK(head(nats)) -> c16(A__HEAD(cons(0, incr(nats)))) MARK(head(pairs)) -> c16(A__HEAD(cons(0, incr(odds)))) MARK(tail(incr(nats))) -> c17(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(incr(z0)))) -> c17(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(pairs))) -> c17(A__TAIL(a__incr(a__pairs)), MARK(incr(pairs))) MARK(tail(incr(odds))) -> c17(A__TAIL(a__incr(a__odds)), MARK(incr(odds))) MARK(tail(incr(head(z0)))) -> c17(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c17(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c17(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c17(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c17(MARK(incr(x0))) MARK(tail(incr(0))) -> c17(A__TAIL(a__incr(0))) MARK(tail(odds)) -> c17(A__TAIL(a__incr(a__pairs)), MARK(odds)) MARK(tail(odds)) -> c17(MARK(odds)) MARK(tail(head(nats))) -> c17(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(incr(z0)))) -> c17(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(pairs))) -> c17(A__TAIL(a__head(a__pairs)), MARK(head(pairs))) MARK(tail(head(odds))) -> c17(A__TAIL(a__head(a__odds)), MARK(head(odds))) MARK(tail(head(head(z0)))) -> c17(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c17(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c17(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c17(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c17(MARK(head(x0))) MARK(tail(head(0))) -> c17(A__TAIL(a__head(0))) MARK(tail(tail(nats))) -> c17(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(incr(z0)))) -> c17(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(pairs))) -> c17(A__TAIL(a__tail(a__pairs)), MARK(tail(pairs))) MARK(tail(tail(odds))) -> c17(A__TAIL(a__tail(a__odds)), MARK(tail(odds))) MARK(tail(tail(head(z0)))) -> c17(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c17(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c17(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c17(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c17(MARK(tail(x0))) MARK(tail(tail(0))) -> c17(A__TAIL(a__tail(0))) MARK(tail(nats)) -> c17(A__TAIL(cons(0, incr(nats)))) MARK(tail(pairs)) -> c17(A__TAIL(cons(0, incr(odds)))) K tuples:none Defined Rule Symbols: a__nats, a__pairs, a__odds, a__incr_1, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__HEAD_1, A__TAIL_1, MARK_1, A__ODDS Compound Symbols: c6_1, c8_1, c10_1, c15_1, c18_1, c20_1, c13_2, c13_1, c16_2, c16_1, c17_2, c17_1, c4_1 ---------------------------------------- (135) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(head(incr(0))) -> c16(A__HEAD(a__incr(0))) by MARK(head(incr(0))) -> c16(A__HEAD(incr(0))) ---------------------------------------- (136) Obligation: Complexity Dependency Tuples Problem Rules: a__nats -> cons(0, incr(nats)) a__nats -> nats a__pairs -> cons(0, incr(odds)) a__pairs -> pairs a__odds -> a__incr(a__pairs) a__odds -> odds a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) 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(nats) -> a__nats mark(incr(z0)) -> a__incr(mark(z0)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(incr(nats)) -> c13(A__INCR(cons(0, incr(nats)))) MARK(incr(pairs)) -> c13(A__INCR(cons(0, incr(odds)))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs)), MARK(incr(pairs))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs)), MARK(odds)) MARK(head(odds)) -> c16(MARK(odds)) MARK(head(head(nats))) -> c16(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(pairs))) -> c16(A__HEAD(a__head(a__pairs)), MARK(head(pairs))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(head(0))) -> c16(A__HEAD(a__head(0))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(head(tail(0))) -> c16(A__HEAD(a__tail(0))) MARK(head(nats)) -> c16(A__HEAD(cons(0, incr(nats)))) MARK(head(pairs)) -> c16(A__HEAD(cons(0, incr(odds)))) MARK(tail(incr(nats))) -> c17(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(incr(z0)))) -> c17(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(pairs))) -> c17(A__TAIL(a__incr(a__pairs)), MARK(incr(pairs))) MARK(tail(incr(odds))) -> c17(A__TAIL(a__incr(a__odds)), MARK(incr(odds))) MARK(tail(incr(head(z0)))) -> c17(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c17(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c17(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c17(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c17(MARK(incr(x0))) MARK(tail(incr(0))) -> c17(A__TAIL(a__incr(0))) MARK(tail(odds)) -> c17(A__TAIL(a__incr(a__pairs)), MARK(odds)) MARK(tail(odds)) -> c17(MARK(odds)) MARK(tail(head(nats))) -> c17(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(incr(z0)))) -> c17(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(pairs))) -> c17(A__TAIL(a__head(a__pairs)), MARK(head(pairs))) MARK(tail(head(odds))) -> c17(A__TAIL(a__head(a__odds)), MARK(head(odds))) MARK(tail(head(head(z0)))) -> c17(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c17(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c17(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c17(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c17(MARK(head(x0))) MARK(tail(head(0))) -> c17(A__TAIL(a__head(0))) MARK(tail(tail(nats))) -> c17(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(incr(z0)))) -> c17(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(pairs))) -> c17(A__TAIL(a__tail(a__pairs)), MARK(tail(pairs))) MARK(tail(tail(odds))) -> c17(A__TAIL(a__tail(a__odds)), MARK(tail(odds))) MARK(tail(tail(head(z0)))) -> c17(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c17(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c17(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c17(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c17(MARK(tail(x0))) MARK(tail(tail(0))) -> c17(A__TAIL(a__tail(0))) MARK(tail(nats)) -> c17(A__TAIL(cons(0, incr(nats)))) MARK(tail(pairs)) -> c17(A__TAIL(cons(0, incr(odds)))) MARK(head(incr(0))) -> c16(A__HEAD(incr(0))) S tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(incr(nats)) -> c13(A__INCR(cons(0, incr(nats)))) MARK(incr(pairs)) -> c13(A__INCR(cons(0, incr(odds)))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs)), MARK(incr(pairs))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs)), MARK(odds)) MARK(head(odds)) -> c16(MARK(odds)) MARK(head(head(nats))) -> c16(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(pairs))) -> c16(A__HEAD(a__head(a__pairs)), MARK(head(pairs))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(head(0))) -> c16(A__HEAD(a__head(0))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(head(tail(0))) -> c16(A__HEAD(a__tail(0))) MARK(head(nats)) -> c16(A__HEAD(cons(0, incr(nats)))) MARK(head(pairs)) -> c16(A__HEAD(cons(0, incr(odds)))) MARK(tail(incr(nats))) -> c17(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(incr(z0)))) -> c17(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(pairs))) -> c17(A__TAIL(a__incr(a__pairs)), MARK(incr(pairs))) MARK(tail(incr(odds))) -> c17(A__TAIL(a__incr(a__odds)), MARK(incr(odds))) MARK(tail(incr(head(z0)))) -> c17(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c17(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c17(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c17(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c17(MARK(incr(x0))) MARK(tail(incr(0))) -> c17(A__TAIL(a__incr(0))) MARK(tail(odds)) -> c17(A__TAIL(a__incr(a__pairs)), MARK(odds)) MARK(tail(odds)) -> c17(MARK(odds)) MARK(tail(head(nats))) -> c17(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(incr(z0)))) -> c17(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(pairs))) -> c17(A__TAIL(a__head(a__pairs)), MARK(head(pairs))) MARK(tail(head(odds))) -> c17(A__TAIL(a__head(a__odds)), MARK(head(odds))) MARK(tail(head(head(z0)))) -> c17(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c17(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c17(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c17(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c17(MARK(head(x0))) MARK(tail(head(0))) -> c17(A__TAIL(a__head(0))) MARK(tail(tail(nats))) -> c17(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(incr(z0)))) -> c17(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(pairs))) -> c17(A__TAIL(a__tail(a__pairs)), MARK(tail(pairs))) MARK(tail(tail(odds))) -> c17(A__TAIL(a__tail(a__odds)), MARK(tail(odds))) MARK(tail(tail(head(z0)))) -> c17(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c17(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c17(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c17(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c17(MARK(tail(x0))) MARK(tail(tail(0))) -> c17(A__TAIL(a__tail(0))) MARK(tail(nats)) -> c17(A__TAIL(cons(0, incr(nats)))) MARK(tail(pairs)) -> c17(A__TAIL(cons(0, incr(odds)))) MARK(head(incr(0))) -> c16(A__HEAD(incr(0))) K tuples:none Defined Rule Symbols: a__nats, a__pairs, a__odds, a__incr_1, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__HEAD_1, A__TAIL_1, MARK_1, A__ODDS Compound Symbols: c6_1, c8_1, c10_1, c15_1, c18_1, c20_1, c13_2, c13_1, c16_2, c16_1, c17_2, c17_1, c4_1 ---------------------------------------- (137) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(head(incr(0))) -> c16(A__HEAD(incr(0))) ---------------------------------------- (138) Obligation: Complexity Dependency Tuples Problem Rules: a__nats -> cons(0, incr(nats)) a__nats -> nats a__pairs -> cons(0, incr(odds)) a__pairs -> pairs a__odds -> a__incr(a__pairs) a__odds -> odds a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) 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(nats) -> a__nats mark(incr(z0)) -> a__incr(mark(z0)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(incr(nats)) -> c13(A__INCR(cons(0, incr(nats)))) MARK(incr(pairs)) -> c13(A__INCR(cons(0, incr(odds)))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs)), MARK(incr(pairs))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs)), MARK(odds)) MARK(head(odds)) -> c16(MARK(odds)) MARK(head(head(nats))) -> c16(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(pairs))) -> c16(A__HEAD(a__head(a__pairs)), MARK(head(pairs))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(head(0))) -> c16(A__HEAD(a__head(0))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(head(tail(0))) -> c16(A__HEAD(a__tail(0))) MARK(head(nats)) -> c16(A__HEAD(cons(0, incr(nats)))) MARK(head(pairs)) -> c16(A__HEAD(cons(0, incr(odds)))) MARK(tail(incr(nats))) -> c17(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(incr(z0)))) -> c17(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(pairs))) -> c17(A__TAIL(a__incr(a__pairs)), MARK(incr(pairs))) MARK(tail(incr(odds))) -> c17(A__TAIL(a__incr(a__odds)), MARK(incr(odds))) MARK(tail(incr(head(z0)))) -> c17(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c17(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c17(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c17(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c17(MARK(incr(x0))) MARK(tail(incr(0))) -> c17(A__TAIL(a__incr(0))) MARK(tail(odds)) -> c17(A__TAIL(a__incr(a__pairs)), MARK(odds)) MARK(tail(odds)) -> c17(MARK(odds)) MARK(tail(head(nats))) -> c17(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(incr(z0)))) -> c17(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(pairs))) -> c17(A__TAIL(a__head(a__pairs)), MARK(head(pairs))) MARK(tail(head(odds))) -> c17(A__TAIL(a__head(a__odds)), MARK(head(odds))) MARK(tail(head(head(z0)))) -> c17(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c17(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c17(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c17(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c17(MARK(head(x0))) MARK(tail(head(0))) -> c17(A__TAIL(a__head(0))) MARK(tail(tail(nats))) -> c17(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(incr(z0)))) -> c17(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(pairs))) -> c17(A__TAIL(a__tail(a__pairs)), MARK(tail(pairs))) MARK(tail(tail(odds))) -> c17(A__TAIL(a__tail(a__odds)), MARK(tail(odds))) MARK(tail(tail(head(z0)))) -> c17(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c17(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c17(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c17(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c17(MARK(tail(x0))) MARK(tail(tail(0))) -> c17(A__TAIL(a__tail(0))) MARK(tail(nats)) -> c17(A__TAIL(cons(0, incr(nats)))) MARK(tail(pairs)) -> c17(A__TAIL(cons(0, incr(odds)))) S tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(incr(nats)) -> c13(A__INCR(cons(0, incr(nats)))) MARK(incr(pairs)) -> c13(A__INCR(cons(0, incr(odds)))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs)), MARK(incr(pairs))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs)), MARK(odds)) MARK(head(odds)) -> c16(MARK(odds)) MARK(head(head(nats))) -> c16(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(pairs))) -> c16(A__HEAD(a__head(a__pairs)), MARK(head(pairs))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(head(0))) -> c16(A__HEAD(a__head(0))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(head(tail(0))) -> c16(A__HEAD(a__tail(0))) MARK(head(nats)) -> c16(A__HEAD(cons(0, incr(nats)))) MARK(head(pairs)) -> c16(A__HEAD(cons(0, incr(odds)))) MARK(tail(incr(nats))) -> c17(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(incr(z0)))) -> c17(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(pairs))) -> c17(A__TAIL(a__incr(a__pairs)), MARK(incr(pairs))) MARK(tail(incr(odds))) -> c17(A__TAIL(a__incr(a__odds)), MARK(incr(odds))) MARK(tail(incr(head(z0)))) -> c17(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c17(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c17(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c17(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c17(MARK(incr(x0))) MARK(tail(incr(0))) -> c17(A__TAIL(a__incr(0))) MARK(tail(odds)) -> c17(A__TAIL(a__incr(a__pairs)), MARK(odds)) MARK(tail(odds)) -> c17(MARK(odds)) MARK(tail(head(nats))) -> c17(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(incr(z0)))) -> c17(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(pairs))) -> c17(A__TAIL(a__head(a__pairs)), MARK(head(pairs))) MARK(tail(head(odds))) -> c17(A__TAIL(a__head(a__odds)), MARK(head(odds))) MARK(tail(head(head(z0)))) -> c17(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c17(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c17(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c17(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c17(MARK(head(x0))) MARK(tail(head(0))) -> c17(A__TAIL(a__head(0))) MARK(tail(tail(nats))) -> c17(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(incr(z0)))) -> c17(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(pairs))) -> c17(A__TAIL(a__tail(a__pairs)), MARK(tail(pairs))) MARK(tail(tail(odds))) -> c17(A__TAIL(a__tail(a__odds)), MARK(tail(odds))) MARK(tail(tail(head(z0)))) -> c17(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c17(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c17(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c17(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c17(MARK(tail(x0))) MARK(tail(tail(0))) -> c17(A__TAIL(a__tail(0))) MARK(tail(nats)) -> c17(A__TAIL(cons(0, incr(nats)))) MARK(tail(pairs)) -> c17(A__TAIL(cons(0, incr(odds)))) K tuples:none Defined Rule Symbols: a__nats, a__pairs, a__odds, a__incr_1, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__HEAD_1, A__TAIL_1, MARK_1, A__ODDS Compound Symbols: c6_1, c8_1, c10_1, c15_1, c18_1, c20_1, c13_2, c13_1, c16_2, c16_1, c17_2, c17_1, c4_1 ---------------------------------------- (139) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(head(head(0))) -> c16(A__HEAD(a__head(0))) by MARK(head(head(0))) -> c16(A__HEAD(head(0))) ---------------------------------------- (140) Obligation: Complexity Dependency Tuples Problem Rules: a__nats -> cons(0, incr(nats)) a__nats -> nats a__pairs -> cons(0, incr(odds)) a__pairs -> pairs a__odds -> a__incr(a__pairs) a__odds -> odds a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) 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(nats) -> a__nats mark(incr(z0)) -> a__incr(mark(z0)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(incr(nats)) -> c13(A__INCR(cons(0, incr(nats)))) MARK(incr(pairs)) -> c13(A__INCR(cons(0, incr(odds)))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs)), MARK(incr(pairs))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs)), MARK(odds)) MARK(head(odds)) -> c16(MARK(odds)) MARK(head(head(nats))) -> c16(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(pairs))) -> c16(A__HEAD(a__head(a__pairs)), MARK(head(pairs))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(head(tail(0))) -> c16(A__HEAD(a__tail(0))) MARK(head(nats)) -> c16(A__HEAD(cons(0, incr(nats)))) MARK(head(pairs)) -> c16(A__HEAD(cons(0, incr(odds)))) MARK(tail(incr(nats))) -> c17(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(incr(z0)))) -> c17(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(pairs))) -> c17(A__TAIL(a__incr(a__pairs)), MARK(incr(pairs))) MARK(tail(incr(odds))) -> c17(A__TAIL(a__incr(a__odds)), MARK(incr(odds))) MARK(tail(incr(head(z0)))) -> c17(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c17(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c17(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c17(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c17(MARK(incr(x0))) MARK(tail(incr(0))) -> c17(A__TAIL(a__incr(0))) MARK(tail(odds)) -> c17(A__TAIL(a__incr(a__pairs)), MARK(odds)) MARK(tail(odds)) -> c17(MARK(odds)) MARK(tail(head(nats))) -> c17(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(incr(z0)))) -> c17(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(pairs))) -> c17(A__TAIL(a__head(a__pairs)), MARK(head(pairs))) MARK(tail(head(odds))) -> c17(A__TAIL(a__head(a__odds)), MARK(head(odds))) MARK(tail(head(head(z0)))) -> c17(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c17(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c17(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c17(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c17(MARK(head(x0))) MARK(tail(head(0))) -> c17(A__TAIL(a__head(0))) MARK(tail(tail(nats))) -> c17(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(incr(z0)))) -> c17(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(pairs))) -> c17(A__TAIL(a__tail(a__pairs)), MARK(tail(pairs))) MARK(tail(tail(odds))) -> c17(A__TAIL(a__tail(a__odds)), MARK(tail(odds))) MARK(tail(tail(head(z0)))) -> c17(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c17(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c17(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c17(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c17(MARK(tail(x0))) MARK(tail(tail(0))) -> c17(A__TAIL(a__tail(0))) MARK(tail(nats)) -> c17(A__TAIL(cons(0, incr(nats)))) MARK(tail(pairs)) -> c17(A__TAIL(cons(0, incr(odds)))) MARK(head(head(0))) -> c16(A__HEAD(head(0))) S tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(incr(nats)) -> c13(A__INCR(cons(0, incr(nats)))) MARK(incr(pairs)) -> c13(A__INCR(cons(0, incr(odds)))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs)), MARK(incr(pairs))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs)), MARK(odds)) MARK(head(odds)) -> c16(MARK(odds)) MARK(head(head(nats))) -> c16(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(pairs))) -> c16(A__HEAD(a__head(a__pairs)), MARK(head(pairs))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(head(tail(0))) -> c16(A__HEAD(a__tail(0))) MARK(head(nats)) -> c16(A__HEAD(cons(0, incr(nats)))) MARK(head(pairs)) -> c16(A__HEAD(cons(0, incr(odds)))) MARK(tail(incr(nats))) -> c17(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(incr(z0)))) -> c17(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(pairs))) -> c17(A__TAIL(a__incr(a__pairs)), MARK(incr(pairs))) MARK(tail(incr(odds))) -> c17(A__TAIL(a__incr(a__odds)), MARK(incr(odds))) MARK(tail(incr(head(z0)))) -> c17(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c17(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c17(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c17(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c17(MARK(incr(x0))) MARK(tail(incr(0))) -> c17(A__TAIL(a__incr(0))) MARK(tail(odds)) -> c17(A__TAIL(a__incr(a__pairs)), MARK(odds)) MARK(tail(odds)) -> c17(MARK(odds)) MARK(tail(head(nats))) -> c17(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(incr(z0)))) -> c17(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(pairs))) -> c17(A__TAIL(a__head(a__pairs)), MARK(head(pairs))) MARK(tail(head(odds))) -> c17(A__TAIL(a__head(a__odds)), MARK(head(odds))) MARK(tail(head(head(z0)))) -> c17(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c17(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c17(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c17(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c17(MARK(head(x0))) MARK(tail(head(0))) -> c17(A__TAIL(a__head(0))) MARK(tail(tail(nats))) -> c17(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(incr(z0)))) -> c17(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(pairs))) -> c17(A__TAIL(a__tail(a__pairs)), MARK(tail(pairs))) MARK(tail(tail(odds))) -> c17(A__TAIL(a__tail(a__odds)), MARK(tail(odds))) MARK(tail(tail(head(z0)))) -> c17(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c17(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c17(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c17(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c17(MARK(tail(x0))) MARK(tail(tail(0))) -> c17(A__TAIL(a__tail(0))) MARK(tail(nats)) -> c17(A__TAIL(cons(0, incr(nats)))) MARK(tail(pairs)) -> c17(A__TAIL(cons(0, incr(odds)))) MARK(head(head(0))) -> c16(A__HEAD(head(0))) K tuples:none Defined Rule Symbols: a__nats, a__pairs, a__odds, a__incr_1, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__HEAD_1, A__TAIL_1, MARK_1, A__ODDS Compound Symbols: c6_1, c8_1, c10_1, c15_1, c18_1, c20_1, c13_2, c13_1, c16_2, c16_1, c17_2, c17_1, c4_1 ---------------------------------------- (141) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(head(head(0))) -> c16(A__HEAD(head(0))) ---------------------------------------- (142) Obligation: Complexity Dependency Tuples Problem Rules: a__nats -> cons(0, incr(nats)) a__nats -> nats a__pairs -> cons(0, incr(odds)) a__pairs -> pairs a__odds -> a__incr(a__pairs) a__odds -> odds a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) 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(nats) -> a__nats mark(incr(z0)) -> a__incr(mark(z0)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(incr(nats)) -> c13(A__INCR(cons(0, incr(nats)))) MARK(incr(pairs)) -> c13(A__INCR(cons(0, incr(odds)))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs)), MARK(incr(pairs))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs)), MARK(odds)) MARK(head(odds)) -> c16(MARK(odds)) MARK(head(head(nats))) -> c16(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(pairs))) -> c16(A__HEAD(a__head(a__pairs)), MARK(head(pairs))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(head(tail(0))) -> c16(A__HEAD(a__tail(0))) MARK(head(nats)) -> c16(A__HEAD(cons(0, incr(nats)))) MARK(head(pairs)) -> c16(A__HEAD(cons(0, incr(odds)))) MARK(tail(incr(nats))) -> c17(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(incr(z0)))) -> c17(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(pairs))) -> c17(A__TAIL(a__incr(a__pairs)), MARK(incr(pairs))) MARK(tail(incr(odds))) -> c17(A__TAIL(a__incr(a__odds)), MARK(incr(odds))) MARK(tail(incr(head(z0)))) -> c17(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c17(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c17(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c17(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c17(MARK(incr(x0))) MARK(tail(incr(0))) -> c17(A__TAIL(a__incr(0))) MARK(tail(odds)) -> c17(A__TAIL(a__incr(a__pairs)), MARK(odds)) MARK(tail(odds)) -> c17(MARK(odds)) MARK(tail(head(nats))) -> c17(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(incr(z0)))) -> c17(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(pairs))) -> c17(A__TAIL(a__head(a__pairs)), MARK(head(pairs))) MARK(tail(head(odds))) -> c17(A__TAIL(a__head(a__odds)), MARK(head(odds))) MARK(tail(head(head(z0)))) -> c17(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c17(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c17(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c17(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c17(MARK(head(x0))) MARK(tail(head(0))) -> c17(A__TAIL(a__head(0))) MARK(tail(tail(nats))) -> c17(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(incr(z0)))) -> c17(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(pairs))) -> c17(A__TAIL(a__tail(a__pairs)), MARK(tail(pairs))) MARK(tail(tail(odds))) -> c17(A__TAIL(a__tail(a__odds)), MARK(tail(odds))) MARK(tail(tail(head(z0)))) -> c17(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c17(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c17(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c17(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c17(MARK(tail(x0))) MARK(tail(tail(0))) -> c17(A__TAIL(a__tail(0))) MARK(tail(nats)) -> c17(A__TAIL(cons(0, incr(nats)))) MARK(tail(pairs)) -> c17(A__TAIL(cons(0, incr(odds)))) S tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(incr(nats)) -> c13(A__INCR(cons(0, incr(nats)))) MARK(incr(pairs)) -> c13(A__INCR(cons(0, incr(odds)))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs)), MARK(incr(pairs))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs)), MARK(odds)) MARK(head(odds)) -> c16(MARK(odds)) MARK(head(head(nats))) -> c16(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(pairs))) -> c16(A__HEAD(a__head(a__pairs)), MARK(head(pairs))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(head(tail(0))) -> c16(A__HEAD(a__tail(0))) MARK(head(nats)) -> c16(A__HEAD(cons(0, incr(nats)))) MARK(head(pairs)) -> c16(A__HEAD(cons(0, incr(odds)))) MARK(tail(incr(nats))) -> c17(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(incr(z0)))) -> c17(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(pairs))) -> c17(A__TAIL(a__incr(a__pairs)), MARK(incr(pairs))) MARK(tail(incr(odds))) -> c17(A__TAIL(a__incr(a__odds)), MARK(incr(odds))) MARK(tail(incr(head(z0)))) -> c17(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c17(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c17(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c17(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c17(MARK(incr(x0))) MARK(tail(incr(0))) -> c17(A__TAIL(a__incr(0))) MARK(tail(odds)) -> c17(A__TAIL(a__incr(a__pairs)), MARK(odds)) MARK(tail(odds)) -> c17(MARK(odds)) MARK(tail(head(nats))) -> c17(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(incr(z0)))) -> c17(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(pairs))) -> c17(A__TAIL(a__head(a__pairs)), MARK(head(pairs))) MARK(tail(head(odds))) -> c17(A__TAIL(a__head(a__odds)), MARK(head(odds))) MARK(tail(head(head(z0)))) -> c17(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c17(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c17(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c17(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c17(MARK(head(x0))) MARK(tail(head(0))) -> c17(A__TAIL(a__head(0))) MARK(tail(tail(nats))) -> c17(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(incr(z0)))) -> c17(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(pairs))) -> c17(A__TAIL(a__tail(a__pairs)), MARK(tail(pairs))) MARK(tail(tail(odds))) -> c17(A__TAIL(a__tail(a__odds)), MARK(tail(odds))) MARK(tail(tail(head(z0)))) -> c17(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c17(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c17(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c17(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c17(MARK(tail(x0))) MARK(tail(tail(0))) -> c17(A__TAIL(a__tail(0))) MARK(tail(nats)) -> c17(A__TAIL(cons(0, incr(nats)))) MARK(tail(pairs)) -> c17(A__TAIL(cons(0, incr(odds)))) K tuples:none Defined Rule Symbols: a__nats, a__pairs, a__odds, a__incr_1, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__HEAD_1, A__TAIL_1, MARK_1, A__ODDS Compound Symbols: c6_1, c8_1, c10_1, c15_1, c18_1, c20_1, c13_2, c13_1, c16_2, c16_1, c17_2, c17_1, c4_1 ---------------------------------------- (143) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(head(tail(0))) -> c16(A__HEAD(a__tail(0))) by MARK(head(tail(0))) -> c16(A__HEAD(tail(0))) ---------------------------------------- (144) Obligation: Complexity Dependency Tuples Problem Rules: a__nats -> cons(0, incr(nats)) a__nats -> nats a__pairs -> cons(0, incr(odds)) a__pairs -> pairs a__odds -> a__incr(a__pairs) a__odds -> odds a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) 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(nats) -> a__nats mark(incr(z0)) -> a__incr(mark(z0)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(incr(nats)) -> c13(A__INCR(cons(0, incr(nats)))) MARK(incr(pairs)) -> c13(A__INCR(cons(0, incr(odds)))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs)), MARK(incr(pairs))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs)), MARK(odds)) MARK(head(odds)) -> c16(MARK(odds)) MARK(head(head(nats))) -> c16(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(pairs))) -> c16(A__HEAD(a__head(a__pairs)), MARK(head(pairs))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(head(nats)) -> c16(A__HEAD(cons(0, incr(nats)))) MARK(head(pairs)) -> c16(A__HEAD(cons(0, incr(odds)))) MARK(tail(incr(nats))) -> c17(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(incr(z0)))) -> c17(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(pairs))) -> c17(A__TAIL(a__incr(a__pairs)), MARK(incr(pairs))) MARK(tail(incr(odds))) -> c17(A__TAIL(a__incr(a__odds)), MARK(incr(odds))) MARK(tail(incr(head(z0)))) -> c17(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c17(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c17(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c17(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c17(MARK(incr(x0))) MARK(tail(incr(0))) -> c17(A__TAIL(a__incr(0))) MARK(tail(odds)) -> c17(A__TAIL(a__incr(a__pairs)), MARK(odds)) MARK(tail(odds)) -> c17(MARK(odds)) MARK(tail(head(nats))) -> c17(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(incr(z0)))) -> c17(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(pairs))) -> c17(A__TAIL(a__head(a__pairs)), MARK(head(pairs))) MARK(tail(head(odds))) -> c17(A__TAIL(a__head(a__odds)), MARK(head(odds))) MARK(tail(head(head(z0)))) -> c17(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c17(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c17(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c17(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c17(MARK(head(x0))) MARK(tail(head(0))) -> c17(A__TAIL(a__head(0))) MARK(tail(tail(nats))) -> c17(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(incr(z0)))) -> c17(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(pairs))) -> c17(A__TAIL(a__tail(a__pairs)), MARK(tail(pairs))) MARK(tail(tail(odds))) -> c17(A__TAIL(a__tail(a__odds)), MARK(tail(odds))) MARK(tail(tail(head(z0)))) -> c17(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c17(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c17(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c17(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c17(MARK(tail(x0))) MARK(tail(tail(0))) -> c17(A__TAIL(a__tail(0))) MARK(tail(nats)) -> c17(A__TAIL(cons(0, incr(nats)))) MARK(tail(pairs)) -> c17(A__TAIL(cons(0, incr(odds)))) MARK(head(tail(0))) -> c16(A__HEAD(tail(0))) S tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(incr(nats)) -> c13(A__INCR(cons(0, incr(nats)))) MARK(incr(pairs)) -> c13(A__INCR(cons(0, incr(odds)))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs)), MARK(incr(pairs))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs)), MARK(odds)) MARK(head(odds)) -> c16(MARK(odds)) MARK(head(head(nats))) -> c16(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(pairs))) -> c16(A__HEAD(a__head(a__pairs)), MARK(head(pairs))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(head(nats)) -> c16(A__HEAD(cons(0, incr(nats)))) MARK(head(pairs)) -> c16(A__HEAD(cons(0, incr(odds)))) MARK(tail(incr(nats))) -> c17(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(incr(z0)))) -> c17(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(pairs))) -> c17(A__TAIL(a__incr(a__pairs)), MARK(incr(pairs))) MARK(tail(incr(odds))) -> c17(A__TAIL(a__incr(a__odds)), MARK(incr(odds))) MARK(tail(incr(head(z0)))) -> c17(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c17(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c17(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c17(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c17(MARK(incr(x0))) MARK(tail(incr(0))) -> c17(A__TAIL(a__incr(0))) MARK(tail(odds)) -> c17(A__TAIL(a__incr(a__pairs)), MARK(odds)) MARK(tail(odds)) -> c17(MARK(odds)) MARK(tail(head(nats))) -> c17(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(incr(z0)))) -> c17(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(pairs))) -> c17(A__TAIL(a__head(a__pairs)), MARK(head(pairs))) MARK(tail(head(odds))) -> c17(A__TAIL(a__head(a__odds)), MARK(head(odds))) MARK(tail(head(head(z0)))) -> c17(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c17(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c17(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c17(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c17(MARK(head(x0))) MARK(tail(head(0))) -> c17(A__TAIL(a__head(0))) MARK(tail(tail(nats))) -> c17(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(incr(z0)))) -> c17(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(pairs))) -> c17(A__TAIL(a__tail(a__pairs)), MARK(tail(pairs))) MARK(tail(tail(odds))) -> c17(A__TAIL(a__tail(a__odds)), MARK(tail(odds))) MARK(tail(tail(head(z0)))) -> c17(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c17(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c17(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c17(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c17(MARK(tail(x0))) MARK(tail(tail(0))) -> c17(A__TAIL(a__tail(0))) MARK(tail(nats)) -> c17(A__TAIL(cons(0, incr(nats)))) MARK(tail(pairs)) -> c17(A__TAIL(cons(0, incr(odds)))) MARK(head(tail(0))) -> c16(A__HEAD(tail(0))) K tuples:none Defined Rule Symbols: a__nats, a__pairs, a__odds, a__incr_1, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__HEAD_1, A__TAIL_1, MARK_1, A__ODDS Compound Symbols: c6_1, c8_1, c10_1, c15_1, c18_1, c20_1, c13_2, c13_1, c16_2, c16_1, c17_2, c17_1, c4_1 ---------------------------------------- (145) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(head(tail(0))) -> c16(A__HEAD(tail(0))) ---------------------------------------- (146) Obligation: Complexity Dependency Tuples Problem Rules: a__nats -> cons(0, incr(nats)) a__nats -> nats a__pairs -> cons(0, incr(odds)) a__pairs -> pairs a__odds -> a__incr(a__pairs) a__odds -> odds a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) 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(nats) -> a__nats mark(incr(z0)) -> a__incr(mark(z0)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(incr(nats)) -> c13(A__INCR(cons(0, incr(nats)))) MARK(incr(pairs)) -> c13(A__INCR(cons(0, incr(odds)))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs)), MARK(incr(pairs))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs)), MARK(odds)) MARK(head(odds)) -> c16(MARK(odds)) MARK(head(head(nats))) -> c16(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(pairs))) -> c16(A__HEAD(a__head(a__pairs)), MARK(head(pairs))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(head(nats)) -> c16(A__HEAD(cons(0, incr(nats)))) MARK(head(pairs)) -> c16(A__HEAD(cons(0, incr(odds)))) MARK(tail(incr(nats))) -> c17(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(incr(z0)))) -> c17(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(pairs))) -> c17(A__TAIL(a__incr(a__pairs)), MARK(incr(pairs))) MARK(tail(incr(odds))) -> c17(A__TAIL(a__incr(a__odds)), MARK(incr(odds))) MARK(tail(incr(head(z0)))) -> c17(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c17(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c17(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c17(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c17(MARK(incr(x0))) MARK(tail(incr(0))) -> c17(A__TAIL(a__incr(0))) MARK(tail(odds)) -> c17(A__TAIL(a__incr(a__pairs)), MARK(odds)) MARK(tail(odds)) -> c17(MARK(odds)) MARK(tail(head(nats))) -> c17(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(incr(z0)))) -> c17(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(pairs))) -> c17(A__TAIL(a__head(a__pairs)), MARK(head(pairs))) MARK(tail(head(odds))) -> c17(A__TAIL(a__head(a__odds)), MARK(head(odds))) MARK(tail(head(head(z0)))) -> c17(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c17(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c17(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c17(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c17(MARK(head(x0))) MARK(tail(head(0))) -> c17(A__TAIL(a__head(0))) MARK(tail(tail(nats))) -> c17(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(incr(z0)))) -> c17(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(pairs))) -> c17(A__TAIL(a__tail(a__pairs)), MARK(tail(pairs))) MARK(tail(tail(odds))) -> c17(A__TAIL(a__tail(a__odds)), MARK(tail(odds))) MARK(tail(tail(head(z0)))) -> c17(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c17(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c17(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c17(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c17(MARK(tail(x0))) MARK(tail(tail(0))) -> c17(A__TAIL(a__tail(0))) MARK(tail(nats)) -> c17(A__TAIL(cons(0, incr(nats)))) MARK(tail(pairs)) -> c17(A__TAIL(cons(0, incr(odds)))) S tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(incr(nats)) -> c13(A__INCR(cons(0, incr(nats)))) MARK(incr(pairs)) -> c13(A__INCR(cons(0, incr(odds)))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs)), MARK(incr(pairs))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs)), MARK(odds)) MARK(head(odds)) -> c16(MARK(odds)) MARK(head(head(nats))) -> c16(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(pairs))) -> c16(A__HEAD(a__head(a__pairs)), MARK(head(pairs))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(head(nats)) -> c16(A__HEAD(cons(0, incr(nats)))) MARK(head(pairs)) -> c16(A__HEAD(cons(0, incr(odds)))) MARK(tail(incr(nats))) -> c17(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(incr(z0)))) -> c17(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(pairs))) -> c17(A__TAIL(a__incr(a__pairs)), MARK(incr(pairs))) MARK(tail(incr(odds))) -> c17(A__TAIL(a__incr(a__odds)), MARK(incr(odds))) MARK(tail(incr(head(z0)))) -> c17(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c17(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c17(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c17(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c17(MARK(incr(x0))) MARK(tail(incr(0))) -> c17(A__TAIL(a__incr(0))) MARK(tail(odds)) -> c17(A__TAIL(a__incr(a__pairs)), MARK(odds)) MARK(tail(odds)) -> c17(MARK(odds)) MARK(tail(head(nats))) -> c17(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(incr(z0)))) -> c17(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(pairs))) -> c17(A__TAIL(a__head(a__pairs)), MARK(head(pairs))) MARK(tail(head(odds))) -> c17(A__TAIL(a__head(a__odds)), MARK(head(odds))) MARK(tail(head(head(z0)))) -> c17(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c17(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c17(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c17(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c17(MARK(head(x0))) MARK(tail(head(0))) -> c17(A__TAIL(a__head(0))) MARK(tail(tail(nats))) -> c17(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(incr(z0)))) -> c17(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(pairs))) -> c17(A__TAIL(a__tail(a__pairs)), MARK(tail(pairs))) MARK(tail(tail(odds))) -> c17(A__TAIL(a__tail(a__odds)), MARK(tail(odds))) MARK(tail(tail(head(z0)))) -> c17(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c17(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c17(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c17(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c17(MARK(tail(x0))) MARK(tail(tail(0))) -> c17(A__TAIL(a__tail(0))) MARK(tail(nats)) -> c17(A__TAIL(cons(0, incr(nats)))) MARK(tail(pairs)) -> c17(A__TAIL(cons(0, incr(odds)))) K tuples:none Defined Rule Symbols: a__nats, a__pairs, a__odds, a__incr_1, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__HEAD_1, A__TAIL_1, MARK_1, A__ODDS Compound Symbols: c6_1, c8_1, c10_1, c15_1, c18_1, c20_1, c13_2, c13_1, c16_2, c16_1, c17_2, c17_1, c4_1 ---------------------------------------- (147) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(tail(incr(0))) -> c17(A__TAIL(a__incr(0))) by MARK(tail(incr(0))) -> c17(A__TAIL(incr(0))) ---------------------------------------- (148) Obligation: Complexity Dependency Tuples Problem Rules: a__nats -> cons(0, incr(nats)) a__nats -> nats a__pairs -> cons(0, incr(odds)) a__pairs -> pairs a__odds -> a__incr(a__pairs) a__odds -> odds a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) 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(nats) -> a__nats mark(incr(z0)) -> a__incr(mark(z0)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(incr(nats)) -> c13(A__INCR(cons(0, incr(nats)))) MARK(incr(pairs)) -> c13(A__INCR(cons(0, incr(odds)))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs)), MARK(incr(pairs))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs)), MARK(odds)) MARK(head(odds)) -> c16(MARK(odds)) MARK(head(head(nats))) -> c16(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(pairs))) -> c16(A__HEAD(a__head(a__pairs)), MARK(head(pairs))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(head(nats)) -> c16(A__HEAD(cons(0, incr(nats)))) MARK(head(pairs)) -> c16(A__HEAD(cons(0, incr(odds)))) MARK(tail(incr(nats))) -> c17(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(incr(z0)))) -> c17(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(pairs))) -> c17(A__TAIL(a__incr(a__pairs)), MARK(incr(pairs))) MARK(tail(incr(odds))) -> c17(A__TAIL(a__incr(a__odds)), MARK(incr(odds))) MARK(tail(incr(head(z0)))) -> c17(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c17(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c17(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c17(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c17(MARK(incr(x0))) MARK(tail(odds)) -> c17(A__TAIL(a__incr(a__pairs)), MARK(odds)) MARK(tail(odds)) -> c17(MARK(odds)) MARK(tail(head(nats))) -> c17(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(incr(z0)))) -> c17(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(pairs))) -> c17(A__TAIL(a__head(a__pairs)), MARK(head(pairs))) MARK(tail(head(odds))) -> c17(A__TAIL(a__head(a__odds)), MARK(head(odds))) MARK(tail(head(head(z0)))) -> c17(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c17(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c17(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c17(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c17(MARK(head(x0))) MARK(tail(head(0))) -> c17(A__TAIL(a__head(0))) MARK(tail(tail(nats))) -> c17(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(incr(z0)))) -> c17(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(pairs))) -> c17(A__TAIL(a__tail(a__pairs)), MARK(tail(pairs))) MARK(tail(tail(odds))) -> c17(A__TAIL(a__tail(a__odds)), MARK(tail(odds))) MARK(tail(tail(head(z0)))) -> c17(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c17(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c17(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c17(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c17(MARK(tail(x0))) MARK(tail(tail(0))) -> c17(A__TAIL(a__tail(0))) MARK(tail(nats)) -> c17(A__TAIL(cons(0, incr(nats)))) MARK(tail(pairs)) -> c17(A__TAIL(cons(0, incr(odds)))) MARK(tail(incr(0))) -> c17(A__TAIL(incr(0))) S tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(incr(nats)) -> c13(A__INCR(cons(0, incr(nats)))) MARK(incr(pairs)) -> c13(A__INCR(cons(0, incr(odds)))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs)), MARK(incr(pairs))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs)), MARK(odds)) MARK(head(odds)) -> c16(MARK(odds)) MARK(head(head(nats))) -> c16(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(pairs))) -> c16(A__HEAD(a__head(a__pairs)), MARK(head(pairs))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(head(nats)) -> c16(A__HEAD(cons(0, incr(nats)))) MARK(head(pairs)) -> c16(A__HEAD(cons(0, incr(odds)))) MARK(tail(incr(nats))) -> c17(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(incr(z0)))) -> c17(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(pairs))) -> c17(A__TAIL(a__incr(a__pairs)), MARK(incr(pairs))) MARK(tail(incr(odds))) -> c17(A__TAIL(a__incr(a__odds)), MARK(incr(odds))) MARK(tail(incr(head(z0)))) -> c17(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c17(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c17(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c17(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c17(MARK(incr(x0))) MARK(tail(odds)) -> c17(A__TAIL(a__incr(a__pairs)), MARK(odds)) MARK(tail(odds)) -> c17(MARK(odds)) MARK(tail(head(nats))) -> c17(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(incr(z0)))) -> c17(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(pairs))) -> c17(A__TAIL(a__head(a__pairs)), MARK(head(pairs))) MARK(tail(head(odds))) -> c17(A__TAIL(a__head(a__odds)), MARK(head(odds))) MARK(tail(head(head(z0)))) -> c17(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c17(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c17(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c17(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c17(MARK(head(x0))) MARK(tail(head(0))) -> c17(A__TAIL(a__head(0))) MARK(tail(tail(nats))) -> c17(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(incr(z0)))) -> c17(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(pairs))) -> c17(A__TAIL(a__tail(a__pairs)), MARK(tail(pairs))) MARK(tail(tail(odds))) -> c17(A__TAIL(a__tail(a__odds)), MARK(tail(odds))) MARK(tail(tail(head(z0)))) -> c17(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c17(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c17(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c17(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c17(MARK(tail(x0))) MARK(tail(tail(0))) -> c17(A__TAIL(a__tail(0))) MARK(tail(nats)) -> c17(A__TAIL(cons(0, incr(nats)))) MARK(tail(pairs)) -> c17(A__TAIL(cons(0, incr(odds)))) MARK(tail(incr(0))) -> c17(A__TAIL(incr(0))) K tuples:none Defined Rule Symbols: a__nats, a__pairs, a__odds, a__incr_1, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__HEAD_1, A__TAIL_1, MARK_1, A__ODDS Compound Symbols: c6_1, c8_1, c10_1, c15_1, c18_1, c20_1, c13_2, c13_1, c16_2, c16_1, c17_2, c17_1, c4_1 ---------------------------------------- (149) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(tail(incr(0))) -> c17(A__TAIL(incr(0))) ---------------------------------------- (150) Obligation: Complexity Dependency Tuples Problem Rules: a__nats -> cons(0, incr(nats)) a__nats -> nats a__pairs -> cons(0, incr(odds)) a__pairs -> pairs a__odds -> a__incr(a__pairs) a__odds -> odds a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) 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(nats) -> a__nats mark(incr(z0)) -> a__incr(mark(z0)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(incr(nats)) -> c13(A__INCR(cons(0, incr(nats)))) MARK(incr(pairs)) -> c13(A__INCR(cons(0, incr(odds)))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs)), MARK(incr(pairs))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs)), MARK(odds)) MARK(head(odds)) -> c16(MARK(odds)) MARK(head(head(nats))) -> c16(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(pairs))) -> c16(A__HEAD(a__head(a__pairs)), MARK(head(pairs))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(head(nats)) -> c16(A__HEAD(cons(0, incr(nats)))) MARK(head(pairs)) -> c16(A__HEAD(cons(0, incr(odds)))) MARK(tail(incr(nats))) -> c17(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(incr(z0)))) -> c17(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(pairs))) -> c17(A__TAIL(a__incr(a__pairs)), MARK(incr(pairs))) MARK(tail(incr(odds))) -> c17(A__TAIL(a__incr(a__odds)), MARK(incr(odds))) MARK(tail(incr(head(z0)))) -> c17(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c17(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c17(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c17(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c17(MARK(incr(x0))) MARK(tail(odds)) -> c17(A__TAIL(a__incr(a__pairs)), MARK(odds)) MARK(tail(odds)) -> c17(MARK(odds)) MARK(tail(head(nats))) -> c17(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(incr(z0)))) -> c17(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(pairs))) -> c17(A__TAIL(a__head(a__pairs)), MARK(head(pairs))) MARK(tail(head(odds))) -> c17(A__TAIL(a__head(a__odds)), MARK(head(odds))) MARK(tail(head(head(z0)))) -> c17(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c17(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c17(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c17(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c17(MARK(head(x0))) MARK(tail(head(0))) -> c17(A__TAIL(a__head(0))) MARK(tail(tail(nats))) -> c17(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(incr(z0)))) -> c17(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(pairs))) -> c17(A__TAIL(a__tail(a__pairs)), MARK(tail(pairs))) MARK(tail(tail(odds))) -> c17(A__TAIL(a__tail(a__odds)), MARK(tail(odds))) MARK(tail(tail(head(z0)))) -> c17(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c17(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c17(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c17(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c17(MARK(tail(x0))) MARK(tail(tail(0))) -> c17(A__TAIL(a__tail(0))) MARK(tail(nats)) -> c17(A__TAIL(cons(0, incr(nats)))) MARK(tail(pairs)) -> c17(A__TAIL(cons(0, incr(odds)))) S tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(incr(nats)) -> c13(A__INCR(cons(0, incr(nats)))) MARK(incr(pairs)) -> c13(A__INCR(cons(0, incr(odds)))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs)), MARK(incr(pairs))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs)), MARK(odds)) MARK(head(odds)) -> c16(MARK(odds)) MARK(head(head(nats))) -> c16(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(pairs))) -> c16(A__HEAD(a__head(a__pairs)), MARK(head(pairs))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(head(nats)) -> c16(A__HEAD(cons(0, incr(nats)))) MARK(head(pairs)) -> c16(A__HEAD(cons(0, incr(odds)))) MARK(tail(incr(nats))) -> c17(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(incr(z0)))) -> c17(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(pairs))) -> c17(A__TAIL(a__incr(a__pairs)), MARK(incr(pairs))) MARK(tail(incr(odds))) -> c17(A__TAIL(a__incr(a__odds)), MARK(incr(odds))) MARK(tail(incr(head(z0)))) -> c17(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c17(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c17(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c17(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c17(MARK(incr(x0))) MARK(tail(odds)) -> c17(A__TAIL(a__incr(a__pairs)), MARK(odds)) MARK(tail(odds)) -> c17(MARK(odds)) MARK(tail(head(nats))) -> c17(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(incr(z0)))) -> c17(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(pairs))) -> c17(A__TAIL(a__head(a__pairs)), MARK(head(pairs))) MARK(tail(head(odds))) -> c17(A__TAIL(a__head(a__odds)), MARK(head(odds))) MARK(tail(head(head(z0)))) -> c17(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c17(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c17(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c17(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c17(MARK(head(x0))) MARK(tail(head(0))) -> c17(A__TAIL(a__head(0))) MARK(tail(tail(nats))) -> c17(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(incr(z0)))) -> c17(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(pairs))) -> c17(A__TAIL(a__tail(a__pairs)), MARK(tail(pairs))) MARK(tail(tail(odds))) -> c17(A__TAIL(a__tail(a__odds)), MARK(tail(odds))) MARK(tail(tail(head(z0)))) -> c17(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c17(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c17(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c17(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c17(MARK(tail(x0))) MARK(tail(tail(0))) -> c17(A__TAIL(a__tail(0))) MARK(tail(nats)) -> c17(A__TAIL(cons(0, incr(nats)))) MARK(tail(pairs)) -> c17(A__TAIL(cons(0, incr(odds)))) K tuples:none Defined Rule Symbols: a__nats, a__pairs, a__odds, a__incr_1, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__HEAD_1, A__TAIL_1, MARK_1, A__ODDS Compound Symbols: c6_1, c8_1, c10_1, c15_1, c18_1, c20_1, c13_2, c13_1, c16_2, c16_1, c17_2, c17_1, c4_1 ---------------------------------------- (151) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(tail(head(0))) -> c17(A__TAIL(a__head(0))) by MARK(tail(head(0))) -> c17(A__TAIL(head(0))) ---------------------------------------- (152) Obligation: Complexity Dependency Tuples Problem Rules: a__nats -> cons(0, incr(nats)) a__nats -> nats a__pairs -> cons(0, incr(odds)) a__pairs -> pairs a__odds -> a__incr(a__pairs) a__odds -> odds a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) 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(nats) -> a__nats mark(incr(z0)) -> a__incr(mark(z0)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(incr(nats)) -> c13(A__INCR(cons(0, incr(nats)))) MARK(incr(pairs)) -> c13(A__INCR(cons(0, incr(odds)))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs)), MARK(incr(pairs))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs)), MARK(odds)) MARK(head(odds)) -> c16(MARK(odds)) MARK(head(head(nats))) -> c16(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(pairs))) -> c16(A__HEAD(a__head(a__pairs)), MARK(head(pairs))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(head(nats)) -> c16(A__HEAD(cons(0, incr(nats)))) MARK(head(pairs)) -> c16(A__HEAD(cons(0, incr(odds)))) MARK(tail(incr(nats))) -> c17(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(incr(z0)))) -> c17(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(pairs))) -> c17(A__TAIL(a__incr(a__pairs)), MARK(incr(pairs))) MARK(tail(incr(odds))) -> c17(A__TAIL(a__incr(a__odds)), MARK(incr(odds))) MARK(tail(incr(head(z0)))) -> c17(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c17(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c17(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c17(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c17(MARK(incr(x0))) MARK(tail(odds)) -> c17(A__TAIL(a__incr(a__pairs)), MARK(odds)) MARK(tail(odds)) -> c17(MARK(odds)) MARK(tail(head(nats))) -> c17(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(incr(z0)))) -> c17(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(pairs))) -> c17(A__TAIL(a__head(a__pairs)), MARK(head(pairs))) MARK(tail(head(odds))) -> c17(A__TAIL(a__head(a__odds)), MARK(head(odds))) MARK(tail(head(head(z0)))) -> c17(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c17(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c17(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c17(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c17(MARK(head(x0))) MARK(tail(tail(nats))) -> c17(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(incr(z0)))) -> c17(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(pairs))) -> c17(A__TAIL(a__tail(a__pairs)), MARK(tail(pairs))) MARK(tail(tail(odds))) -> c17(A__TAIL(a__tail(a__odds)), MARK(tail(odds))) MARK(tail(tail(head(z0)))) -> c17(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c17(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c17(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c17(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c17(MARK(tail(x0))) MARK(tail(tail(0))) -> c17(A__TAIL(a__tail(0))) MARK(tail(nats)) -> c17(A__TAIL(cons(0, incr(nats)))) MARK(tail(pairs)) -> c17(A__TAIL(cons(0, incr(odds)))) MARK(tail(head(0))) -> c17(A__TAIL(head(0))) S tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(incr(nats)) -> c13(A__INCR(cons(0, incr(nats)))) MARK(incr(pairs)) -> c13(A__INCR(cons(0, incr(odds)))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs)), MARK(incr(pairs))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs)), MARK(odds)) MARK(head(odds)) -> c16(MARK(odds)) MARK(head(head(nats))) -> c16(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(pairs))) -> c16(A__HEAD(a__head(a__pairs)), MARK(head(pairs))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(head(nats)) -> c16(A__HEAD(cons(0, incr(nats)))) MARK(head(pairs)) -> c16(A__HEAD(cons(0, incr(odds)))) MARK(tail(incr(nats))) -> c17(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(incr(z0)))) -> c17(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(pairs))) -> c17(A__TAIL(a__incr(a__pairs)), MARK(incr(pairs))) MARK(tail(incr(odds))) -> c17(A__TAIL(a__incr(a__odds)), MARK(incr(odds))) MARK(tail(incr(head(z0)))) -> c17(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c17(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c17(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c17(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c17(MARK(incr(x0))) MARK(tail(odds)) -> c17(A__TAIL(a__incr(a__pairs)), MARK(odds)) MARK(tail(odds)) -> c17(MARK(odds)) MARK(tail(head(nats))) -> c17(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(incr(z0)))) -> c17(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(pairs))) -> c17(A__TAIL(a__head(a__pairs)), MARK(head(pairs))) MARK(tail(head(odds))) -> c17(A__TAIL(a__head(a__odds)), MARK(head(odds))) MARK(tail(head(head(z0)))) -> c17(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c17(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c17(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c17(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c17(MARK(head(x0))) MARK(tail(tail(nats))) -> c17(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(incr(z0)))) -> c17(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(pairs))) -> c17(A__TAIL(a__tail(a__pairs)), MARK(tail(pairs))) MARK(tail(tail(odds))) -> c17(A__TAIL(a__tail(a__odds)), MARK(tail(odds))) MARK(tail(tail(head(z0)))) -> c17(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c17(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c17(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c17(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c17(MARK(tail(x0))) MARK(tail(tail(0))) -> c17(A__TAIL(a__tail(0))) MARK(tail(nats)) -> c17(A__TAIL(cons(0, incr(nats)))) MARK(tail(pairs)) -> c17(A__TAIL(cons(0, incr(odds)))) MARK(tail(head(0))) -> c17(A__TAIL(head(0))) K tuples:none Defined Rule Symbols: a__nats, a__pairs, a__odds, a__incr_1, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__HEAD_1, A__TAIL_1, MARK_1, A__ODDS Compound Symbols: c6_1, c8_1, c10_1, c15_1, c18_1, c20_1, c13_2, c13_1, c16_2, c16_1, c17_2, c17_1, c4_1 ---------------------------------------- (153) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(tail(head(0))) -> c17(A__TAIL(head(0))) ---------------------------------------- (154) Obligation: Complexity Dependency Tuples Problem Rules: a__nats -> cons(0, incr(nats)) a__nats -> nats a__pairs -> cons(0, incr(odds)) a__pairs -> pairs a__odds -> a__incr(a__pairs) a__odds -> odds a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) 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(nats) -> a__nats mark(incr(z0)) -> a__incr(mark(z0)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(incr(nats)) -> c13(A__INCR(cons(0, incr(nats)))) MARK(incr(pairs)) -> c13(A__INCR(cons(0, incr(odds)))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs)), MARK(incr(pairs))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs)), MARK(odds)) MARK(head(odds)) -> c16(MARK(odds)) MARK(head(head(nats))) -> c16(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(pairs))) -> c16(A__HEAD(a__head(a__pairs)), MARK(head(pairs))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(head(nats)) -> c16(A__HEAD(cons(0, incr(nats)))) MARK(head(pairs)) -> c16(A__HEAD(cons(0, incr(odds)))) MARK(tail(incr(nats))) -> c17(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(incr(z0)))) -> c17(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(pairs))) -> c17(A__TAIL(a__incr(a__pairs)), MARK(incr(pairs))) MARK(tail(incr(odds))) -> c17(A__TAIL(a__incr(a__odds)), MARK(incr(odds))) MARK(tail(incr(head(z0)))) -> c17(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c17(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c17(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c17(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c17(MARK(incr(x0))) MARK(tail(odds)) -> c17(A__TAIL(a__incr(a__pairs)), MARK(odds)) MARK(tail(odds)) -> c17(MARK(odds)) MARK(tail(head(nats))) -> c17(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(incr(z0)))) -> c17(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(pairs))) -> c17(A__TAIL(a__head(a__pairs)), MARK(head(pairs))) MARK(tail(head(odds))) -> c17(A__TAIL(a__head(a__odds)), MARK(head(odds))) MARK(tail(head(head(z0)))) -> c17(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c17(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c17(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c17(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c17(MARK(head(x0))) MARK(tail(tail(nats))) -> c17(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(incr(z0)))) -> c17(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(pairs))) -> c17(A__TAIL(a__tail(a__pairs)), MARK(tail(pairs))) MARK(tail(tail(odds))) -> c17(A__TAIL(a__tail(a__odds)), MARK(tail(odds))) MARK(tail(tail(head(z0)))) -> c17(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c17(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c17(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c17(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c17(MARK(tail(x0))) MARK(tail(tail(0))) -> c17(A__TAIL(a__tail(0))) MARK(tail(nats)) -> c17(A__TAIL(cons(0, incr(nats)))) MARK(tail(pairs)) -> c17(A__TAIL(cons(0, incr(odds)))) S tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(incr(nats)) -> c13(A__INCR(cons(0, incr(nats)))) MARK(incr(pairs)) -> c13(A__INCR(cons(0, incr(odds)))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs)), MARK(incr(pairs))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs)), MARK(odds)) MARK(head(odds)) -> c16(MARK(odds)) MARK(head(head(nats))) -> c16(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(pairs))) -> c16(A__HEAD(a__head(a__pairs)), MARK(head(pairs))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(head(nats)) -> c16(A__HEAD(cons(0, incr(nats)))) MARK(head(pairs)) -> c16(A__HEAD(cons(0, incr(odds)))) MARK(tail(incr(nats))) -> c17(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(incr(z0)))) -> c17(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(pairs))) -> c17(A__TAIL(a__incr(a__pairs)), MARK(incr(pairs))) MARK(tail(incr(odds))) -> c17(A__TAIL(a__incr(a__odds)), MARK(incr(odds))) MARK(tail(incr(head(z0)))) -> c17(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c17(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c17(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c17(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c17(MARK(incr(x0))) MARK(tail(odds)) -> c17(A__TAIL(a__incr(a__pairs)), MARK(odds)) MARK(tail(odds)) -> c17(MARK(odds)) MARK(tail(head(nats))) -> c17(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(incr(z0)))) -> c17(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(pairs))) -> c17(A__TAIL(a__head(a__pairs)), MARK(head(pairs))) MARK(tail(head(odds))) -> c17(A__TAIL(a__head(a__odds)), MARK(head(odds))) MARK(tail(head(head(z0)))) -> c17(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c17(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c17(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c17(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c17(MARK(head(x0))) MARK(tail(tail(nats))) -> c17(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(incr(z0)))) -> c17(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(pairs))) -> c17(A__TAIL(a__tail(a__pairs)), MARK(tail(pairs))) MARK(tail(tail(odds))) -> c17(A__TAIL(a__tail(a__odds)), MARK(tail(odds))) MARK(tail(tail(head(z0)))) -> c17(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c17(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c17(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c17(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c17(MARK(tail(x0))) MARK(tail(tail(0))) -> c17(A__TAIL(a__tail(0))) MARK(tail(nats)) -> c17(A__TAIL(cons(0, incr(nats)))) MARK(tail(pairs)) -> c17(A__TAIL(cons(0, incr(odds)))) K tuples:none Defined Rule Symbols: a__nats, a__pairs, a__odds, a__incr_1, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__HEAD_1, A__TAIL_1, MARK_1, A__ODDS Compound Symbols: c6_1, c8_1, c10_1, c15_1, c18_1, c20_1, c13_2, c13_1, c16_2, c16_1, c17_2, c17_1, c4_1 ---------------------------------------- (155) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(tail(tail(0))) -> c17(A__TAIL(a__tail(0))) by MARK(tail(tail(0))) -> c17(A__TAIL(tail(0))) ---------------------------------------- (156) Obligation: Complexity Dependency Tuples Problem Rules: a__nats -> cons(0, incr(nats)) a__nats -> nats a__pairs -> cons(0, incr(odds)) a__pairs -> pairs a__odds -> a__incr(a__pairs) a__odds -> odds a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) 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(nats) -> a__nats mark(incr(z0)) -> a__incr(mark(z0)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(incr(nats)) -> c13(A__INCR(cons(0, incr(nats)))) MARK(incr(pairs)) -> c13(A__INCR(cons(0, incr(odds)))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs)), MARK(incr(pairs))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs)), MARK(odds)) MARK(head(odds)) -> c16(MARK(odds)) MARK(head(head(nats))) -> c16(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(pairs))) -> c16(A__HEAD(a__head(a__pairs)), MARK(head(pairs))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(head(nats)) -> c16(A__HEAD(cons(0, incr(nats)))) MARK(head(pairs)) -> c16(A__HEAD(cons(0, incr(odds)))) MARK(tail(incr(nats))) -> c17(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(incr(z0)))) -> c17(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(pairs))) -> c17(A__TAIL(a__incr(a__pairs)), MARK(incr(pairs))) MARK(tail(incr(odds))) -> c17(A__TAIL(a__incr(a__odds)), MARK(incr(odds))) MARK(tail(incr(head(z0)))) -> c17(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c17(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c17(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c17(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c17(MARK(incr(x0))) MARK(tail(odds)) -> c17(A__TAIL(a__incr(a__pairs)), MARK(odds)) MARK(tail(odds)) -> c17(MARK(odds)) MARK(tail(head(nats))) -> c17(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(incr(z0)))) -> c17(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(pairs))) -> c17(A__TAIL(a__head(a__pairs)), MARK(head(pairs))) MARK(tail(head(odds))) -> c17(A__TAIL(a__head(a__odds)), MARK(head(odds))) MARK(tail(head(head(z0)))) -> c17(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c17(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c17(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c17(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c17(MARK(head(x0))) MARK(tail(tail(nats))) -> c17(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(incr(z0)))) -> c17(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(pairs))) -> c17(A__TAIL(a__tail(a__pairs)), MARK(tail(pairs))) MARK(tail(tail(odds))) -> c17(A__TAIL(a__tail(a__odds)), MARK(tail(odds))) MARK(tail(tail(head(z0)))) -> c17(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c17(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c17(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c17(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c17(MARK(tail(x0))) MARK(tail(nats)) -> c17(A__TAIL(cons(0, incr(nats)))) MARK(tail(pairs)) -> c17(A__TAIL(cons(0, incr(odds)))) MARK(tail(tail(0))) -> c17(A__TAIL(tail(0))) S tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(incr(nats)) -> c13(A__INCR(cons(0, incr(nats)))) MARK(incr(pairs)) -> c13(A__INCR(cons(0, incr(odds)))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs)), MARK(incr(pairs))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs)), MARK(odds)) MARK(head(odds)) -> c16(MARK(odds)) MARK(head(head(nats))) -> c16(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(pairs))) -> c16(A__HEAD(a__head(a__pairs)), MARK(head(pairs))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(head(nats)) -> c16(A__HEAD(cons(0, incr(nats)))) MARK(head(pairs)) -> c16(A__HEAD(cons(0, incr(odds)))) MARK(tail(incr(nats))) -> c17(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(incr(z0)))) -> c17(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(pairs))) -> c17(A__TAIL(a__incr(a__pairs)), MARK(incr(pairs))) MARK(tail(incr(odds))) -> c17(A__TAIL(a__incr(a__odds)), MARK(incr(odds))) MARK(tail(incr(head(z0)))) -> c17(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c17(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c17(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c17(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c17(MARK(incr(x0))) MARK(tail(odds)) -> c17(A__TAIL(a__incr(a__pairs)), MARK(odds)) MARK(tail(odds)) -> c17(MARK(odds)) MARK(tail(head(nats))) -> c17(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(incr(z0)))) -> c17(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(pairs))) -> c17(A__TAIL(a__head(a__pairs)), MARK(head(pairs))) MARK(tail(head(odds))) -> c17(A__TAIL(a__head(a__odds)), MARK(head(odds))) MARK(tail(head(head(z0)))) -> c17(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c17(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c17(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c17(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c17(MARK(head(x0))) MARK(tail(tail(nats))) -> c17(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(incr(z0)))) -> c17(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(pairs))) -> c17(A__TAIL(a__tail(a__pairs)), MARK(tail(pairs))) MARK(tail(tail(odds))) -> c17(A__TAIL(a__tail(a__odds)), MARK(tail(odds))) MARK(tail(tail(head(z0)))) -> c17(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c17(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c17(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c17(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c17(MARK(tail(x0))) MARK(tail(nats)) -> c17(A__TAIL(cons(0, incr(nats)))) MARK(tail(pairs)) -> c17(A__TAIL(cons(0, incr(odds)))) MARK(tail(tail(0))) -> c17(A__TAIL(tail(0))) K tuples:none Defined Rule Symbols: a__nats, a__pairs, a__odds, a__incr_1, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__HEAD_1, A__TAIL_1, MARK_1, A__ODDS Compound Symbols: c6_1, c8_1, c10_1, c15_1, c18_1, c20_1, c13_2, c13_1, c16_2, c16_1, c17_2, c17_1, c4_1 ---------------------------------------- (157) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(tail(tail(0))) -> c17(A__TAIL(tail(0))) ---------------------------------------- (158) Obligation: Complexity Dependency Tuples Problem Rules: a__nats -> cons(0, incr(nats)) a__nats -> nats a__pairs -> cons(0, incr(odds)) a__pairs -> pairs a__odds -> a__incr(a__pairs) a__odds -> odds a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) 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(nats) -> a__nats mark(incr(z0)) -> a__incr(mark(z0)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(incr(nats)) -> c13(A__INCR(cons(0, incr(nats)))) MARK(incr(pairs)) -> c13(A__INCR(cons(0, incr(odds)))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs)), MARK(incr(pairs))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs)), MARK(odds)) MARK(head(odds)) -> c16(MARK(odds)) MARK(head(head(nats))) -> c16(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(pairs))) -> c16(A__HEAD(a__head(a__pairs)), MARK(head(pairs))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(head(nats)) -> c16(A__HEAD(cons(0, incr(nats)))) MARK(head(pairs)) -> c16(A__HEAD(cons(0, incr(odds)))) MARK(tail(incr(nats))) -> c17(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(incr(z0)))) -> c17(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(pairs))) -> c17(A__TAIL(a__incr(a__pairs)), MARK(incr(pairs))) MARK(tail(incr(odds))) -> c17(A__TAIL(a__incr(a__odds)), MARK(incr(odds))) MARK(tail(incr(head(z0)))) -> c17(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c17(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c17(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c17(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c17(MARK(incr(x0))) MARK(tail(odds)) -> c17(A__TAIL(a__incr(a__pairs)), MARK(odds)) MARK(tail(odds)) -> c17(MARK(odds)) MARK(tail(head(nats))) -> c17(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(incr(z0)))) -> c17(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(pairs))) -> c17(A__TAIL(a__head(a__pairs)), MARK(head(pairs))) MARK(tail(head(odds))) -> c17(A__TAIL(a__head(a__odds)), MARK(head(odds))) MARK(tail(head(head(z0)))) -> c17(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c17(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c17(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c17(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c17(MARK(head(x0))) MARK(tail(tail(nats))) -> c17(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(incr(z0)))) -> c17(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(pairs))) -> c17(A__TAIL(a__tail(a__pairs)), MARK(tail(pairs))) MARK(tail(tail(odds))) -> c17(A__TAIL(a__tail(a__odds)), MARK(tail(odds))) MARK(tail(tail(head(z0)))) -> c17(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c17(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c17(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c17(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c17(MARK(tail(x0))) MARK(tail(nats)) -> c17(A__TAIL(cons(0, incr(nats)))) MARK(tail(pairs)) -> c17(A__TAIL(cons(0, incr(odds)))) S tuples: A__INCR(cons(z0, z1)) -> c6(MARK(z0)) A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(incr(nats)) -> c13(A__INCR(cons(0, incr(nats)))) MARK(incr(pairs)) -> c13(A__INCR(cons(0, incr(odds)))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs)), MARK(incr(pairs))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs)), MARK(odds)) MARK(head(odds)) -> c16(MARK(odds)) MARK(head(head(nats))) -> c16(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(pairs))) -> c16(A__HEAD(a__head(a__pairs)), MARK(head(pairs))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(head(nats)) -> c16(A__HEAD(cons(0, incr(nats)))) MARK(head(pairs)) -> c16(A__HEAD(cons(0, incr(odds)))) MARK(tail(incr(nats))) -> c17(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(incr(z0)))) -> c17(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(pairs))) -> c17(A__TAIL(a__incr(a__pairs)), MARK(incr(pairs))) MARK(tail(incr(odds))) -> c17(A__TAIL(a__incr(a__odds)), MARK(incr(odds))) MARK(tail(incr(head(z0)))) -> c17(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c17(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c17(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c17(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c17(MARK(incr(x0))) MARK(tail(odds)) -> c17(A__TAIL(a__incr(a__pairs)), MARK(odds)) MARK(tail(odds)) -> c17(MARK(odds)) MARK(tail(head(nats))) -> c17(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(incr(z0)))) -> c17(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(pairs))) -> c17(A__TAIL(a__head(a__pairs)), MARK(head(pairs))) MARK(tail(head(odds))) -> c17(A__TAIL(a__head(a__odds)), MARK(head(odds))) MARK(tail(head(head(z0)))) -> c17(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c17(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c17(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c17(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c17(MARK(head(x0))) MARK(tail(tail(nats))) -> c17(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(incr(z0)))) -> c17(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(pairs))) -> c17(A__TAIL(a__tail(a__pairs)), MARK(tail(pairs))) MARK(tail(tail(odds))) -> c17(A__TAIL(a__tail(a__odds)), MARK(tail(odds))) MARK(tail(tail(head(z0)))) -> c17(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c17(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c17(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c17(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c17(MARK(tail(x0))) MARK(tail(nats)) -> c17(A__TAIL(cons(0, incr(nats)))) MARK(tail(pairs)) -> c17(A__TAIL(cons(0, incr(odds)))) K tuples:none Defined Rule Symbols: a__nats, a__pairs, a__odds, a__incr_1, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__INCR_1, A__HEAD_1, A__TAIL_1, MARK_1, A__ODDS Compound Symbols: c6_1, c8_1, c10_1, c15_1, c18_1, c20_1, c13_2, c13_1, c16_2, c16_1, c17_2, c17_1, c4_1 ---------------------------------------- (159) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace A__INCR(cons(z0, z1)) -> c6(MARK(z0)) by A__INCR(cons(odds, z1)) -> c6(MARK(odds)) A__INCR(cons(cons(y0, y1), z1)) -> c6(MARK(cons(y0, y1))) A__INCR(cons(s(y0), z1)) -> c6(MARK(s(y0))) A__INCR(cons(incr(cons(y0, y1)), z1)) -> c6(MARK(incr(cons(y0, y1)))) A__INCR(cons(incr(s(y0)), z1)) -> c6(MARK(incr(s(y0)))) A__INCR(cons(head(cons(y0, y1)), z1)) -> c6(MARK(head(cons(y0, y1)))) A__INCR(cons(head(s(y0)), z1)) -> c6(MARK(head(s(y0)))) A__INCR(cons(tail(cons(y0, y1)), z1)) -> c6(MARK(tail(cons(y0, y1)))) A__INCR(cons(tail(s(y0)), z1)) -> c6(MARK(tail(s(y0)))) A__INCR(cons(incr(incr(nats)), z1)) -> c6(MARK(incr(incr(nats)))) A__INCR(cons(incr(incr(incr(y0))), z1)) -> c6(MARK(incr(incr(incr(y0))))) A__INCR(cons(incr(incr(pairs)), z1)) -> c6(MARK(incr(incr(pairs)))) A__INCR(cons(incr(incr(odds)), z1)) -> c6(MARK(incr(incr(odds)))) A__INCR(cons(incr(incr(head(y0))), z1)) -> c6(MARK(incr(incr(head(y0))))) A__INCR(cons(incr(incr(tail(y0))), z1)) -> c6(MARK(incr(incr(tail(y0))))) A__INCR(cons(incr(incr(cons(y0, y1))), z1)) -> c6(MARK(incr(incr(cons(y0, y1))))) A__INCR(cons(incr(incr(s(y0))), z1)) -> c6(MARK(incr(incr(s(y0))))) A__INCR(cons(incr(incr(y0)), z1)) -> c6(MARK(incr(incr(y0)))) A__INCR(cons(incr(odds), z1)) -> c6(MARK(incr(odds))) A__INCR(cons(incr(head(nats)), z1)) -> c6(MARK(incr(head(nats)))) A__INCR(cons(incr(head(incr(y0))), z1)) -> c6(MARK(incr(head(incr(y0))))) A__INCR(cons(incr(head(pairs)), z1)) -> c6(MARK(incr(head(pairs)))) A__INCR(cons(incr(head(odds)), z1)) -> c6(MARK(incr(head(odds)))) A__INCR(cons(incr(head(head(y0))), z1)) -> c6(MARK(incr(head(head(y0))))) A__INCR(cons(incr(head(tail(y0))), z1)) -> c6(MARK(incr(head(tail(y0))))) A__INCR(cons(incr(head(cons(y0, y1))), z1)) -> c6(MARK(incr(head(cons(y0, y1))))) A__INCR(cons(incr(head(s(y0))), z1)) -> c6(MARK(incr(head(s(y0))))) A__INCR(cons(incr(head(y0)), z1)) -> c6(MARK(incr(head(y0)))) A__INCR(cons(incr(tail(nats)), z1)) -> c6(MARK(incr(tail(nats)))) A__INCR(cons(incr(tail(incr(y0))), z1)) -> c6(MARK(incr(tail(incr(y0))))) A__INCR(cons(incr(tail(pairs)), z1)) -> c6(MARK(incr(tail(pairs)))) A__INCR(cons(incr(tail(odds)), z1)) -> c6(MARK(incr(tail(odds)))) A__INCR(cons(incr(tail(head(y0))), z1)) -> c6(MARK(incr(tail(head(y0))))) A__INCR(cons(incr(tail(tail(y0))), z1)) -> c6(MARK(incr(tail(tail(y0))))) A__INCR(cons(incr(tail(cons(y0, y1))), z1)) -> c6(MARK(incr(tail(cons(y0, y1))))) A__INCR(cons(incr(tail(s(y0))), z1)) -> c6(MARK(incr(tail(s(y0))))) A__INCR(cons(incr(tail(y0)), z1)) -> c6(MARK(incr(tail(y0)))) A__INCR(cons(incr(nats), z1)) -> c6(MARK(incr(nats))) A__INCR(cons(incr(pairs), z1)) -> c6(MARK(incr(pairs))) A__INCR(cons(head(incr(nats)), z1)) -> c6(MARK(head(incr(nats)))) A__INCR(cons(head(incr(incr(y0))), z1)) -> c6(MARK(head(incr(incr(y0))))) A__INCR(cons(head(incr(pairs)), z1)) -> c6(MARK(head(incr(pairs)))) A__INCR(cons(head(incr(odds)), z1)) -> c6(MARK(head(incr(odds)))) A__INCR(cons(head(incr(head(y0))), z1)) -> c6(MARK(head(incr(head(y0))))) A__INCR(cons(head(incr(tail(y0))), z1)) -> c6(MARK(head(incr(tail(y0))))) A__INCR(cons(head(incr(cons(y0, y1))), z1)) -> c6(MARK(head(incr(cons(y0, y1))))) A__INCR(cons(head(incr(s(y0))), z1)) -> c6(MARK(head(incr(s(y0))))) A__INCR(cons(head(incr(y0)), z1)) -> c6(MARK(head(incr(y0)))) A__INCR(cons(head(odds), z1)) -> c6(MARK(head(odds))) A__INCR(cons(head(head(nats)), z1)) -> c6(MARK(head(head(nats)))) A__INCR(cons(head(head(incr(y0))), z1)) -> c6(MARK(head(head(incr(y0))))) A__INCR(cons(head(head(pairs)), z1)) -> c6(MARK(head(head(pairs)))) A__INCR(cons(head(head(odds)), z1)) -> c6(MARK(head(head(odds)))) A__INCR(cons(head(head(head(y0))), z1)) -> c6(MARK(head(head(head(y0))))) A__INCR(cons(head(head(tail(y0))), z1)) -> c6(MARK(head(head(tail(y0))))) A__INCR(cons(head(head(cons(y0, y1))), z1)) -> c6(MARK(head(head(cons(y0, y1))))) A__INCR(cons(head(head(s(y0))), z1)) -> c6(MARK(head(head(s(y0))))) A__INCR(cons(head(head(y0)), z1)) -> c6(MARK(head(head(y0)))) A__INCR(cons(head(tail(nats)), z1)) -> c6(MARK(head(tail(nats)))) A__INCR(cons(head(tail(incr(y0))), z1)) -> c6(MARK(head(tail(incr(y0))))) A__INCR(cons(head(tail(pairs)), z1)) -> c6(MARK(head(tail(pairs)))) A__INCR(cons(head(tail(odds)), z1)) -> c6(MARK(head(tail(odds)))) A__INCR(cons(head(tail(head(y0))), z1)) -> c6(MARK(head(tail(head(y0))))) A__INCR(cons(head(tail(tail(y0))), z1)) -> c6(MARK(head(tail(tail(y0))))) A__INCR(cons(head(tail(cons(y0, y1))), z1)) -> c6(MARK(head(tail(cons(y0, y1))))) A__INCR(cons(head(tail(s(y0))), z1)) -> c6(MARK(head(tail(s(y0))))) A__INCR(cons(head(tail(y0)), z1)) -> c6(MARK(head(tail(y0)))) A__INCR(cons(head(nats), z1)) -> c6(MARK(head(nats))) A__INCR(cons(head(pairs), z1)) -> c6(MARK(head(pairs))) A__INCR(cons(tail(incr(nats)), z1)) -> c6(MARK(tail(incr(nats)))) A__INCR(cons(tail(incr(incr(y0))), z1)) -> c6(MARK(tail(incr(incr(y0))))) A__INCR(cons(tail(incr(pairs)), z1)) -> c6(MARK(tail(incr(pairs)))) A__INCR(cons(tail(incr(odds)), z1)) -> c6(MARK(tail(incr(odds)))) A__INCR(cons(tail(incr(head(y0))), z1)) -> c6(MARK(tail(incr(head(y0))))) A__INCR(cons(tail(incr(tail(y0))), z1)) -> c6(MARK(tail(incr(tail(y0))))) A__INCR(cons(tail(incr(cons(y0, y1))), z1)) -> c6(MARK(tail(incr(cons(y0, y1))))) A__INCR(cons(tail(incr(s(y0))), z1)) -> c6(MARK(tail(incr(s(y0))))) A__INCR(cons(tail(incr(y0)), z1)) -> c6(MARK(tail(incr(y0)))) A__INCR(cons(tail(odds), z1)) -> c6(MARK(tail(odds))) A__INCR(cons(tail(head(nats)), z1)) -> c6(MARK(tail(head(nats)))) A__INCR(cons(tail(head(incr(y0))), z1)) -> c6(MARK(tail(head(incr(y0))))) A__INCR(cons(tail(head(pairs)), z1)) -> c6(MARK(tail(head(pairs)))) A__INCR(cons(tail(head(odds)), z1)) -> c6(MARK(tail(head(odds)))) A__INCR(cons(tail(head(head(y0))), z1)) -> c6(MARK(tail(head(head(y0))))) A__INCR(cons(tail(head(tail(y0))), z1)) -> c6(MARK(tail(head(tail(y0))))) A__INCR(cons(tail(head(cons(y0, y1))), z1)) -> c6(MARK(tail(head(cons(y0, y1))))) A__INCR(cons(tail(head(s(y0))), z1)) -> c6(MARK(tail(head(s(y0))))) A__INCR(cons(tail(head(y0)), z1)) -> c6(MARK(tail(head(y0)))) A__INCR(cons(tail(tail(nats)), z1)) -> c6(MARK(tail(tail(nats)))) A__INCR(cons(tail(tail(incr(y0))), z1)) -> c6(MARK(tail(tail(incr(y0))))) A__INCR(cons(tail(tail(pairs)), z1)) -> c6(MARK(tail(tail(pairs)))) A__INCR(cons(tail(tail(odds)), z1)) -> c6(MARK(tail(tail(odds)))) A__INCR(cons(tail(tail(head(y0))), z1)) -> c6(MARK(tail(tail(head(y0))))) A__INCR(cons(tail(tail(tail(y0))), z1)) -> c6(MARK(tail(tail(tail(y0))))) A__INCR(cons(tail(tail(cons(y0, y1))), z1)) -> c6(MARK(tail(tail(cons(y0, y1))))) A__INCR(cons(tail(tail(s(y0))), z1)) -> c6(MARK(tail(tail(s(y0))))) A__INCR(cons(tail(tail(y0)), z1)) -> c6(MARK(tail(tail(y0)))) A__INCR(cons(tail(nats), z1)) -> c6(MARK(tail(nats))) A__INCR(cons(tail(pairs), z1)) -> c6(MARK(tail(pairs))) ---------------------------------------- (160) Obligation: Complexity Dependency Tuples Problem Rules: a__nats -> cons(0, incr(nats)) a__nats -> nats a__pairs -> cons(0, incr(odds)) a__pairs -> pairs a__odds -> a__incr(a__pairs) a__odds -> odds a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) 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(nats) -> a__nats mark(incr(z0)) -> a__incr(mark(z0)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(incr(nats)) -> c13(A__INCR(cons(0, incr(nats)))) MARK(incr(pairs)) -> c13(A__INCR(cons(0, incr(odds)))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs)), MARK(incr(pairs))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs)), MARK(odds)) MARK(head(odds)) -> c16(MARK(odds)) MARK(head(head(nats))) -> c16(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(pairs))) -> c16(A__HEAD(a__head(a__pairs)), MARK(head(pairs))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(head(nats)) -> c16(A__HEAD(cons(0, incr(nats)))) MARK(head(pairs)) -> c16(A__HEAD(cons(0, incr(odds)))) MARK(tail(incr(nats))) -> c17(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(incr(z0)))) -> c17(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(pairs))) -> c17(A__TAIL(a__incr(a__pairs)), MARK(incr(pairs))) MARK(tail(incr(odds))) -> c17(A__TAIL(a__incr(a__odds)), MARK(incr(odds))) MARK(tail(incr(head(z0)))) -> c17(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c17(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c17(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c17(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c17(MARK(incr(x0))) MARK(tail(odds)) -> c17(A__TAIL(a__incr(a__pairs)), MARK(odds)) MARK(tail(odds)) -> c17(MARK(odds)) MARK(tail(head(nats))) -> c17(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(incr(z0)))) -> c17(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(pairs))) -> c17(A__TAIL(a__head(a__pairs)), MARK(head(pairs))) MARK(tail(head(odds))) -> c17(A__TAIL(a__head(a__odds)), MARK(head(odds))) MARK(tail(head(head(z0)))) -> c17(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c17(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c17(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c17(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c17(MARK(head(x0))) MARK(tail(tail(nats))) -> c17(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(incr(z0)))) -> c17(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(pairs))) -> c17(A__TAIL(a__tail(a__pairs)), MARK(tail(pairs))) MARK(tail(tail(odds))) -> c17(A__TAIL(a__tail(a__odds)), MARK(tail(odds))) MARK(tail(tail(head(z0)))) -> c17(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c17(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c17(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c17(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c17(MARK(tail(x0))) MARK(tail(nats)) -> c17(A__TAIL(cons(0, incr(nats)))) MARK(tail(pairs)) -> c17(A__TAIL(cons(0, incr(odds)))) A__INCR(cons(odds, z1)) -> c6(MARK(odds)) A__INCR(cons(cons(y0, y1), z1)) -> c6(MARK(cons(y0, y1))) A__INCR(cons(s(y0), z1)) -> c6(MARK(s(y0))) A__INCR(cons(incr(cons(y0, y1)), z1)) -> c6(MARK(incr(cons(y0, y1)))) A__INCR(cons(incr(s(y0)), z1)) -> c6(MARK(incr(s(y0)))) A__INCR(cons(head(cons(y0, y1)), z1)) -> c6(MARK(head(cons(y0, y1)))) A__INCR(cons(head(s(y0)), z1)) -> c6(MARK(head(s(y0)))) A__INCR(cons(tail(cons(y0, y1)), z1)) -> c6(MARK(tail(cons(y0, y1)))) A__INCR(cons(tail(s(y0)), z1)) -> c6(MARK(tail(s(y0)))) A__INCR(cons(incr(incr(nats)), z1)) -> c6(MARK(incr(incr(nats)))) A__INCR(cons(incr(incr(incr(y0))), z1)) -> c6(MARK(incr(incr(incr(y0))))) A__INCR(cons(incr(incr(pairs)), z1)) -> c6(MARK(incr(incr(pairs)))) A__INCR(cons(incr(incr(odds)), z1)) -> c6(MARK(incr(incr(odds)))) A__INCR(cons(incr(incr(head(y0))), z1)) -> c6(MARK(incr(incr(head(y0))))) A__INCR(cons(incr(incr(tail(y0))), z1)) -> c6(MARK(incr(incr(tail(y0))))) A__INCR(cons(incr(incr(cons(y0, y1))), z1)) -> c6(MARK(incr(incr(cons(y0, y1))))) A__INCR(cons(incr(incr(s(y0))), z1)) -> c6(MARK(incr(incr(s(y0))))) A__INCR(cons(incr(incr(y0)), z1)) -> c6(MARK(incr(incr(y0)))) A__INCR(cons(incr(odds), z1)) -> c6(MARK(incr(odds))) A__INCR(cons(incr(head(nats)), z1)) -> c6(MARK(incr(head(nats)))) A__INCR(cons(incr(head(incr(y0))), z1)) -> c6(MARK(incr(head(incr(y0))))) A__INCR(cons(incr(head(pairs)), z1)) -> c6(MARK(incr(head(pairs)))) A__INCR(cons(incr(head(odds)), z1)) -> c6(MARK(incr(head(odds)))) A__INCR(cons(incr(head(head(y0))), z1)) -> c6(MARK(incr(head(head(y0))))) A__INCR(cons(incr(head(tail(y0))), z1)) -> c6(MARK(incr(head(tail(y0))))) A__INCR(cons(incr(head(cons(y0, y1))), z1)) -> c6(MARK(incr(head(cons(y0, y1))))) A__INCR(cons(incr(head(s(y0))), z1)) -> c6(MARK(incr(head(s(y0))))) A__INCR(cons(incr(head(y0)), z1)) -> c6(MARK(incr(head(y0)))) A__INCR(cons(incr(tail(nats)), z1)) -> c6(MARK(incr(tail(nats)))) A__INCR(cons(incr(tail(incr(y0))), z1)) -> c6(MARK(incr(tail(incr(y0))))) A__INCR(cons(incr(tail(pairs)), z1)) -> c6(MARK(incr(tail(pairs)))) A__INCR(cons(incr(tail(odds)), z1)) -> c6(MARK(incr(tail(odds)))) A__INCR(cons(incr(tail(head(y0))), z1)) -> c6(MARK(incr(tail(head(y0))))) A__INCR(cons(incr(tail(tail(y0))), z1)) -> c6(MARK(incr(tail(tail(y0))))) A__INCR(cons(incr(tail(cons(y0, y1))), z1)) -> c6(MARK(incr(tail(cons(y0, y1))))) A__INCR(cons(incr(tail(s(y0))), z1)) -> c6(MARK(incr(tail(s(y0))))) A__INCR(cons(incr(tail(y0)), z1)) -> c6(MARK(incr(tail(y0)))) A__INCR(cons(incr(nats), z1)) -> c6(MARK(incr(nats))) A__INCR(cons(incr(pairs), z1)) -> c6(MARK(incr(pairs))) A__INCR(cons(head(incr(nats)), z1)) -> c6(MARK(head(incr(nats)))) A__INCR(cons(head(incr(incr(y0))), z1)) -> c6(MARK(head(incr(incr(y0))))) A__INCR(cons(head(incr(pairs)), z1)) -> c6(MARK(head(incr(pairs)))) A__INCR(cons(head(incr(odds)), z1)) -> c6(MARK(head(incr(odds)))) A__INCR(cons(head(incr(head(y0))), z1)) -> c6(MARK(head(incr(head(y0))))) A__INCR(cons(head(incr(tail(y0))), z1)) -> c6(MARK(head(incr(tail(y0))))) A__INCR(cons(head(incr(cons(y0, y1))), z1)) -> c6(MARK(head(incr(cons(y0, y1))))) A__INCR(cons(head(incr(s(y0))), z1)) -> c6(MARK(head(incr(s(y0))))) A__INCR(cons(head(incr(y0)), z1)) -> c6(MARK(head(incr(y0)))) A__INCR(cons(head(odds), z1)) -> c6(MARK(head(odds))) A__INCR(cons(head(head(nats)), z1)) -> c6(MARK(head(head(nats)))) A__INCR(cons(head(head(incr(y0))), z1)) -> c6(MARK(head(head(incr(y0))))) A__INCR(cons(head(head(pairs)), z1)) -> c6(MARK(head(head(pairs)))) A__INCR(cons(head(head(odds)), z1)) -> c6(MARK(head(head(odds)))) A__INCR(cons(head(head(head(y0))), z1)) -> c6(MARK(head(head(head(y0))))) A__INCR(cons(head(head(tail(y0))), z1)) -> c6(MARK(head(head(tail(y0))))) A__INCR(cons(head(head(cons(y0, y1))), z1)) -> c6(MARK(head(head(cons(y0, y1))))) A__INCR(cons(head(head(s(y0))), z1)) -> c6(MARK(head(head(s(y0))))) A__INCR(cons(head(head(y0)), z1)) -> c6(MARK(head(head(y0)))) A__INCR(cons(head(tail(nats)), z1)) -> c6(MARK(head(tail(nats)))) A__INCR(cons(head(tail(incr(y0))), z1)) -> c6(MARK(head(tail(incr(y0))))) A__INCR(cons(head(tail(pairs)), z1)) -> c6(MARK(head(tail(pairs)))) A__INCR(cons(head(tail(odds)), z1)) -> c6(MARK(head(tail(odds)))) A__INCR(cons(head(tail(head(y0))), z1)) -> c6(MARK(head(tail(head(y0))))) A__INCR(cons(head(tail(tail(y0))), z1)) -> c6(MARK(head(tail(tail(y0))))) A__INCR(cons(head(tail(cons(y0, y1))), z1)) -> c6(MARK(head(tail(cons(y0, y1))))) A__INCR(cons(head(tail(s(y0))), z1)) -> c6(MARK(head(tail(s(y0))))) A__INCR(cons(head(tail(y0)), z1)) -> c6(MARK(head(tail(y0)))) A__INCR(cons(head(nats), z1)) -> c6(MARK(head(nats))) A__INCR(cons(head(pairs), z1)) -> c6(MARK(head(pairs))) A__INCR(cons(tail(incr(nats)), z1)) -> c6(MARK(tail(incr(nats)))) A__INCR(cons(tail(incr(incr(y0))), z1)) -> c6(MARK(tail(incr(incr(y0))))) A__INCR(cons(tail(incr(pairs)), z1)) -> c6(MARK(tail(incr(pairs)))) A__INCR(cons(tail(incr(odds)), z1)) -> c6(MARK(tail(incr(odds)))) A__INCR(cons(tail(incr(head(y0))), z1)) -> c6(MARK(tail(incr(head(y0))))) A__INCR(cons(tail(incr(tail(y0))), z1)) -> c6(MARK(tail(incr(tail(y0))))) A__INCR(cons(tail(incr(cons(y0, y1))), z1)) -> c6(MARK(tail(incr(cons(y0, y1))))) A__INCR(cons(tail(incr(s(y0))), z1)) -> c6(MARK(tail(incr(s(y0))))) A__INCR(cons(tail(incr(y0)), z1)) -> c6(MARK(tail(incr(y0)))) A__INCR(cons(tail(odds), z1)) -> c6(MARK(tail(odds))) A__INCR(cons(tail(head(nats)), z1)) -> c6(MARK(tail(head(nats)))) A__INCR(cons(tail(head(incr(y0))), z1)) -> c6(MARK(tail(head(incr(y0))))) A__INCR(cons(tail(head(pairs)), z1)) -> c6(MARK(tail(head(pairs)))) A__INCR(cons(tail(head(odds)), z1)) -> c6(MARK(tail(head(odds)))) A__INCR(cons(tail(head(head(y0))), z1)) -> c6(MARK(tail(head(head(y0))))) A__INCR(cons(tail(head(tail(y0))), z1)) -> c6(MARK(tail(head(tail(y0))))) A__INCR(cons(tail(head(cons(y0, y1))), z1)) -> c6(MARK(tail(head(cons(y0, y1))))) A__INCR(cons(tail(head(s(y0))), z1)) -> c6(MARK(tail(head(s(y0))))) A__INCR(cons(tail(head(y0)), z1)) -> c6(MARK(tail(head(y0)))) A__INCR(cons(tail(tail(nats)), z1)) -> c6(MARK(tail(tail(nats)))) A__INCR(cons(tail(tail(incr(y0))), z1)) -> c6(MARK(tail(tail(incr(y0))))) A__INCR(cons(tail(tail(pairs)), z1)) -> c6(MARK(tail(tail(pairs)))) A__INCR(cons(tail(tail(odds)), z1)) -> c6(MARK(tail(tail(odds)))) A__INCR(cons(tail(tail(head(y0))), z1)) -> c6(MARK(tail(tail(head(y0))))) A__INCR(cons(tail(tail(tail(y0))), z1)) -> c6(MARK(tail(tail(tail(y0))))) A__INCR(cons(tail(tail(cons(y0, y1))), z1)) -> c6(MARK(tail(tail(cons(y0, y1))))) A__INCR(cons(tail(tail(s(y0))), z1)) -> c6(MARK(tail(tail(s(y0))))) A__INCR(cons(tail(tail(y0)), z1)) -> c6(MARK(tail(tail(y0)))) A__INCR(cons(tail(nats), z1)) -> c6(MARK(tail(nats))) A__INCR(cons(tail(pairs), z1)) -> c6(MARK(tail(pairs))) S tuples: A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(odds) -> c15(A__ODDS) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c17(MARK(s(z0))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(odds)) -> c13(MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(incr(nats)) -> c13(A__INCR(cons(0, incr(nats)))) MARK(incr(pairs)) -> c13(A__INCR(cons(0, incr(odds)))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs)), MARK(incr(pairs))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs)), MARK(odds)) MARK(head(odds)) -> c16(MARK(odds)) MARK(head(head(nats))) -> c16(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(pairs))) -> c16(A__HEAD(a__head(a__pairs)), MARK(head(pairs))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(head(nats)) -> c16(A__HEAD(cons(0, incr(nats)))) MARK(head(pairs)) -> c16(A__HEAD(cons(0, incr(odds)))) MARK(tail(incr(nats))) -> c17(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(incr(z0)))) -> c17(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(pairs))) -> c17(A__TAIL(a__incr(a__pairs)), MARK(incr(pairs))) MARK(tail(incr(odds))) -> c17(A__TAIL(a__incr(a__odds)), MARK(incr(odds))) MARK(tail(incr(head(z0)))) -> c17(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c17(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c17(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c17(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c17(MARK(incr(x0))) MARK(tail(odds)) -> c17(A__TAIL(a__incr(a__pairs)), MARK(odds)) MARK(tail(odds)) -> c17(MARK(odds)) MARK(tail(head(nats))) -> c17(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(incr(z0)))) -> c17(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(pairs))) -> c17(A__TAIL(a__head(a__pairs)), MARK(head(pairs))) MARK(tail(head(odds))) -> c17(A__TAIL(a__head(a__odds)), MARK(head(odds))) MARK(tail(head(head(z0)))) -> c17(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c17(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c17(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c17(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c17(MARK(head(x0))) MARK(tail(tail(nats))) -> c17(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(incr(z0)))) -> c17(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(pairs))) -> c17(A__TAIL(a__tail(a__pairs)), MARK(tail(pairs))) MARK(tail(tail(odds))) -> c17(A__TAIL(a__tail(a__odds)), MARK(tail(odds))) MARK(tail(tail(head(z0)))) -> c17(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c17(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c17(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c17(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c17(MARK(tail(x0))) MARK(tail(nats)) -> c17(A__TAIL(cons(0, incr(nats)))) MARK(tail(pairs)) -> c17(A__TAIL(cons(0, incr(odds)))) A__INCR(cons(odds, z1)) -> c6(MARK(odds)) A__INCR(cons(cons(y0, y1), z1)) -> c6(MARK(cons(y0, y1))) A__INCR(cons(s(y0), z1)) -> c6(MARK(s(y0))) A__INCR(cons(incr(cons(y0, y1)), z1)) -> c6(MARK(incr(cons(y0, y1)))) A__INCR(cons(incr(s(y0)), z1)) -> c6(MARK(incr(s(y0)))) A__INCR(cons(head(cons(y0, y1)), z1)) -> c6(MARK(head(cons(y0, y1)))) A__INCR(cons(head(s(y0)), z1)) -> c6(MARK(head(s(y0)))) A__INCR(cons(tail(cons(y0, y1)), z1)) -> c6(MARK(tail(cons(y0, y1)))) A__INCR(cons(tail(s(y0)), z1)) -> c6(MARK(tail(s(y0)))) A__INCR(cons(incr(incr(nats)), z1)) -> c6(MARK(incr(incr(nats)))) A__INCR(cons(incr(incr(incr(y0))), z1)) -> c6(MARK(incr(incr(incr(y0))))) A__INCR(cons(incr(incr(pairs)), z1)) -> c6(MARK(incr(incr(pairs)))) A__INCR(cons(incr(incr(odds)), z1)) -> c6(MARK(incr(incr(odds)))) A__INCR(cons(incr(incr(head(y0))), z1)) -> c6(MARK(incr(incr(head(y0))))) A__INCR(cons(incr(incr(tail(y0))), z1)) -> c6(MARK(incr(incr(tail(y0))))) A__INCR(cons(incr(incr(cons(y0, y1))), z1)) -> c6(MARK(incr(incr(cons(y0, y1))))) A__INCR(cons(incr(incr(s(y0))), z1)) -> c6(MARK(incr(incr(s(y0))))) A__INCR(cons(incr(incr(y0)), z1)) -> c6(MARK(incr(incr(y0)))) A__INCR(cons(incr(odds), z1)) -> c6(MARK(incr(odds))) A__INCR(cons(incr(head(nats)), z1)) -> c6(MARK(incr(head(nats)))) A__INCR(cons(incr(head(incr(y0))), z1)) -> c6(MARK(incr(head(incr(y0))))) A__INCR(cons(incr(head(pairs)), z1)) -> c6(MARK(incr(head(pairs)))) A__INCR(cons(incr(head(odds)), z1)) -> c6(MARK(incr(head(odds)))) A__INCR(cons(incr(head(head(y0))), z1)) -> c6(MARK(incr(head(head(y0))))) A__INCR(cons(incr(head(tail(y0))), z1)) -> c6(MARK(incr(head(tail(y0))))) A__INCR(cons(incr(head(cons(y0, y1))), z1)) -> c6(MARK(incr(head(cons(y0, y1))))) A__INCR(cons(incr(head(s(y0))), z1)) -> c6(MARK(incr(head(s(y0))))) A__INCR(cons(incr(head(y0)), z1)) -> c6(MARK(incr(head(y0)))) A__INCR(cons(incr(tail(nats)), z1)) -> c6(MARK(incr(tail(nats)))) A__INCR(cons(incr(tail(incr(y0))), z1)) -> c6(MARK(incr(tail(incr(y0))))) A__INCR(cons(incr(tail(pairs)), z1)) -> c6(MARK(incr(tail(pairs)))) A__INCR(cons(incr(tail(odds)), z1)) -> c6(MARK(incr(tail(odds)))) A__INCR(cons(incr(tail(head(y0))), z1)) -> c6(MARK(incr(tail(head(y0))))) A__INCR(cons(incr(tail(tail(y0))), z1)) -> c6(MARK(incr(tail(tail(y0))))) A__INCR(cons(incr(tail(cons(y0, y1))), z1)) -> c6(MARK(incr(tail(cons(y0, y1))))) A__INCR(cons(incr(tail(s(y0))), z1)) -> c6(MARK(incr(tail(s(y0))))) A__INCR(cons(incr(tail(y0)), z1)) -> c6(MARK(incr(tail(y0)))) A__INCR(cons(incr(nats), z1)) -> c6(MARK(incr(nats))) A__INCR(cons(incr(pairs), z1)) -> c6(MARK(incr(pairs))) A__INCR(cons(head(incr(nats)), z1)) -> c6(MARK(head(incr(nats)))) A__INCR(cons(head(incr(incr(y0))), z1)) -> c6(MARK(head(incr(incr(y0))))) A__INCR(cons(head(incr(pairs)), z1)) -> c6(MARK(head(incr(pairs)))) A__INCR(cons(head(incr(odds)), z1)) -> c6(MARK(head(incr(odds)))) A__INCR(cons(head(incr(head(y0))), z1)) -> c6(MARK(head(incr(head(y0))))) A__INCR(cons(head(incr(tail(y0))), z1)) -> c6(MARK(head(incr(tail(y0))))) A__INCR(cons(head(incr(cons(y0, y1))), z1)) -> c6(MARK(head(incr(cons(y0, y1))))) A__INCR(cons(head(incr(s(y0))), z1)) -> c6(MARK(head(incr(s(y0))))) A__INCR(cons(head(incr(y0)), z1)) -> c6(MARK(head(incr(y0)))) A__INCR(cons(head(odds), z1)) -> c6(MARK(head(odds))) A__INCR(cons(head(head(nats)), z1)) -> c6(MARK(head(head(nats)))) A__INCR(cons(head(head(incr(y0))), z1)) -> c6(MARK(head(head(incr(y0))))) A__INCR(cons(head(head(pairs)), z1)) -> c6(MARK(head(head(pairs)))) A__INCR(cons(head(head(odds)), z1)) -> c6(MARK(head(head(odds)))) A__INCR(cons(head(head(head(y0))), z1)) -> c6(MARK(head(head(head(y0))))) A__INCR(cons(head(head(tail(y0))), z1)) -> c6(MARK(head(head(tail(y0))))) A__INCR(cons(head(head(cons(y0, y1))), z1)) -> c6(MARK(head(head(cons(y0, y1))))) A__INCR(cons(head(head(s(y0))), z1)) -> c6(MARK(head(head(s(y0))))) A__INCR(cons(head(head(y0)), z1)) -> c6(MARK(head(head(y0)))) A__INCR(cons(head(tail(nats)), z1)) -> c6(MARK(head(tail(nats)))) A__INCR(cons(head(tail(incr(y0))), z1)) -> c6(MARK(head(tail(incr(y0))))) A__INCR(cons(head(tail(pairs)), z1)) -> c6(MARK(head(tail(pairs)))) A__INCR(cons(head(tail(odds)), z1)) -> c6(MARK(head(tail(odds)))) A__INCR(cons(head(tail(head(y0))), z1)) -> c6(MARK(head(tail(head(y0))))) A__INCR(cons(head(tail(tail(y0))), z1)) -> c6(MARK(head(tail(tail(y0))))) A__INCR(cons(head(tail(cons(y0, y1))), z1)) -> c6(MARK(head(tail(cons(y0, y1))))) A__INCR(cons(head(tail(s(y0))), z1)) -> c6(MARK(head(tail(s(y0))))) A__INCR(cons(head(tail(y0)), z1)) -> c6(MARK(head(tail(y0)))) A__INCR(cons(head(nats), z1)) -> c6(MARK(head(nats))) A__INCR(cons(head(pairs), z1)) -> c6(MARK(head(pairs))) A__INCR(cons(tail(incr(nats)), z1)) -> c6(MARK(tail(incr(nats)))) A__INCR(cons(tail(incr(incr(y0))), z1)) -> c6(MARK(tail(incr(incr(y0))))) A__INCR(cons(tail(incr(pairs)), z1)) -> c6(MARK(tail(incr(pairs)))) A__INCR(cons(tail(incr(odds)), z1)) -> c6(MARK(tail(incr(odds)))) A__INCR(cons(tail(incr(head(y0))), z1)) -> c6(MARK(tail(incr(head(y0))))) A__INCR(cons(tail(incr(tail(y0))), z1)) -> c6(MARK(tail(incr(tail(y0))))) A__INCR(cons(tail(incr(cons(y0, y1))), z1)) -> c6(MARK(tail(incr(cons(y0, y1))))) A__INCR(cons(tail(incr(s(y0))), z1)) -> c6(MARK(tail(incr(s(y0))))) A__INCR(cons(tail(incr(y0)), z1)) -> c6(MARK(tail(incr(y0)))) A__INCR(cons(tail(odds), z1)) -> c6(MARK(tail(odds))) A__INCR(cons(tail(head(nats)), z1)) -> c6(MARK(tail(head(nats)))) A__INCR(cons(tail(head(incr(y0))), z1)) -> c6(MARK(tail(head(incr(y0))))) A__INCR(cons(tail(head(pairs)), z1)) -> c6(MARK(tail(head(pairs)))) A__INCR(cons(tail(head(odds)), z1)) -> c6(MARK(tail(head(odds)))) A__INCR(cons(tail(head(head(y0))), z1)) -> c6(MARK(tail(head(head(y0))))) A__INCR(cons(tail(head(tail(y0))), z1)) -> c6(MARK(tail(head(tail(y0))))) A__INCR(cons(tail(head(cons(y0, y1))), z1)) -> c6(MARK(tail(head(cons(y0, y1))))) A__INCR(cons(tail(head(s(y0))), z1)) -> c6(MARK(tail(head(s(y0))))) A__INCR(cons(tail(head(y0)), z1)) -> c6(MARK(tail(head(y0)))) A__INCR(cons(tail(tail(nats)), z1)) -> c6(MARK(tail(tail(nats)))) A__INCR(cons(tail(tail(incr(y0))), z1)) -> c6(MARK(tail(tail(incr(y0))))) A__INCR(cons(tail(tail(pairs)), z1)) -> c6(MARK(tail(tail(pairs)))) A__INCR(cons(tail(tail(odds)), z1)) -> c6(MARK(tail(tail(odds)))) A__INCR(cons(tail(tail(head(y0))), z1)) -> c6(MARK(tail(tail(head(y0))))) A__INCR(cons(tail(tail(tail(y0))), z1)) -> c6(MARK(tail(tail(tail(y0))))) A__INCR(cons(tail(tail(cons(y0, y1))), z1)) -> c6(MARK(tail(tail(cons(y0, y1))))) A__INCR(cons(tail(tail(s(y0))), z1)) -> c6(MARK(tail(tail(s(y0))))) A__INCR(cons(tail(tail(y0)), z1)) -> c6(MARK(tail(tail(y0)))) A__INCR(cons(tail(nats), z1)) -> c6(MARK(tail(nats))) A__INCR(cons(tail(pairs), z1)) -> c6(MARK(tail(pairs))) K tuples:none Defined Rule Symbols: a__nats, a__pairs, a__odds, a__incr_1, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__HEAD_1, A__TAIL_1, MARK_1, A__ODDS, A__INCR_1 Compound Symbols: c8_1, c10_1, c15_1, c18_1, c20_1, c13_2, c13_1, c16_2, c16_1, c17_2, c17_1, c4_1, c6_1 ---------------------------------------- (161) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 10 trailing nodes: MARK(odds) -> c15(A__ODDS) A__INCR(cons(odds, z1)) -> c6(MARK(odds)) MARK(tail(odds)) -> c17(MARK(odds)) MARK(incr(nats)) -> c13(A__INCR(cons(0, incr(nats)))) A__ODDS -> c4(A__INCR(cons(0, incr(odds)))) MARK(incr(pairs)) -> c13(A__INCR(cons(0, incr(odds)))) A__INCR(cons(incr(nats), z1)) -> c6(MARK(incr(nats))) MARK(head(odds)) -> c16(MARK(odds)) A__INCR(cons(incr(pairs), z1)) -> c6(MARK(incr(pairs))) MARK(incr(odds)) -> c13(MARK(odds)) ---------------------------------------- (162) Obligation: Complexity Dependency Tuples Problem Rules: a__nats -> cons(0, incr(nats)) a__nats -> nats a__pairs -> cons(0, incr(odds)) a__pairs -> pairs a__odds -> a__incr(a__pairs) a__odds -> odds a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) 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(nats) -> a__nats mark(incr(z0)) -> a__incr(mark(z0)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c17(MARK(s(z0))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs)), MARK(incr(pairs))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs)), MARK(odds)) MARK(head(head(nats))) -> c16(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(pairs))) -> c16(A__HEAD(a__head(a__pairs)), MARK(head(pairs))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(head(nats)) -> c16(A__HEAD(cons(0, incr(nats)))) MARK(head(pairs)) -> c16(A__HEAD(cons(0, incr(odds)))) MARK(tail(incr(nats))) -> c17(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(incr(z0)))) -> c17(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(pairs))) -> c17(A__TAIL(a__incr(a__pairs)), MARK(incr(pairs))) MARK(tail(incr(odds))) -> c17(A__TAIL(a__incr(a__odds)), MARK(incr(odds))) MARK(tail(incr(head(z0)))) -> c17(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c17(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c17(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c17(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c17(MARK(incr(x0))) MARK(tail(odds)) -> c17(A__TAIL(a__incr(a__pairs)), MARK(odds)) MARK(tail(head(nats))) -> c17(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(incr(z0)))) -> c17(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(pairs))) -> c17(A__TAIL(a__head(a__pairs)), MARK(head(pairs))) MARK(tail(head(odds))) -> c17(A__TAIL(a__head(a__odds)), MARK(head(odds))) MARK(tail(head(head(z0)))) -> c17(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c17(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c17(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c17(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c17(MARK(head(x0))) MARK(tail(tail(nats))) -> c17(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(incr(z0)))) -> c17(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(pairs))) -> c17(A__TAIL(a__tail(a__pairs)), MARK(tail(pairs))) MARK(tail(tail(odds))) -> c17(A__TAIL(a__tail(a__odds)), MARK(tail(odds))) MARK(tail(tail(head(z0)))) -> c17(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c17(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c17(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c17(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c17(MARK(tail(x0))) MARK(tail(nats)) -> c17(A__TAIL(cons(0, incr(nats)))) MARK(tail(pairs)) -> c17(A__TAIL(cons(0, incr(odds)))) A__INCR(cons(cons(y0, y1), z1)) -> c6(MARK(cons(y0, y1))) A__INCR(cons(s(y0), z1)) -> c6(MARK(s(y0))) A__INCR(cons(incr(cons(y0, y1)), z1)) -> c6(MARK(incr(cons(y0, y1)))) A__INCR(cons(incr(s(y0)), z1)) -> c6(MARK(incr(s(y0)))) A__INCR(cons(head(cons(y0, y1)), z1)) -> c6(MARK(head(cons(y0, y1)))) A__INCR(cons(head(s(y0)), z1)) -> c6(MARK(head(s(y0)))) A__INCR(cons(tail(cons(y0, y1)), z1)) -> c6(MARK(tail(cons(y0, y1)))) A__INCR(cons(tail(s(y0)), z1)) -> c6(MARK(tail(s(y0)))) A__INCR(cons(incr(incr(nats)), z1)) -> c6(MARK(incr(incr(nats)))) A__INCR(cons(incr(incr(incr(y0))), z1)) -> c6(MARK(incr(incr(incr(y0))))) A__INCR(cons(incr(incr(pairs)), z1)) -> c6(MARK(incr(incr(pairs)))) A__INCR(cons(incr(incr(odds)), z1)) -> c6(MARK(incr(incr(odds)))) A__INCR(cons(incr(incr(head(y0))), z1)) -> c6(MARK(incr(incr(head(y0))))) A__INCR(cons(incr(incr(tail(y0))), z1)) -> c6(MARK(incr(incr(tail(y0))))) A__INCR(cons(incr(incr(cons(y0, y1))), z1)) -> c6(MARK(incr(incr(cons(y0, y1))))) A__INCR(cons(incr(incr(s(y0))), z1)) -> c6(MARK(incr(incr(s(y0))))) A__INCR(cons(incr(incr(y0)), z1)) -> c6(MARK(incr(incr(y0)))) A__INCR(cons(incr(odds), z1)) -> c6(MARK(incr(odds))) A__INCR(cons(incr(head(nats)), z1)) -> c6(MARK(incr(head(nats)))) A__INCR(cons(incr(head(incr(y0))), z1)) -> c6(MARK(incr(head(incr(y0))))) A__INCR(cons(incr(head(pairs)), z1)) -> c6(MARK(incr(head(pairs)))) A__INCR(cons(incr(head(odds)), z1)) -> c6(MARK(incr(head(odds)))) A__INCR(cons(incr(head(head(y0))), z1)) -> c6(MARK(incr(head(head(y0))))) A__INCR(cons(incr(head(tail(y0))), z1)) -> c6(MARK(incr(head(tail(y0))))) A__INCR(cons(incr(head(cons(y0, y1))), z1)) -> c6(MARK(incr(head(cons(y0, y1))))) A__INCR(cons(incr(head(s(y0))), z1)) -> c6(MARK(incr(head(s(y0))))) A__INCR(cons(incr(head(y0)), z1)) -> c6(MARK(incr(head(y0)))) A__INCR(cons(incr(tail(nats)), z1)) -> c6(MARK(incr(tail(nats)))) A__INCR(cons(incr(tail(incr(y0))), z1)) -> c6(MARK(incr(tail(incr(y0))))) A__INCR(cons(incr(tail(pairs)), z1)) -> c6(MARK(incr(tail(pairs)))) A__INCR(cons(incr(tail(odds)), z1)) -> c6(MARK(incr(tail(odds)))) A__INCR(cons(incr(tail(head(y0))), z1)) -> c6(MARK(incr(tail(head(y0))))) A__INCR(cons(incr(tail(tail(y0))), z1)) -> c6(MARK(incr(tail(tail(y0))))) A__INCR(cons(incr(tail(cons(y0, y1))), z1)) -> c6(MARK(incr(tail(cons(y0, y1))))) A__INCR(cons(incr(tail(s(y0))), z1)) -> c6(MARK(incr(tail(s(y0))))) A__INCR(cons(incr(tail(y0)), z1)) -> c6(MARK(incr(tail(y0)))) A__INCR(cons(head(incr(nats)), z1)) -> c6(MARK(head(incr(nats)))) A__INCR(cons(head(incr(incr(y0))), z1)) -> c6(MARK(head(incr(incr(y0))))) A__INCR(cons(head(incr(pairs)), z1)) -> c6(MARK(head(incr(pairs)))) A__INCR(cons(head(incr(odds)), z1)) -> c6(MARK(head(incr(odds)))) A__INCR(cons(head(incr(head(y0))), z1)) -> c6(MARK(head(incr(head(y0))))) A__INCR(cons(head(incr(tail(y0))), z1)) -> c6(MARK(head(incr(tail(y0))))) A__INCR(cons(head(incr(cons(y0, y1))), z1)) -> c6(MARK(head(incr(cons(y0, y1))))) A__INCR(cons(head(incr(s(y0))), z1)) -> c6(MARK(head(incr(s(y0))))) A__INCR(cons(head(incr(y0)), z1)) -> c6(MARK(head(incr(y0)))) A__INCR(cons(head(odds), z1)) -> c6(MARK(head(odds))) A__INCR(cons(head(head(nats)), z1)) -> c6(MARK(head(head(nats)))) A__INCR(cons(head(head(incr(y0))), z1)) -> c6(MARK(head(head(incr(y0))))) A__INCR(cons(head(head(pairs)), z1)) -> c6(MARK(head(head(pairs)))) A__INCR(cons(head(head(odds)), z1)) -> c6(MARK(head(head(odds)))) A__INCR(cons(head(head(head(y0))), z1)) -> c6(MARK(head(head(head(y0))))) A__INCR(cons(head(head(tail(y0))), z1)) -> c6(MARK(head(head(tail(y0))))) A__INCR(cons(head(head(cons(y0, y1))), z1)) -> c6(MARK(head(head(cons(y0, y1))))) A__INCR(cons(head(head(s(y0))), z1)) -> c6(MARK(head(head(s(y0))))) A__INCR(cons(head(head(y0)), z1)) -> c6(MARK(head(head(y0)))) A__INCR(cons(head(tail(nats)), z1)) -> c6(MARK(head(tail(nats)))) A__INCR(cons(head(tail(incr(y0))), z1)) -> c6(MARK(head(tail(incr(y0))))) A__INCR(cons(head(tail(pairs)), z1)) -> c6(MARK(head(tail(pairs)))) A__INCR(cons(head(tail(odds)), z1)) -> c6(MARK(head(tail(odds)))) A__INCR(cons(head(tail(head(y0))), z1)) -> c6(MARK(head(tail(head(y0))))) A__INCR(cons(head(tail(tail(y0))), z1)) -> c6(MARK(head(tail(tail(y0))))) A__INCR(cons(head(tail(cons(y0, y1))), z1)) -> c6(MARK(head(tail(cons(y0, y1))))) A__INCR(cons(head(tail(s(y0))), z1)) -> c6(MARK(head(tail(s(y0))))) A__INCR(cons(head(tail(y0)), z1)) -> c6(MARK(head(tail(y0)))) A__INCR(cons(head(nats), z1)) -> c6(MARK(head(nats))) A__INCR(cons(head(pairs), z1)) -> c6(MARK(head(pairs))) A__INCR(cons(tail(incr(nats)), z1)) -> c6(MARK(tail(incr(nats)))) A__INCR(cons(tail(incr(incr(y0))), z1)) -> c6(MARK(tail(incr(incr(y0))))) A__INCR(cons(tail(incr(pairs)), z1)) -> c6(MARK(tail(incr(pairs)))) A__INCR(cons(tail(incr(odds)), z1)) -> c6(MARK(tail(incr(odds)))) A__INCR(cons(tail(incr(head(y0))), z1)) -> c6(MARK(tail(incr(head(y0))))) A__INCR(cons(tail(incr(tail(y0))), z1)) -> c6(MARK(tail(incr(tail(y0))))) A__INCR(cons(tail(incr(cons(y0, y1))), z1)) -> c6(MARK(tail(incr(cons(y0, y1))))) A__INCR(cons(tail(incr(s(y0))), z1)) -> c6(MARK(tail(incr(s(y0))))) A__INCR(cons(tail(incr(y0)), z1)) -> c6(MARK(tail(incr(y0)))) A__INCR(cons(tail(odds), z1)) -> c6(MARK(tail(odds))) A__INCR(cons(tail(head(nats)), z1)) -> c6(MARK(tail(head(nats)))) A__INCR(cons(tail(head(incr(y0))), z1)) -> c6(MARK(tail(head(incr(y0))))) A__INCR(cons(tail(head(pairs)), z1)) -> c6(MARK(tail(head(pairs)))) A__INCR(cons(tail(head(odds)), z1)) -> c6(MARK(tail(head(odds)))) A__INCR(cons(tail(head(head(y0))), z1)) -> c6(MARK(tail(head(head(y0))))) A__INCR(cons(tail(head(tail(y0))), z1)) -> c6(MARK(tail(head(tail(y0))))) A__INCR(cons(tail(head(cons(y0, y1))), z1)) -> c6(MARK(tail(head(cons(y0, y1))))) A__INCR(cons(tail(head(s(y0))), z1)) -> c6(MARK(tail(head(s(y0))))) A__INCR(cons(tail(head(y0)), z1)) -> c6(MARK(tail(head(y0)))) A__INCR(cons(tail(tail(nats)), z1)) -> c6(MARK(tail(tail(nats)))) A__INCR(cons(tail(tail(incr(y0))), z1)) -> c6(MARK(tail(tail(incr(y0))))) A__INCR(cons(tail(tail(pairs)), z1)) -> c6(MARK(tail(tail(pairs)))) A__INCR(cons(tail(tail(odds)), z1)) -> c6(MARK(tail(tail(odds)))) A__INCR(cons(tail(tail(head(y0))), z1)) -> c6(MARK(tail(tail(head(y0))))) A__INCR(cons(tail(tail(tail(y0))), z1)) -> c6(MARK(tail(tail(tail(y0))))) A__INCR(cons(tail(tail(cons(y0, y1))), z1)) -> c6(MARK(tail(tail(cons(y0, y1))))) A__INCR(cons(tail(tail(s(y0))), z1)) -> c6(MARK(tail(tail(s(y0))))) A__INCR(cons(tail(tail(y0)), z1)) -> c6(MARK(tail(tail(y0)))) A__INCR(cons(tail(nats), z1)) -> c6(MARK(tail(nats))) A__INCR(cons(tail(pairs), z1)) -> c6(MARK(tail(pairs))) S tuples: A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c17(MARK(s(z0))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats)), MARK(incr(nats))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs)), MARK(incr(pairs))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs)), MARK(odds)) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats)), MARK(incr(nats))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs)), MARK(incr(pairs))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs)), MARK(odds)) MARK(head(head(nats))) -> c16(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(pairs))) -> c16(A__HEAD(a__head(a__pairs)), MARK(head(pairs))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(head(nats)) -> c16(A__HEAD(cons(0, incr(nats)))) MARK(head(pairs)) -> c16(A__HEAD(cons(0, incr(odds)))) MARK(tail(incr(nats))) -> c17(A__TAIL(a__incr(a__nats)), MARK(incr(nats))) MARK(tail(incr(incr(z0)))) -> c17(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(pairs))) -> c17(A__TAIL(a__incr(a__pairs)), MARK(incr(pairs))) MARK(tail(incr(odds))) -> c17(A__TAIL(a__incr(a__odds)), MARK(incr(odds))) MARK(tail(incr(head(z0)))) -> c17(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c17(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c17(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c17(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c17(MARK(incr(x0))) MARK(tail(odds)) -> c17(A__TAIL(a__incr(a__pairs)), MARK(odds)) MARK(tail(head(nats))) -> c17(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(incr(z0)))) -> c17(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(pairs))) -> c17(A__TAIL(a__head(a__pairs)), MARK(head(pairs))) MARK(tail(head(odds))) -> c17(A__TAIL(a__head(a__odds)), MARK(head(odds))) MARK(tail(head(head(z0)))) -> c17(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c17(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c17(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c17(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c17(MARK(head(x0))) MARK(tail(tail(nats))) -> c17(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(incr(z0)))) -> c17(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(pairs))) -> c17(A__TAIL(a__tail(a__pairs)), MARK(tail(pairs))) MARK(tail(tail(odds))) -> c17(A__TAIL(a__tail(a__odds)), MARK(tail(odds))) MARK(tail(tail(head(z0)))) -> c17(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c17(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c17(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c17(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c17(MARK(tail(x0))) MARK(tail(nats)) -> c17(A__TAIL(cons(0, incr(nats)))) MARK(tail(pairs)) -> c17(A__TAIL(cons(0, incr(odds)))) A__INCR(cons(cons(y0, y1), z1)) -> c6(MARK(cons(y0, y1))) A__INCR(cons(s(y0), z1)) -> c6(MARK(s(y0))) A__INCR(cons(incr(cons(y0, y1)), z1)) -> c6(MARK(incr(cons(y0, y1)))) A__INCR(cons(incr(s(y0)), z1)) -> c6(MARK(incr(s(y0)))) A__INCR(cons(head(cons(y0, y1)), z1)) -> c6(MARK(head(cons(y0, y1)))) A__INCR(cons(head(s(y0)), z1)) -> c6(MARK(head(s(y0)))) A__INCR(cons(tail(cons(y0, y1)), z1)) -> c6(MARK(tail(cons(y0, y1)))) A__INCR(cons(tail(s(y0)), z1)) -> c6(MARK(tail(s(y0)))) A__INCR(cons(incr(incr(nats)), z1)) -> c6(MARK(incr(incr(nats)))) A__INCR(cons(incr(incr(incr(y0))), z1)) -> c6(MARK(incr(incr(incr(y0))))) A__INCR(cons(incr(incr(pairs)), z1)) -> c6(MARK(incr(incr(pairs)))) A__INCR(cons(incr(incr(odds)), z1)) -> c6(MARK(incr(incr(odds)))) A__INCR(cons(incr(incr(head(y0))), z1)) -> c6(MARK(incr(incr(head(y0))))) A__INCR(cons(incr(incr(tail(y0))), z1)) -> c6(MARK(incr(incr(tail(y0))))) A__INCR(cons(incr(incr(cons(y0, y1))), z1)) -> c6(MARK(incr(incr(cons(y0, y1))))) A__INCR(cons(incr(incr(s(y0))), z1)) -> c6(MARK(incr(incr(s(y0))))) A__INCR(cons(incr(incr(y0)), z1)) -> c6(MARK(incr(incr(y0)))) A__INCR(cons(incr(odds), z1)) -> c6(MARK(incr(odds))) A__INCR(cons(incr(head(nats)), z1)) -> c6(MARK(incr(head(nats)))) A__INCR(cons(incr(head(incr(y0))), z1)) -> c6(MARK(incr(head(incr(y0))))) A__INCR(cons(incr(head(pairs)), z1)) -> c6(MARK(incr(head(pairs)))) A__INCR(cons(incr(head(odds)), z1)) -> c6(MARK(incr(head(odds)))) A__INCR(cons(incr(head(head(y0))), z1)) -> c6(MARK(incr(head(head(y0))))) A__INCR(cons(incr(head(tail(y0))), z1)) -> c6(MARK(incr(head(tail(y0))))) A__INCR(cons(incr(head(cons(y0, y1))), z1)) -> c6(MARK(incr(head(cons(y0, y1))))) A__INCR(cons(incr(head(s(y0))), z1)) -> c6(MARK(incr(head(s(y0))))) A__INCR(cons(incr(head(y0)), z1)) -> c6(MARK(incr(head(y0)))) A__INCR(cons(incr(tail(nats)), z1)) -> c6(MARK(incr(tail(nats)))) A__INCR(cons(incr(tail(incr(y0))), z1)) -> c6(MARK(incr(tail(incr(y0))))) A__INCR(cons(incr(tail(pairs)), z1)) -> c6(MARK(incr(tail(pairs)))) A__INCR(cons(incr(tail(odds)), z1)) -> c6(MARK(incr(tail(odds)))) A__INCR(cons(incr(tail(head(y0))), z1)) -> c6(MARK(incr(tail(head(y0))))) A__INCR(cons(incr(tail(tail(y0))), z1)) -> c6(MARK(incr(tail(tail(y0))))) A__INCR(cons(incr(tail(cons(y0, y1))), z1)) -> c6(MARK(incr(tail(cons(y0, y1))))) A__INCR(cons(incr(tail(s(y0))), z1)) -> c6(MARK(incr(tail(s(y0))))) A__INCR(cons(incr(tail(y0)), z1)) -> c6(MARK(incr(tail(y0)))) A__INCR(cons(head(incr(nats)), z1)) -> c6(MARK(head(incr(nats)))) A__INCR(cons(head(incr(incr(y0))), z1)) -> c6(MARK(head(incr(incr(y0))))) A__INCR(cons(head(incr(pairs)), z1)) -> c6(MARK(head(incr(pairs)))) A__INCR(cons(head(incr(odds)), z1)) -> c6(MARK(head(incr(odds)))) A__INCR(cons(head(incr(head(y0))), z1)) -> c6(MARK(head(incr(head(y0))))) A__INCR(cons(head(incr(tail(y0))), z1)) -> c6(MARK(head(incr(tail(y0))))) A__INCR(cons(head(incr(cons(y0, y1))), z1)) -> c6(MARK(head(incr(cons(y0, y1))))) A__INCR(cons(head(incr(s(y0))), z1)) -> c6(MARK(head(incr(s(y0))))) A__INCR(cons(head(incr(y0)), z1)) -> c6(MARK(head(incr(y0)))) A__INCR(cons(head(odds), z1)) -> c6(MARK(head(odds))) A__INCR(cons(head(head(nats)), z1)) -> c6(MARK(head(head(nats)))) A__INCR(cons(head(head(incr(y0))), z1)) -> c6(MARK(head(head(incr(y0))))) A__INCR(cons(head(head(pairs)), z1)) -> c6(MARK(head(head(pairs)))) A__INCR(cons(head(head(odds)), z1)) -> c6(MARK(head(head(odds)))) A__INCR(cons(head(head(head(y0))), z1)) -> c6(MARK(head(head(head(y0))))) A__INCR(cons(head(head(tail(y0))), z1)) -> c6(MARK(head(head(tail(y0))))) A__INCR(cons(head(head(cons(y0, y1))), z1)) -> c6(MARK(head(head(cons(y0, y1))))) A__INCR(cons(head(head(s(y0))), z1)) -> c6(MARK(head(head(s(y0))))) A__INCR(cons(head(head(y0)), z1)) -> c6(MARK(head(head(y0)))) A__INCR(cons(head(tail(nats)), z1)) -> c6(MARK(head(tail(nats)))) A__INCR(cons(head(tail(incr(y0))), z1)) -> c6(MARK(head(tail(incr(y0))))) A__INCR(cons(head(tail(pairs)), z1)) -> c6(MARK(head(tail(pairs)))) A__INCR(cons(head(tail(odds)), z1)) -> c6(MARK(head(tail(odds)))) A__INCR(cons(head(tail(head(y0))), z1)) -> c6(MARK(head(tail(head(y0))))) A__INCR(cons(head(tail(tail(y0))), z1)) -> c6(MARK(head(tail(tail(y0))))) A__INCR(cons(head(tail(cons(y0, y1))), z1)) -> c6(MARK(head(tail(cons(y0, y1))))) A__INCR(cons(head(tail(s(y0))), z1)) -> c6(MARK(head(tail(s(y0))))) A__INCR(cons(head(tail(y0)), z1)) -> c6(MARK(head(tail(y0)))) A__INCR(cons(head(nats), z1)) -> c6(MARK(head(nats))) A__INCR(cons(head(pairs), z1)) -> c6(MARK(head(pairs))) A__INCR(cons(tail(incr(nats)), z1)) -> c6(MARK(tail(incr(nats)))) A__INCR(cons(tail(incr(incr(y0))), z1)) -> c6(MARK(tail(incr(incr(y0))))) A__INCR(cons(tail(incr(pairs)), z1)) -> c6(MARK(tail(incr(pairs)))) A__INCR(cons(tail(incr(odds)), z1)) -> c6(MARK(tail(incr(odds)))) A__INCR(cons(tail(incr(head(y0))), z1)) -> c6(MARK(tail(incr(head(y0))))) A__INCR(cons(tail(incr(tail(y0))), z1)) -> c6(MARK(tail(incr(tail(y0))))) A__INCR(cons(tail(incr(cons(y0, y1))), z1)) -> c6(MARK(tail(incr(cons(y0, y1))))) A__INCR(cons(tail(incr(s(y0))), z1)) -> c6(MARK(tail(incr(s(y0))))) A__INCR(cons(tail(incr(y0)), z1)) -> c6(MARK(tail(incr(y0)))) A__INCR(cons(tail(odds), z1)) -> c6(MARK(tail(odds))) A__INCR(cons(tail(head(nats)), z1)) -> c6(MARK(tail(head(nats)))) A__INCR(cons(tail(head(incr(y0))), z1)) -> c6(MARK(tail(head(incr(y0))))) A__INCR(cons(tail(head(pairs)), z1)) -> c6(MARK(tail(head(pairs)))) A__INCR(cons(tail(head(odds)), z1)) -> c6(MARK(tail(head(odds)))) A__INCR(cons(tail(head(head(y0))), z1)) -> c6(MARK(tail(head(head(y0))))) A__INCR(cons(tail(head(tail(y0))), z1)) -> c6(MARK(tail(head(tail(y0))))) A__INCR(cons(tail(head(cons(y0, y1))), z1)) -> c6(MARK(tail(head(cons(y0, y1))))) A__INCR(cons(tail(head(s(y0))), z1)) -> c6(MARK(tail(head(s(y0))))) A__INCR(cons(tail(head(y0)), z1)) -> c6(MARK(tail(head(y0)))) A__INCR(cons(tail(tail(nats)), z1)) -> c6(MARK(tail(tail(nats)))) A__INCR(cons(tail(tail(incr(y0))), z1)) -> c6(MARK(tail(tail(incr(y0))))) A__INCR(cons(tail(tail(pairs)), z1)) -> c6(MARK(tail(tail(pairs)))) A__INCR(cons(tail(tail(odds)), z1)) -> c6(MARK(tail(tail(odds)))) A__INCR(cons(tail(tail(head(y0))), z1)) -> c6(MARK(tail(tail(head(y0))))) A__INCR(cons(tail(tail(tail(y0))), z1)) -> c6(MARK(tail(tail(tail(y0))))) A__INCR(cons(tail(tail(cons(y0, y1))), z1)) -> c6(MARK(tail(tail(cons(y0, y1))))) A__INCR(cons(tail(tail(s(y0))), z1)) -> c6(MARK(tail(tail(s(y0))))) A__INCR(cons(tail(tail(y0)), z1)) -> c6(MARK(tail(tail(y0)))) A__INCR(cons(tail(nats), z1)) -> c6(MARK(tail(nats))) A__INCR(cons(tail(pairs), z1)) -> c6(MARK(tail(pairs))) K tuples:none Defined Rule Symbols: a__nats, a__pairs, a__odds, a__incr_1, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__HEAD_1, A__TAIL_1, MARK_1, A__INCR_1 Compound Symbols: c8_1, c10_1, c18_1, c20_1, c13_2, c13_1, c16_2, c16_1, c17_2, c17_1, c6_1 ---------------------------------------- (163) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 9 trailing tuple parts ---------------------------------------- (164) Obligation: Complexity Dependency Tuples Problem Rules: a__nats -> cons(0, incr(nats)) a__nats -> nats a__pairs -> cons(0, incr(odds)) a__pairs -> pairs a__odds -> a__incr(a__pairs) a__odds -> odds a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) 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(nats) -> a__nats mark(incr(z0)) -> a__incr(mark(z0)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c17(MARK(s(z0))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(head(nats))) -> c16(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(pairs))) -> c16(A__HEAD(a__head(a__pairs)), MARK(head(pairs))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(head(nats)) -> c16(A__HEAD(cons(0, incr(nats)))) MARK(head(pairs)) -> c16(A__HEAD(cons(0, incr(odds)))) MARK(tail(incr(incr(z0)))) -> c17(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(odds))) -> c17(A__TAIL(a__incr(a__odds)), MARK(incr(odds))) MARK(tail(incr(head(z0)))) -> c17(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c17(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c17(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c17(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c17(MARK(incr(x0))) MARK(tail(head(nats))) -> c17(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(incr(z0)))) -> c17(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(pairs))) -> c17(A__TAIL(a__head(a__pairs)), MARK(head(pairs))) MARK(tail(head(odds))) -> c17(A__TAIL(a__head(a__odds)), MARK(head(odds))) MARK(tail(head(head(z0)))) -> c17(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c17(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c17(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c17(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c17(MARK(head(x0))) MARK(tail(tail(nats))) -> c17(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(incr(z0)))) -> c17(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(pairs))) -> c17(A__TAIL(a__tail(a__pairs)), MARK(tail(pairs))) MARK(tail(tail(odds))) -> c17(A__TAIL(a__tail(a__odds)), MARK(tail(odds))) MARK(tail(tail(head(z0)))) -> c17(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c17(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c17(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c17(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c17(MARK(tail(x0))) MARK(tail(nats)) -> c17(A__TAIL(cons(0, incr(nats)))) MARK(tail(pairs)) -> c17(A__TAIL(cons(0, incr(odds)))) A__INCR(cons(cons(y0, y1), z1)) -> c6(MARK(cons(y0, y1))) A__INCR(cons(s(y0), z1)) -> c6(MARK(s(y0))) A__INCR(cons(incr(cons(y0, y1)), z1)) -> c6(MARK(incr(cons(y0, y1)))) A__INCR(cons(incr(s(y0)), z1)) -> c6(MARK(incr(s(y0)))) A__INCR(cons(head(cons(y0, y1)), z1)) -> c6(MARK(head(cons(y0, y1)))) A__INCR(cons(head(s(y0)), z1)) -> c6(MARK(head(s(y0)))) A__INCR(cons(tail(cons(y0, y1)), z1)) -> c6(MARK(tail(cons(y0, y1)))) A__INCR(cons(tail(s(y0)), z1)) -> c6(MARK(tail(s(y0)))) A__INCR(cons(incr(incr(nats)), z1)) -> c6(MARK(incr(incr(nats)))) A__INCR(cons(incr(incr(incr(y0))), z1)) -> c6(MARK(incr(incr(incr(y0))))) A__INCR(cons(incr(incr(pairs)), z1)) -> c6(MARK(incr(incr(pairs)))) A__INCR(cons(incr(incr(odds)), z1)) -> c6(MARK(incr(incr(odds)))) A__INCR(cons(incr(incr(head(y0))), z1)) -> c6(MARK(incr(incr(head(y0))))) A__INCR(cons(incr(incr(tail(y0))), z1)) -> c6(MARK(incr(incr(tail(y0))))) A__INCR(cons(incr(incr(cons(y0, y1))), z1)) -> c6(MARK(incr(incr(cons(y0, y1))))) A__INCR(cons(incr(incr(s(y0))), z1)) -> c6(MARK(incr(incr(s(y0))))) A__INCR(cons(incr(incr(y0)), z1)) -> c6(MARK(incr(incr(y0)))) A__INCR(cons(incr(odds), z1)) -> c6(MARK(incr(odds))) A__INCR(cons(incr(head(nats)), z1)) -> c6(MARK(incr(head(nats)))) A__INCR(cons(incr(head(incr(y0))), z1)) -> c6(MARK(incr(head(incr(y0))))) A__INCR(cons(incr(head(pairs)), z1)) -> c6(MARK(incr(head(pairs)))) A__INCR(cons(incr(head(odds)), z1)) -> c6(MARK(incr(head(odds)))) A__INCR(cons(incr(head(head(y0))), z1)) -> c6(MARK(incr(head(head(y0))))) A__INCR(cons(incr(head(tail(y0))), z1)) -> c6(MARK(incr(head(tail(y0))))) A__INCR(cons(incr(head(cons(y0, y1))), z1)) -> c6(MARK(incr(head(cons(y0, y1))))) A__INCR(cons(incr(head(s(y0))), z1)) -> c6(MARK(incr(head(s(y0))))) A__INCR(cons(incr(head(y0)), z1)) -> c6(MARK(incr(head(y0)))) A__INCR(cons(incr(tail(nats)), z1)) -> c6(MARK(incr(tail(nats)))) A__INCR(cons(incr(tail(incr(y0))), z1)) -> c6(MARK(incr(tail(incr(y0))))) A__INCR(cons(incr(tail(pairs)), z1)) -> c6(MARK(incr(tail(pairs)))) A__INCR(cons(incr(tail(odds)), z1)) -> c6(MARK(incr(tail(odds)))) A__INCR(cons(incr(tail(head(y0))), z1)) -> c6(MARK(incr(tail(head(y0))))) A__INCR(cons(incr(tail(tail(y0))), z1)) -> c6(MARK(incr(tail(tail(y0))))) A__INCR(cons(incr(tail(cons(y0, y1))), z1)) -> c6(MARK(incr(tail(cons(y0, y1))))) A__INCR(cons(incr(tail(s(y0))), z1)) -> c6(MARK(incr(tail(s(y0))))) A__INCR(cons(incr(tail(y0)), z1)) -> c6(MARK(incr(tail(y0)))) A__INCR(cons(head(incr(nats)), z1)) -> c6(MARK(head(incr(nats)))) A__INCR(cons(head(incr(incr(y0))), z1)) -> c6(MARK(head(incr(incr(y0))))) A__INCR(cons(head(incr(pairs)), z1)) -> c6(MARK(head(incr(pairs)))) A__INCR(cons(head(incr(odds)), z1)) -> c6(MARK(head(incr(odds)))) A__INCR(cons(head(incr(head(y0))), z1)) -> c6(MARK(head(incr(head(y0))))) A__INCR(cons(head(incr(tail(y0))), z1)) -> c6(MARK(head(incr(tail(y0))))) A__INCR(cons(head(incr(cons(y0, y1))), z1)) -> c6(MARK(head(incr(cons(y0, y1))))) A__INCR(cons(head(incr(s(y0))), z1)) -> c6(MARK(head(incr(s(y0))))) A__INCR(cons(head(incr(y0)), z1)) -> c6(MARK(head(incr(y0)))) A__INCR(cons(head(odds), z1)) -> c6(MARK(head(odds))) A__INCR(cons(head(head(nats)), z1)) -> c6(MARK(head(head(nats)))) A__INCR(cons(head(head(incr(y0))), z1)) -> c6(MARK(head(head(incr(y0))))) A__INCR(cons(head(head(pairs)), z1)) -> c6(MARK(head(head(pairs)))) A__INCR(cons(head(head(odds)), z1)) -> c6(MARK(head(head(odds)))) A__INCR(cons(head(head(head(y0))), z1)) -> c6(MARK(head(head(head(y0))))) A__INCR(cons(head(head(tail(y0))), z1)) -> c6(MARK(head(head(tail(y0))))) A__INCR(cons(head(head(cons(y0, y1))), z1)) -> c6(MARK(head(head(cons(y0, y1))))) A__INCR(cons(head(head(s(y0))), z1)) -> c6(MARK(head(head(s(y0))))) A__INCR(cons(head(head(y0)), z1)) -> c6(MARK(head(head(y0)))) A__INCR(cons(head(tail(nats)), z1)) -> c6(MARK(head(tail(nats)))) A__INCR(cons(head(tail(incr(y0))), z1)) -> c6(MARK(head(tail(incr(y0))))) A__INCR(cons(head(tail(pairs)), z1)) -> c6(MARK(head(tail(pairs)))) A__INCR(cons(head(tail(odds)), z1)) -> c6(MARK(head(tail(odds)))) A__INCR(cons(head(tail(head(y0))), z1)) -> c6(MARK(head(tail(head(y0))))) A__INCR(cons(head(tail(tail(y0))), z1)) -> c6(MARK(head(tail(tail(y0))))) A__INCR(cons(head(tail(cons(y0, y1))), z1)) -> c6(MARK(head(tail(cons(y0, y1))))) A__INCR(cons(head(tail(s(y0))), z1)) -> c6(MARK(head(tail(s(y0))))) A__INCR(cons(head(tail(y0)), z1)) -> c6(MARK(head(tail(y0)))) A__INCR(cons(head(nats), z1)) -> c6(MARK(head(nats))) A__INCR(cons(head(pairs), z1)) -> c6(MARK(head(pairs))) A__INCR(cons(tail(incr(nats)), z1)) -> c6(MARK(tail(incr(nats)))) A__INCR(cons(tail(incr(incr(y0))), z1)) -> c6(MARK(tail(incr(incr(y0))))) A__INCR(cons(tail(incr(pairs)), z1)) -> c6(MARK(tail(incr(pairs)))) A__INCR(cons(tail(incr(odds)), z1)) -> c6(MARK(tail(incr(odds)))) A__INCR(cons(tail(incr(head(y0))), z1)) -> c6(MARK(tail(incr(head(y0))))) A__INCR(cons(tail(incr(tail(y0))), z1)) -> c6(MARK(tail(incr(tail(y0))))) A__INCR(cons(tail(incr(cons(y0, y1))), z1)) -> c6(MARK(tail(incr(cons(y0, y1))))) A__INCR(cons(tail(incr(s(y0))), z1)) -> c6(MARK(tail(incr(s(y0))))) A__INCR(cons(tail(incr(y0)), z1)) -> c6(MARK(tail(incr(y0)))) A__INCR(cons(tail(odds), z1)) -> c6(MARK(tail(odds))) A__INCR(cons(tail(head(nats)), z1)) -> c6(MARK(tail(head(nats)))) A__INCR(cons(tail(head(incr(y0))), z1)) -> c6(MARK(tail(head(incr(y0))))) A__INCR(cons(tail(head(pairs)), z1)) -> c6(MARK(tail(head(pairs)))) A__INCR(cons(tail(head(odds)), z1)) -> c6(MARK(tail(head(odds)))) A__INCR(cons(tail(head(head(y0))), z1)) -> c6(MARK(tail(head(head(y0))))) A__INCR(cons(tail(head(tail(y0))), z1)) -> c6(MARK(tail(head(tail(y0))))) A__INCR(cons(tail(head(cons(y0, y1))), z1)) -> c6(MARK(tail(head(cons(y0, y1))))) A__INCR(cons(tail(head(s(y0))), z1)) -> c6(MARK(tail(head(s(y0))))) A__INCR(cons(tail(head(y0)), z1)) -> c6(MARK(tail(head(y0)))) A__INCR(cons(tail(tail(nats)), z1)) -> c6(MARK(tail(tail(nats)))) A__INCR(cons(tail(tail(incr(y0))), z1)) -> c6(MARK(tail(tail(incr(y0))))) A__INCR(cons(tail(tail(pairs)), z1)) -> c6(MARK(tail(tail(pairs)))) A__INCR(cons(tail(tail(odds)), z1)) -> c6(MARK(tail(tail(odds)))) A__INCR(cons(tail(tail(head(y0))), z1)) -> c6(MARK(tail(tail(head(y0))))) A__INCR(cons(tail(tail(tail(y0))), z1)) -> c6(MARK(tail(tail(tail(y0))))) A__INCR(cons(tail(tail(cons(y0, y1))), z1)) -> c6(MARK(tail(tail(cons(y0, y1))))) A__INCR(cons(tail(tail(s(y0))), z1)) -> c6(MARK(tail(tail(s(y0))))) A__INCR(cons(tail(tail(y0)), z1)) -> c6(MARK(tail(tail(y0)))) A__INCR(cons(tail(nats), z1)) -> c6(MARK(tail(nats))) A__INCR(cons(tail(pairs), z1)) -> c6(MARK(tail(pairs))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs))) MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs))) MARK(tail(incr(nats))) -> c17(A__TAIL(a__incr(a__nats))) MARK(tail(incr(pairs))) -> c17(A__TAIL(a__incr(a__pairs))) MARK(tail(odds)) -> c17(A__TAIL(a__incr(a__pairs))) S tuples: A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c17(MARK(s(z0))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(head(nats))) -> c16(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(pairs))) -> c16(A__HEAD(a__head(a__pairs)), MARK(head(pairs))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(head(nats)) -> c16(A__HEAD(cons(0, incr(nats)))) MARK(head(pairs)) -> c16(A__HEAD(cons(0, incr(odds)))) MARK(tail(incr(incr(z0)))) -> c17(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(odds))) -> c17(A__TAIL(a__incr(a__odds)), MARK(incr(odds))) MARK(tail(incr(head(z0)))) -> c17(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c17(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c17(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c17(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c17(MARK(incr(x0))) MARK(tail(head(nats))) -> c17(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(incr(z0)))) -> c17(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(pairs))) -> c17(A__TAIL(a__head(a__pairs)), MARK(head(pairs))) MARK(tail(head(odds))) -> c17(A__TAIL(a__head(a__odds)), MARK(head(odds))) MARK(tail(head(head(z0)))) -> c17(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c17(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c17(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c17(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c17(MARK(head(x0))) MARK(tail(tail(nats))) -> c17(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(incr(z0)))) -> c17(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(pairs))) -> c17(A__TAIL(a__tail(a__pairs)), MARK(tail(pairs))) MARK(tail(tail(odds))) -> c17(A__TAIL(a__tail(a__odds)), MARK(tail(odds))) MARK(tail(tail(head(z0)))) -> c17(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c17(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c17(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c17(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c17(MARK(tail(x0))) MARK(tail(nats)) -> c17(A__TAIL(cons(0, incr(nats)))) MARK(tail(pairs)) -> c17(A__TAIL(cons(0, incr(odds)))) A__INCR(cons(cons(y0, y1), z1)) -> c6(MARK(cons(y0, y1))) A__INCR(cons(s(y0), z1)) -> c6(MARK(s(y0))) A__INCR(cons(incr(cons(y0, y1)), z1)) -> c6(MARK(incr(cons(y0, y1)))) A__INCR(cons(incr(s(y0)), z1)) -> c6(MARK(incr(s(y0)))) A__INCR(cons(head(cons(y0, y1)), z1)) -> c6(MARK(head(cons(y0, y1)))) A__INCR(cons(head(s(y0)), z1)) -> c6(MARK(head(s(y0)))) A__INCR(cons(tail(cons(y0, y1)), z1)) -> c6(MARK(tail(cons(y0, y1)))) A__INCR(cons(tail(s(y0)), z1)) -> c6(MARK(tail(s(y0)))) A__INCR(cons(incr(incr(nats)), z1)) -> c6(MARK(incr(incr(nats)))) A__INCR(cons(incr(incr(incr(y0))), z1)) -> c6(MARK(incr(incr(incr(y0))))) A__INCR(cons(incr(incr(pairs)), z1)) -> c6(MARK(incr(incr(pairs)))) A__INCR(cons(incr(incr(odds)), z1)) -> c6(MARK(incr(incr(odds)))) A__INCR(cons(incr(incr(head(y0))), z1)) -> c6(MARK(incr(incr(head(y0))))) A__INCR(cons(incr(incr(tail(y0))), z1)) -> c6(MARK(incr(incr(tail(y0))))) A__INCR(cons(incr(incr(cons(y0, y1))), z1)) -> c6(MARK(incr(incr(cons(y0, y1))))) A__INCR(cons(incr(incr(s(y0))), z1)) -> c6(MARK(incr(incr(s(y0))))) A__INCR(cons(incr(incr(y0)), z1)) -> c6(MARK(incr(incr(y0)))) A__INCR(cons(incr(odds), z1)) -> c6(MARK(incr(odds))) A__INCR(cons(incr(head(nats)), z1)) -> c6(MARK(incr(head(nats)))) A__INCR(cons(incr(head(incr(y0))), z1)) -> c6(MARK(incr(head(incr(y0))))) A__INCR(cons(incr(head(pairs)), z1)) -> c6(MARK(incr(head(pairs)))) A__INCR(cons(incr(head(odds)), z1)) -> c6(MARK(incr(head(odds)))) A__INCR(cons(incr(head(head(y0))), z1)) -> c6(MARK(incr(head(head(y0))))) A__INCR(cons(incr(head(tail(y0))), z1)) -> c6(MARK(incr(head(tail(y0))))) A__INCR(cons(incr(head(cons(y0, y1))), z1)) -> c6(MARK(incr(head(cons(y0, y1))))) A__INCR(cons(incr(head(s(y0))), z1)) -> c6(MARK(incr(head(s(y0))))) A__INCR(cons(incr(head(y0)), z1)) -> c6(MARK(incr(head(y0)))) A__INCR(cons(incr(tail(nats)), z1)) -> c6(MARK(incr(tail(nats)))) A__INCR(cons(incr(tail(incr(y0))), z1)) -> c6(MARK(incr(tail(incr(y0))))) A__INCR(cons(incr(tail(pairs)), z1)) -> c6(MARK(incr(tail(pairs)))) A__INCR(cons(incr(tail(odds)), z1)) -> c6(MARK(incr(tail(odds)))) A__INCR(cons(incr(tail(head(y0))), z1)) -> c6(MARK(incr(tail(head(y0))))) A__INCR(cons(incr(tail(tail(y0))), z1)) -> c6(MARK(incr(tail(tail(y0))))) A__INCR(cons(incr(tail(cons(y0, y1))), z1)) -> c6(MARK(incr(tail(cons(y0, y1))))) A__INCR(cons(incr(tail(s(y0))), z1)) -> c6(MARK(incr(tail(s(y0))))) A__INCR(cons(incr(tail(y0)), z1)) -> c6(MARK(incr(tail(y0)))) A__INCR(cons(head(incr(nats)), z1)) -> c6(MARK(head(incr(nats)))) A__INCR(cons(head(incr(incr(y0))), z1)) -> c6(MARK(head(incr(incr(y0))))) A__INCR(cons(head(incr(pairs)), z1)) -> c6(MARK(head(incr(pairs)))) A__INCR(cons(head(incr(odds)), z1)) -> c6(MARK(head(incr(odds)))) A__INCR(cons(head(incr(head(y0))), z1)) -> c6(MARK(head(incr(head(y0))))) A__INCR(cons(head(incr(tail(y0))), z1)) -> c6(MARK(head(incr(tail(y0))))) A__INCR(cons(head(incr(cons(y0, y1))), z1)) -> c6(MARK(head(incr(cons(y0, y1))))) A__INCR(cons(head(incr(s(y0))), z1)) -> c6(MARK(head(incr(s(y0))))) A__INCR(cons(head(incr(y0)), z1)) -> c6(MARK(head(incr(y0)))) A__INCR(cons(head(odds), z1)) -> c6(MARK(head(odds))) A__INCR(cons(head(head(nats)), z1)) -> c6(MARK(head(head(nats)))) A__INCR(cons(head(head(incr(y0))), z1)) -> c6(MARK(head(head(incr(y0))))) A__INCR(cons(head(head(pairs)), z1)) -> c6(MARK(head(head(pairs)))) A__INCR(cons(head(head(odds)), z1)) -> c6(MARK(head(head(odds)))) A__INCR(cons(head(head(head(y0))), z1)) -> c6(MARK(head(head(head(y0))))) A__INCR(cons(head(head(tail(y0))), z1)) -> c6(MARK(head(head(tail(y0))))) A__INCR(cons(head(head(cons(y0, y1))), z1)) -> c6(MARK(head(head(cons(y0, y1))))) A__INCR(cons(head(head(s(y0))), z1)) -> c6(MARK(head(head(s(y0))))) A__INCR(cons(head(head(y0)), z1)) -> c6(MARK(head(head(y0)))) A__INCR(cons(head(tail(nats)), z1)) -> c6(MARK(head(tail(nats)))) A__INCR(cons(head(tail(incr(y0))), z1)) -> c6(MARK(head(tail(incr(y0))))) A__INCR(cons(head(tail(pairs)), z1)) -> c6(MARK(head(tail(pairs)))) A__INCR(cons(head(tail(odds)), z1)) -> c6(MARK(head(tail(odds)))) A__INCR(cons(head(tail(head(y0))), z1)) -> c6(MARK(head(tail(head(y0))))) A__INCR(cons(head(tail(tail(y0))), z1)) -> c6(MARK(head(tail(tail(y0))))) A__INCR(cons(head(tail(cons(y0, y1))), z1)) -> c6(MARK(head(tail(cons(y0, y1))))) A__INCR(cons(head(tail(s(y0))), z1)) -> c6(MARK(head(tail(s(y0))))) A__INCR(cons(head(tail(y0)), z1)) -> c6(MARK(head(tail(y0)))) A__INCR(cons(head(nats), z1)) -> c6(MARK(head(nats))) A__INCR(cons(head(pairs), z1)) -> c6(MARK(head(pairs))) A__INCR(cons(tail(incr(nats)), z1)) -> c6(MARK(tail(incr(nats)))) A__INCR(cons(tail(incr(incr(y0))), z1)) -> c6(MARK(tail(incr(incr(y0))))) A__INCR(cons(tail(incr(pairs)), z1)) -> c6(MARK(tail(incr(pairs)))) A__INCR(cons(tail(incr(odds)), z1)) -> c6(MARK(tail(incr(odds)))) A__INCR(cons(tail(incr(head(y0))), z1)) -> c6(MARK(tail(incr(head(y0))))) A__INCR(cons(tail(incr(tail(y0))), z1)) -> c6(MARK(tail(incr(tail(y0))))) A__INCR(cons(tail(incr(cons(y0, y1))), z1)) -> c6(MARK(tail(incr(cons(y0, y1))))) A__INCR(cons(tail(incr(s(y0))), z1)) -> c6(MARK(tail(incr(s(y0))))) A__INCR(cons(tail(incr(y0)), z1)) -> c6(MARK(tail(incr(y0)))) A__INCR(cons(tail(odds), z1)) -> c6(MARK(tail(odds))) A__INCR(cons(tail(head(nats)), z1)) -> c6(MARK(tail(head(nats)))) A__INCR(cons(tail(head(incr(y0))), z1)) -> c6(MARK(tail(head(incr(y0))))) A__INCR(cons(tail(head(pairs)), z1)) -> c6(MARK(tail(head(pairs)))) A__INCR(cons(tail(head(odds)), z1)) -> c6(MARK(tail(head(odds)))) A__INCR(cons(tail(head(head(y0))), z1)) -> c6(MARK(tail(head(head(y0))))) A__INCR(cons(tail(head(tail(y0))), z1)) -> c6(MARK(tail(head(tail(y0))))) A__INCR(cons(tail(head(cons(y0, y1))), z1)) -> c6(MARK(tail(head(cons(y0, y1))))) A__INCR(cons(tail(head(s(y0))), z1)) -> c6(MARK(tail(head(s(y0))))) A__INCR(cons(tail(head(y0)), z1)) -> c6(MARK(tail(head(y0)))) A__INCR(cons(tail(tail(nats)), z1)) -> c6(MARK(tail(tail(nats)))) A__INCR(cons(tail(tail(incr(y0))), z1)) -> c6(MARK(tail(tail(incr(y0))))) A__INCR(cons(tail(tail(pairs)), z1)) -> c6(MARK(tail(tail(pairs)))) A__INCR(cons(tail(tail(odds)), z1)) -> c6(MARK(tail(tail(odds)))) A__INCR(cons(tail(tail(head(y0))), z1)) -> c6(MARK(tail(tail(head(y0))))) A__INCR(cons(tail(tail(tail(y0))), z1)) -> c6(MARK(tail(tail(tail(y0))))) A__INCR(cons(tail(tail(cons(y0, y1))), z1)) -> c6(MARK(tail(tail(cons(y0, y1))))) A__INCR(cons(tail(tail(s(y0))), z1)) -> c6(MARK(tail(tail(s(y0))))) A__INCR(cons(tail(tail(y0)), z1)) -> c6(MARK(tail(tail(y0)))) A__INCR(cons(tail(nats), z1)) -> c6(MARK(tail(nats))) A__INCR(cons(tail(pairs), z1)) -> c6(MARK(tail(pairs))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs))) MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs))) MARK(tail(incr(nats))) -> c17(A__TAIL(a__incr(a__nats))) MARK(tail(incr(pairs))) -> c17(A__TAIL(a__incr(a__pairs))) MARK(tail(odds)) -> c17(A__TAIL(a__incr(a__pairs))) K tuples:none Defined Rule Symbols: a__nats, a__pairs, a__odds, a__incr_1, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__HEAD_1, A__TAIL_1, MARK_1, A__INCR_1 Compound Symbols: c8_1, c10_1, c18_1, c20_1, c13_2, c13_1, c16_2, c16_1, c17_2, c17_1, c6_1 ---------------------------------------- (165) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(incr(cons(z0, z1))) -> c13(A__INCR(cons(mark(z0), z1)), MARK(cons(z0, z1))) by MARK(incr(cons(nats, x1))) -> c13(A__INCR(cons(a__nats, x1)), MARK(cons(nats, x1))) MARK(incr(cons(incr(z0), x1))) -> c13(A__INCR(cons(a__incr(mark(z0)), x1)), MARK(cons(incr(z0), x1))) MARK(incr(cons(pairs, x1))) -> c13(A__INCR(cons(a__pairs, x1)), MARK(cons(pairs, x1))) MARK(incr(cons(odds, x1))) -> c13(A__INCR(cons(a__odds, x1)), MARK(cons(odds, x1))) MARK(incr(cons(head(z0), x1))) -> c13(A__INCR(cons(a__head(mark(z0)), x1)), MARK(cons(head(z0), x1))) MARK(incr(cons(tail(z0), x1))) -> c13(A__INCR(cons(a__tail(mark(z0)), x1)), MARK(cons(tail(z0), x1))) MARK(incr(cons(cons(z0, z1), x1))) -> c13(A__INCR(cons(cons(mark(z0), z1), x1)), MARK(cons(cons(z0, z1), x1))) MARK(incr(cons(0, x1))) -> c13(A__INCR(cons(0, x1)), MARK(cons(0, x1))) MARK(incr(cons(s(z0), x1))) -> c13(A__INCR(cons(s(mark(z0)), x1)), MARK(cons(s(z0), x1))) MARK(incr(cons(x0, x1))) -> c13(MARK(cons(x0, x1))) ---------------------------------------- (166) Obligation: Complexity Dependency Tuples Problem Rules: a__nats -> cons(0, incr(nats)) a__nats -> nats a__pairs -> cons(0, incr(odds)) a__pairs -> pairs a__odds -> a__incr(a__pairs) a__odds -> odds a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) 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(nats) -> a__nats mark(incr(z0)) -> a__incr(mark(z0)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c17(MARK(s(z0))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(head(nats))) -> c16(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(pairs))) -> c16(A__HEAD(a__head(a__pairs)), MARK(head(pairs))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(head(nats)) -> c16(A__HEAD(cons(0, incr(nats)))) MARK(head(pairs)) -> c16(A__HEAD(cons(0, incr(odds)))) MARK(tail(incr(incr(z0)))) -> c17(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(odds))) -> c17(A__TAIL(a__incr(a__odds)), MARK(incr(odds))) MARK(tail(incr(head(z0)))) -> c17(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c17(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c17(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c17(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c17(MARK(incr(x0))) MARK(tail(head(nats))) -> c17(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(incr(z0)))) -> c17(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(pairs))) -> c17(A__TAIL(a__head(a__pairs)), MARK(head(pairs))) MARK(tail(head(odds))) -> c17(A__TAIL(a__head(a__odds)), MARK(head(odds))) MARK(tail(head(head(z0)))) -> c17(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c17(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c17(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c17(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c17(MARK(head(x0))) MARK(tail(tail(nats))) -> c17(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(incr(z0)))) -> c17(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(pairs))) -> c17(A__TAIL(a__tail(a__pairs)), MARK(tail(pairs))) MARK(tail(tail(odds))) -> c17(A__TAIL(a__tail(a__odds)), MARK(tail(odds))) MARK(tail(tail(head(z0)))) -> c17(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c17(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c17(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c17(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c17(MARK(tail(x0))) MARK(tail(nats)) -> c17(A__TAIL(cons(0, incr(nats)))) MARK(tail(pairs)) -> c17(A__TAIL(cons(0, incr(odds)))) A__INCR(cons(cons(y0, y1), z1)) -> c6(MARK(cons(y0, y1))) A__INCR(cons(s(y0), z1)) -> c6(MARK(s(y0))) A__INCR(cons(incr(cons(y0, y1)), z1)) -> c6(MARK(incr(cons(y0, y1)))) A__INCR(cons(incr(s(y0)), z1)) -> c6(MARK(incr(s(y0)))) A__INCR(cons(head(cons(y0, y1)), z1)) -> c6(MARK(head(cons(y0, y1)))) A__INCR(cons(head(s(y0)), z1)) -> c6(MARK(head(s(y0)))) A__INCR(cons(tail(cons(y0, y1)), z1)) -> c6(MARK(tail(cons(y0, y1)))) A__INCR(cons(tail(s(y0)), z1)) -> c6(MARK(tail(s(y0)))) A__INCR(cons(incr(incr(nats)), z1)) -> c6(MARK(incr(incr(nats)))) A__INCR(cons(incr(incr(incr(y0))), z1)) -> c6(MARK(incr(incr(incr(y0))))) A__INCR(cons(incr(incr(pairs)), z1)) -> c6(MARK(incr(incr(pairs)))) A__INCR(cons(incr(incr(odds)), z1)) -> c6(MARK(incr(incr(odds)))) A__INCR(cons(incr(incr(head(y0))), z1)) -> c6(MARK(incr(incr(head(y0))))) A__INCR(cons(incr(incr(tail(y0))), z1)) -> c6(MARK(incr(incr(tail(y0))))) A__INCR(cons(incr(incr(cons(y0, y1))), z1)) -> c6(MARK(incr(incr(cons(y0, y1))))) A__INCR(cons(incr(incr(s(y0))), z1)) -> c6(MARK(incr(incr(s(y0))))) A__INCR(cons(incr(incr(y0)), z1)) -> c6(MARK(incr(incr(y0)))) A__INCR(cons(incr(odds), z1)) -> c6(MARK(incr(odds))) A__INCR(cons(incr(head(nats)), z1)) -> c6(MARK(incr(head(nats)))) A__INCR(cons(incr(head(incr(y0))), z1)) -> c6(MARK(incr(head(incr(y0))))) A__INCR(cons(incr(head(pairs)), z1)) -> c6(MARK(incr(head(pairs)))) A__INCR(cons(incr(head(odds)), z1)) -> c6(MARK(incr(head(odds)))) A__INCR(cons(incr(head(head(y0))), z1)) -> c6(MARK(incr(head(head(y0))))) A__INCR(cons(incr(head(tail(y0))), z1)) -> c6(MARK(incr(head(tail(y0))))) A__INCR(cons(incr(head(cons(y0, y1))), z1)) -> c6(MARK(incr(head(cons(y0, y1))))) A__INCR(cons(incr(head(s(y0))), z1)) -> c6(MARK(incr(head(s(y0))))) A__INCR(cons(incr(head(y0)), z1)) -> c6(MARK(incr(head(y0)))) A__INCR(cons(incr(tail(nats)), z1)) -> c6(MARK(incr(tail(nats)))) A__INCR(cons(incr(tail(incr(y0))), z1)) -> c6(MARK(incr(tail(incr(y0))))) A__INCR(cons(incr(tail(pairs)), z1)) -> c6(MARK(incr(tail(pairs)))) A__INCR(cons(incr(tail(odds)), z1)) -> c6(MARK(incr(tail(odds)))) A__INCR(cons(incr(tail(head(y0))), z1)) -> c6(MARK(incr(tail(head(y0))))) A__INCR(cons(incr(tail(tail(y0))), z1)) -> c6(MARK(incr(tail(tail(y0))))) A__INCR(cons(incr(tail(cons(y0, y1))), z1)) -> c6(MARK(incr(tail(cons(y0, y1))))) A__INCR(cons(incr(tail(s(y0))), z1)) -> c6(MARK(incr(tail(s(y0))))) A__INCR(cons(incr(tail(y0)), z1)) -> c6(MARK(incr(tail(y0)))) A__INCR(cons(head(incr(nats)), z1)) -> c6(MARK(head(incr(nats)))) A__INCR(cons(head(incr(incr(y0))), z1)) -> c6(MARK(head(incr(incr(y0))))) A__INCR(cons(head(incr(pairs)), z1)) -> c6(MARK(head(incr(pairs)))) A__INCR(cons(head(incr(odds)), z1)) -> c6(MARK(head(incr(odds)))) A__INCR(cons(head(incr(head(y0))), z1)) -> c6(MARK(head(incr(head(y0))))) A__INCR(cons(head(incr(tail(y0))), z1)) -> c6(MARK(head(incr(tail(y0))))) A__INCR(cons(head(incr(cons(y0, y1))), z1)) -> c6(MARK(head(incr(cons(y0, y1))))) A__INCR(cons(head(incr(s(y0))), z1)) -> c6(MARK(head(incr(s(y0))))) A__INCR(cons(head(incr(y0)), z1)) -> c6(MARK(head(incr(y0)))) A__INCR(cons(head(odds), z1)) -> c6(MARK(head(odds))) A__INCR(cons(head(head(nats)), z1)) -> c6(MARK(head(head(nats)))) A__INCR(cons(head(head(incr(y0))), z1)) -> c6(MARK(head(head(incr(y0))))) A__INCR(cons(head(head(pairs)), z1)) -> c6(MARK(head(head(pairs)))) A__INCR(cons(head(head(odds)), z1)) -> c6(MARK(head(head(odds)))) A__INCR(cons(head(head(head(y0))), z1)) -> c6(MARK(head(head(head(y0))))) A__INCR(cons(head(head(tail(y0))), z1)) -> c6(MARK(head(head(tail(y0))))) A__INCR(cons(head(head(cons(y0, y1))), z1)) -> c6(MARK(head(head(cons(y0, y1))))) A__INCR(cons(head(head(s(y0))), z1)) -> c6(MARK(head(head(s(y0))))) A__INCR(cons(head(head(y0)), z1)) -> c6(MARK(head(head(y0)))) A__INCR(cons(head(tail(nats)), z1)) -> c6(MARK(head(tail(nats)))) A__INCR(cons(head(tail(incr(y0))), z1)) -> c6(MARK(head(tail(incr(y0))))) A__INCR(cons(head(tail(pairs)), z1)) -> c6(MARK(head(tail(pairs)))) A__INCR(cons(head(tail(odds)), z1)) -> c6(MARK(head(tail(odds)))) A__INCR(cons(head(tail(head(y0))), z1)) -> c6(MARK(head(tail(head(y0))))) A__INCR(cons(head(tail(tail(y0))), z1)) -> c6(MARK(head(tail(tail(y0))))) A__INCR(cons(head(tail(cons(y0, y1))), z1)) -> c6(MARK(head(tail(cons(y0, y1))))) A__INCR(cons(head(tail(s(y0))), z1)) -> c6(MARK(head(tail(s(y0))))) A__INCR(cons(head(tail(y0)), z1)) -> c6(MARK(head(tail(y0)))) A__INCR(cons(head(nats), z1)) -> c6(MARK(head(nats))) A__INCR(cons(head(pairs), z1)) -> c6(MARK(head(pairs))) A__INCR(cons(tail(incr(nats)), z1)) -> c6(MARK(tail(incr(nats)))) A__INCR(cons(tail(incr(incr(y0))), z1)) -> c6(MARK(tail(incr(incr(y0))))) A__INCR(cons(tail(incr(pairs)), z1)) -> c6(MARK(tail(incr(pairs)))) A__INCR(cons(tail(incr(odds)), z1)) -> c6(MARK(tail(incr(odds)))) A__INCR(cons(tail(incr(head(y0))), z1)) -> c6(MARK(tail(incr(head(y0))))) A__INCR(cons(tail(incr(tail(y0))), z1)) -> c6(MARK(tail(incr(tail(y0))))) A__INCR(cons(tail(incr(cons(y0, y1))), z1)) -> c6(MARK(tail(incr(cons(y0, y1))))) A__INCR(cons(tail(incr(s(y0))), z1)) -> c6(MARK(tail(incr(s(y0))))) A__INCR(cons(tail(incr(y0)), z1)) -> c6(MARK(tail(incr(y0)))) A__INCR(cons(tail(odds), z1)) -> c6(MARK(tail(odds))) A__INCR(cons(tail(head(nats)), z1)) -> c6(MARK(tail(head(nats)))) A__INCR(cons(tail(head(incr(y0))), z1)) -> c6(MARK(tail(head(incr(y0))))) A__INCR(cons(tail(head(pairs)), z1)) -> c6(MARK(tail(head(pairs)))) A__INCR(cons(tail(head(odds)), z1)) -> c6(MARK(tail(head(odds)))) A__INCR(cons(tail(head(head(y0))), z1)) -> c6(MARK(tail(head(head(y0))))) A__INCR(cons(tail(head(tail(y0))), z1)) -> c6(MARK(tail(head(tail(y0))))) A__INCR(cons(tail(head(cons(y0, y1))), z1)) -> c6(MARK(tail(head(cons(y0, y1))))) A__INCR(cons(tail(head(s(y0))), z1)) -> c6(MARK(tail(head(s(y0))))) A__INCR(cons(tail(head(y0)), z1)) -> c6(MARK(tail(head(y0)))) A__INCR(cons(tail(tail(nats)), z1)) -> c6(MARK(tail(tail(nats)))) A__INCR(cons(tail(tail(incr(y0))), z1)) -> c6(MARK(tail(tail(incr(y0))))) A__INCR(cons(tail(tail(pairs)), z1)) -> c6(MARK(tail(tail(pairs)))) A__INCR(cons(tail(tail(odds)), z1)) -> c6(MARK(tail(tail(odds)))) A__INCR(cons(tail(tail(head(y0))), z1)) -> c6(MARK(tail(tail(head(y0))))) A__INCR(cons(tail(tail(tail(y0))), z1)) -> c6(MARK(tail(tail(tail(y0))))) A__INCR(cons(tail(tail(cons(y0, y1))), z1)) -> c6(MARK(tail(tail(cons(y0, y1))))) A__INCR(cons(tail(tail(s(y0))), z1)) -> c6(MARK(tail(tail(s(y0))))) A__INCR(cons(tail(tail(y0)), z1)) -> c6(MARK(tail(tail(y0)))) A__INCR(cons(tail(nats), z1)) -> c6(MARK(tail(nats))) A__INCR(cons(tail(pairs), z1)) -> c6(MARK(tail(pairs))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs))) MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs))) MARK(tail(incr(nats))) -> c17(A__TAIL(a__incr(a__nats))) MARK(tail(incr(pairs))) -> c17(A__TAIL(a__incr(a__pairs))) MARK(tail(odds)) -> c17(A__TAIL(a__incr(a__pairs))) MARK(incr(cons(nats, x1))) -> c13(A__INCR(cons(a__nats, x1)), MARK(cons(nats, x1))) MARK(incr(cons(incr(z0), x1))) -> c13(A__INCR(cons(a__incr(mark(z0)), x1)), MARK(cons(incr(z0), x1))) MARK(incr(cons(pairs, x1))) -> c13(A__INCR(cons(a__pairs, x1)), MARK(cons(pairs, x1))) MARK(incr(cons(odds, x1))) -> c13(A__INCR(cons(a__odds, x1)), MARK(cons(odds, x1))) MARK(incr(cons(head(z0), x1))) -> c13(A__INCR(cons(a__head(mark(z0)), x1)), MARK(cons(head(z0), x1))) MARK(incr(cons(tail(z0), x1))) -> c13(A__INCR(cons(a__tail(mark(z0)), x1)), MARK(cons(tail(z0), x1))) MARK(incr(cons(cons(z0, z1), x1))) -> c13(A__INCR(cons(cons(mark(z0), z1), x1)), MARK(cons(cons(z0, z1), x1))) MARK(incr(cons(0, x1))) -> c13(A__INCR(cons(0, x1)), MARK(cons(0, x1))) MARK(incr(cons(s(z0), x1))) -> c13(A__INCR(cons(s(mark(z0)), x1)), MARK(cons(s(z0), x1))) MARK(incr(cons(x0, x1))) -> c13(MARK(cons(x0, x1))) S tuples: A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c17(MARK(s(z0))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(head(nats))) -> c16(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(pairs))) -> c16(A__HEAD(a__head(a__pairs)), MARK(head(pairs))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(head(nats)) -> c16(A__HEAD(cons(0, incr(nats)))) MARK(head(pairs)) -> c16(A__HEAD(cons(0, incr(odds)))) MARK(tail(incr(incr(z0)))) -> c17(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(odds))) -> c17(A__TAIL(a__incr(a__odds)), MARK(incr(odds))) MARK(tail(incr(head(z0)))) -> c17(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c17(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c17(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c17(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c17(MARK(incr(x0))) MARK(tail(head(nats))) -> c17(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(incr(z0)))) -> c17(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(pairs))) -> c17(A__TAIL(a__head(a__pairs)), MARK(head(pairs))) MARK(tail(head(odds))) -> c17(A__TAIL(a__head(a__odds)), MARK(head(odds))) MARK(tail(head(head(z0)))) -> c17(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c17(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c17(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c17(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c17(MARK(head(x0))) MARK(tail(tail(nats))) -> c17(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(incr(z0)))) -> c17(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(pairs))) -> c17(A__TAIL(a__tail(a__pairs)), MARK(tail(pairs))) MARK(tail(tail(odds))) -> c17(A__TAIL(a__tail(a__odds)), MARK(tail(odds))) MARK(tail(tail(head(z0)))) -> c17(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c17(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c17(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c17(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c17(MARK(tail(x0))) MARK(tail(nats)) -> c17(A__TAIL(cons(0, incr(nats)))) MARK(tail(pairs)) -> c17(A__TAIL(cons(0, incr(odds)))) A__INCR(cons(cons(y0, y1), z1)) -> c6(MARK(cons(y0, y1))) A__INCR(cons(s(y0), z1)) -> c6(MARK(s(y0))) A__INCR(cons(incr(cons(y0, y1)), z1)) -> c6(MARK(incr(cons(y0, y1)))) A__INCR(cons(incr(s(y0)), z1)) -> c6(MARK(incr(s(y0)))) A__INCR(cons(head(cons(y0, y1)), z1)) -> c6(MARK(head(cons(y0, y1)))) A__INCR(cons(head(s(y0)), z1)) -> c6(MARK(head(s(y0)))) A__INCR(cons(tail(cons(y0, y1)), z1)) -> c6(MARK(tail(cons(y0, y1)))) A__INCR(cons(tail(s(y0)), z1)) -> c6(MARK(tail(s(y0)))) A__INCR(cons(incr(incr(nats)), z1)) -> c6(MARK(incr(incr(nats)))) A__INCR(cons(incr(incr(incr(y0))), z1)) -> c6(MARK(incr(incr(incr(y0))))) A__INCR(cons(incr(incr(pairs)), z1)) -> c6(MARK(incr(incr(pairs)))) A__INCR(cons(incr(incr(odds)), z1)) -> c6(MARK(incr(incr(odds)))) A__INCR(cons(incr(incr(head(y0))), z1)) -> c6(MARK(incr(incr(head(y0))))) A__INCR(cons(incr(incr(tail(y0))), z1)) -> c6(MARK(incr(incr(tail(y0))))) A__INCR(cons(incr(incr(cons(y0, y1))), z1)) -> c6(MARK(incr(incr(cons(y0, y1))))) A__INCR(cons(incr(incr(s(y0))), z1)) -> c6(MARK(incr(incr(s(y0))))) A__INCR(cons(incr(incr(y0)), z1)) -> c6(MARK(incr(incr(y0)))) A__INCR(cons(incr(odds), z1)) -> c6(MARK(incr(odds))) A__INCR(cons(incr(head(nats)), z1)) -> c6(MARK(incr(head(nats)))) A__INCR(cons(incr(head(incr(y0))), z1)) -> c6(MARK(incr(head(incr(y0))))) A__INCR(cons(incr(head(pairs)), z1)) -> c6(MARK(incr(head(pairs)))) A__INCR(cons(incr(head(odds)), z1)) -> c6(MARK(incr(head(odds)))) A__INCR(cons(incr(head(head(y0))), z1)) -> c6(MARK(incr(head(head(y0))))) A__INCR(cons(incr(head(tail(y0))), z1)) -> c6(MARK(incr(head(tail(y0))))) A__INCR(cons(incr(head(cons(y0, y1))), z1)) -> c6(MARK(incr(head(cons(y0, y1))))) A__INCR(cons(incr(head(s(y0))), z1)) -> c6(MARK(incr(head(s(y0))))) A__INCR(cons(incr(head(y0)), z1)) -> c6(MARK(incr(head(y0)))) A__INCR(cons(incr(tail(nats)), z1)) -> c6(MARK(incr(tail(nats)))) A__INCR(cons(incr(tail(incr(y0))), z1)) -> c6(MARK(incr(tail(incr(y0))))) A__INCR(cons(incr(tail(pairs)), z1)) -> c6(MARK(incr(tail(pairs)))) A__INCR(cons(incr(tail(odds)), z1)) -> c6(MARK(incr(tail(odds)))) A__INCR(cons(incr(tail(head(y0))), z1)) -> c6(MARK(incr(tail(head(y0))))) A__INCR(cons(incr(tail(tail(y0))), z1)) -> c6(MARK(incr(tail(tail(y0))))) A__INCR(cons(incr(tail(cons(y0, y1))), z1)) -> c6(MARK(incr(tail(cons(y0, y1))))) A__INCR(cons(incr(tail(s(y0))), z1)) -> c6(MARK(incr(tail(s(y0))))) A__INCR(cons(incr(tail(y0)), z1)) -> c6(MARK(incr(tail(y0)))) A__INCR(cons(head(incr(nats)), z1)) -> c6(MARK(head(incr(nats)))) A__INCR(cons(head(incr(incr(y0))), z1)) -> c6(MARK(head(incr(incr(y0))))) A__INCR(cons(head(incr(pairs)), z1)) -> c6(MARK(head(incr(pairs)))) A__INCR(cons(head(incr(odds)), z1)) -> c6(MARK(head(incr(odds)))) A__INCR(cons(head(incr(head(y0))), z1)) -> c6(MARK(head(incr(head(y0))))) A__INCR(cons(head(incr(tail(y0))), z1)) -> c6(MARK(head(incr(tail(y0))))) A__INCR(cons(head(incr(cons(y0, y1))), z1)) -> c6(MARK(head(incr(cons(y0, y1))))) A__INCR(cons(head(incr(s(y0))), z1)) -> c6(MARK(head(incr(s(y0))))) A__INCR(cons(head(incr(y0)), z1)) -> c6(MARK(head(incr(y0)))) A__INCR(cons(head(odds), z1)) -> c6(MARK(head(odds))) A__INCR(cons(head(head(nats)), z1)) -> c6(MARK(head(head(nats)))) A__INCR(cons(head(head(incr(y0))), z1)) -> c6(MARK(head(head(incr(y0))))) A__INCR(cons(head(head(pairs)), z1)) -> c6(MARK(head(head(pairs)))) A__INCR(cons(head(head(odds)), z1)) -> c6(MARK(head(head(odds)))) A__INCR(cons(head(head(head(y0))), z1)) -> c6(MARK(head(head(head(y0))))) A__INCR(cons(head(head(tail(y0))), z1)) -> c6(MARK(head(head(tail(y0))))) A__INCR(cons(head(head(cons(y0, y1))), z1)) -> c6(MARK(head(head(cons(y0, y1))))) A__INCR(cons(head(head(s(y0))), z1)) -> c6(MARK(head(head(s(y0))))) A__INCR(cons(head(head(y0)), z1)) -> c6(MARK(head(head(y0)))) A__INCR(cons(head(tail(nats)), z1)) -> c6(MARK(head(tail(nats)))) A__INCR(cons(head(tail(incr(y0))), z1)) -> c6(MARK(head(tail(incr(y0))))) A__INCR(cons(head(tail(pairs)), z1)) -> c6(MARK(head(tail(pairs)))) A__INCR(cons(head(tail(odds)), z1)) -> c6(MARK(head(tail(odds)))) A__INCR(cons(head(tail(head(y0))), z1)) -> c6(MARK(head(tail(head(y0))))) A__INCR(cons(head(tail(tail(y0))), z1)) -> c6(MARK(head(tail(tail(y0))))) A__INCR(cons(head(tail(cons(y0, y1))), z1)) -> c6(MARK(head(tail(cons(y0, y1))))) A__INCR(cons(head(tail(s(y0))), z1)) -> c6(MARK(head(tail(s(y0))))) A__INCR(cons(head(tail(y0)), z1)) -> c6(MARK(head(tail(y0)))) A__INCR(cons(head(nats), z1)) -> c6(MARK(head(nats))) A__INCR(cons(head(pairs), z1)) -> c6(MARK(head(pairs))) A__INCR(cons(tail(incr(nats)), z1)) -> c6(MARK(tail(incr(nats)))) A__INCR(cons(tail(incr(incr(y0))), z1)) -> c6(MARK(tail(incr(incr(y0))))) A__INCR(cons(tail(incr(pairs)), z1)) -> c6(MARK(tail(incr(pairs)))) A__INCR(cons(tail(incr(odds)), z1)) -> c6(MARK(tail(incr(odds)))) A__INCR(cons(tail(incr(head(y0))), z1)) -> c6(MARK(tail(incr(head(y0))))) A__INCR(cons(tail(incr(tail(y0))), z1)) -> c6(MARK(tail(incr(tail(y0))))) A__INCR(cons(tail(incr(cons(y0, y1))), z1)) -> c6(MARK(tail(incr(cons(y0, y1))))) A__INCR(cons(tail(incr(s(y0))), z1)) -> c6(MARK(tail(incr(s(y0))))) A__INCR(cons(tail(incr(y0)), z1)) -> c6(MARK(tail(incr(y0)))) A__INCR(cons(tail(odds), z1)) -> c6(MARK(tail(odds))) A__INCR(cons(tail(head(nats)), z1)) -> c6(MARK(tail(head(nats)))) A__INCR(cons(tail(head(incr(y0))), z1)) -> c6(MARK(tail(head(incr(y0))))) A__INCR(cons(tail(head(pairs)), z1)) -> c6(MARK(tail(head(pairs)))) A__INCR(cons(tail(head(odds)), z1)) -> c6(MARK(tail(head(odds)))) A__INCR(cons(tail(head(head(y0))), z1)) -> c6(MARK(tail(head(head(y0))))) A__INCR(cons(tail(head(tail(y0))), z1)) -> c6(MARK(tail(head(tail(y0))))) A__INCR(cons(tail(head(cons(y0, y1))), z1)) -> c6(MARK(tail(head(cons(y0, y1))))) A__INCR(cons(tail(head(s(y0))), z1)) -> c6(MARK(tail(head(s(y0))))) A__INCR(cons(tail(head(y0)), z1)) -> c6(MARK(tail(head(y0)))) A__INCR(cons(tail(tail(nats)), z1)) -> c6(MARK(tail(tail(nats)))) A__INCR(cons(tail(tail(incr(y0))), z1)) -> c6(MARK(tail(tail(incr(y0))))) A__INCR(cons(tail(tail(pairs)), z1)) -> c6(MARK(tail(tail(pairs)))) A__INCR(cons(tail(tail(odds)), z1)) -> c6(MARK(tail(tail(odds)))) A__INCR(cons(tail(tail(head(y0))), z1)) -> c6(MARK(tail(tail(head(y0))))) A__INCR(cons(tail(tail(tail(y0))), z1)) -> c6(MARK(tail(tail(tail(y0))))) A__INCR(cons(tail(tail(cons(y0, y1))), z1)) -> c6(MARK(tail(tail(cons(y0, y1))))) A__INCR(cons(tail(tail(s(y0))), z1)) -> c6(MARK(tail(tail(s(y0))))) A__INCR(cons(tail(tail(y0)), z1)) -> c6(MARK(tail(tail(y0)))) A__INCR(cons(tail(nats), z1)) -> c6(MARK(tail(nats))) A__INCR(cons(tail(pairs), z1)) -> c6(MARK(tail(pairs))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs))) MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs))) MARK(tail(incr(nats))) -> c17(A__TAIL(a__incr(a__nats))) MARK(tail(incr(pairs))) -> c17(A__TAIL(a__incr(a__pairs))) MARK(tail(odds)) -> c17(A__TAIL(a__incr(a__pairs))) MARK(incr(cons(nats, x1))) -> c13(A__INCR(cons(a__nats, x1)), MARK(cons(nats, x1))) MARK(incr(cons(incr(z0), x1))) -> c13(A__INCR(cons(a__incr(mark(z0)), x1)), MARK(cons(incr(z0), x1))) MARK(incr(cons(pairs, x1))) -> c13(A__INCR(cons(a__pairs, x1)), MARK(cons(pairs, x1))) MARK(incr(cons(odds, x1))) -> c13(A__INCR(cons(a__odds, x1)), MARK(cons(odds, x1))) MARK(incr(cons(head(z0), x1))) -> c13(A__INCR(cons(a__head(mark(z0)), x1)), MARK(cons(head(z0), x1))) MARK(incr(cons(tail(z0), x1))) -> c13(A__INCR(cons(a__tail(mark(z0)), x1)), MARK(cons(tail(z0), x1))) MARK(incr(cons(cons(z0, z1), x1))) -> c13(A__INCR(cons(cons(mark(z0), z1), x1)), MARK(cons(cons(z0, z1), x1))) MARK(incr(cons(0, x1))) -> c13(A__INCR(cons(0, x1)), MARK(cons(0, x1))) MARK(incr(cons(s(z0), x1))) -> c13(A__INCR(cons(s(mark(z0)), x1)), MARK(cons(s(z0), x1))) MARK(incr(cons(x0, x1))) -> c13(MARK(cons(x0, x1))) K tuples:none Defined Rule Symbols: a__nats, a__pairs, a__odds, a__incr_1, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__HEAD_1, A__TAIL_1, MARK_1, A__INCR_1 Compound Symbols: c8_1, c10_1, c18_1, c20_1, c13_1, c16_2, c16_1, c17_2, c17_1, c13_2, c6_1 ---------------------------------------- (167) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (168) Obligation: Complexity Dependency Tuples Problem Rules: a__nats -> cons(0, incr(nats)) a__nats -> nats a__pairs -> cons(0, incr(odds)) a__pairs -> pairs a__odds -> a__incr(a__pairs) a__odds -> odds a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) 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(nats) -> a__nats mark(incr(z0)) -> a__incr(mark(z0)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c17(MARK(s(z0))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(head(nats))) -> c16(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(pairs))) -> c16(A__HEAD(a__head(a__pairs)), MARK(head(pairs))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(head(nats)) -> c16(A__HEAD(cons(0, incr(nats)))) MARK(head(pairs)) -> c16(A__HEAD(cons(0, incr(odds)))) MARK(tail(incr(incr(z0)))) -> c17(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(odds))) -> c17(A__TAIL(a__incr(a__odds)), MARK(incr(odds))) MARK(tail(incr(head(z0)))) -> c17(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c17(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c17(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c17(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c17(MARK(incr(x0))) MARK(tail(head(nats))) -> c17(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(incr(z0)))) -> c17(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(pairs))) -> c17(A__TAIL(a__head(a__pairs)), MARK(head(pairs))) MARK(tail(head(odds))) -> c17(A__TAIL(a__head(a__odds)), MARK(head(odds))) MARK(tail(head(head(z0)))) -> c17(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c17(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c17(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c17(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c17(MARK(head(x0))) MARK(tail(tail(nats))) -> c17(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(incr(z0)))) -> c17(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(pairs))) -> c17(A__TAIL(a__tail(a__pairs)), MARK(tail(pairs))) MARK(tail(tail(odds))) -> c17(A__TAIL(a__tail(a__odds)), MARK(tail(odds))) MARK(tail(tail(head(z0)))) -> c17(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c17(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c17(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c17(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c17(MARK(tail(x0))) MARK(tail(nats)) -> c17(A__TAIL(cons(0, incr(nats)))) MARK(tail(pairs)) -> c17(A__TAIL(cons(0, incr(odds)))) A__INCR(cons(cons(y0, y1), z1)) -> c6(MARK(cons(y0, y1))) A__INCR(cons(s(y0), z1)) -> c6(MARK(s(y0))) A__INCR(cons(incr(cons(y0, y1)), z1)) -> c6(MARK(incr(cons(y0, y1)))) A__INCR(cons(incr(s(y0)), z1)) -> c6(MARK(incr(s(y0)))) A__INCR(cons(head(cons(y0, y1)), z1)) -> c6(MARK(head(cons(y0, y1)))) A__INCR(cons(head(s(y0)), z1)) -> c6(MARK(head(s(y0)))) A__INCR(cons(tail(cons(y0, y1)), z1)) -> c6(MARK(tail(cons(y0, y1)))) A__INCR(cons(tail(s(y0)), z1)) -> c6(MARK(tail(s(y0)))) A__INCR(cons(incr(incr(nats)), z1)) -> c6(MARK(incr(incr(nats)))) A__INCR(cons(incr(incr(incr(y0))), z1)) -> c6(MARK(incr(incr(incr(y0))))) A__INCR(cons(incr(incr(pairs)), z1)) -> c6(MARK(incr(incr(pairs)))) A__INCR(cons(incr(incr(odds)), z1)) -> c6(MARK(incr(incr(odds)))) A__INCR(cons(incr(incr(head(y0))), z1)) -> c6(MARK(incr(incr(head(y0))))) A__INCR(cons(incr(incr(tail(y0))), z1)) -> c6(MARK(incr(incr(tail(y0))))) A__INCR(cons(incr(incr(cons(y0, y1))), z1)) -> c6(MARK(incr(incr(cons(y0, y1))))) A__INCR(cons(incr(incr(s(y0))), z1)) -> c6(MARK(incr(incr(s(y0))))) A__INCR(cons(incr(incr(y0)), z1)) -> c6(MARK(incr(incr(y0)))) A__INCR(cons(incr(odds), z1)) -> c6(MARK(incr(odds))) A__INCR(cons(incr(head(nats)), z1)) -> c6(MARK(incr(head(nats)))) A__INCR(cons(incr(head(incr(y0))), z1)) -> c6(MARK(incr(head(incr(y0))))) A__INCR(cons(incr(head(pairs)), z1)) -> c6(MARK(incr(head(pairs)))) A__INCR(cons(incr(head(odds)), z1)) -> c6(MARK(incr(head(odds)))) A__INCR(cons(incr(head(head(y0))), z1)) -> c6(MARK(incr(head(head(y0))))) A__INCR(cons(incr(head(tail(y0))), z1)) -> c6(MARK(incr(head(tail(y0))))) A__INCR(cons(incr(head(cons(y0, y1))), z1)) -> c6(MARK(incr(head(cons(y0, y1))))) A__INCR(cons(incr(head(s(y0))), z1)) -> c6(MARK(incr(head(s(y0))))) A__INCR(cons(incr(head(y0)), z1)) -> c6(MARK(incr(head(y0)))) A__INCR(cons(incr(tail(nats)), z1)) -> c6(MARK(incr(tail(nats)))) A__INCR(cons(incr(tail(incr(y0))), z1)) -> c6(MARK(incr(tail(incr(y0))))) A__INCR(cons(incr(tail(pairs)), z1)) -> c6(MARK(incr(tail(pairs)))) A__INCR(cons(incr(tail(odds)), z1)) -> c6(MARK(incr(tail(odds)))) A__INCR(cons(incr(tail(head(y0))), z1)) -> c6(MARK(incr(tail(head(y0))))) A__INCR(cons(incr(tail(tail(y0))), z1)) -> c6(MARK(incr(tail(tail(y0))))) A__INCR(cons(incr(tail(cons(y0, y1))), z1)) -> c6(MARK(incr(tail(cons(y0, y1))))) A__INCR(cons(incr(tail(s(y0))), z1)) -> c6(MARK(incr(tail(s(y0))))) A__INCR(cons(incr(tail(y0)), z1)) -> c6(MARK(incr(tail(y0)))) A__INCR(cons(head(incr(nats)), z1)) -> c6(MARK(head(incr(nats)))) A__INCR(cons(head(incr(incr(y0))), z1)) -> c6(MARK(head(incr(incr(y0))))) A__INCR(cons(head(incr(pairs)), z1)) -> c6(MARK(head(incr(pairs)))) A__INCR(cons(head(incr(odds)), z1)) -> c6(MARK(head(incr(odds)))) A__INCR(cons(head(incr(head(y0))), z1)) -> c6(MARK(head(incr(head(y0))))) A__INCR(cons(head(incr(tail(y0))), z1)) -> c6(MARK(head(incr(tail(y0))))) A__INCR(cons(head(incr(cons(y0, y1))), z1)) -> c6(MARK(head(incr(cons(y0, y1))))) A__INCR(cons(head(incr(s(y0))), z1)) -> c6(MARK(head(incr(s(y0))))) A__INCR(cons(head(incr(y0)), z1)) -> c6(MARK(head(incr(y0)))) A__INCR(cons(head(odds), z1)) -> c6(MARK(head(odds))) A__INCR(cons(head(head(nats)), z1)) -> c6(MARK(head(head(nats)))) A__INCR(cons(head(head(incr(y0))), z1)) -> c6(MARK(head(head(incr(y0))))) A__INCR(cons(head(head(pairs)), z1)) -> c6(MARK(head(head(pairs)))) A__INCR(cons(head(head(odds)), z1)) -> c6(MARK(head(head(odds)))) A__INCR(cons(head(head(head(y0))), z1)) -> c6(MARK(head(head(head(y0))))) A__INCR(cons(head(head(tail(y0))), z1)) -> c6(MARK(head(head(tail(y0))))) A__INCR(cons(head(head(cons(y0, y1))), z1)) -> c6(MARK(head(head(cons(y0, y1))))) A__INCR(cons(head(head(s(y0))), z1)) -> c6(MARK(head(head(s(y0))))) A__INCR(cons(head(head(y0)), z1)) -> c6(MARK(head(head(y0)))) A__INCR(cons(head(tail(nats)), z1)) -> c6(MARK(head(tail(nats)))) A__INCR(cons(head(tail(incr(y0))), z1)) -> c6(MARK(head(tail(incr(y0))))) A__INCR(cons(head(tail(pairs)), z1)) -> c6(MARK(head(tail(pairs)))) A__INCR(cons(head(tail(odds)), z1)) -> c6(MARK(head(tail(odds)))) A__INCR(cons(head(tail(head(y0))), z1)) -> c6(MARK(head(tail(head(y0))))) A__INCR(cons(head(tail(tail(y0))), z1)) -> c6(MARK(head(tail(tail(y0))))) A__INCR(cons(head(tail(cons(y0, y1))), z1)) -> c6(MARK(head(tail(cons(y0, y1))))) A__INCR(cons(head(tail(s(y0))), z1)) -> c6(MARK(head(tail(s(y0))))) A__INCR(cons(head(tail(y0)), z1)) -> c6(MARK(head(tail(y0)))) A__INCR(cons(head(nats), z1)) -> c6(MARK(head(nats))) A__INCR(cons(head(pairs), z1)) -> c6(MARK(head(pairs))) A__INCR(cons(tail(incr(nats)), z1)) -> c6(MARK(tail(incr(nats)))) A__INCR(cons(tail(incr(incr(y0))), z1)) -> c6(MARK(tail(incr(incr(y0))))) A__INCR(cons(tail(incr(pairs)), z1)) -> c6(MARK(tail(incr(pairs)))) A__INCR(cons(tail(incr(odds)), z1)) -> c6(MARK(tail(incr(odds)))) A__INCR(cons(tail(incr(head(y0))), z1)) -> c6(MARK(tail(incr(head(y0))))) A__INCR(cons(tail(incr(tail(y0))), z1)) -> c6(MARK(tail(incr(tail(y0))))) A__INCR(cons(tail(incr(cons(y0, y1))), z1)) -> c6(MARK(tail(incr(cons(y0, y1))))) A__INCR(cons(tail(incr(s(y0))), z1)) -> c6(MARK(tail(incr(s(y0))))) A__INCR(cons(tail(incr(y0)), z1)) -> c6(MARK(tail(incr(y0)))) A__INCR(cons(tail(odds), z1)) -> c6(MARK(tail(odds))) A__INCR(cons(tail(head(nats)), z1)) -> c6(MARK(tail(head(nats)))) A__INCR(cons(tail(head(incr(y0))), z1)) -> c6(MARK(tail(head(incr(y0))))) A__INCR(cons(tail(head(pairs)), z1)) -> c6(MARK(tail(head(pairs)))) A__INCR(cons(tail(head(odds)), z1)) -> c6(MARK(tail(head(odds)))) A__INCR(cons(tail(head(head(y0))), z1)) -> c6(MARK(tail(head(head(y0))))) A__INCR(cons(tail(head(tail(y0))), z1)) -> c6(MARK(tail(head(tail(y0))))) A__INCR(cons(tail(head(cons(y0, y1))), z1)) -> c6(MARK(tail(head(cons(y0, y1))))) A__INCR(cons(tail(head(s(y0))), z1)) -> c6(MARK(tail(head(s(y0))))) A__INCR(cons(tail(head(y0)), z1)) -> c6(MARK(tail(head(y0)))) A__INCR(cons(tail(tail(nats)), z1)) -> c6(MARK(tail(tail(nats)))) A__INCR(cons(tail(tail(incr(y0))), z1)) -> c6(MARK(tail(tail(incr(y0))))) A__INCR(cons(tail(tail(pairs)), z1)) -> c6(MARK(tail(tail(pairs)))) A__INCR(cons(tail(tail(odds)), z1)) -> c6(MARK(tail(tail(odds)))) A__INCR(cons(tail(tail(head(y0))), z1)) -> c6(MARK(tail(tail(head(y0))))) A__INCR(cons(tail(tail(tail(y0))), z1)) -> c6(MARK(tail(tail(tail(y0))))) A__INCR(cons(tail(tail(cons(y0, y1))), z1)) -> c6(MARK(tail(tail(cons(y0, y1))))) A__INCR(cons(tail(tail(s(y0))), z1)) -> c6(MARK(tail(tail(s(y0))))) A__INCR(cons(tail(tail(y0)), z1)) -> c6(MARK(tail(tail(y0)))) A__INCR(cons(tail(nats), z1)) -> c6(MARK(tail(nats))) A__INCR(cons(tail(pairs), z1)) -> c6(MARK(tail(pairs))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs))) MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs))) MARK(tail(incr(nats))) -> c17(A__TAIL(a__incr(a__nats))) MARK(tail(incr(pairs))) -> c17(A__TAIL(a__incr(a__pairs))) MARK(tail(odds)) -> c17(A__TAIL(a__incr(a__pairs))) MARK(incr(cons(nats, x1))) -> c13(A__INCR(cons(a__nats, x1)), MARK(cons(nats, x1))) MARK(incr(cons(incr(z0), x1))) -> c13(A__INCR(cons(a__incr(mark(z0)), x1)), MARK(cons(incr(z0), x1))) MARK(incr(cons(pairs, x1))) -> c13(A__INCR(cons(a__pairs, x1)), MARK(cons(pairs, x1))) MARK(incr(cons(odds, x1))) -> c13(A__INCR(cons(a__odds, x1)), MARK(cons(odds, x1))) MARK(incr(cons(head(z0), x1))) -> c13(A__INCR(cons(a__head(mark(z0)), x1)), MARK(cons(head(z0), x1))) MARK(incr(cons(tail(z0), x1))) -> c13(A__INCR(cons(a__tail(mark(z0)), x1)), MARK(cons(tail(z0), x1))) MARK(incr(cons(cons(z0, z1), x1))) -> c13(A__INCR(cons(cons(mark(z0), z1), x1)), MARK(cons(cons(z0, z1), x1))) MARK(incr(cons(s(z0), x1))) -> c13(A__INCR(cons(s(mark(z0)), x1)), MARK(cons(s(z0), x1))) MARK(incr(cons(x0, x1))) -> c13(MARK(cons(x0, x1))) MARK(incr(cons(0, x1))) -> c13(MARK(cons(0, x1))) S tuples: A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c17(MARK(s(z0))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(head(nats))) -> c16(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(pairs))) -> c16(A__HEAD(a__head(a__pairs)), MARK(head(pairs))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(head(nats)) -> c16(A__HEAD(cons(0, incr(nats)))) MARK(head(pairs)) -> c16(A__HEAD(cons(0, incr(odds)))) MARK(tail(incr(incr(z0)))) -> c17(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(odds))) -> c17(A__TAIL(a__incr(a__odds)), MARK(incr(odds))) MARK(tail(incr(head(z0)))) -> c17(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c17(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c17(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c17(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c17(MARK(incr(x0))) MARK(tail(head(nats))) -> c17(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(incr(z0)))) -> c17(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(pairs))) -> c17(A__TAIL(a__head(a__pairs)), MARK(head(pairs))) MARK(tail(head(odds))) -> c17(A__TAIL(a__head(a__odds)), MARK(head(odds))) MARK(tail(head(head(z0)))) -> c17(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c17(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c17(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c17(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c17(MARK(head(x0))) MARK(tail(tail(nats))) -> c17(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(incr(z0)))) -> c17(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(pairs))) -> c17(A__TAIL(a__tail(a__pairs)), MARK(tail(pairs))) MARK(tail(tail(odds))) -> c17(A__TAIL(a__tail(a__odds)), MARK(tail(odds))) MARK(tail(tail(head(z0)))) -> c17(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c17(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c17(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c17(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c17(MARK(tail(x0))) MARK(tail(nats)) -> c17(A__TAIL(cons(0, incr(nats)))) MARK(tail(pairs)) -> c17(A__TAIL(cons(0, incr(odds)))) A__INCR(cons(cons(y0, y1), z1)) -> c6(MARK(cons(y0, y1))) A__INCR(cons(s(y0), z1)) -> c6(MARK(s(y0))) A__INCR(cons(incr(cons(y0, y1)), z1)) -> c6(MARK(incr(cons(y0, y1)))) A__INCR(cons(incr(s(y0)), z1)) -> c6(MARK(incr(s(y0)))) A__INCR(cons(head(cons(y0, y1)), z1)) -> c6(MARK(head(cons(y0, y1)))) A__INCR(cons(head(s(y0)), z1)) -> c6(MARK(head(s(y0)))) A__INCR(cons(tail(cons(y0, y1)), z1)) -> c6(MARK(tail(cons(y0, y1)))) A__INCR(cons(tail(s(y0)), z1)) -> c6(MARK(tail(s(y0)))) A__INCR(cons(incr(incr(nats)), z1)) -> c6(MARK(incr(incr(nats)))) A__INCR(cons(incr(incr(incr(y0))), z1)) -> c6(MARK(incr(incr(incr(y0))))) A__INCR(cons(incr(incr(pairs)), z1)) -> c6(MARK(incr(incr(pairs)))) A__INCR(cons(incr(incr(odds)), z1)) -> c6(MARK(incr(incr(odds)))) A__INCR(cons(incr(incr(head(y0))), z1)) -> c6(MARK(incr(incr(head(y0))))) A__INCR(cons(incr(incr(tail(y0))), z1)) -> c6(MARK(incr(incr(tail(y0))))) A__INCR(cons(incr(incr(cons(y0, y1))), z1)) -> c6(MARK(incr(incr(cons(y0, y1))))) A__INCR(cons(incr(incr(s(y0))), z1)) -> c6(MARK(incr(incr(s(y0))))) A__INCR(cons(incr(incr(y0)), z1)) -> c6(MARK(incr(incr(y0)))) A__INCR(cons(incr(odds), z1)) -> c6(MARK(incr(odds))) A__INCR(cons(incr(head(nats)), z1)) -> c6(MARK(incr(head(nats)))) A__INCR(cons(incr(head(incr(y0))), z1)) -> c6(MARK(incr(head(incr(y0))))) A__INCR(cons(incr(head(pairs)), z1)) -> c6(MARK(incr(head(pairs)))) A__INCR(cons(incr(head(odds)), z1)) -> c6(MARK(incr(head(odds)))) A__INCR(cons(incr(head(head(y0))), z1)) -> c6(MARK(incr(head(head(y0))))) A__INCR(cons(incr(head(tail(y0))), z1)) -> c6(MARK(incr(head(tail(y0))))) A__INCR(cons(incr(head(cons(y0, y1))), z1)) -> c6(MARK(incr(head(cons(y0, y1))))) A__INCR(cons(incr(head(s(y0))), z1)) -> c6(MARK(incr(head(s(y0))))) A__INCR(cons(incr(head(y0)), z1)) -> c6(MARK(incr(head(y0)))) A__INCR(cons(incr(tail(nats)), z1)) -> c6(MARK(incr(tail(nats)))) A__INCR(cons(incr(tail(incr(y0))), z1)) -> c6(MARK(incr(tail(incr(y0))))) A__INCR(cons(incr(tail(pairs)), z1)) -> c6(MARK(incr(tail(pairs)))) A__INCR(cons(incr(tail(odds)), z1)) -> c6(MARK(incr(tail(odds)))) A__INCR(cons(incr(tail(head(y0))), z1)) -> c6(MARK(incr(tail(head(y0))))) A__INCR(cons(incr(tail(tail(y0))), z1)) -> c6(MARK(incr(tail(tail(y0))))) A__INCR(cons(incr(tail(cons(y0, y1))), z1)) -> c6(MARK(incr(tail(cons(y0, y1))))) A__INCR(cons(incr(tail(s(y0))), z1)) -> c6(MARK(incr(tail(s(y0))))) A__INCR(cons(incr(tail(y0)), z1)) -> c6(MARK(incr(tail(y0)))) A__INCR(cons(head(incr(nats)), z1)) -> c6(MARK(head(incr(nats)))) A__INCR(cons(head(incr(incr(y0))), z1)) -> c6(MARK(head(incr(incr(y0))))) A__INCR(cons(head(incr(pairs)), z1)) -> c6(MARK(head(incr(pairs)))) A__INCR(cons(head(incr(odds)), z1)) -> c6(MARK(head(incr(odds)))) A__INCR(cons(head(incr(head(y0))), z1)) -> c6(MARK(head(incr(head(y0))))) A__INCR(cons(head(incr(tail(y0))), z1)) -> c6(MARK(head(incr(tail(y0))))) A__INCR(cons(head(incr(cons(y0, y1))), z1)) -> c6(MARK(head(incr(cons(y0, y1))))) A__INCR(cons(head(incr(s(y0))), z1)) -> c6(MARK(head(incr(s(y0))))) A__INCR(cons(head(incr(y0)), z1)) -> c6(MARK(head(incr(y0)))) A__INCR(cons(head(odds), z1)) -> c6(MARK(head(odds))) A__INCR(cons(head(head(nats)), z1)) -> c6(MARK(head(head(nats)))) A__INCR(cons(head(head(incr(y0))), z1)) -> c6(MARK(head(head(incr(y0))))) A__INCR(cons(head(head(pairs)), z1)) -> c6(MARK(head(head(pairs)))) A__INCR(cons(head(head(odds)), z1)) -> c6(MARK(head(head(odds)))) A__INCR(cons(head(head(head(y0))), z1)) -> c6(MARK(head(head(head(y0))))) A__INCR(cons(head(head(tail(y0))), z1)) -> c6(MARK(head(head(tail(y0))))) A__INCR(cons(head(head(cons(y0, y1))), z1)) -> c6(MARK(head(head(cons(y0, y1))))) A__INCR(cons(head(head(s(y0))), z1)) -> c6(MARK(head(head(s(y0))))) A__INCR(cons(head(head(y0)), z1)) -> c6(MARK(head(head(y0)))) A__INCR(cons(head(tail(nats)), z1)) -> c6(MARK(head(tail(nats)))) A__INCR(cons(head(tail(incr(y0))), z1)) -> c6(MARK(head(tail(incr(y0))))) A__INCR(cons(head(tail(pairs)), z1)) -> c6(MARK(head(tail(pairs)))) A__INCR(cons(head(tail(odds)), z1)) -> c6(MARK(head(tail(odds)))) A__INCR(cons(head(tail(head(y0))), z1)) -> c6(MARK(head(tail(head(y0))))) A__INCR(cons(head(tail(tail(y0))), z1)) -> c6(MARK(head(tail(tail(y0))))) A__INCR(cons(head(tail(cons(y0, y1))), z1)) -> c6(MARK(head(tail(cons(y0, y1))))) A__INCR(cons(head(tail(s(y0))), z1)) -> c6(MARK(head(tail(s(y0))))) A__INCR(cons(head(tail(y0)), z1)) -> c6(MARK(head(tail(y0)))) A__INCR(cons(head(nats), z1)) -> c6(MARK(head(nats))) A__INCR(cons(head(pairs), z1)) -> c6(MARK(head(pairs))) A__INCR(cons(tail(incr(nats)), z1)) -> c6(MARK(tail(incr(nats)))) A__INCR(cons(tail(incr(incr(y0))), z1)) -> c6(MARK(tail(incr(incr(y0))))) A__INCR(cons(tail(incr(pairs)), z1)) -> c6(MARK(tail(incr(pairs)))) A__INCR(cons(tail(incr(odds)), z1)) -> c6(MARK(tail(incr(odds)))) A__INCR(cons(tail(incr(head(y0))), z1)) -> c6(MARK(tail(incr(head(y0))))) A__INCR(cons(tail(incr(tail(y0))), z1)) -> c6(MARK(tail(incr(tail(y0))))) A__INCR(cons(tail(incr(cons(y0, y1))), z1)) -> c6(MARK(tail(incr(cons(y0, y1))))) A__INCR(cons(tail(incr(s(y0))), z1)) -> c6(MARK(tail(incr(s(y0))))) A__INCR(cons(tail(incr(y0)), z1)) -> c6(MARK(tail(incr(y0)))) A__INCR(cons(tail(odds), z1)) -> c6(MARK(tail(odds))) A__INCR(cons(tail(head(nats)), z1)) -> c6(MARK(tail(head(nats)))) A__INCR(cons(tail(head(incr(y0))), z1)) -> c6(MARK(tail(head(incr(y0))))) A__INCR(cons(tail(head(pairs)), z1)) -> c6(MARK(tail(head(pairs)))) A__INCR(cons(tail(head(odds)), z1)) -> c6(MARK(tail(head(odds)))) A__INCR(cons(tail(head(head(y0))), z1)) -> c6(MARK(tail(head(head(y0))))) A__INCR(cons(tail(head(tail(y0))), z1)) -> c6(MARK(tail(head(tail(y0))))) A__INCR(cons(tail(head(cons(y0, y1))), z1)) -> c6(MARK(tail(head(cons(y0, y1))))) A__INCR(cons(tail(head(s(y0))), z1)) -> c6(MARK(tail(head(s(y0))))) A__INCR(cons(tail(head(y0)), z1)) -> c6(MARK(tail(head(y0)))) A__INCR(cons(tail(tail(nats)), z1)) -> c6(MARK(tail(tail(nats)))) A__INCR(cons(tail(tail(incr(y0))), z1)) -> c6(MARK(tail(tail(incr(y0))))) A__INCR(cons(tail(tail(pairs)), z1)) -> c6(MARK(tail(tail(pairs)))) A__INCR(cons(tail(tail(odds)), z1)) -> c6(MARK(tail(tail(odds)))) A__INCR(cons(tail(tail(head(y0))), z1)) -> c6(MARK(tail(tail(head(y0))))) A__INCR(cons(tail(tail(tail(y0))), z1)) -> c6(MARK(tail(tail(tail(y0))))) A__INCR(cons(tail(tail(cons(y0, y1))), z1)) -> c6(MARK(tail(tail(cons(y0, y1))))) A__INCR(cons(tail(tail(s(y0))), z1)) -> c6(MARK(tail(tail(s(y0))))) A__INCR(cons(tail(tail(y0)), z1)) -> c6(MARK(tail(tail(y0)))) A__INCR(cons(tail(nats), z1)) -> c6(MARK(tail(nats))) A__INCR(cons(tail(pairs), z1)) -> c6(MARK(tail(pairs))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs))) MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs))) MARK(tail(incr(nats))) -> c17(A__TAIL(a__incr(a__nats))) MARK(tail(incr(pairs))) -> c17(A__TAIL(a__incr(a__pairs))) MARK(tail(odds)) -> c17(A__TAIL(a__incr(a__pairs))) MARK(incr(cons(nats, x1))) -> c13(A__INCR(cons(a__nats, x1)), MARK(cons(nats, x1))) MARK(incr(cons(incr(z0), x1))) -> c13(A__INCR(cons(a__incr(mark(z0)), x1)), MARK(cons(incr(z0), x1))) MARK(incr(cons(pairs, x1))) -> c13(A__INCR(cons(a__pairs, x1)), MARK(cons(pairs, x1))) MARK(incr(cons(odds, x1))) -> c13(A__INCR(cons(a__odds, x1)), MARK(cons(odds, x1))) MARK(incr(cons(head(z0), x1))) -> c13(A__INCR(cons(a__head(mark(z0)), x1)), MARK(cons(head(z0), x1))) MARK(incr(cons(tail(z0), x1))) -> c13(A__INCR(cons(a__tail(mark(z0)), x1)), MARK(cons(tail(z0), x1))) MARK(incr(cons(cons(z0, z1), x1))) -> c13(A__INCR(cons(cons(mark(z0), z1), x1)), MARK(cons(cons(z0, z1), x1))) MARK(incr(cons(s(z0), x1))) -> c13(A__INCR(cons(s(mark(z0)), x1)), MARK(cons(s(z0), x1))) MARK(incr(cons(x0, x1))) -> c13(MARK(cons(x0, x1))) MARK(incr(cons(0, x1))) -> c13(MARK(cons(0, x1))) K tuples:none Defined Rule Symbols: a__nats, a__pairs, a__odds, a__incr_1, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__HEAD_1, A__TAIL_1, MARK_1, A__INCR_1 Compound Symbols: c8_1, c10_1, c18_1, c20_1, c13_1, c16_2, c16_1, c17_2, c17_1, c13_2, c6_1 ---------------------------------------- (169) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace A__HEAD(cons(z0, z1)) -> c8(MARK(z0)) by A__HEAD(cons(cons(y0, y1), z1)) -> c8(MARK(cons(y0, y1))) A__HEAD(cons(s(y0), z1)) -> c8(MARK(s(y0))) A__HEAD(cons(incr(s(y0)), z1)) -> c8(MARK(incr(s(y0)))) A__HEAD(cons(head(cons(y0, y1)), z1)) -> c8(MARK(head(cons(y0, y1)))) A__HEAD(cons(head(s(y0)), z1)) -> c8(MARK(head(s(y0)))) A__HEAD(cons(tail(cons(y0, y1)), z1)) -> c8(MARK(tail(cons(y0, y1)))) A__HEAD(cons(tail(s(y0)), z1)) -> c8(MARK(tail(s(y0)))) A__HEAD(cons(incr(incr(incr(y0))), z1)) -> c8(MARK(incr(incr(incr(y0))))) A__HEAD(cons(incr(incr(odds)), z1)) -> c8(MARK(incr(incr(odds)))) A__HEAD(cons(incr(incr(head(y0))), z1)) -> c8(MARK(incr(incr(head(y0))))) A__HEAD(cons(incr(incr(tail(y0))), z1)) -> c8(MARK(incr(incr(tail(y0))))) A__HEAD(cons(incr(incr(cons(y0, y1))), z1)) -> c8(MARK(incr(incr(cons(y0, y1))))) A__HEAD(cons(incr(incr(s(y0))), z1)) -> c8(MARK(incr(incr(s(y0))))) A__HEAD(cons(incr(incr(y0)), z1)) -> c8(MARK(incr(incr(y0)))) A__HEAD(cons(incr(head(nats)), z1)) -> c8(MARK(incr(head(nats)))) A__HEAD(cons(incr(head(incr(y0))), z1)) -> c8(MARK(incr(head(incr(y0))))) A__HEAD(cons(incr(head(pairs)), z1)) -> c8(MARK(incr(head(pairs)))) A__HEAD(cons(incr(head(odds)), z1)) -> c8(MARK(incr(head(odds)))) A__HEAD(cons(incr(head(head(y0))), z1)) -> c8(MARK(incr(head(head(y0))))) A__HEAD(cons(incr(head(tail(y0))), z1)) -> c8(MARK(incr(head(tail(y0))))) A__HEAD(cons(incr(head(cons(y0, y1))), z1)) -> c8(MARK(incr(head(cons(y0, y1))))) A__HEAD(cons(incr(head(s(y0))), z1)) -> c8(MARK(incr(head(s(y0))))) A__HEAD(cons(incr(head(y0)), z1)) -> c8(MARK(incr(head(y0)))) A__HEAD(cons(incr(tail(nats)), z1)) -> c8(MARK(incr(tail(nats)))) A__HEAD(cons(incr(tail(incr(y0))), z1)) -> c8(MARK(incr(tail(incr(y0))))) A__HEAD(cons(incr(tail(pairs)), z1)) -> c8(MARK(incr(tail(pairs)))) A__HEAD(cons(incr(tail(odds)), z1)) -> c8(MARK(incr(tail(odds)))) A__HEAD(cons(incr(tail(head(y0))), z1)) -> c8(MARK(incr(tail(head(y0))))) A__HEAD(cons(incr(tail(tail(y0))), z1)) -> c8(MARK(incr(tail(tail(y0))))) A__HEAD(cons(incr(tail(cons(y0, y1))), z1)) -> c8(MARK(incr(tail(cons(y0, y1))))) A__HEAD(cons(incr(tail(s(y0))), z1)) -> c8(MARK(incr(tail(s(y0))))) A__HEAD(cons(incr(tail(y0)), z1)) -> c8(MARK(incr(tail(y0)))) A__HEAD(cons(head(incr(incr(y0))), z1)) -> c8(MARK(head(incr(incr(y0))))) A__HEAD(cons(head(incr(odds)), z1)) -> c8(MARK(head(incr(odds)))) A__HEAD(cons(head(incr(head(y0))), z1)) -> c8(MARK(head(incr(head(y0))))) A__HEAD(cons(head(incr(tail(y0))), z1)) -> c8(MARK(head(incr(tail(y0))))) A__HEAD(cons(head(incr(cons(y0, y1))), z1)) -> c8(MARK(head(incr(cons(y0, y1))))) A__HEAD(cons(head(incr(s(y0))), z1)) -> c8(MARK(head(incr(s(y0))))) A__HEAD(cons(head(incr(y0)), z1)) -> c8(MARK(head(incr(y0)))) A__HEAD(cons(head(head(nats)), z1)) -> c8(MARK(head(head(nats)))) A__HEAD(cons(head(head(incr(y0))), z1)) -> c8(MARK(head(head(incr(y0))))) A__HEAD(cons(head(head(pairs)), z1)) -> c8(MARK(head(head(pairs)))) A__HEAD(cons(head(head(odds)), z1)) -> c8(MARK(head(head(odds)))) A__HEAD(cons(head(head(head(y0))), z1)) -> c8(MARK(head(head(head(y0))))) A__HEAD(cons(head(head(tail(y0))), z1)) -> c8(MARK(head(head(tail(y0))))) A__HEAD(cons(head(head(cons(y0, y1))), z1)) -> c8(MARK(head(head(cons(y0, y1))))) A__HEAD(cons(head(head(s(y0))), z1)) -> c8(MARK(head(head(s(y0))))) A__HEAD(cons(head(head(y0)), z1)) -> c8(MARK(head(head(y0)))) A__HEAD(cons(head(tail(nats)), z1)) -> c8(MARK(head(tail(nats)))) A__HEAD(cons(head(tail(incr(y0))), z1)) -> c8(MARK(head(tail(incr(y0))))) A__HEAD(cons(head(tail(pairs)), z1)) -> c8(MARK(head(tail(pairs)))) A__HEAD(cons(head(tail(odds)), z1)) -> c8(MARK(head(tail(odds)))) A__HEAD(cons(head(tail(head(y0))), z1)) -> c8(MARK(head(tail(head(y0))))) A__HEAD(cons(head(tail(tail(y0))), z1)) -> c8(MARK(head(tail(tail(y0))))) A__HEAD(cons(head(tail(cons(y0, y1))), z1)) -> c8(MARK(head(tail(cons(y0, y1))))) A__HEAD(cons(head(tail(s(y0))), z1)) -> c8(MARK(head(tail(s(y0))))) A__HEAD(cons(head(tail(y0)), z1)) -> c8(MARK(head(tail(y0)))) A__HEAD(cons(head(nats), z1)) -> c8(MARK(head(nats))) A__HEAD(cons(head(pairs), z1)) -> c8(MARK(head(pairs))) A__HEAD(cons(tail(incr(incr(y0))), z1)) -> c8(MARK(tail(incr(incr(y0))))) A__HEAD(cons(tail(incr(odds)), z1)) -> c8(MARK(tail(incr(odds)))) A__HEAD(cons(tail(incr(head(y0))), z1)) -> c8(MARK(tail(incr(head(y0))))) A__HEAD(cons(tail(incr(tail(y0))), z1)) -> c8(MARK(tail(incr(tail(y0))))) A__HEAD(cons(tail(incr(cons(y0, y1))), z1)) -> c8(MARK(tail(incr(cons(y0, y1))))) A__HEAD(cons(tail(incr(s(y0))), z1)) -> c8(MARK(tail(incr(s(y0))))) A__HEAD(cons(tail(incr(y0)), z1)) -> c8(MARK(tail(incr(y0)))) A__HEAD(cons(tail(head(nats)), z1)) -> c8(MARK(tail(head(nats)))) A__HEAD(cons(tail(head(incr(y0))), z1)) -> c8(MARK(tail(head(incr(y0))))) A__HEAD(cons(tail(head(pairs)), z1)) -> c8(MARK(tail(head(pairs)))) A__HEAD(cons(tail(head(odds)), z1)) -> c8(MARK(tail(head(odds)))) A__HEAD(cons(tail(head(head(y0))), z1)) -> c8(MARK(tail(head(head(y0))))) A__HEAD(cons(tail(head(tail(y0))), z1)) -> c8(MARK(tail(head(tail(y0))))) A__HEAD(cons(tail(head(cons(y0, y1))), z1)) -> c8(MARK(tail(head(cons(y0, y1))))) A__HEAD(cons(tail(head(s(y0))), z1)) -> c8(MARK(tail(head(s(y0))))) A__HEAD(cons(tail(head(y0)), z1)) -> c8(MARK(tail(head(y0)))) A__HEAD(cons(tail(tail(nats)), z1)) -> c8(MARK(tail(tail(nats)))) A__HEAD(cons(tail(tail(incr(y0))), z1)) -> c8(MARK(tail(tail(incr(y0))))) A__HEAD(cons(tail(tail(pairs)), z1)) -> c8(MARK(tail(tail(pairs)))) A__HEAD(cons(tail(tail(odds)), z1)) -> c8(MARK(tail(tail(odds)))) A__HEAD(cons(tail(tail(head(y0))), z1)) -> c8(MARK(tail(tail(head(y0))))) A__HEAD(cons(tail(tail(tail(y0))), z1)) -> c8(MARK(tail(tail(tail(y0))))) A__HEAD(cons(tail(tail(cons(y0, y1))), z1)) -> c8(MARK(tail(tail(cons(y0, y1))))) A__HEAD(cons(tail(tail(s(y0))), z1)) -> c8(MARK(tail(tail(s(y0))))) A__HEAD(cons(tail(tail(y0)), z1)) -> c8(MARK(tail(tail(y0)))) A__HEAD(cons(tail(nats), z1)) -> c8(MARK(tail(nats))) A__HEAD(cons(tail(pairs), z1)) -> c8(MARK(tail(pairs))) A__HEAD(cons(incr(incr(nats)), z1)) -> c8(MARK(incr(incr(nats)))) A__HEAD(cons(incr(incr(pairs)), z1)) -> c8(MARK(incr(incr(pairs)))) A__HEAD(cons(incr(odds), z1)) -> c8(MARK(incr(odds))) A__HEAD(cons(head(incr(nats)), z1)) -> c8(MARK(head(incr(nats)))) A__HEAD(cons(head(incr(pairs)), z1)) -> c8(MARK(head(incr(pairs)))) A__HEAD(cons(head(odds), z1)) -> c8(MARK(head(odds))) A__HEAD(cons(tail(incr(nats)), z1)) -> c8(MARK(tail(incr(nats)))) A__HEAD(cons(tail(incr(pairs)), z1)) -> c8(MARK(tail(incr(pairs)))) A__HEAD(cons(tail(odds), z1)) -> c8(MARK(tail(odds))) A__HEAD(cons(incr(cons(nats, y0)), z1)) -> c8(MARK(incr(cons(nats, y0)))) A__HEAD(cons(incr(cons(incr(y0), y1)), z1)) -> c8(MARK(incr(cons(incr(y0), y1)))) A__HEAD(cons(incr(cons(pairs, y0)), z1)) -> c8(MARK(incr(cons(pairs, y0)))) A__HEAD(cons(incr(cons(odds, y0)), z1)) -> c8(MARK(incr(cons(odds, y0)))) A__HEAD(cons(incr(cons(head(y0), y1)), z1)) -> c8(MARK(incr(cons(head(y0), y1)))) A__HEAD(cons(incr(cons(tail(y0), y1)), z1)) -> c8(MARK(incr(cons(tail(y0), y1)))) A__HEAD(cons(incr(cons(cons(y0, y1), y2)), z1)) -> c8(MARK(incr(cons(cons(y0, y1), y2)))) A__HEAD(cons(incr(cons(s(y0), y1)), z1)) -> c8(MARK(incr(cons(s(y0), y1)))) A__HEAD(cons(incr(cons(y0, y1)), z1)) -> c8(MARK(incr(cons(y0, y1)))) A__HEAD(cons(incr(cons(0, y0)), z1)) -> c8(MARK(incr(cons(0, y0)))) ---------------------------------------- (170) Obligation: Complexity Dependency Tuples Problem Rules: a__nats -> cons(0, incr(nats)) a__nats -> nats a__pairs -> cons(0, incr(odds)) a__pairs -> pairs a__odds -> a__incr(a__pairs) a__odds -> odds a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) 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(nats) -> a__nats mark(incr(z0)) -> a__incr(mark(z0)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c17(MARK(s(z0))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(head(nats))) -> c16(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(pairs))) -> c16(A__HEAD(a__head(a__pairs)), MARK(head(pairs))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(head(nats)) -> c16(A__HEAD(cons(0, incr(nats)))) MARK(head(pairs)) -> c16(A__HEAD(cons(0, incr(odds)))) MARK(tail(incr(incr(z0)))) -> c17(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(odds))) -> c17(A__TAIL(a__incr(a__odds)), MARK(incr(odds))) MARK(tail(incr(head(z0)))) -> c17(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c17(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c17(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c17(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c17(MARK(incr(x0))) MARK(tail(head(nats))) -> c17(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(incr(z0)))) -> c17(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(pairs))) -> c17(A__TAIL(a__head(a__pairs)), MARK(head(pairs))) MARK(tail(head(odds))) -> c17(A__TAIL(a__head(a__odds)), MARK(head(odds))) MARK(tail(head(head(z0)))) -> c17(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c17(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c17(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c17(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c17(MARK(head(x0))) MARK(tail(tail(nats))) -> c17(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(incr(z0)))) -> c17(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(pairs))) -> c17(A__TAIL(a__tail(a__pairs)), MARK(tail(pairs))) MARK(tail(tail(odds))) -> c17(A__TAIL(a__tail(a__odds)), MARK(tail(odds))) MARK(tail(tail(head(z0)))) -> c17(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c17(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c17(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c17(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c17(MARK(tail(x0))) MARK(tail(nats)) -> c17(A__TAIL(cons(0, incr(nats)))) MARK(tail(pairs)) -> c17(A__TAIL(cons(0, incr(odds)))) A__INCR(cons(cons(y0, y1), z1)) -> c6(MARK(cons(y0, y1))) A__INCR(cons(s(y0), z1)) -> c6(MARK(s(y0))) A__INCR(cons(incr(cons(y0, y1)), z1)) -> c6(MARK(incr(cons(y0, y1)))) A__INCR(cons(incr(s(y0)), z1)) -> c6(MARK(incr(s(y0)))) A__INCR(cons(head(cons(y0, y1)), z1)) -> c6(MARK(head(cons(y0, y1)))) A__INCR(cons(head(s(y0)), z1)) -> c6(MARK(head(s(y0)))) A__INCR(cons(tail(cons(y0, y1)), z1)) -> c6(MARK(tail(cons(y0, y1)))) A__INCR(cons(tail(s(y0)), z1)) -> c6(MARK(tail(s(y0)))) A__INCR(cons(incr(incr(nats)), z1)) -> c6(MARK(incr(incr(nats)))) A__INCR(cons(incr(incr(incr(y0))), z1)) -> c6(MARK(incr(incr(incr(y0))))) A__INCR(cons(incr(incr(pairs)), z1)) -> c6(MARK(incr(incr(pairs)))) A__INCR(cons(incr(incr(odds)), z1)) -> c6(MARK(incr(incr(odds)))) A__INCR(cons(incr(incr(head(y0))), z1)) -> c6(MARK(incr(incr(head(y0))))) A__INCR(cons(incr(incr(tail(y0))), z1)) -> c6(MARK(incr(incr(tail(y0))))) A__INCR(cons(incr(incr(cons(y0, y1))), z1)) -> c6(MARK(incr(incr(cons(y0, y1))))) A__INCR(cons(incr(incr(s(y0))), z1)) -> c6(MARK(incr(incr(s(y0))))) A__INCR(cons(incr(incr(y0)), z1)) -> c6(MARK(incr(incr(y0)))) A__INCR(cons(incr(odds), z1)) -> c6(MARK(incr(odds))) A__INCR(cons(incr(head(nats)), z1)) -> c6(MARK(incr(head(nats)))) A__INCR(cons(incr(head(incr(y0))), z1)) -> c6(MARK(incr(head(incr(y0))))) A__INCR(cons(incr(head(pairs)), z1)) -> c6(MARK(incr(head(pairs)))) A__INCR(cons(incr(head(odds)), z1)) -> c6(MARK(incr(head(odds)))) A__INCR(cons(incr(head(head(y0))), z1)) -> c6(MARK(incr(head(head(y0))))) A__INCR(cons(incr(head(tail(y0))), z1)) -> c6(MARK(incr(head(tail(y0))))) A__INCR(cons(incr(head(cons(y0, y1))), z1)) -> c6(MARK(incr(head(cons(y0, y1))))) A__INCR(cons(incr(head(s(y0))), z1)) -> c6(MARK(incr(head(s(y0))))) A__INCR(cons(incr(head(y0)), z1)) -> c6(MARK(incr(head(y0)))) A__INCR(cons(incr(tail(nats)), z1)) -> c6(MARK(incr(tail(nats)))) A__INCR(cons(incr(tail(incr(y0))), z1)) -> c6(MARK(incr(tail(incr(y0))))) A__INCR(cons(incr(tail(pairs)), z1)) -> 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c6(MARK(head(incr(s(y0))))) A__INCR(cons(head(incr(y0)), z1)) -> c6(MARK(head(incr(y0)))) A__INCR(cons(head(odds), z1)) -> c6(MARK(head(odds))) A__INCR(cons(head(head(nats)), z1)) -> c6(MARK(head(head(nats)))) A__INCR(cons(head(head(incr(y0))), z1)) -> c6(MARK(head(head(incr(y0))))) A__INCR(cons(head(head(pairs)), z1)) -> c6(MARK(head(head(pairs)))) A__INCR(cons(head(head(odds)), z1)) -> c6(MARK(head(head(odds)))) A__INCR(cons(head(head(head(y0))), z1)) -> c6(MARK(head(head(head(y0))))) A__INCR(cons(head(head(tail(y0))), z1)) -> c6(MARK(head(head(tail(y0))))) A__INCR(cons(head(head(cons(y0, y1))), z1)) -> c6(MARK(head(head(cons(y0, y1))))) A__INCR(cons(head(head(s(y0))), z1)) -> c6(MARK(head(head(s(y0))))) A__INCR(cons(head(head(y0)), z1)) -> c6(MARK(head(head(y0)))) A__INCR(cons(head(tail(nats)), z1)) -> c6(MARK(head(tail(nats)))) A__INCR(cons(head(tail(incr(y0))), z1)) -> c6(MARK(head(tail(incr(y0))))) A__INCR(cons(head(tail(pairs)), z1)) -> c6(MARK(head(tail(pairs)))) 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c8(MARK(head(head(head(y0))))) A__HEAD(cons(head(head(tail(y0))), z1)) -> c8(MARK(head(head(tail(y0))))) A__HEAD(cons(head(head(cons(y0, y1))), z1)) -> c8(MARK(head(head(cons(y0, y1))))) A__HEAD(cons(head(head(s(y0))), z1)) -> c8(MARK(head(head(s(y0))))) A__HEAD(cons(head(head(y0)), z1)) -> c8(MARK(head(head(y0)))) A__HEAD(cons(head(tail(nats)), z1)) -> c8(MARK(head(tail(nats)))) A__HEAD(cons(head(tail(incr(y0))), z1)) -> c8(MARK(head(tail(incr(y0))))) A__HEAD(cons(head(tail(pairs)), z1)) -> c8(MARK(head(tail(pairs)))) A__HEAD(cons(head(tail(odds)), z1)) -> c8(MARK(head(tail(odds)))) A__HEAD(cons(head(tail(head(y0))), z1)) -> c8(MARK(head(tail(head(y0))))) A__HEAD(cons(head(tail(tail(y0))), z1)) -> c8(MARK(head(tail(tail(y0))))) A__HEAD(cons(head(tail(cons(y0, y1))), z1)) -> c8(MARK(head(tail(cons(y0, y1))))) A__HEAD(cons(head(tail(s(y0))), z1)) -> c8(MARK(head(tail(s(y0))))) A__HEAD(cons(head(tail(y0)), z1)) -> c8(MARK(head(tail(y0)))) A__HEAD(cons(head(nats), z1)) -> 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A__HEAD(cons(incr(cons(cons(y0, y1), y2)), z1)) -> c8(MARK(incr(cons(cons(y0, y1), y2)))) A__HEAD(cons(incr(cons(s(y0), y1)), z1)) -> c8(MARK(incr(cons(s(y0), y1)))) A__HEAD(cons(incr(cons(y0, y1)), z1)) -> c8(MARK(incr(cons(y0, y1)))) A__HEAD(cons(incr(cons(0, y0)), z1)) -> c8(MARK(incr(cons(0, y0)))) S tuples: A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c17(MARK(s(z0))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats)), MARK(head(nats))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) 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A__INCR(cons(incr(tail(incr(y0))), z1)) -> c6(MARK(incr(tail(incr(y0))))) A__INCR(cons(incr(tail(pairs)), z1)) -> c6(MARK(incr(tail(pairs)))) A__INCR(cons(incr(tail(odds)), z1)) -> c6(MARK(incr(tail(odds)))) A__INCR(cons(incr(tail(head(y0))), z1)) -> c6(MARK(incr(tail(head(y0))))) A__INCR(cons(incr(tail(tail(y0))), z1)) -> c6(MARK(incr(tail(tail(y0))))) A__INCR(cons(incr(tail(cons(y0, y1))), z1)) -> c6(MARK(incr(tail(cons(y0, y1))))) A__INCR(cons(incr(tail(s(y0))), z1)) -> c6(MARK(incr(tail(s(y0))))) A__INCR(cons(incr(tail(y0)), z1)) -> c6(MARK(incr(tail(y0)))) A__INCR(cons(head(incr(nats)), z1)) -> c6(MARK(head(incr(nats)))) A__INCR(cons(head(incr(incr(y0))), z1)) -> c6(MARK(head(incr(incr(y0))))) A__INCR(cons(head(incr(pairs)), z1)) -> c6(MARK(head(incr(pairs)))) A__INCR(cons(head(incr(odds)), z1)) -> c6(MARK(head(incr(odds)))) A__INCR(cons(head(incr(head(y0))), z1)) -> c6(MARK(head(incr(head(y0))))) A__INCR(cons(head(incr(tail(y0))), z1)) -> c6(MARK(head(incr(tail(y0))))) 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A__HEAD(cons(head(tail(s(y0))), z1)) -> c8(MARK(head(tail(s(y0))))) A__HEAD(cons(head(tail(y0)), z1)) -> c8(MARK(head(tail(y0)))) A__HEAD(cons(head(nats), z1)) -> c8(MARK(head(nats))) A__HEAD(cons(head(pairs), z1)) -> c8(MARK(head(pairs))) A__HEAD(cons(tail(incr(incr(y0))), z1)) -> c8(MARK(tail(incr(incr(y0))))) A__HEAD(cons(tail(incr(odds)), z1)) -> c8(MARK(tail(incr(odds)))) A__HEAD(cons(tail(incr(head(y0))), z1)) -> c8(MARK(tail(incr(head(y0))))) A__HEAD(cons(tail(incr(tail(y0))), z1)) -> c8(MARK(tail(incr(tail(y0))))) A__HEAD(cons(tail(incr(cons(y0, y1))), z1)) -> c8(MARK(tail(incr(cons(y0, y1))))) A__HEAD(cons(tail(incr(s(y0))), z1)) -> c8(MARK(tail(incr(s(y0))))) A__HEAD(cons(tail(incr(y0)), z1)) -> c8(MARK(tail(incr(y0)))) A__HEAD(cons(tail(head(nats)), z1)) -> c8(MARK(tail(head(nats)))) A__HEAD(cons(tail(head(incr(y0))), z1)) -> c8(MARK(tail(head(incr(y0))))) A__HEAD(cons(tail(head(pairs)), z1)) -> c8(MARK(tail(head(pairs)))) A__HEAD(cons(tail(head(odds)), z1)) -> c8(MARK(tail(head(odds)))) A__HEAD(cons(tail(head(head(y0))), z1)) -> c8(MARK(tail(head(head(y0))))) A__HEAD(cons(tail(head(tail(y0))), z1)) -> c8(MARK(tail(head(tail(y0))))) A__HEAD(cons(tail(head(cons(y0, y1))), z1)) -> c8(MARK(tail(head(cons(y0, y1))))) A__HEAD(cons(tail(head(s(y0))), z1)) -> c8(MARK(tail(head(s(y0))))) A__HEAD(cons(tail(head(y0)), z1)) -> c8(MARK(tail(head(y0)))) A__HEAD(cons(tail(tail(nats)), z1)) -> c8(MARK(tail(tail(nats)))) A__HEAD(cons(tail(tail(incr(y0))), z1)) -> c8(MARK(tail(tail(incr(y0))))) A__HEAD(cons(tail(tail(pairs)), z1)) -> c8(MARK(tail(tail(pairs)))) A__HEAD(cons(tail(tail(odds)), z1)) -> c8(MARK(tail(tail(odds)))) A__HEAD(cons(tail(tail(head(y0))), z1)) -> c8(MARK(tail(tail(head(y0))))) A__HEAD(cons(tail(tail(tail(y0))), z1)) -> c8(MARK(tail(tail(tail(y0))))) A__HEAD(cons(tail(tail(cons(y0, y1))), z1)) -> c8(MARK(tail(tail(cons(y0, y1))))) A__HEAD(cons(tail(tail(s(y0))), z1)) -> c8(MARK(tail(tail(s(y0))))) A__HEAD(cons(tail(tail(y0)), z1)) -> c8(MARK(tail(tail(y0)))) A__HEAD(cons(tail(nats), z1)) -> c8(MARK(tail(nats))) A__HEAD(cons(tail(pairs), z1)) -> c8(MARK(tail(pairs))) A__HEAD(cons(incr(incr(nats)), z1)) -> c8(MARK(incr(incr(nats)))) A__HEAD(cons(incr(incr(pairs)), z1)) -> c8(MARK(incr(incr(pairs)))) A__HEAD(cons(incr(odds), z1)) -> c8(MARK(incr(odds))) A__HEAD(cons(head(incr(nats)), z1)) -> c8(MARK(head(incr(nats)))) A__HEAD(cons(head(incr(pairs)), z1)) -> c8(MARK(head(incr(pairs)))) A__HEAD(cons(head(odds), z1)) -> c8(MARK(head(odds))) A__HEAD(cons(tail(incr(nats)), z1)) -> c8(MARK(tail(incr(nats)))) A__HEAD(cons(tail(incr(pairs)), z1)) -> c8(MARK(tail(incr(pairs)))) A__HEAD(cons(tail(odds), z1)) -> c8(MARK(tail(odds))) A__HEAD(cons(incr(cons(nats, y0)), z1)) -> c8(MARK(incr(cons(nats, y0)))) A__HEAD(cons(incr(cons(incr(y0), y1)), z1)) -> c8(MARK(incr(cons(incr(y0), y1)))) A__HEAD(cons(incr(cons(pairs, y0)), z1)) -> c8(MARK(incr(cons(pairs, y0)))) A__HEAD(cons(incr(cons(odds, y0)), z1)) -> c8(MARK(incr(cons(odds, y0)))) A__HEAD(cons(incr(cons(head(y0), y1)), z1)) -> c8(MARK(incr(cons(head(y0), y1)))) A__HEAD(cons(incr(cons(tail(y0), y1)), z1)) -> c8(MARK(incr(cons(tail(y0), y1)))) A__HEAD(cons(incr(cons(cons(y0, y1), y2)), z1)) -> c8(MARK(incr(cons(cons(y0, y1), y2)))) A__HEAD(cons(incr(cons(s(y0), y1)), z1)) -> c8(MARK(incr(cons(s(y0), y1)))) A__HEAD(cons(incr(cons(y0, y1)), z1)) -> c8(MARK(incr(cons(y0, y1)))) A__HEAD(cons(incr(cons(0, y0)), z1)) -> c8(MARK(incr(cons(0, y0)))) K tuples:none Defined Rule Symbols: a__nats, a__pairs, a__odds, a__incr_1, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__TAIL_1, MARK_1, A__INCR_1, A__HEAD_1 Compound Symbols: c10_1, c18_1, c20_1, c13_1, c16_2, c16_1, c17_2, c17_1, c13_2, c6_1, c8_1 ---------------------------------------- (171) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 6 trailing nodes: A__INCR(cons(head(pairs), z1)) -> c6(MARK(head(pairs))) MARK(head(pairs)) -> c16(A__HEAD(cons(0, incr(odds)))) MARK(head(nats)) -> c16(A__HEAD(cons(0, incr(nats)))) A__HEAD(cons(head(nats), z1)) -> c8(MARK(head(nats))) A__INCR(cons(head(nats), z1)) -> c6(MARK(head(nats))) A__HEAD(cons(head(pairs), z1)) -> c8(MARK(head(pairs))) ---------------------------------------- (172) Obligation: Complexity Dependency Tuples Problem Rules: a__nats -> cons(0, incr(nats)) a__nats -> nats a__pairs -> cons(0, incr(odds)) a__pairs -> pairs a__odds -> a__incr(a__pairs) a__odds -> odds a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) 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(nats) -> a__nats mark(incr(z0)) -> a__incr(mark(z0)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) 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c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(head(nats))) -> c16(A__HEAD(a__head(a__nats)), MARK(head(nats))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(pairs))) -> c16(A__HEAD(a__head(a__pairs)), MARK(head(pairs))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> 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A__INCR(cons(tail(head(odds)), z1)) -> c6(MARK(tail(head(odds)))) A__INCR(cons(tail(head(head(y0))), z1)) -> c6(MARK(tail(head(head(y0))))) A__INCR(cons(tail(head(tail(y0))), z1)) -> c6(MARK(tail(head(tail(y0))))) A__INCR(cons(tail(head(cons(y0, y1))), z1)) -> c6(MARK(tail(head(cons(y0, y1))))) A__INCR(cons(tail(head(s(y0))), z1)) -> c6(MARK(tail(head(s(y0))))) A__INCR(cons(tail(head(y0)), z1)) -> c6(MARK(tail(head(y0)))) A__INCR(cons(tail(tail(nats)), z1)) -> c6(MARK(tail(tail(nats)))) A__INCR(cons(tail(tail(incr(y0))), z1)) -> c6(MARK(tail(tail(incr(y0))))) A__INCR(cons(tail(tail(pairs)), z1)) -> c6(MARK(tail(tail(pairs)))) A__INCR(cons(tail(tail(odds)), z1)) -> c6(MARK(tail(tail(odds)))) A__INCR(cons(tail(tail(head(y0))), z1)) -> c6(MARK(tail(tail(head(y0))))) A__INCR(cons(tail(tail(tail(y0))), z1)) -> c6(MARK(tail(tail(tail(y0))))) A__INCR(cons(tail(tail(cons(y0, y1))), z1)) -> c6(MARK(tail(tail(cons(y0, y1))))) A__INCR(cons(tail(tail(s(y0))), z1)) -> c6(MARK(tail(tail(s(y0))))) A__INCR(cons(tail(tail(y0)), z1)) -> c6(MARK(tail(tail(y0)))) A__INCR(cons(tail(nats), z1)) -> c6(MARK(tail(nats))) A__INCR(cons(tail(pairs), z1)) -> c6(MARK(tail(pairs))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs))) MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs))) MARK(tail(incr(nats))) -> c17(A__TAIL(a__incr(a__nats))) MARK(tail(incr(pairs))) -> c17(A__TAIL(a__incr(a__pairs))) MARK(tail(odds)) -> c17(A__TAIL(a__incr(a__pairs))) MARK(incr(cons(nats, x1))) -> c13(A__INCR(cons(a__nats, x1)), MARK(cons(nats, x1))) MARK(incr(cons(incr(z0), x1))) -> c13(A__INCR(cons(a__incr(mark(z0)), x1)), MARK(cons(incr(z0), x1))) MARK(incr(cons(pairs, x1))) -> c13(A__INCR(cons(a__pairs, x1)), MARK(cons(pairs, x1))) MARK(incr(cons(odds, x1))) -> c13(A__INCR(cons(a__odds, x1)), MARK(cons(odds, x1))) MARK(incr(cons(head(z0), x1))) -> c13(A__INCR(cons(a__head(mark(z0)), x1)), MARK(cons(head(z0), x1))) MARK(incr(cons(tail(z0), x1))) -> c13(A__INCR(cons(a__tail(mark(z0)), x1)), MARK(cons(tail(z0), x1))) MARK(incr(cons(cons(z0, z1), x1))) -> c13(A__INCR(cons(cons(mark(z0), z1), x1)), MARK(cons(cons(z0, z1), x1))) MARK(incr(cons(s(z0), x1))) -> c13(A__INCR(cons(s(mark(z0)), x1)), MARK(cons(s(z0), x1))) MARK(incr(cons(x0, x1))) -> c13(MARK(cons(x0, x1))) MARK(incr(cons(0, x1))) -> c13(MARK(cons(0, x1))) A__HEAD(cons(cons(y0, y1), z1)) -> c8(MARK(cons(y0, y1))) A__HEAD(cons(s(y0), z1)) -> c8(MARK(s(y0))) A__HEAD(cons(incr(s(y0)), z1)) -> c8(MARK(incr(s(y0)))) A__HEAD(cons(head(cons(y0, y1)), z1)) -> c8(MARK(head(cons(y0, y1)))) A__HEAD(cons(head(s(y0)), z1)) -> c8(MARK(head(s(y0)))) A__HEAD(cons(tail(cons(y0, y1)), z1)) -> c8(MARK(tail(cons(y0, y1)))) A__HEAD(cons(tail(s(y0)), z1)) -> c8(MARK(tail(s(y0)))) A__HEAD(cons(incr(incr(incr(y0))), z1)) -> c8(MARK(incr(incr(incr(y0))))) A__HEAD(cons(incr(incr(odds)), z1)) -> c8(MARK(incr(incr(odds)))) A__HEAD(cons(incr(incr(head(y0))), z1)) -> c8(MARK(incr(incr(head(y0))))) A__HEAD(cons(incr(incr(tail(y0))), z1)) -> c8(MARK(incr(incr(tail(y0))))) A__HEAD(cons(incr(incr(cons(y0, y1))), z1)) -> c8(MARK(incr(incr(cons(y0, y1))))) A__HEAD(cons(incr(incr(s(y0))), z1)) -> c8(MARK(incr(incr(s(y0))))) A__HEAD(cons(incr(incr(y0)), z1)) -> c8(MARK(incr(incr(y0)))) A__HEAD(cons(incr(head(nats)), z1)) -> c8(MARK(incr(head(nats)))) A__HEAD(cons(incr(head(incr(y0))), z1)) -> c8(MARK(incr(head(incr(y0))))) A__HEAD(cons(incr(head(pairs)), z1)) -> c8(MARK(incr(head(pairs)))) A__HEAD(cons(incr(head(odds)), z1)) -> c8(MARK(incr(head(odds)))) A__HEAD(cons(incr(head(head(y0))), z1)) -> c8(MARK(incr(head(head(y0))))) A__HEAD(cons(incr(head(tail(y0))), z1)) -> c8(MARK(incr(head(tail(y0))))) A__HEAD(cons(incr(head(cons(y0, y1))), z1)) -> c8(MARK(incr(head(cons(y0, 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A__HEAD(cons(head(incr(tail(y0))), z1)) -> c8(MARK(head(incr(tail(y0))))) A__HEAD(cons(head(incr(cons(y0, y1))), z1)) -> c8(MARK(head(incr(cons(y0, y1))))) A__HEAD(cons(head(incr(s(y0))), z1)) -> c8(MARK(head(incr(s(y0))))) A__HEAD(cons(head(incr(y0)), z1)) -> c8(MARK(head(incr(y0)))) A__HEAD(cons(head(head(nats)), z1)) -> c8(MARK(head(head(nats)))) A__HEAD(cons(head(head(incr(y0))), z1)) -> c8(MARK(head(head(incr(y0))))) A__HEAD(cons(head(head(pairs)), z1)) -> c8(MARK(head(head(pairs)))) A__HEAD(cons(head(head(odds)), z1)) -> c8(MARK(head(head(odds)))) A__HEAD(cons(head(head(head(y0))), z1)) -> c8(MARK(head(head(head(y0))))) A__HEAD(cons(head(head(tail(y0))), z1)) -> c8(MARK(head(head(tail(y0))))) A__HEAD(cons(head(head(cons(y0, y1))), z1)) -> c8(MARK(head(head(cons(y0, y1))))) A__HEAD(cons(head(head(s(y0))), z1)) -> c8(MARK(head(head(s(y0))))) A__HEAD(cons(head(head(y0)), z1)) -> c8(MARK(head(head(y0)))) A__HEAD(cons(head(tail(nats)), z1)) -> c8(MARK(head(tail(nats)))) A__HEAD(cons(head(tail(incr(y0))), z1)) -> c8(MARK(head(tail(incr(y0))))) A__HEAD(cons(head(tail(pairs)), z1)) -> c8(MARK(head(tail(pairs)))) A__HEAD(cons(head(tail(odds)), z1)) -> c8(MARK(head(tail(odds)))) A__HEAD(cons(head(tail(head(y0))), z1)) -> c8(MARK(head(tail(head(y0))))) A__HEAD(cons(head(tail(tail(y0))), z1)) -> c8(MARK(head(tail(tail(y0))))) A__HEAD(cons(head(tail(cons(y0, y1))), z1)) -> c8(MARK(head(tail(cons(y0, y1))))) A__HEAD(cons(head(tail(s(y0))), z1)) -> c8(MARK(head(tail(s(y0))))) A__HEAD(cons(head(tail(y0)), z1)) -> c8(MARK(head(tail(y0)))) A__HEAD(cons(tail(incr(incr(y0))), z1)) -> c8(MARK(tail(incr(incr(y0))))) A__HEAD(cons(tail(incr(odds)), z1)) -> c8(MARK(tail(incr(odds)))) A__HEAD(cons(tail(incr(head(y0))), z1)) -> c8(MARK(tail(incr(head(y0))))) A__HEAD(cons(tail(incr(tail(y0))), z1)) -> c8(MARK(tail(incr(tail(y0))))) A__HEAD(cons(tail(incr(cons(y0, y1))), z1)) -> c8(MARK(tail(incr(cons(y0, y1))))) A__HEAD(cons(tail(incr(s(y0))), z1)) -> 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A__HEAD(cons(tail(tail(head(y0))), z1)) -> c8(MARK(tail(tail(head(y0))))) A__HEAD(cons(tail(tail(tail(y0))), z1)) -> c8(MARK(tail(tail(tail(y0))))) A__HEAD(cons(tail(tail(cons(y0, y1))), z1)) -> c8(MARK(tail(tail(cons(y0, y1))))) A__HEAD(cons(tail(tail(s(y0))), z1)) -> c8(MARK(tail(tail(s(y0))))) A__HEAD(cons(tail(tail(y0)), z1)) -> c8(MARK(tail(tail(y0)))) A__HEAD(cons(tail(nats), z1)) -> c8(MARK(tail(nats))) A__HEAD(cons(tail(pairs), z1)) -> c8(MARK(tail(pairs))) A__HEAD(cons(incr(incr(nats)), z1)) -> c8(MARK(incr(incr(nats)))) A__HEAD(cons(incr(incr(pairs)), z1)) -> c8(MARK(incr(incr(pairs)))) A__HEAD(cons(incr(odds), z1)) -> c8(MARK(incr(odds))) A__HEAD(cons(head(incr(nats)), z1)) -> c8(MARK(head(incr(nats)))) A__HEAD(cons(head(incr(pairs)), z1)) -> c8(MARK(head(incr(pairs)))) A__HEAD(cons(head(odds), z1)) -> c8(MARK(head(odds))) A__HEAD(cons(tail(incr(nats)), z1)) -> c8(MARK(tail(incr(nats)))) A__HEAD(cons(tail(incr(pairs)), z1)) -> c8(MARK(tail(incr(pairs)))) A__HEAD(cons(tail(odds), z1)) -> c8(MARK(tail(odds))) A__HEAD(cons(incr(cons(nats, y0)), z1)) -> c8(MARK(incr(cons(nats, y0)))) A__HEAD(cons(incr(cons(incr(y0), y1)), z1)) -> c8(MARK(incr(cons(incr(y0), y1)))) A__HEAD(cons(incr(cons(pairs, y0)), z1)) -> c8(MARK(incr(cons(pairs, y0)))) A__HEAD(cons(incr(cons(odds, y0)), z1)) -> c8(MARK(incr(cons(odds, y0)))) A__HEAD(cons(incr(cons(head(y0), y1)), z1)) -> c8(MARK(incr(cons(head(y0), y1)))) A__HEAD(cons(incr(cons(tail(y0), y1)), z1)) -> c8(MARK(incr(cons(tail(y0), y1)))) A__HEAD(cons(incr(cons(cons(y0, y1), y2)), z1)) -> c8(MARK(incr(cons(cons(y0, y1), y2)))) A__HEAD(cons(incr(cons(s(y0), y1)), z1)) -> c8(MARK(incr(cons(s(y0), y1)))) A__HEAD(cons(incr(cons(y0, y1)), z1)) -> c8(MARK(incr(cons(y0, y1)))) A__HEAD(cons(incr(cons(0, y0)), z1)) -> c8(MARK(incr(cons(0, y0)))) S tuples: A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(cons(z0, 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c13(A__INCR(a__head(a__pairs)), MARK(head(pairs))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> 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c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(pairs))) -> c16(A__HEAD(a__head(a__pairs)), MARK(head(pairs))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> 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c17(A__TAIL(a__head(a__nats)), MARK(head(nats))) MARK(tail(head(incr(z0)))) -> c17(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(pairs))) -> c17(A__TAIL(a__head(a__pairs)), MARK(head(pairs))) MARK(tail(head(odds))) -> c17(A__TAIL(a__head(a__odds)), MARK(head(odds))) MARK(tail(head(head(z0)))) -> c17(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c17(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c17(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c17(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c17(MARK(head(x0))) MARK(tail(tail(nats))) -> c17(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(incr(z0)))) -> c17(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(pairs))) -> c17(A__TAIL(a__tail(a__pairs)), MARK(tail(pairs))) MARK(tail(tail(odds))) -> 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A__INCR(cons(tail(s(y0)), z1)) -> c6(MARK(tail(s(y0)))) A__INCR(cons(incr(incr(nats)), z1)) -> c6(MARK(incr(incr(nats)))) A__INCR(cons(incr(incr(incr(y0))), z1)) -> c6(MARK(incr(incr(incr(y0))))) A__INCR(cons(incr(incr(pairs)), z1)) -> c6(MARK(incr(incr(pairs)))) A__INCR(cons(incr(incr(odds)), z1)) -> c6(MARK(incr(incr(odds)))) A__INCR(cons(incr(incr(head(y0))), z1)) -> c6(MARK(incr(incr(head(y0))))) A__INCR(cons(incr(incr(tail(y0))), z1)) -> c6(MARK(incr(incr(tail(y0))))) A__INCR(cons(incr(incr(cons(y0, y1))), z1)) -> c6(MARK(incr(incr(cons(y0, y1))))) A__INCR(cons(incr(incr(s(y0))), z1)) -> c6(MARK(incr(incr(s(y0))))) A__INCR(cons(incr(incr(y0)), z1)) -> c6(MARK(incr(incr(y0)))) A__INCR(cons(incr(odds), z1)) -> c6(MARK(incr(odds))) A__INCR(cons(incr(head(nats)), z1)) -> c6(MARK(incr(head(nats)))) A__INCR(cons(incr(head(incr(y0))), z1)) -> c6(MARK(incr(head(incr(y0))))) A__INCR(cons(incr(head(pairs)), z1)) -> c6(MARK(incr(head(pairs)))) A__INCR(cons(incr(head(odds)), z1)) -> c6(MARK(incr(head(odds)))) A__INCR(cons(incr(head(head(y0))), z1)) -> c6(MARK(incr(head(head(y0))))) A__INCR(cons(incr(head(tail(y0))), z1)) -> c6(MARK(incr(head(tail(y0))))) A__INCR(cons(incr(head(cons(y0, y1))), z1)) -> c6(MARK(incr(head(cons(y0, y1))))) A__INCR(cons(incr(head(s(y0))), z1)) -> c6(MARK(incr(head(s(y0))))) A__INCR(cons(incr(head(y0)), z1)) -> c6(MARK(incr(head(y0)))) A__INCR(cons(incr(tail(nats)), z1)) -> c6(MARK(incr(tail(nats)))) A__INCR(cons(incr(tail(incr(y0))), z1)) -> c6(MARK(incr(tail(incr(y0))))) A__INCR(cons(incr(tail(pairs)), z1)) -> c6(MARK(incr(tail(pairs)))) A__INCR(cons(incr(tail(odds)), z1)) -> c6(MARK(incr(tail(odds)))) A__INCR(cons(incr(tail(head(y0))), z1)) -> c6(MARK(incr(tail(head(y0))))) A__INCR(cons(incr(tail(tail(y0))), z1)) -> c6(MARK(incr(tail(tail(y0))))) A__INCR(cons(incr(tail(cons(y0, y1))), z1)) -> c6(MARK(incr(tail(cons(y0, y1))))) A__INCR(cons(incr(tail(s(y0))), z1)) -> c6(MARK(incr(tail(s(y0))))) A__INCR(cons(incr(tail(y0)), z1)) -> c6(MARK(incr(tail(y0)))) A__INCR(cons(head(incr(nats)), z1)) -> c6(MARK(head(incr(nats)))) A__INCR(cons(head(incr(incr(y0))), z1)) -> c6(MARK(head(incr(incr(y0))))) A__INCR(cons(head(incr(pairs)), z1)) -> c6(MARK(head(incr(pairs)))) A__INCR(cons(head(incr(odds)), z1)) -> c6(MARK(head(incr(odds)))) A__INCR(cons(head(incr(head(y0))), z1)) -> c6(MARK(head(incr(head(y0))))) A__INCR(cons(head(incr(tail(y0))), z1)) -> c6(MARK(head(incr(tail(y0))))) A__INCR(cons(head(incr(cons(y0, y1))), z1)) -> c6(MARK(head(incr(cons(y0, y1))))) A__INCR(cons(head(incr(s(y0))), z1)) -> c6(MARK(head(incr(s(y0))))) A__INCR(cons(head(incr(y0)), z1)) -> c6(MARK(head(incr(y0)))) A__INCR(cons(head(odds), z1)) -> c6(MARK(head(odds))) A__INCR(cons(head(head(nats)), z1)) -> c6(MARK(head(head(nats)))) A__INCR(cons(head(head(incr(y0))), z1)) -> c6(MARK(head(head(incr(y0))))) A__INCR(cons(head(head(pairs)), z1)) -> c6(MARK(head(head(pairs)))) A__INCR(cons(head(head(odds)), z1)) -> c6(MARK(head(head(odds)))) A__INCR(cons(head(head(head(y0))), z1)) -> c6(MARK(head(head(head(y0))))) A__INCR(cons(head(head(tail(y0))), z1)) -> c6(MARK(head(head(tail(y0))))) A__INCR(cons(head(head(cons(y0, y1))), z1)) -> c6(MARK(head(head(cons(y0, y1))))) A__INCR(cons(head(head(s(y0))), z1)) -> c6(MARK(head(head(s(y0))))) A__INCR(cons(head(head(y0)), z1)) -> c6(MARK(head(head(y0)))) A__INCR(cons(head(tail(nats)), z1)) -> c6(MARK(head(tail(nats)))) A__INCR(cons(head(tail(incr(y0))), z1)) -> c6(MARK(head(tail(incr(y0))))) A__INCR(cons(head(tail(pairs)), z1)) -> c6(MARK(head(tail(pairs)))) A__INCR(cons(head(tail(odds)), z1)) -> c6(MARK(head(tail(odds)))) A__INCR(cons(head(tail(head(y0))), z1)) -> c6(MARK(head(tail(head(y0))))) A__INCR(cons(head(tail(tail(y0))), z1)) -> c6(MARK(head(tail(tail(y0))))) A__INCR(cons(head(tail(cons(y0, y1))), z1)) -> c6(MARK(head(tail(cons(y0, y1))))) A__INCR(cons(head(tail(s(y0))), z1)) -> c6(MARK(head(tail(s(y0))))) A__INCR(cons(head(tail(y0)), z1)) -> c6(MARK(head(tail(y0)))) 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c8(MARK(incr(head(y0)))) A__HEAD(cons(incr(tail(nats)), z1)) -> c8(MARK(incr(tail(nats)))) A__HEAD(cons(incr(tail(incr(y0))), z1)) -> c8(MARK(incr(tail(incr(y0))))) A__HEAD(cons(incr(tail(pairs)), z1)) -> c8(MARK(incr(tail(pairs)))) A__HEAD(cons(incr(tail(odds)), z1)) -> c8(MARK(incr(tail(odds)))) A__HEAD(cons(incr(tail(head(y0))), z1)) -> c8(MARK(incr(tail(head(y0))))) A__HEAD(cons(incr(tail(tail(y0))), z1)) -> c8(MARK(incr(tail(tail(y0))))) A__HEAD(cons(incr(tail(cons(y0, y1))), z1)) -> c8(MARK(incr(tail(cons(y0, y1))))) A__HEAD(cons(incr(tail(s(y0))), z1)) -> c8(MARK(incr(tail(s(y0))))) A__HEAD(cons(incr(tail(y0)), z1)) -> c8(MARK(incr(tail(y0)))) A__HEAD(cons(head(incr(incr(y0))), z1)) -> c8(MARK(head(incr(incr(y0))))) A__HEAD(cons(head(incr(odds)), z1)) -> c8(MARK(head(incr(odds)))) A__HEAD(cons(head(incr(head(y0))), z1)) -> c8(MARK(head(incr(head(y0))))) A__HEAD(cons(head(incr(tail(y0))), z1)) -> c8(MARK(head(incr(tail(y0))))) A__HEAD(cons(head(incr(cons(y0, y1))), z1)) -> 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c8(MARK(head(tail(pairs)))) A__HEAD(cons(head(tail(odds)), z1)) -> c8(MARK(head(tail(odds)))) A__HEAD(cons(head(tail(head(y0))), z1)) -> c8(MARK(head(tail(head(y0))))) A__HEAD(cons(head(tail(tail(y0))), z1)) -> c8(MARK(head(tail(tail(y0))))) A__HEAD(cons(head(tail(cons(y0, y1))), z1)) -> c8(MARK(head(tail(cons(y0, y1))))) A__HEAD(cons(head(tail(s(y0))), z1)) -> c8(MARK(head(tail(s(y0))))) A__HEAD(cons(head(tail(y0)), z1)) -> c8(MARK(head(tail(y0)))) A__HEAD(cons(tail(incr(incr(y0))), z1)) -> c8(MARK(tail(incr(incr(y0))))) A__HEAD(cons(tail(incr(odds)), z1)) -> c8(MARK(tail(incr(odds)))) A__HEAD(cons(tail(incr(head(y0))), z1)) -> c8(MARK(tail(incr(head(y0))))) A__HEAD(cons(tail(incr(tail(y0))), z1)) -> c8(MARK(tail(incr(tail(y0))))) A__HEAD(cons(tail(incr(cons(y0, y1))), z1)) -> c8(MARK(tail(incr(cons(y0, y1))))) A__HEAD(cons(tail(incr(s(y0))), z1)) -> c8(MARK(tail(incr(s(y0))))) A__HEAD(cons(tail(incr(y0)), z1)) -> c8(MARK(tail(incr(y0)))) A__HEAD(cons(tail(head(nats)), z1)) -> c8(MARK(tail(head(nats)))) A__HEAD(cons(tail(head(incr(y0))), z1)) -> c8(MARK(tail(head(incr(y0))))) A__HEAD(cons(tail(head(pairs)), z1)) -> c8(MARK(tail(head(pairs)))) A__HEAD(cons(tail(head(odds)), z1)) -> c8(MARK(tail(head(odds)))) A__HEAD(cons(tail(head(head(y0))), z1)) -> c8(MARK(tail(head(head(y0))))) A__HEAD(cons(tail(head(tail(y0))), z1)) -> c8(MARK(tail(head(tail(y0))))) A__HEAD(cons(tail(head(cons(y0, y1))), z1)) -> c8(MARK(tail(head(cons(y0, y1))))) A__HEAD(cons(tail(head(s(y0))), z1)) -> c8(MARK(tail(head(s(y0))))) A__HEAD(cons(tail(head(y0)), z1)) -> c8(MARK(tail(head(y0)))) A__HEAD(cons(tail(tail(nats)), z1)) -> c8(MARK(tail(tail(nats)))) A__HEAD(cons(tail(tail(incr(y0))), z1)) -> c8(MARK(tail(tail(incr(y0))))) A__HEAD(cons(tail(tail(pairs)), z1)) -> c8(MARK(tail(tail(pairs)))) A__HEAD(cons(tail(tail(odds)), z1)) -> c8(MARK(tail(tail(odds)))) A__HEAD(cons(tail(tail(head(y0))), z1)) -> c8(MARK(tail(tail(head(y0))))) A__HEAD(cons(tail(tail(tail(y0))), z1)) -> c8(MARK(tail(tail(tail(y0))))) A__HEAD(cons(tail(tail(cons(y0, y1))), z1)) -> c8(MARK(tail(tail(cons(y0, y1))))) A__HEAD(cons(tail(tail(s(y0))), z1)) -> c8(MARK(tail(tail(s(y0))))) A__HEAD(cons(tail(tail(y0)), z1)) -> c8(MARK(tail(tail(y0)))) A__HEAD(cons(tail(nats), z1)) -> c8(MARK(tail(nats))) A__HEAD(cons(tail(pairs), z1)) -> c8(MARK(tail(pairs))) A__HEAD(cons(incr(incr(nats)), z1)) -> c8(MARK(incr(incr(nats)))) A__HEAD(cons(incr(incr(pairs)), z1)) -> c8(MARK(incr(incr(pairs)))) A__HEAD(cons(incr(odds), z1)) -> c8(MARK(incr(odds))) A__HEAD(cons(head(incr(nats)), z1)) -> c8(MARK(head(incr(nats)))) A__HEAD(cons(head(incr(pairs)), z1)) -> c8(MARK(head(incr(pairs)))) A__HEAD(cons(head(odds), z1)) -> c8(MARK(head(odds))) A__HEAD(cons(tail(incr(nats)), z1)) -> c8(MARK(tail(incr(nats)))) A__HEAD(cons(tail(incr(pairs)), z1)) -> c8(MARK(tail(incr(pairs)))) A__HEAD(cons(tail(odds), z1)) -> c8(MARK(tail(odds))) A__HEAD(cons(incr(cons(nats, y0)), z1)) -> c8(MARK(incr(cons(nats, y0)))) A__HEAD(cons(incr(cons(incr(y0), y1)), z1)) -> c8(MARK(incr(cons(incr(y0), y1)))) A__HEAD(cons(incr(cons(pairs, y0)), z1)) -> c8(MARK(incr(cons(pairs, y0)))) A__HEAD(cons(incr(cons(odds, y0)), z1)) -> c8(MARK(incr(cons(odds, y0)))) A__HEAD(cons(incr(cons(head(y0), y1)), z1)) -> c8(MARK(incr(cons(head(y0), y1)))) A__HEAD(cons(incr(cons(tail(y0), y1)), z1)) -> c8(MARK(incr(cons(tail(y0), y1)))) A__HEAD(cons(incr(cons(cons(y0, y1), y2)), z1)) -> c8(MARK(incr(cons(cons(y0, y1), y2)))) A__HEAD(cons(incr(cons(s(y0), y1)), z1)) -> c8(MARK(incr(cons(s(y0), y1)))) A__HEAD(cons(incr(cons(y0, y1)), z1)) -> c8(MARK(incr(cons(y0, y1)))) A__HEAD(cons(incr(cons(0, y0)), z1)) -> c8(MARK(incr(cons(0, y0)))) K tuples:none Defined Rule Symbols: a__nats, a__pairs, a__odds, a__incr_1, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__TAIL_1, MARK_1, A__INCR_1, A__HEAD_1 Compound Symbols: c10_1, c18_1, c20_1, c13_1, c16_2, c16_1, c17_2, c17_1, c13_2, c6_1, c8_1 ---------------------------------------- (173) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 6 trailing tuple parts ---------------------------------------- (174) Obligation: Complexity Dependency Tuples Problem Rules: a__nats -> cons(0, incr(nats)) a__nats -> nats a__pairs -> cons(0, incr(odds)) a__pairs -> pairs a__odds -> a__incr(a__pairs) a__odds -> odds a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) 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(nats) -> a__nats mark(incr(z0)) -> a__incr(mark(z0)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c17(MARK(s(z0))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) 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MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> 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c6(MARK(head(tail(s(y0))))) A__INCR(cons(head(tail(y0)), z1)) -> c6(MARK(head(tail(y0)))) A__INCR(cons(tail(incr(nats)), z1)) -> c6(MARK(tail(incr(nats)))) A__INCR(cons(tail(incr(incr(y0))), z1)) -> c6(MARK(tail(incr(incr(y0))))) A__INCR(cons(tail(incr(pairs)), z1)) -> c6(MARK(tail(incr(pairs)))) A__INCR(cons(tail(incr(odds)), z1)) -> c6(MARK(tail(incr(odds)))) A__INCR(cons(tail(incr(head(y0))), z1)) -> c6(MARK(tail(incr(head(y0))))) A__INCR(cons(tail(incr(tail(y0))), z1)) -> c6(MARK(tail(incr(tail(y0))))) A__INCR(cons(tail(incr(cons(y0, y1))), z1)) -> c6(MARK(tail(incr(cons(y0, y1))))) A__INCR(cons(tail(incr(s(y0))), z1)) -> c6(MARK(tail(incr(s(y0))))) A__INCR(cons(tail(incr(y0)), z1)) -> c6(MARK(tail(incr(y0)))) A__INCR(cons(tail(odds), z1)) -> c6(MARK(tail(odds))) A__INCR(cons(tail(head(nats)), z1)) -> c6(MARK(tail(head(nats)))) A__INCR(cons(tail(head(incr(y0))), z1)) -> c6(MARK(tail(head(incr(y0))))) A__INCR(cons(tail(head(pairs)), z1)) -> c6(MARK(tail(head(pairs)))) 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A__INCR(cons(tail(tail(y0)), z1)) -> c6(MARK(tail(tail(y0)))) A__INCR(cons(tail(nats), z1)) -> c6(MARK(tail(nats))) A__INCR(cons(tail(pairs), z1)) -> c6(MARK(tail(pairs))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs))) MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs))) MARK(tail(incr(nats))) -> c17(A__TAIL(a__incr(a__nats))) MARK(tail(incr(pairs))) -> c17(A__TAIL(a__incr(a__pairs))) MARK(tail(odds)) -> c17(A__TAIL(a__incr(a__pairs))) MARK(incr(cons(nats, x1))) -> c13(A__INCR(cons(a__nats, x1)), MARK(cons(nats, x1))) MARK(incr(cons(incr(z0), x1))) -> c13(A__INCR(cons(a__incr(mark(z0)), x1)), MARK(cons(incr(z0), x1))) MARK(incr(cons(pairs, x1))) -> c13(A__INCR(cons(a__pairs, x1)), MARK(cons(pairs, x1))) MARK(incr(cons(odds, x1))) -> c13(A__INCR(cons(a__odds, x1)), MARK(cons(odds, x1))) MARK(incr(cons(head(z0), x1))) -> c13(A__INCR(cons(a__head(mark(z0)), x1)), MARK(cons(head(z0), x1))) MARK(incr(cons(tail(z0), x1))) -> c13(A__INCR(cons(a__tail(mark(z0)), x1)), MARK(cons(tail(z0), x1))) MARK(incr(cons(cons(z0, z1), x1))) -> c13(A__INCR(cons(cons(mark(z0), z1), x1)), MARK(cons(cons(z0, z1), x1))) MARK(incr(cons(s(z0), x1))) -> c13(A__INCR(cons(s(mark(z0)), x1)), MARK(cons(s(z0), x1))) MARK(incr(cons(x0, x1))) -> c13(MARK(cons(x0, x1))) MARK(incr(cons(0, x1))) -> c13(MARK(cons(0, x1))) A__HEAD(cons(cons(y0, y1), z1)) -> c8(MARK(cons(y0, y1))) A__HEAD(cons(s(y0), z1)) -> c8(MARK(s(y0))) A__HEAD(cons(incr(s(y0)), z1)) -> c8(MARK(incr(s(y0)))) A__HEAD(cons(head(cons(y0, y1)), z1)) -> c8(MARK(head(cons(y0, y1)))) A__HEAD(cons(head(s(y0)), z1)) -> c8(MARK(head(s(y0)))) A__HEAD(cons(tail(cons(y0, y1)), z1)) -> c8(MARK(tail(cons(y0, y1)))) A__HEAD(cons(tail(s(y0)), z1)) -> c8(MARK(tail(s(y0)))) A__HEAD(cons(incr(incr(incr(y0))), z1)) -> c8(MARK(incr(incr(incr(y0))))) A__HEAD(cons(incr(incr(odds)), z1)) -> c8(MARK(incr(incr(odds)))) A__HEAD(cons(incr(incr(head(y0))), z1)) -> c8(MARK(incr(incr(head(y0))))) A__HEAD(cons(incr(incr(tail(y0))), z1)) -> c8(MARK(incr(incr(tail(y0))))) A__HEAD(cons(incr(incr(cons(y0, y1))), z1)) -> c8(MARK(incr(incr(cons(y0, y1))))) A__HEAD(cons(incr(incr(s(y0))), z1)) -> c8(MARK(incr(incr(s(y0))))) A__HEAD(cons(incr(incr(y0)), z1)) -> c8(MARK(incr(incr(y0)))) A__HEAD(cons(incr(head(nats)), z1)) -> c8(MARK(incr(head(nats)))) A__HEAD(cons(incr(head(incr(y0))), z1)) -> c8(MARK(incr(head(incr(y0))))) A__HEAD(cons(incr(head(pairs)), z1)) -> c8(MARK(incr(head(pairs)))) A__HEAD(cons(incr(head(odds)), z1)) -> c8(MARK(incr(head(odds)))) A__HEAD(cons(incr(head(head(y0))), z1)) -> c8(MARK(incr(head(head(y0))))) A__HEAD(cons(incr(head(tail(y0))), z1)) -> c8(MARK(incr(head(tail(y0))))) A__HEAD(cons(incr(head(cons(y0, y1))), z1)) -> c8(MARK(incr(head(cons(y0, y1))))) A__HEAD(cons(incr(head(s(y0))), z1)) -> c8(MARK(incr(head(s(y0))))) A__HEAD(cons(incr(head(y0)), z1)) -> c8(MARK(incr(head(y0)))) A__HEAD(cons(incr(tail(nats)), z1)) -> c8(MARK(incr(tail(nats)))) A__HEAD(cons(incr(tail(incr(y0))), z1)) -> c8(MARK(incr(tail(incr(y0))))) A__HEAD(cons(incr(tail(pairs)), z1)) -> c8(MARK(incr(tail(pairs)))) A__HEAD(cons(incr(tail(odds)), z1)) -> c8(MARK(incr(tail(odds)))) A__HEAD(cons(incr(tail(head(y0))), z1)) -> c8(MARK(incr(tail(head(y0))))) A__HEAD(cons(incr(tail(tail(y0))), z1)) -> c8(MARK(incr(tail(tail(y0))))) A__HEAD(cons(incr(tail(cons(y0, y1))), z1)) -> c8(MARK(incr(tail(cons(y0, y1))))) A__HEAD(cons(incr(tail(s(y0))), z1)) -> c8(MARK(incr(tail(s(y0))))) A__HEAD(cons(incr(tail(y0)), z1)) -> c8(MARK(incr(tail(y0)))) A__HEAD(cons(head(incr(incr(y0))), z1)) -> c8(MARK(head(incr(incr(y0))))) A__HEAD(cons(head(incr(odds)), z1)) -> c8(MARK(head(incr(odds)))) A__HEAD(cons(head(incr(head(y0))), z1)) -> c8(MARK(head(incr(head(y0))))) A__HEAD(cons(head(incr(tail(y0))), z1)) -> c8(MARK(head(incr(tail(y0))))) A__HEAD(cons(head(incr(cons(y0, y1))), z1)) -> c8(MARK(head(incr(cons(y0, y1))))) A__HEAD(cons(head(incr(s(y0))), z1)) -> c8(MARK(head(incr(s(y0))))) A__HEAD(cons(head(incr(y0)), z1)) -> c8(MARK(head(incr(y0)))) A__HEAD(cons(head(head(nats)), z1)) -> c8(MARK(head(head(nats)))) A__HEAD(cons(head(head(incr(y0))), z1)) -> c8(MARK(head(head(incr(y0))))) A__HEAD(cons(head(head(pairs)), z1)) -> c8(MARK(head(head(pairs)))) A__HEAD(cons(head(head(odds)), z1)) -> c8(MARK(head(head(odds)))) A__HEAD(cons(head(head(head(y0))), z1)) -> c8(MARK(head(head(head(y0))))) A__HEAD(cons(head(head(tail(y0))), z1)) -> c8(MARK(head(head(tail(y0))))) A__HEAD(cons(head(head(cons(y0, y1))), z1)) -> c8(MARK(head(head(cons(y0, y1))))) A__HEAD(cons(head(head(s(y0))), z1)) -> c8(MARK(head(head(s(y0))))) A__HEAD(cons(head(head(y0)), z1)) -> c8(MARK(head(head(y0)))) A__HEAD(cons(head(tail(nats)), z1)) -> c8(MARK(head(tail(nats)))) A__HEAD(cons(head(tail(incr(y0))), z1)) -> c8(MARK(head(tail(incr(y0))))) A__HEAD(cons(head(tail(pairs)), z1)) -> c8(MARK(head(tail(pairs)))) A__HEAD(cons(head(tail(odds)), z1)) -> c8(MARK(head(tail(odds)))) A__HEAD(cons(head(tail(head(y0))), z1)) -> c8(MARK(head(tail(head(y0))))) A__HEAD(cons(head(tail(tail(y0))), z1)) -> c8(MARK(head(tail(tail(y0))))) A__HEAD(cons(head(tail(cons(y0, y1))), z1)) -> c8(MARK(head(tail(cons(y0, y1))))) A__HEAD(cons(head(tail(s(y0))), z1)) -> c8(MARK(head(tail(s(y0))))) A__HEAD(cons(head(tail(y0)), z1)) -> c8(MARK(head(tail(y0)))) A__HEAD(cons(tail(incr(incr(y0))), z1)) -> c8(MARK(tail(incr(incr(y0))))) A__HEAD(cons(tail(incr(odds)), z1)) -> c8(MARK(tail(incr(odds)))) A__HEAD(cons(tail(incr(head(y0))), z1)) -> c8(MARK(tail(incr(head(y0))))) A__HEAD(cons(tail(incr(tail(y0))), z1)) -> c8(MARK(tail(incr(tail(y0))))) A__HEAD(cons(tail(incr(cons(y0, y1))), z1)) -> c8(MARK(tail(incr(cons(y0, y1))))) A__HEAD(cons(tail(incr(s(y0))), z1)) -> c8(MARK(tail(incr(s(y0))))) A__HEAD(cons(tail(incr(y0)), z1)) -> c8(MARK(tail(incr(y0)))) A__HEAD(cons(tail(head(nats)), z1)) -> c8(MARK(tail(head(nats)))) A__HEAD(cons(tail(head(incr(y0))), z1)) -> c8(MARK(tail(head(incr(y0))))) A__HEAD(cons(tail(head(pairs)), z1)) -> c8(MARK(tail(head(pairs)))) A__HEAD(cons(tail(head(odds)), z1)) -> c8(MARK(tail(head(odds)))) A__HEAD(cons(tail(head(head(y0))), z1)) -> c8(MARK(tail(head(head(y0))))) A__HEAD(cons(tail(head(tail(y0))), z1)) -> c8(MARK(tail(head(tail(y0))))) A__HEAD(cons(tail(head(cons(y0, y1))), z1)) -> c8(MARK(tail(head(cons(y0, y1))))) A__HEAD(cons(tail(head(s(y0))), z1)) -> c8(MARK(tail(head(s(y0))))) A__HEAD(cons(tail(head(y0)), z1)) -> c8(MARK(tail(head(y0)))) A__HEAD(cons(tail(tail(nats)), z1)) -> c8(MARK(tail(tail(nats)))) A__HEAD(cons(tail(tail(incr(y0))), z1)) -> c8(MARK(tail(tail(incr(y0))))) A__HEAD(cons(tail(tail(pairs)), z1)) -> c8(MARK(tail(tail(pairs)))) A__HEAD(cons(tail(tail(odds)), z1)) -> c8(MARK(tail(tail(odds)))) A__HEAD(cons(tail(tail(head(y0))), z1)) -> c8(MARK(tail(tail(head(y0))))) A__HEAD(cons(tail(tail(tail(y0))), z1)) -> c8(MARK(tail(tail(tail(y0))))) A__HEAD(cons(tail(tail(cons(y0, y1))), z1)) -> c8(MARK(tail(tail(cons(y0, y1))))) A__HEAD(cons(tail(tail(s(y0))), z1)) -> c8(MARK(tail(tail(s(y0))))) A__HEAD(cons(tail(tail(y0)), z1)) -> c8(MARK(tail(tail(y0)))) A__HEAD(cons(tail(nats), z1)) -> c8(MARK(tail(nats))) A__HEAD(cons(tail(pairs), z1)) -> c8(MARK(tail(pairs))) A__HEAD(cons(incr(incr(nats)), z1)) -> c8(MARK(incr(incr(nats)))) A__HEAD(cons(incr(incr(pairs)), z1)) -> c8(MARK(incr(incr(pairs)))) A__HEAD(cons(incr(odds), z1)) -> c8(MARK(incr(odds))) A__HEAD(cons(head(incr(nats)), z1)) -> c8(MARK(head(incr(nats)))) A__HEAD(cons(head(incr(pairs)), z1)) -> c8(MARK(head(incr(pairs)))) A__HEAD(cons(head(odds), z1)) -> c8(MARK(head(odds))) A__HEAD(cons(tail(incr(nats)), z1)) -> c8(MARK(tail(incr(nats)))) A__HEAD(cons(tail(incr(pairs)), z1)) -> c8(MARK(tail(incr(pairs)))) A__HEAD(cons(tail(odds), z1)) -> c8(MARK(tail(odds))) A__HEAD(cons(incr(cons(nats, y0)), z1)) -> c8(MARK(incr(cons(nats, y0)))) A__HEAD(cons(incr(cons(incr(y0), y1)), z1)) -> c8(MARK(incr(cons(incr(y0), y1)))) A__HEAD(cons(incr(cons(pairs, y0)), z1)) -> c8(MARK(incr(cons(pairs, y0)))) A__HEAD(cons(incr(cons(odds, y0)), z1)) -> c8(MARK(incr(cons(odds, y0)))) A__HEAD(cons(incr(cons(head(y0), y1)), z1)) -> c8(MARK(incr(cons(head(y0), y1)))) A__HEAD(cons(incr(cons(tail(y0), y1)), z1)) -> c8(MARK(incr(cons(tail(y0), y1)))) A__HEAD(cons(incr(cons(cons(y0, y1), y2)), z1)) -> c8(MARK(incr(cons(cons(y0, y1), y2)))) A__HEAD(cons(incr(cons(s(y0), y1)), z1)) -> c8(MARK(incr(cons(s(y0), y1)))) A__HEAD(cons(incr(cons(y0, y1)), z1)) -> c8(MARK(incr(cons(y0, y1)))) A__HEAD(cons(incr(cons(0, y0)), z1)) -> c8(MARK(incr(cons(0, y0)))) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs))) MARK(head(head(nats))) -> c16(A__HEAD(a__head(a__nats))) MARK(head(head(pairs))) -> c16(A__HEAD(a__head(a__pairs))) MARK(tail(head(nats))) -> c17(A__TAIL(a__head(a__nats))) MARK(tail(head(pairs))) -> c17(A__TAIL(a__head(a__pairs))) K tuples:none Defined Rule Symbols: a__nats, a__pairs, a__odds, a__incr_1, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__TAIL_1, MARK_1, A__INCR_1, A__HEAD_1 Compound Symbols: c10_1, c18_1, c20_1, c13_1, c16_2, c16_1, c17_2, c17_1, c13_2, c6_1, c8_1 ---------------------------------------- (175) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(head(cons(z0, z1))) -> c16(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) by MARK(head(cons(nats, x1))) -> c16(A__HEAD(cons(a__nats, x1)), MARK(cons(nats, x1))) MARK(head(cons(incr(z0), x1))) -> c16(A__HEAD(cons(a__incr(mark(z0)), x1)), MARK(cons(incr(z0), x1))) MARK(head(cons(pairs, x1))) -> c16(A__HEAD(cons(a__pairs, x1)), MARK(cons(pairs, x1))) MARK(head(cons(odds, x1))) -> c16(A__HEAD(cons(a__odds, x1)), MARK(cons(odds, x1))) MARK(head(cons(head(z0), x1))) -> c16(A__HEAD(cons(a__head(mark(z0)), x1)), MARK(cons(head(z0), x1))) MARK(head(cons(tail(z0), x1))) -> c16(A__HEAD(cons(a__tail(mark(z0)), x1)), MARK(cons(tail(z0), x1))) MARK(head(cons(cons(z0, z1), x1))) -> c16(A__HEAD(cons(cons(mark(z0), z1), x1)), MARK(cons(cons(z0, z1), x1))) MARK(head(cons(0, x1))) -> c16(A__HEAD(cons(0, x1)), MARK(cons(0, x1))) MARK(head(cons(s(z0), x1))) -> c16(A__HEAD(cons(s(mark(z0)), x1)), MARK(cons(s(z0), x1))) MARK(head(cons(x0, x1))) -> c16(MARK(cons(x0, x1))) ---------------------------------------- (176) Obligation: Complexity Dependency Tuples Problem Rules: a__nats -> cons(0, incr(nats)) a__nats -> nats a__pairs -> cons(0, incr(odds)) a__pairs -> pairs a__odds -> a__incr(a__pairs) a__odds -> odds a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) 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(nats) -> a__nats mark(incr(z0)) -> a__incr(mark(z0)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c17(MARK(s(z0))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> 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A__HEAD(cons(tail(head(pairs)), z1)) -> c8(MARK(tail(head(pairs)))) A__HEAD(cons(tail(head(odds)), z1)) -> c8(MARK(tail(head(odds)))) A__HEAD(cons(tail(head(head(y0))), z1)) -> c8(MARK(tail(head(head(y0))))) A__HEAD(cons(tail(head(tail(y0))), z1)) -> c8(MARK(tail(head(tail(y0))))) A__HEAD(cons(tail(head(cons(y0, y1))), z1)) -> c8(MARK(tail(head(cons(y0, y1))))) A__HEAD(cons(tail(head(s(y0))), z1)) -> c8(MARK(tail(head(s(y0))))) A__HEAD(cons(tail(head(y0)), z1)) -> c8(MARK(tail(head(y0)))) A__HEAD(cons(tail(tail(nats)), z1)) -> c8(MARK(tail(tail(nats)))) A__HEAD(cons(tail(tail(incr(y0))), z1)) -> c8(MARK(tail(tail(incr(y0))))) A__HEAD(cons(tail(tail(pairs)), z1)) -> c8(MARK(tail(tail(pairs)))) A__HEAD(cons(tail(tail(odds)), z1)) -> c8(MARK(tail(tail(odds)))) A__HEAD(cons(tail(tail(head(y0))), z1)) -> c8(MARK(tail(tail(head(y0))))) A__HEAD(cons(tail(tail(tail(y0))), z1)) -> c8(MARK(tail(tail(tail(y0))))) A__HEAD(cons(tail(tail(cons(y0, y1))), z1)) -> c8(MARK(tail(tail(cons(y0, y1))))) A__HEAD(cons(tail(tail(s(y0))), z1)) -> c8(MARK(tail(tail(s(y0))))) A__HEAD(cons(tail(tail(y0)), z1)) -> c8(MARK(tail(tail(y0)))) A__HEAD(cons(tail(nats), z1)) -> c8(MARK(tail(nats))) A__HEAD(cons(tail(pairs), z1)) -> c8(MARK(tail(pairs))) A__HEAD(cons(incr(incr(nats)), z1)) -> c8(MARK(incr(incr(nats)))) A__HEAD(cons(incr(incr(pairs)), z1)) -> c8(MARK(incr(incr(pairs)))) A__HEAD(cons(incr(odds), z1)) -> c8(MARK(incr(odds))) A__HEAD(cons(head(incr(nats)), z1)) -> c8(MARK(head(incr(nats)))) A__HEAD(cons(head(incr(pairs)), z1)) -> c8(MARK(head(incr(pairs)))) A__HEAD(cons(head(odds), z1)) -> c8(MARK(head(odds))) A__HEAD(cons(tail(incr(nats)), z1)) -> c8(MARK(tail(incr(nats)))) A__HEAD(cons(tail(incr(pairs)), z1)) -> c8(MARK(tail(incr(pairs)))) A__HEAD(cons(tail(odds), z1)) -> c8(MARK(tail(odds))) A__HEAD(cons(incr(cons(nats, y0)), z1)) -> c8(MARK(incr(cons(nats, y0)))) A__HEAD(cons(incr(cons(incr(y0), y1)), z1)) -> c8(MARK(incr(cons(incr(y0), y1)))) A__HEAD(cons(incr(cons(pairs, y0)), z1)) -> c8(MARK(incr(cons(pairs, y0)))) A__HEAD(cons(incr(cons(odds, y0)), z1)) -> c8(MARK(incr(cons(odds, y0)))) A__HEAD(cons(incr(cons(head(y0), y1)), z1)) -> c8(MARK(incr(cons(head(y0), y1)))) A__HEAD(cons(incr(cons(tail(y0), y1)), z1)) -> c8(MARK(incr(cons(tail(y0), y1)))) A__HEAD(cons(incr(cons(cons(y0, y1), y2)), z1)) -> c8(MARK(incr(cons(cons(y0, y1), y2)))) A__HEAD(cons(incr(cons(s(y0), y1)), z1)) -> c8(MARK(incr(cons(s(y0), y1)))) A__HEAD(cons(incr(cons(y0, y1)), z1)) -> c8(MARK(incr(cons(y0, y1)))) A__HEAD(cons(incr(cons(0, y0)), z1)) -> c8(MARK(incr(cons(0, y0)))) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs))) MARK(head(head(nats))) -> c16(A__HEAD(a__head(a__nats))) MARK(head(head(pairs))) -> c16(A__HEAD(a__head(a__pairs))) MARK(tail(head(nats))) -> c17(A__TAIL(a__head(a__nats))) MARK(tail(head(pairs))) -> c17(A__TAIL(a__head(a__pairs))) MARK(head(cons(nats, x1))) -> c16(A__HEAD(cons(a__nats, x1)), MARK(cons(nats, x1))) MARK(head(cons(incr(z0), x1))) -> c16(A__HEAD(cons(a__incr(mark(z0)), x1)), MARK(cons(incr(z0), x1))) MARK(head(cons(pairs, x1))) -> c16(A__HEAD(cons(a__pairs, x1)), MARK(cons(pairs, x1))) MARK(head(cons(odds, x1))) -> c16(A__HEAD(cons(a__odds, x1)), MARK(cons(odds, x1))) MARK(head(cons(head(z0), x1))) -> c16(A__HEAD(cons(a__head(mark(z0)), x1)), MARK(cons(head(z0), x1))) MARK(head(cons(tail(z0), x1))) -> c16(A__HEAD(cons(a__tail(mark(z0)), x1)), MARK(cons(tail(z0), x1))) MARK(head(cons(cons(z0, z1), x1))) -> c16(A__HEAD(cons(cons(mark(z0), z1), x1)), MARK(cons(cons(z0, z1), x1))) MARK(head(cons(0, x1))) -> c16(A__HEAD(cons(0, x1)), MARK(cons(0, x1))) MARK(head(cons(s(z0), x1))) -> c16(A__HEAD(cons(s(mark(z0)), x1)), MARK(cons(s(z0), x1))) MARK(head(cons(x0, x1))) -> c16(MARK(cons(x0, x1))) S tuples: A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c17(MARK(s(z0))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(tail(incr(incr(z0)))) -> c17(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(odds))) -> c17(A__TAIL(a__incr(a__odds)), MARK(incr(odds))) MARK(tail(incr(head(z0)))) -> c17(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c17(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c17(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c17(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c17(MARK(incr(x0))) MARK(tail(head(incr(z0)))) -> c17(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(odds))) -> c17(A__TAIL(a__head(a__odds)), MARK(head(odds))) MARK(tail(head(head(z0)))) -> c17(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c17(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c17(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, 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y1), z1)) -> c6(MARK(cons(y0, y1))) A__INCR(cons(s(y0), z1)) -> c6(MARK(s(y0))) A__INCR(cons(incr(cons(y0, y1)), z1)) -> c6(MARK(incr(cons(y0, y1)))) A__INCR(cons(incr(s(y0)), z1)) -> c6(MARK(incr(s(y0)))) A__INCR(cons(head(cons(y0, y1)), z1)) -> c6(MARK(head(cons(y0, y1)))) A__INCR(cons(head(s(y0)), z1)) -> c6(MARK(head(s(y0)))) A__INCR(cons(tail(cons(y0, y1)), z1)) -> c6(MARK(tail(cons(y0, y1)))) A__INCR(cons(tail(s(y0)), z1)) -> c6(MARK(tail(s(y0)))) A__INCR(cons(incr(incr(nats)), z1)) -> c6(MARK(incr(incr(nats)))) A__INCR(cons(incr(incr(incr(y0))), z1)) -> c6(MARK(incr(incr(incr(y0))))) A__INCR(cons(incr(incr(pairs)), z1)) -> c6(MARK(incr(incr(pairs)))) A__INCR(cons(incr(incr(odds)), z1)) -> c6(MARK(incr(incr(odds)))) A__INCR(cons(incr(incr(head(y0))), z1)) -> c6(MARK(incr(incr(head(y0))))) A__INCR(cons(incr(incr(tail(y0))), z1)) -> c6(MARK(incr(incr(tail(y0))))) A__INCR(cons(incr(incr(cons(y0, y1))), z1)) -> c6(MARK(incr(incr(cons(y0, y1))))) A__INCR(cons(incr(incr(s(y0))), z1)) -> c6(MARK(incr(incr(s(y0))))) A__INCR(cons(incr(incr(y0)), z1)) -> c6(MARK(incr(incr(y0)))) A__INCR(cons(incr(odds), z1)) -> c6(MARK(incr(odds))) A__INCR(cons(incr(head(nats)), z1)) -> c6(MARK(incr(head(nats)))) A__INCR(cons(incr(head(incr(y0))), z1)) -> c6(MARK(incr(head(incr(y0))))) A__INCR(cons(incr(head(pairs)), z1)) -> c6(MARK(incr(head(pairs)))) A__INCR(cons(incr(head(odds)), z1)) -> c6(MARK(incr(head(odds)))) A__INCR(cons(incr(head(head(y0))), z1)) -> c6(MARK(incr(head(head(y0))))) A__INCR(cons(incr(head(tail(y0))), z1)) -> c6(MARK(incr(head(tail(y0))))) A__INCR(cons(incr(head(cons(y0, y1))), z1)) -> c6(MARK(incr(head(cons(y0, y1))))) A__INCR(cons(incr(head(s(y0))), z1)) -> c6(MARK(incr(head(s(y0))))) A__INCR(cons(incr(head(y0)), z1)) -> c6(MARK(incr(head(y0)))) A__INCR(cons(incr(tail(nats)), z1)) -> c6(MARK(incr(tail(nats)))) A__INCR(cons(incr(tail(incr(y0))), z1)) -> c6(MARK(incr(tail(incr(y0))))) A__INCR(cons(incr(tail(pairs)), z1)) -> c6(MARK(incr(tail(pairs)))) A__INCR(cons(incr(tail(odds)), z1)) -> c6(MARK(incr(tail(odds)))) A__INCR(cons(incr(tail(head(y0))), z1)) -> c6(MARK(incr(tail(head(y0))))) A__INCR(cons(incr(tail(tail(y0))), z1)) -> c6(MARK(incr(tail(tail(y0))))) A__INCR(cons(incr(tail(cons(y0, y1))), z1)) -> c6(MARK(incr(tail(cons(y0, y1))))) A__INCR(cons(incr(tail(s(y0))), z1)) -> c6(MARK(incr(tail(s(y0))))) A__INCR(cons(incr(tail(y0)), z1)) -> c6(MARK(incr(tail(y0)))) A__INCR(cons(head(incr(nats)), z1)) -> c6(MARK(head(incr(nats)))) A__INCR(cons(head(incr(incr(y0))), z1)) -> c6(MARK(head(incr(incr(y0))))) A__INCR(cons(head(incr(pairs)), z1)) -> c6(MARK(head(incr(pairs)))) A__INCR(cons(head(incr(odds)), z1)) -> c6(MARK(head(incr(odds)))) A__INCR(cons(head(incr(head(y0))), z1)) -> c6(MARK(head(incr(head(y0))))) A__INCR(cons(head(incr(tail(y0))), z1)) -> c6(MARK(head(incr(tail(y0))))) A__INCR(cons(head(incr(cons(y0, y1))), z1)) -> c6(MARK(head(incr(cons(y0, y1))))) A__INCR(cons(head(incr(s(y0))), z1)) -> c6(MARK(head(incr(s(y0))))) A__INCR(cons(head(incr(y0)), z1)) -> c6(MARK(head(incr(y0)))) A__INCR(cons(head(odds), z1)) -> c6(MARK(head(odds))) A__INCR(cons(head(head(nats)), z1)) -> c6(MARK(head(head(nats)))) A__INCR(cons(head(head(incr(y0))), z1)) -> c6(MARK(head(head(incr(y0))))) A__INCR(cons(head(head(pairs)), z1)) -> c6(MARK(head(head(pairs)))) A__INCR(cons(head(head(odds)), z1)) -> c6(MARK(head(head(odds)))) A__INCR(cons(head(head(head(y0))), z1)) -> c6(MARK(head(head(head(y0))))) A__INCR(cons(head(head(tail(y0))), z1)) -> c6(MARK(head(head(tail(y0))))) A__INCR(cons(head(head(cons(y0, y1))), z1)) -> c6(MARK(head(head(cons(y0, y1))))) A__INCR(cons(head(head(s(y0))), z1)) -> c6(MARK(head(head(s(y0))))) A__INCR(cons(head(head(y0)), z1)) -> c6(MARK(head(head(y0)))) A__INCR(cons(head(tail(nats)), z1)) -> c6(MARK(head(tail(nats)))) A__INCR(cons(head(tail(incr(y0))), z1)) -> c6(MARK(head(tail(incr(y0))))) A__INCR(cons(head(tail(pairs)), z1)) -> c6(MARK(head(tail(pairs)))) A__INCR(cons(head(tail(odds)), z1)) -> c6(MARK(head(tail(odds)))) A__INCR(cons(head(tail(head(y0))), z1)) -> c6(MARK(head(tail(head(y0))))) A__INCR(cons(head(tail(tail(y0))), z1)) -> c6(MARK(head(tail(tail(y0))))) A__INCR(cons(head(tail(cons(y0, y1))), z1)) -> c6(MARK(head(tail(cons(y0, y1))))) A__INCR(cons(head(tail(s(y0))), z1)) -> c6(MARK(head(tail(s(y0))))) A__INCR(cons(head(tail(y0)), z1)) -> c6(MARK(head(tail(y0)))) A__INCR(cons(tail(incr(nats)), z1)) -> c6(MARK(tail(incr(nats)))) A__INCR(cons(tail(incr(incr(y0))), z1)) -> c6(MARK(tail(incr(incr(y0))))) A__INCR(cons(tail(incr(pairs)), z1)) -> c6(MARK(tail(incr(pairs)))) A__INCR(cons(tail(incr(odds)), z1)) -> c6(MARK(tail(incr(odds)))) A__INCR(cons(tail(incr(head(y0))), z1)) -> c6(MARK(tail(incr(head(y0))))) A__INCR(cons(tail(incr(tail(y0))), z1)) -> c6(MARK(tail(incr(tail(y0))))) A__INCR(cons(tail(incr(cons(y0, y1))), z1)) -> c6(MARK(tail(incr(cons(y0, y1))))) A__INCR(cons(tail(incr(s(y0))), z1)) -> c6(MARK(tail(incr(s(y0))))) A__INCR(cons(tail(incr(y0)), z1)) -> c6(MARK(tail(incr(y0)))) A__INCR(cons(tail(odds), z1)) -> c6(MARK(tail(odds))) A__INCR(cons(tail(head(nats)), z1)) -> c6(MARK(tail(head(nats)))) A__INCR(cons(tail(head(incr(y0))), z1)) -> c6(MARK(tail(head(incr(y0))))) A__INCR(cons(tail(head(pairs)), z1)) -> c6(MARK(tail(head(pairs)))) A__INCR(cons(tail(head(odds)), z1)) -> c6(MARK(tail(head(odds)))) A__INCR(cons(tail(head(head(y0))), z1)) -> c6(MARK(tail(head(head(y0))))) A__INCR(cons(tail(head(tail(y0))), z1)) -> c6(MARK(tail(head(tail(y0))))) A__INCR(cons(tail(head(cons(y0, y1))), z1)) -> c6(MARK(tail(head(cons(y0, y1))))) A__INCR(cons(tail(head(s(y0))), z1)) -> c6(MARK(tail(head(s(y0))))) A__INCR(cons(tail(head(y0)), z1)) -> c6(MARK(tail(head(y0)))) A__INCR(cons(tail(tail(nats)), z1)) -> c6(MARK(tail(tail(nats)))) A__INCR(cons(tail(tail(incr(y0))), z1)) -> c6(MARK(tail(tail(incr(y0))))) A__INCR(cons(tail(tail(pairs)), z1)) -> c6(MARK(tail(tail(pairs)))) A__INCR(cons(tail(tail(odds)), z1)) -> c6(MARK(tail(tail(odds)))) A__INCR(cons(tail(tail(head(y0))), z1)) -> c6(MARK(tail(tail(head(y0))))) A__INCR(cons(tail(tail(tail(y0))), z1)) -> c6(MARK(tail(tail(tail(y0))))) A__INCR(cons(tail(tail(cons(y0, y1))), z1)) -> c6(MARK(tail(tail(cons(y0, y1))))) A__INCR(cons(tail(tail(s(y0))), z1)) -> c6(MARK(tail(tail(s(y0))))) A__INCR(cons(tail(tail(y0)), z1)) -> c6(MARK(tail(tail(y0)))) A__INCR(cons(tail(nats), z1)) -> c6(MARK(tail(nats))) A__INCR(cons(tail(pairs), z1)) -> c6(MARK(tail(pairs))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs))) MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs))) MARK(tail(incr(nats))) -> c17(A__TAIL(a__incr(a__nats))) MARK(tail(incr(pairs))) -> c17(A__TAIL(a__incr(a__pairs))) MARK(tail(odds)) -> c17(A__TAIL(a__incr(a__pairs))) MARK(incr(cons(nats, x1))) -> c13(A__INCR(cons(a__nats, x1)), MARK(cons(nats, x1))) MARK(incr(cons(incr(z0), x1))) -> c13(A__INCR(cons(a__incr(mark(z0)), x1)), MARK(cons(incr(z0), x1))) MARK(incr(cons(pairs, x1))) -> c13(A__INCR(cons(a__pairs, x1)), MARK(cons(pairs, x1))) MARK(incr(cons(odds, x1))) -> c13(A__INCR(cons(a__odds, x1)), MARK(cons(odds, x1))) MARK(incr(cons(head(z0), x1))) -> c13(A__INCR(cons(a__head(mark(z0)), x1)), MARK(cons(head(z0), x1))) MARK(incr(cons(tail(z0), x1))) -> c13(A__INCR(cons(a__tail(mark(z0)), x1)), MARK(cons(tail(z0), x1))) MARK(incr(cons(cons(z0, z1), x1))) -> c13(A__INCR(cons(cons(mark(z0), z1), x1)), MARK(cons(cons(z0, z1), x1))) MARK(incr(cons(s(z0), x1))) -> c13(A__INCR(cons(s(mark(z0)), x1)), MARK(cons(s(z0), x1))) MARK(incr(cons(x0, x1))) -> c13(MARK(cons(x0, x1))) MARK(incr(cons(0, x1))) -> c13(MARK(cons(0, x1))) A__HEAD(cons(cons(y0, y1), z1)) -> c8(MARK(cons(y0, y1))) A__HEAD(cons(s(y0), z1)) -> c8(MARK(s(y0))) A__HEAD(cons(incr(s(y0)), z1)) -> c8(MARK(incr(s(y0)))) A__HEAD(cons(head(cons(y0, y1)), z1)) -> c8(MARK(head(cons(y0, y1)))) A__HEAD(cons(head(s(y0)), z1)) -> c8(MARK(head(s(y0)))) A__HEAD(cons(tail(cons(y0, y1)), z1)) -> c8(MARK(tail(cons(y0, y1)))) A__HEAD(cons(tail(s(y0)), z1)) -> c8(MARK(tail(s(y0)))) A__HEAD(cons(incr(incr(incr(y0))), z1)) -> c8(MARK(incr(incr(incr(y0))))) A__HEAD(cons(incr(incr(odds)), z1)) -> c8(MARK(incr(incr(odds)))) A__HEAD(cons(incr(incr(head(y0))), z1)) -> c8(MARK(incr(incr(head(y0))))) A__HEAD(cons(incr(incr(tail(y0))), z1)) -> c8(MARK(incr(incr(tail(y0))))) A__HEAD(cons(incr(incr(cons(y0, y1))), z1)) -> c8(MARK(incr(incr(cons(y0, y1))))) A__HEAD(cons(incr(incr(s(y0))), z1)) -> c8(MARK(incr(incr(s(y0))))) A__HEAD(cons(incr(incr(y0)), z1)) -> c8(MARK(incr(incr(y0)))) A__HEAD(cons(incr(head(nats)), z1)) -> c8(MARK(incr(head(nats)))) A__HEAD(cons(incr(head(incr(y0))), z1)) -> c8(MARK(incr(head(incr(y0))))) A__HEAD(cons(incr(head(pairs)), z1)) -> c8(MARK(incr(head(pairs)))) A__HEAD(cons(incr(head(odds)), z1)) -> c8(MARK(incr(head(odds)))) A__HEAD(cons(incr(head(head(y0))), z1)) -> c8(MARK(incr(head(head(y0))))) A__HEAD(cons(incr(head(tail(y0))), z1)) -> c8(MARK(incr(head(tail(y0))))) A__HEAD(cons(incr(head(cons(y0, y1))), z1)) -> c8(MARK(incr(head(cons(y0, y1))))) A__HEAD(cons(incr(head(s(y0))), z1)) -> c8(MARK(incr(head(s(y0))))) A__HEAD(cons(incr(head(y0)), z1)) -> c8(MARK(incr(head(y0)))) A__HEAD(cons(incr(tail(nats)), z1)) -> c8(MARK(incr(tail(nats)))) A__HEAD(cons(incr(tail(incr(y0))), z1)) -> c8(MARK(incr(tail(incr(y0))))) A__HEAD(cons(incr(tail(pairs)), z1)) -> c8(MARK(incr(tail(pairs)))) A__HEAD(cons(incr(tail(odds)), z1)) -> c8(MARK(incr(tail(odds)))) A__HEAD(cons(incr(tail(head(y0))), z1)) -> c8(MARK(incr(tail(head(y0))))) A__HEAD(cons(incr(tail(tail(y0))), z1)) -> c8(MARK(incr(tail(tail(y0))))) A__HEAD(cons(incr(tail(cons(y0, y1))), z1)) -> c8(MARK(incr(tail(cons(y0, y1))))) A__HEAD(cons(incr(tail(s(y0))), z1)) -> c8(MARK(incr(tail(s(y0))))) A__HEAD(cons(incr(tail(y0)), z1)) -> c8(MARK(incr(tail(y0)))) A__HEAD(cons(head(incr(incr(y0))), z1)) -> c8(MARK(head(incr(incr(y0))))) A__HEAD(cons(head(incr(odds)), z1)) -> c8(MARK(head(incr(odds)))) A__HEAD(cons(head(incr(head(y0))), z1)) -> c8(MARK(head(incr(head(y0))))) A__HEAD(cons(head(incr(tail(y0))), z1)) -> c8(MARK(head(incr(tail(y0))))) A__HEAD(cons(head(incr(cons(y0, y1))), z1)) -> c8(MARK(head(incr(cons(y0, y1))))) A__HEAD(cons(head(incr(s(y0))), z1)) -> c8(MARK(head(incr(s(y0))))) A__HEAD(cons(head(incr(y0)), z1)) -> c8(MARK(head(incr(y0)))) A__HEAD(cons(head(head(nats)), z1)) -> c8(MARK(head(head(nats)))) A__HEAD(cons(head(head(incr(y0))), z1)) -> c8(MARK(head(head(incr(y0))))) A__HEAD(cons(head(head(pairs)), z1)) -> c8(MARK(head(head(pairs)))) A__HEAD(cons(head(head(odds)), z1)) -> c8(MARK(head(head(odds)))) A__HEAD(cons(head(head(head(y0))), z1)) -> c8(MARK(head(head(head(y0))))) A__HEAD(cons(head(head(tail(y0))), z1)) -> c8(MARK(head(head(tail(y0))))) A__HEAD(cons(head(head(cons(y0, y1))), z1)) -> c8(MARK(head(head(cons(y0, y1))))) A__HEAD(cons(head(head(s(y0))), z1)) -> c8(MARK(head(head(s(y0))))) A__HEAD(cons(head(head(y0)), z1)) -> c8(MARK(head(head(y0)))) A__HEAD(cons(head(tail(nats)), z1)) -> c8(MARK(head(tail(nats)))) A__HEAD(cons(head(tail(incr(y0))), z1)) -> c8(MARK(head(tail(incr(y0))))) A__HEAD(cons(head(tail(pairs)), z1)) -> c8(MARK(head(tail(pairs)))) A__HEAD(cons(head(tail(odds)), z1)) -> c8(MARK(head(tail(odds)))) A__HEAD(cons(head(tail(head(y0))), z1)) -> c8(MARK(head(tail(head(y0))))) A__HEAD(cons(head(tail(tail(y0))), z1)) -> c8(MARK(head(tail(tail(y0))))) A__HEAD(cons(head(tail(cons(y0, y1))), z1)) -> c8(MARK(head(tail(cons(y0, y1))))) A__HEAD(cons(head(tail(s(y0))), z1)) -> c8(MARK(head(tail(s(y0))))) A__HEAD(cons(head(tail(y0)), z1)) -> c8(MARK(head(tail(y0)))) A__HEAD(cons(tail(incr(incr(y0))), z1)) -> c8(MARK(tail(incr(incr(y0))))) A__HEAD(cons(tail(incr(odds)), z1)) -> c8(MARK(tail(incr(odds)))) A__HEAD(cons(tail(incr(head(y0))), z1)) -> c8(MARK(tail(incr(head(y0))))) A__HEAD(cons(tail(incr(tail(y0))), z1)) -> c8(MARK(tail(incr(tail(y0))))) A__HEAD(cons(tail(incr(cons(y0, y1))), z1)) -> c8(MARK(tail(incr(cons(y0, y1))))) A__HEAD(cons(tail(incr(s(y0))), z1)) -> c8(MARK(tail(incr(s(y0))))) A__HEAD(cons(tail(incr(y0)), z1)) -> c8(MARK(tail(incr(y0)))) A__HEAD(cons(tail(head(nats)), z1)) -> c8(MARK(tail(head(nats)))) A__HEAD(cons(tail(head(incr(y0))), z1)) -> c8(MARK(tail(head(incr(y0))))) A__HEAD(cons(tail(head(pairs)), z1)) -> c8(MARK(tail(head(pairs)))) A__HEAD(cons(tail(head(odds)), z1)) -> c8(MARK(tail(head(odds)))) A__HEAD(cons(tail(head(head(y0))), z1)) -> c8(MARK(tail(head(head(y0))))) A__HEAD(cons(tail(head(tail(y0))), z1)) -> c8(MARK(tail(head(tail(y0))))) A__HEAD(cons(tail(head(cons(y0, y1))), z1)) -> c8(MARK(tail(head(cons(y0, y1))))) A__HEAD(cons(tail(head(s(y0))), z1)) -> c8(MARK(tail(head(s(y0))))) A__HEAD(cons(tail(head(y0)), z1)) -> c8(MARK(tail(head(y0)))) A__HEAD(cons(tail(tail(nats)), z1)) -> c8(MARK(tail(tail(nats)))) A__HEAD(cons(tail(tail(incr(y0))), z1)) -> c8(MARK(tail(tail(incr(y0))))) A__HEAD(cons(tail(tail(pairs)), z1)) -> c8(MARK(tail(tail(pairs)))) A__HEAD(cons(tail(tail(odds)), z1)) -> c8(MARK(tail(tail(odds)))) A__HEAD(cons(tail(tail(head(y0))), z1)) -> c8(MARK(tail(tail(head(y0))))) A__HEAD(cons(tail(tail(tail(y0))), z1)) -> c8(MARK(tail(tail(tail(y0))))) A__HEAD(cons(tail(tail(cons(y0, y1))), z1)) -> c8(MARK(tail(tail(cons(y0, y1))))) A__HEAD(cons(tail(tail(s(y0))), z1)) -> c8(MARK(tail(tail(s(y0))))) A__HEAD(cons(tail(tail(y0)), z1)) -> c8(MARK(tail(tail(y0)))) A__HEAD(cons(tail(nats), z1)) -> c8(MARK(tail(nats))) A__HEAD(cons(tail(pairs), z1)) -> c8(MARK(tail(pairs))) A__HEAD(cons(incr(incr(nats)), z1)) -> c8(MARK(incr(incr(nats)))) A__HEAD(cons(incr(incr(pairs)), z1)) -> c8(MARK(incr(incr(pairs)))) A__HEAD(cons(incr(odds), z1)) -> c8(MARK(incr(odds))) A__HEAD(cons(head(incr(nats)), z1)) -> c8(MARK(head(incr(nats)))) A__HEAD(cons(head(incr(pairs)), z1)) -> c8(MARK(head(incr(pairs)))) A__HEAD(cons(head(odds), z1)) -> c8(MARK(head(odds))) A__HEAD(cons(tail(incr(nats)), z1)) -> c8(MARK(tail(incr(nats)))) A__HEAD(cons(tail(incr(pairs)), z1)) -> c8(MARK(tail(incr(pairs)))) A__HEAD(cons(tail(odds), z1)) -> c8(MARK(tail(odds))) A__HEAD(cons(incr(cons(nats, y0)), z1)) -> c8(MARK(incr(cons(nats, y0)))) A__HEAD(cons(incr(cons(incr(y0), y1)), z1)) -> c8(MARK(incr(cons(incr(y0), y1)))) A__HEAD(cons(incr(cons(pairs, y0)), z1)) -> c8(MARK(incr(cons(pairs, y0)))) A__HEAD(cons(incr(cons(odds, y0)), z1)) -> c8(MARK(incr(cons(odds, y0)))) A__HEAD(cons(incr(cons(head(y0), y1)), z1)) -> c8(MARK(incr(cons(head(y0), y1)))) A__HEAD(cons(incr(cons(tail(y0), y1)), z1)) -> c8(MARK(incr(cons(tail(y0), y1)))) A__HEAD(cons(incr(cons(cons(y0, y1), y2)), z1)) -> c8(MARK(incr(cons(cons(y0, y1), y2)))) A__HEAD(cons(incr(cons(s(y0), y1)), z1)) -> c8(MARK(incr(cons(s(y0), y1)))) A__HEAD(cons(incr(cons(y0, y1)), z1)) -> c8(MARK(incr(cons(y0, y1)))) A__HEAD(cons(incr(cons(0, y0)), z1)) -> c8(MARK(incr(cons(0, y0)))) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs))) MARK(head(head(nats))) -> c16(A__HEAD(a__head(a__nats))) MARK(head(head(pairs))) -> c16(A__HEAD(a__head(a__pairs))) MARK(tail(head(nats))) -> c17(A__TAIL(a__head(a__nats))) MARK(tail(head(pairs))) -> c17(A__TAIL(a__head(a__pairs))) MARK(head(cons(nats, x1))) -> c16(A__HEAD(cons(a__nats, x1)), MARK(cons(nats, x1))) MARK(head(cons(incr(z0), x1))) -> c16(A__HEAD(cons(a__incr(mark(z0)), x1)), MARK(cons(incr(z0), x1))) MARK(head(cons(pairs, x1))) -> c16(A__HEAD(cons(a__pairs, x1)), MARK(cons(pairs, x1))) MARK(head(cons(odds, x1))) -> c16(A__HEAD(cons(a__odds, x1)), MARK(cons(odds, x1))) MARK(head(cons(head(z0), x1))) -> c16(A__HEAD(cons(a__head(mark(z0)), x1)), MARK(cons(head(z0), x1))) MARK(head(cons(tail(z0), x1))) -> c16(A__HEAD(cons(a__tail(mark(z0)), x1)), MARK(cons(tail(z0), x1))) MARK(head(cons(cons(z0, z1), x1))) -> c16(A__HEAD(cons(cons(mark(z0), z1), x1)), MARK(cons(cons(z0, z1), x1))) MARK(head(cons(0, x1))) -> c16(A__HEAD(cons(0, x1)), MARK(cons(0, x1))) MARK(head(cons(s(z0), x1))) -> c16(A__HEAD(cons(s(mark(z0)), x1)), MARK(cons(s(z0), x1))) MARK(head(cons(x0, x1))) -> c16(MARK(cons(x0, x1))) K tuples:none Defined Rule Symbols: a__nats, a__pairs, a__odds, a__incr_1, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__TAIL_1, MARK_1, A__INCR_1, A__HEAD_1 Compound Symbols: c10_1, c18_1, c20_1, c13_1, c16_1, c17_2, c17_1, c13_2, c16_2, c6_1, c8_1 ---------------------------------------- (177) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (178) Obligation: Complexity Dependency Tuples Problem Rules: a__nats -> cons(0, incr(nats)) a__nats -> nats a__pairs -> cons(0, incr(odds)) a__pairs -> pairs a__odds -> a__incr(a__pairs) a__odds -> odds a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) 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(nats) -> a__nats mark(incr(z0)) -> a__incr(mark(z0)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c17(MARK(s(z0))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(tail(incr(incr(z0)))) -> c17(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(odds))) -> c17(A__TAIL(a__incr(a__odds)), MARK(incr(odds))) MARK(tail(incr(head(z0)))) -> c17(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c17(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c17(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c17(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c17(MARK(incr(x0))) MARK(tail(head(incr(z0)))) -> c17(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(odds))) -> c17(A__TAIL(a__head(a__odds)), MARK(head(odds))) MARK(tail(head(head(z0)))) -> c17(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c17(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c17(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c17(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c17(MARK(head(x0))) MARK(tail(tail(nats))) -> c17(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(incr(z0)))) -> c17(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(pairs))) -> c17(A__TAIL(a__tail(a__pairs)), MARK(tail(pairs))) MARK(tail(tail(odds))) -> c17(A__TAIL(a__tail(a__odds)), MARK(tail(odds))) MARK(tail(tail(head(z0)))) -> c17(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c17(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c17(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c17(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c17(MARK(tail(x0))) MARK(tail(nats)) -> c17(A__TAIL(cons(0, incr(nats)))) MARK(tail(pairs)) -> c17(A__TAIL(cons(0, incr(odds)))) A__INCR(cons(cons(y0, y1), z1)) -> c6(MARK(cons(y0, y1))) A__INCR(cons(s(y0), z1)) -> c6(MARK(s(y0))) A__INCR(cons(incr(cons(y0, y1)), z1)) -> c6(MARK(incr(cons(y0, y1)))) A__INCR(cons(incr(s(y0)), z1)) -> c6(MARK(incr(s(y0)))) A__INCR(cons(head(cons(y0, y1)), z1)) -> c6(MARK(head(cons(y0, y1)))) A__INCR(cons(head(s(y0)), z1)) -> c6(MARK(head(s(y0)))) A__INCR(cons(tail(cons(y0, y1)), z1)) -> c6(MARK(tail(cons(y0, y1)))) A__INCR(cons(tail(s(y0)), z1)) -> c6(MARK(tail(s(y0)))) A__INCR(cons(incr(incr(nats)), z1)) -> c6(MARK(incr(incr(nats)))) A__INCR(cons(incr(incr(incr(y0))), z1)) -> c6(MARK(incr(incr(incr(y0))))) A__INCR(cons(incr(incr(pairs)), z1)) -> c6(MARK(incr(incr(pairs)))) A__INCR(cons(incr(incr(odds)), z1)) -> c6(MARK(incr(incr(odds)))) A__INCR(cons(incr(incr(head(y0))), z1)) -> c6(MARK(incr(incr(head(y0))))) A__INCR(cons(incr(incr(tail(y0))), z1)) -> c6(MARK(incr(incr(tail(y0))))) A__INCR(cons(incr(incr(cons(y0, y1))), z1)) -> c6(MARK(incr(incr(cons(y0, y1))))) A__INCR(cons(incr(incr(s(y0))), z1)) -> c6(MARK(incr(incr(s(y0))))) A__INCR(cons(incr(incr(y0)), z1)) -> c6(MARK(incr(incr(y0)))) A__INCR(cons(incr(odds), z1)) -> c6(MARK(incr(odds))) A__INCR(cons(incr(head(nats)), z1)) -> c6(MARK(incr(head(nats)))) A__INCR(cons(incr(head(incr(y0))), z1)) -> c6(MARK(incr(head(incr(y0))))) A__INCR(cons(incr(head(pairs)), z1)) -> c6(MARK(incr(head(pairs)))) A__INCR(cons(incr(head(odds)), z1)) -> c6(MARK(incr(head(odds)))) A__INCR(cons(incr(head(head(y0))), z1)) -> c6(MARK(incr(head(head(y0))))) A__INCR(cons(incr(head(tail(y0))), z1)) -> c6(MARK(incr(head(tail(y0))))) A__INCR(cons(incr(head(cons(y0, y1))), z1)) -> c6(MARK(incr(head(cons(y0, y1))))) A__INCR(cons(incr(head(s(y0))), z1)) -> c6(MARK(incr(head(s(y0))))) A__INCR(cons(incr(head(y0)), z1)) -> c6(MARK(incr(head(y0)))) A__INCR(cons(incr(tail(nats)), z1)) -> c6(MARK(incr(tail(nats)))) A__INCR(cons(incr(tail(incr(y0))), z1)) -> c6(MARK(incr(tail(incr(y0))))) A__INCR(cons(incr(tail(pairs)), z1)) -> c6(MARK(incr(tail(pairs)))) A__INCR(cons(incr(tail(odds)), z1)) -> c6(MARK(incr(tail(odds)))) A__INCR(cons(incr(tail(head(y0))), z1)) -> c6(MARK(incr(tail(head(y0))))) A__INCR(cons(incr(tail(tail(y0))), z1)) -> c6(MARK(incr(tail(tail(y0))))) A__INCR(cons(incr(tail(cons(y0, y1))), z1)) -> c6(MARK(incr(tail(cons(y0, y1))))) A__INCR(cons(incr(tail(s(y0))), z1)) -> c6(MARK(incr(tail(s(y0))))) A__INCR(cons(incr(tail(y0)), z1)) -> c6(MARK(incr(tail(y0)))) A__INCR(cons(head(incr(nats)), z1)) -> c6(MARK(head(incr(nats)))) A__INCR(cons(head(incr(incr(y0))), z1)) -> c6(MARK(head(incr(incr(y0))))) A__INCR(cons(head(incr(pairs)), z1)) -> c6(MARK(head(incr(pairs)))) A__INCR(cons(head(incr(odds)), z1)) -> c6(MARK(head(incr(odds)))) A__INCR(cons(head(incr(head(y0))), z1)) -> c6(MARK(head(incr(head(y0))))) A__INCR(cons(head(incr(tail(y0))), z1)) -> c6(MARK(head(incr(tail(y0))))) A__INCR(cons(head(incr(cons(y0, y1))), z1)) -> c6(MARK(head(incr(cons(y0, y1))))) A__INCR(cons(head(incr(s(y0))), z1)) -> c6(MARK(head(incr(s(y0))))) A__INCR(cons(head(incr(y0)), z1)) -> c6(MARK(head(incr(y0)))) A__INCR(cons(head(odds), z1)) -> c6(MARK(head(odds))) A__INCR(cons(head(head(nats)), z1)) -> c6(MARK(head(head(nats)))) A__INCR(cons(head(head(incr(y0))), z1)) -> c6(MARK(head(head(incr(y0))))) A__INCR(cons(head(head(pairs)), z1)) -> c6(MARK(head(head(pairs)))) A__INCR(cons(head(head(odds)), z1)) -> c6(MARK(head(head(odds)))) A__INCR(cons(head(head(head(y0))), z1)) -> c6(MARK(head(head(head(y0))))) A__INCR(cons(head(head(tail(y0))), z1)) -> c6(MARK(head(head(tail(y0))))) A__INCR(cons(head(head(cons(y0, y1))), z1)) -> c6(MARK(head(head(cons(y0, y1))))) A__INCR(cons(head(head(s(y0))), z1)) -> c6(MARK(head(head(s(y0))))) A__INCR(cons(head(head(y0)), z1)) -> c6(MARK(head(head(y0)))) A__INCR(cons(head(tail(nats)), z1)) -> c6(MARK(head(tail(nats)))) A__INCR(cons(head(tail(incr(y0))), z1)) -> c6(MARK(head(tail(incr(y0))))) A__INCR(cons(head(tail(pairs)), z1)) -> c6(MARK(head(tail(pairs)))) A__INCR(cons(head(tail(odds)), z1)) -> c6(MARK(head(tail(odds)))) A__INCR(cons(head(tail(head(y0))), z1)) -> c6(MARK(head(tail(head(y0))))) A__INCR(cons(head(tail(tail(y0))), z1)) -> c6(MARK(head(tail(tail(y0))))) A__INCR(cons(head(tail(cons(y0, y1))), z1)) -> c6(MARK(head(tail(cons(y0, y1))))) A__INCR(cons(head(tail(s(y0))), z1)) -> c6(MARK(head(tail(s(y0))))) A__INCR(cons(head(tail(y0)), z1)) -> c6(MARK(head(tail(y0)))) A__INCR(cons(tail(incr(nats)), z1)) -> c6(MARK(tail(incr(nats)))) A__INCR(cons(tail(incr(incr(y0))), z1)) -> c6(MARK(tail(incr(incr(y0))))) A__INCR(cons(tail(incr(pairs)), z1)) -> c6(MARK(tail(incr(pairs)))) A__INCR(cons(tail(incr(odds)), z1)) -> c6(MARK(tail(incr(odds)))) A__INCR(cons(tail(incr(head(y0))), z1)) -> c6(MARK(tail(incr(head(y0))))) A__INCR(cons(tail(incr(tail(y0))), z1)) -> c6(MARK(tail(incr(tail(y0))))) A__INCR(cons(tail(incr(cons(y0, y1))), z1)) -> c6(MARK(tail(incr(cons(y0, y1))))) A__INCR(cons(tail(incr(s(y0))), z1)) -> c6(MARK(tail(incr(s(y0))))) A__INCR(cons(tail(incr(y0)), z1)) -> c6(MARK(tail(incr(y0)))) A__INCR(cons(tail(odds), z1)) -> c6(MARK(tail(odds))) A__INCR(cons(tail(head(nats)), z1)) -> c6(MARK(tail(head(nats)))) A__INCR(cons(tail(head(incr(y0))), z1)) -> c6(MARK(tail(head(incr(y0))))) A__INCR(cons(tail(head(pairs)), z1)) -> c6(MARK(tail(head(pairs)))) A__INCR(cons(tail(head(odds)), z1)) -> c6(MARK(tail(head(odds)))) A__INCR(cons(tail(head(head(y0))), z1)) -> c6(MARK(tail(head(head(y0))))) A__INCR(cons(tail(head(tail(y0))), z1)) -> c6(MARK(tail(head(tail(y0))))) A__INCR(cons(tail(head(cons(y0, y1))), z1)) -> c6(MARK(tail(head(cons(y0, y1))))) A__INCR(cons(tail(head(s(y0))), z1)) -> c6(MARK(tail(head(s(y0))))) A__INCR(cons(tail(head(y0)), z1)) -> c6(MARK(tail(head(y0)))) A__INCR(cons(tail(tail(nats)), z1)) -> c6(MARK(tail(tail(nats)))) A__INCR(cons(tail(tail(incr(y0))), z1)) -> c6(MARK(tail(tail(incr(y0))))) A__INCR(cons(tail(tail(pairs)), z1)) -> c6(MARK(tail(tail(pairs)))) A__INCR(cons(tail(tail(odds)), z1)) -> c6(MARK(tail(tail(odds)))) A__INCR(cons(tail(tail(head(y0))), z1)) -> c6(MARK(tail(tail(head(y0))))) A__INCR(cons(tail(tail(tail(y0))), z1)) -> c6(MARK(tail(tail(tail(y0))))) A__INCR(cons(tail(tail(cons(y0, y1))), z1)) -> c6(MARK(tail(tail(cons(y0, y1))))) A__INCR(cons(tail(tail(s(y0))), z1)) -> c6(MARK(tail(tail(s(y0))))) A__INCR(cons(tail(tail(y0)), z1)) -> c6(MARK(tail(tail(y0)))) A__INCR(cons(tail(nats), z1)) -> c6(MARK(tail(nats))) A__INCR(cons(tail(pairs), z1)) -> c6(MARK(tail(pairs))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs))) MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs))) MARK(tail(incr(nats))) -> c17(A__TAIL(a__incr(a__nats))) MARK(tail(incr(pairs))) -> c17(A__TAIL(a__incr(a__pairs))) MARK(tail(odds)) -> c17(A__TAIL(a__incr(a__pairs))) MARK(incr(cons(nats, x1))) -> c13(A__INCR(cons(a__nats, x1)), MARK(cons(nats, x1))) MARK(incr(cons(incr(z0), x1))) -> c13(A__INCR(cons(a__incr(mark(z0)), x1)), MARK(cons(incr(z0), x1))) MARK(incr(cons(pairs, x1))) -> c13(A__INCR(cons(a__pairs, x1)), MARK(cons(pairs, x1))) MARK(incr(cons(odds, x1))) -> c13(A__INCR(cons(a__odds, x1)), MARK(cons(odds, x1))) MARK(incr(cons(head(z0), x1))) -> 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A__HEAD(cons(tail(head(y0)), z1)) -> c8(MARK(tail(head(y0)))) A__HEAD(cons(tail(tail(nats)), z1)) -> c8(MARK(tail(tail(nats)))) A__HEAD(cons(tail(tail(incr(y0))), z1)) -> c8(MARK(tail(tail(incr(y0))))) A__HEAD(cons(tail(tail(pairs)), z1)) -> c8(MARK(tail(tail(pairs)))) A__HEAD(cons(tail(tail(odds)), z1)) -> c8(MARK(tail(tail(odds)))) A__HEAD(cons(tail(tail(head(y0))), z1)) -> c8(MARK(tail(tail(head(y0))))) A__HEAD(cons(tail(tail(tail(y0))), z1)) -> c8(MARK(tail(tail(tail(y0))))) A__HEAD(cons(tail(tail(cons(y0, y1))), z1)) -> c8(MARK(tail(tail(cons(y0, y1))))) A__HEAD(cons(tail(tail(s(y0))), z1)) -> c8(MARK(tail(tail(s(y0))))) A__HEAD(cons(tail(tail(y0)), z1)) -> c8(MARK(tail(tail(y0)))) A__HEAD(cons(tail(nats), z1)) -> c8(MARK(tail(nats))) A__HEAD(cons(tail(pairs), z1)) -> c8(MARK(tail(pairs))) A__HEAD(cons(incr(incr(nats)), z1)) -> c8(MARK(incr(incr(nats)))) A__HEAD(cons(incr(incr(pairs)), z1)) -> c8(MARK(incr(incr(pairs)))) A__HEAD(cons(incr(odds), z1)) -> c8(MARK(incr(odds))) A__HEAD(cons(head(incr(nats)), z1)) -> c8(MARK(head(incr(nats)))) A__HEAD(cons(head(incr(pairs)), z1)) -> c8(MARK(head(incr(pairs)))) A__HEAD(cons(head(odds), z1)) -> c8(MARK(head(odds))) A__HEAD(cons(tail(incr(nats)), z1)) -> c8(MARK(tail(incr(nats)))) A__HEAD(cons(tail(incr(pairs)), z1)) -> c8(MARK(tail(incr(pairs)))) A__HEAD(cons(tail(odds), z1)) -> c8(MARK(tail(odds))) A__HEAD(cons(incr(cons(nats, y0)), z1)) -> c8(MARK(incr(cons(nats, y0)))) A__HEAD(cons(incr(cons(incr(y0), y1)), z1)) -> c8(MARK(incr(cons(incr(y0), y1)))) A__HEAD(cons(incr(cons(pairs, y0)), z1)) -> c8(MARK(incr(cons(pairs, y0)))) A__HEAD(cons(incr(cons(odds, y0)), z1)) -> c8(MARK(incr(cons(odds, y0)))) A__HEAD(cons(incr(cons(head(y0), y1)), z1)) -> c8(MARK(incr(cons(head(y0), y1)))) A__HEAD(cons(incr(cons(tail(y0), y1)), z1)) -> c8(MARK(incr(cons(tail(y0), y1)))) A__HEAD(cons(incr(cons(cons(y0, y1), y2)), z1)) -> c8(MARK(incr(cons(cons(y0, y1), y2)))) A__HEAD(cons(incr(cons(s(y0), y1)), z1)) -> c8(MARK(incr(cons(s(y0), y1)))) A__HEAD(cons(incr(cons(y0, y1)), z1)) -> c8(MARK(incr(cons(y0, y1)))) A__HEAD(cons(incr(cons(0, y0)), z1)) -> c8(MARK(incr(cons(0, y0)))) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs))) MARK(head(head(nats))) -> c16(A__HEAD(a__head(a__nats))) MARK(head(head(pairs))) -> c16(A__HEAD(a__head(a__pairs))) MARK(tail(head(nats))) -> c17(A__TAIL(a__head(a__nats))) MARK(tail(head(pairs))) -> c17(A__TAIL(a__head(a__pairs))) MARK(head(cons(nats, x1))) -> c16(A__HEAD(cons(a__nats, x1)), MARK(cons(nats, x1))) MARK(head(cons(incr(z0), x1))) -> c16(A__HEAD(cons(a__incr(mark(z0)), x1)), MARK(cons(incr(z0), x1))) MARK(head(cons(pairs, x1))) -> c16(A__HEAD(cons(a__pairs, x1)), MARK(cons(pairs, x1))) MARK(head(cons(odds, x1))) -> c16(A__HEAD(cons(a__odds, x1)), MARK(cons(odds, x1))) MARK(head(cons(head(z0), x1))) -> c16(A__HEAD(cons(a__head(mark(z0)), x1)), MARK(cons(head(z0), x1))) MARK(head(cons(tail(z0), x1))) -> c16(A__HEAD(cons(a__tail(mark(z0)), x1)), MARK(cons(tail(z0), x1))) MARK(head(cons(cons(z0, z1), x1))) -> c16(A__HEAD(cons(cons(mark(z0), z1), x1)), MARK(cons(cons(z0, z1), x1))) MARK(head(cons(s(z0), x1))) -> c16(A__HEAD(cons(s(mark(z0)), x1)), MARK(cons(s(z0), x1))) MARK(head(cons(x0, x1))) -> c16(MARK(cons(x0, x1))) MARK(head(cons(0, x1))) -> c16(MARK(cons(0, x1))) K tuples:none Defined Rule Symbols: a__nats, a__pairs, a__odds, a__incr_1, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: A__TAIL_1, MARK_1, A__INCR_1, A__HEAD_1 Compound Symbols: c10_1, c18_1, c20_1, c13_1, c16_1, c17_2, c17_1, c13_2, c16_2, c6_1, c8_1 ---------------------------------------- (179) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace A__TAIL(cons(z0, z1)) -> c10(MARK(z1)) by A__TAIL(cons(z0, cons(y0, y1))) -> c10(MARK(cons(y0, y1))) A__TAIL(cons(z0, s(y0))) -> c10(MARK(s(y0))) A__TAIL(cons(z0, incr(s(y0)))) -> c10(MARK(incr(s(y0)))) A__TAIL(cons(z0, head(s(y0)))) -> c10(MARK(head(s(y0)))) A__TAIL(cons(z0, tail(cons(y0, y1)))) -> c10(MARK(tail(cons(y0, y1)))) A__TAIL(cons(z0, tail(s(y0)))) -> c10(MARK(tail(s(y0)))) A__TAIL(cons(z0, incr(incr(incr(y0))))) -> c10(MARK(incr(incr(incr(y0))))) A__TAIL(cons(z0, incr(incr(odds)))) -> c10(MARK(incr(incr(odds)))) A__TAIL(cons(z0, incr(incr(head(y0))))) -> c10(MARK(incr(incr(head(y0))))) A__TAIL(cons(z0, incr(incr(tail(y0))))) -> c10(MARK(incr(incr(tail(y0))))) A__TAIL(cons(z0, incr(incr(cons(y0, y1))))) -> c10(MARK(incr(incr(cons(y0, y1))))) A__TAIL(cons(z0, incr(incr(s(y0))))) -> c10(MARK(incr(incr(s(y0))))) A__TAIL(cons(z0, incr(incr(y0)))) -> c10(MARK(incr(incr(y0)))) A__TAIL(cons(z0, incr(head(incr(y0))))) -> c10(MARK(incr(head(incr(y0))))) A__TAIL(cons(z0, incr(head(odds)))) -> c10(MARK(incr(head(odds)))) A__TAIL(cons(z0, incr(head(head(y0))))) -> c10(MARK(incr(head(head(y0))))) A__TAIL(cons(z0, incr(head(tail(y0))))) -> c10(MARK(incr(head(tail(y0))))) A__TAIL(cons(z0, incr(head(cons(y0, y1))))) -> c10(MARK(incr(head(cons(y0, y1))))) A__TAIL(cons(z0, incr(head(s(y0))))) -> c10(MARK(incr(head(s(y0))))) A__TAIL(cons(z0, incr(head(y0)))) -> c10(MARK(incr(head(y0)))) A__TAIL(cons(z0, incr(tail(nats)))) -> c10(MARK(incr(tail(nats)))) A__TAIL(cons(z0, incr(tail(incr(y0))))) -> c10(MARK(incr(tail(incr(y0))))) A__TAIL(cons(z0, incr(tail(pairs)))) -> c10(MARK(incr(tail(pairs)))) A__TAIL(cons(z0, incr(tail(odds)))) -> c10(MARK(incr(tail(odds)))) A__TAIL(cons(z0, incr(tail(head(y0))))) -> c10(MARK(incr(tail(head(y0))))) A__TAIL(cons(z0, incr(tail(tail(y0))))) -> c10(MARK(incr(tail(tail(y0))))) A__TAIL(cons(z0, incr(tail(cons(y0, y1))))) -> c10(MARK(incr(tail(cons(y0, y1))))) A__TAIL(cons(z0, incr(tail(s(y0))))) -> c10(MARK(incr(tail(s(y0))))) A__TAIL(cons(z0, incr(tail(y0)))) -> c10(MARK(incr(tail(y0)))) A__TAIL(cons(z0, head(incr(incr(y0))))) -> c10(MARK(head(incr(incr(y0))))) A__TAIL(cons(z0, head(incr(odds)))) -> c10(MARK(head(incr(odds)))) A__TAIL(cons(z0, head(incr(head(y0))))) -> c10(MARK(head(incr(head(y0))))) A__TAIL(cons(z0, head(incr(tail(y0))))) -> c10(MARK(head(incr(tail(y0))))) A__TAIL(cons(z0, head(incr(cons(y0, y1))))) -> c10(MARK(head(incr(cons(y0, y1))))) A__TAIL(cons(z0, head(incr(s(y0))))) -> c10(MARK(head(incr(s(y0))))) A__TAIL(cons(z0, head(incr(y0)))) -> c10(MARK(head(incr(y0)))) A__TAIL(cons(z0, head(head(incr(y0))))) -> c10(MARK(head(head(incr(y0))))) A__TAIL(cons(z0, head(head(odds)))) -> c10(MARK(head(head(odds)))) A__TAIL(cons(z0, head(head(head(y0))))) -> c10(MARK(head(head(head(y0))))) A__TAIL(cons(z0, head(head(tail(y0))))) -> c10(MARK(head(head(tail(y0))))) A__TAIL(cons(z0, head(head(cons(y0, y1))))) -> c10(MARK(head(head(cons(y0, y1))))) A__TAIL(cons(z0, head(head(s(y0))))) -> c10(MARK(head(head(s(y0))))) A__TAIL(cons(z0, head(head(y0)))) -> c10(MARK(head(head(y0)))) A__TAIL(cons(z0, head(tail(nats)))) -> c10(MARK(head(tail(nats)))) A__TAIL(cons(z0, head(tail(incr(y0))))) -> c10(MARK(head(tail(incr(y0))))) A__TAIL(cons(z0, head(tail(pairs)))) -> c10(MARK(head(tail(pairs)))) A__TAIL(cons(z0, head(tail(odds)))) -> c10(MARK(head(tail(odds)))) A__TAIL(cons(z0, head(tail(head(y0))))) -> c10(MARK(head(tail(head(y0))))) A__TAIL(cons(z0, head(tail(tail(y0))))) -> c10(MARK(head(tail(tail(y0))))) A__TAIL(cons(z0, head(tail(cons(y0, y1))))) -> c10(MARK(head(tail(cons(y0, y1))))) A__TAIL(cons(z0, head(tail(s(y0))))) -> c10(MARK(head(tail(s(y0))))) A__TAIL(cons(z0, head(tail(y0)))) -> c10(MARK(head(tail(y0)))) A__TAIL(cons(z0, tail(incr(incr(y0))))) -> c10(MARK(tail(incr(incr(y0))))) A__TAIL(cons(z0, tail(incr(odds)))) -> c10(MARK(tail(incr(odds)))) A__TAIL(cons(z0, tail(incr(head(y0))))) -> c10(MARK(tail(incr(head(y0))))) A__TAIL(cons(z0, tail(incr(tail(y0))))) -> c10(MARK(tail(incr(tail(y0))))) A__TAIL(cons(z0, tail(incr(cons(y0, y1))))) -> c10(MARK(tail(incr(cons(y0, y1))))) A__TAIL(cons(z0, tail(incr(s(y0))))) -> c10(MARK(tail(incr(s(y0))))) A__TAIL(cons(z0, tail(incr(y0)))) -> c10(MARK(tail(incr(y0)))) A__TAIL(cons(z0, tail(head(incr(y0))))) -> c10(MARK(tail(head(incr(y0))))) A__TAIL(cons(z0, tail(head(odds)))) -> c10(MARK(tail(head(odds)))) A__TAIL(cons(z0, tail(head(head(y0))))) -> c10(MARK(tail(head(head(y0))))) A__TAIL(cons(z0, tail(head(tail(y0))))) -> c10(MARK(tail(head(tail(y0))))) A__TAIL(cons(z0, tail(head(cons(y0, y1))))) -> c10(MARK(tail(head(cons(y0, y1))))) A__TAIL(cons(z0, tail(head(s(y0))))) -> c10(MARK(tail(head(s(y0))))) A__TAIL(cons(z0, tail(head(y0)))) -> c10(MARK(tail(head(y0)))) A__TAIL(cons(z0, tail(tail(nats)))) -> c10(MARK(tail(tail(nats)))) A__TAIL(cons(z0, tail(tail(incr(y0))))) -> c10(MARK(tail(tail(incr(y0))))) A__TAIL(cons(z0, tail(tail(pairs)))) -> c10(MARK(tail(tail(pairs)))) A__TAIL(cons(z0, tail(tail(odds)))) -> c10(MARK(tail(tail(odds)))) A__TAIL(cons(z0, tail(tail(head(y0))))) -> c10(MARK(tail(tail(head(y0))))) A__TAIL(cons(z0, tail(tail(tail(y0))))) -> c10(MARK(tail(tail(tail(y0))))) A__TAIL(cons(z0, tail(tail(cons(y0, y1))))) -> c10(MARK(tail(tail(cons(y0, y1))))) A__TAIL(cons(z0, tail(tail(s(y0))))) -> c10(MARK(tail(tail(s(y0))))) A__TAIL(cons(z0, tail(tail(y0)))) -> c10(MARK(tail(tail(y0)))) A__TAIL(cons(z0, tail(nats))) -> c10(MARK(tail(nats))) A__TAIL(cons(z0, tail(pairs))) -> c10(MARK(tail(pairs))) A__TAIL(cons(z0, incr(incr(nats)))) -> c10(MARK(incr(incr(nats)))) A__TAIL(cons(z0, incr(incr(pairs)))) -> c10(MARK(incr(incr(pairs)))) A__TAIL(cons(z0, incr(odds))) -> c10(MARK(incr(odds))) A__TAIL(cons(z0, head(incr(nats)))) -> c10(MARK(head(incr(nats)))) A__TAIL(cons(z0, head(incr(pairs)))) -> c10(MARK(head(incr(pairs)))) A__TAIL(cons(z0, head(odds))) -> c10(MARK(head(odds))) A__TAIL(cons(z0, tail(incr(nats)))) -> c10(MARK(tail(incr(nats)))) A__TAIL(cons(z0, tail(incr(pairs)))) -> c10(MARK(tail(incr(pairs)))) A__TAIL(cons(z0, tail(odds))) -> c10(MARK(tail(odds))) A__TAIL(cons(z0, incr(cons(nats, y0)))) -> c10(MARK(incr(cons(nats, y0)))) A__TAIL(cons(z0, incr(cons(incr(y0), y1)))) -> c10(MARK(incr(cons(incr(y0), y1)))) A__TAIL(cons(z0, incr(cons(pairs, y0)))) -> c10(MARK(incr(cons(pairs, y0)))) A__TAIL(cons(z0, incr(cons(odds, y0)))) -> c10(MARK(incr(cons(odds, y0)))) A__TAIL(cons(z0, incr(cons(head(y0), y1)))) -> c10(MARK(incr(cons(head(y0), y1)))) A__TAIL(cons(z0, incr(cons(tail(y0), y1)))) -> c10(MARK(incr(cons(tail(y0), y1)))) A__TAIL(cons(z0, incr(cons(cons(y0, y1), y2)))) -> c10(MARK(incr(cons(cons(y0, y1), y2)))) A__TAIL(cons(z0, incr(cons(s(y0), y1)))) -> c10(MARK(incr(cons(s(y0), y1)))) A__TAIL(cons(z0, incr(cons(y0, y1)))) -> c10(MARK(incr(cons(y0, y1)))) A__TAIL(cons(z0, incr(cons(0, y0)))) -> c10(MARK(incr(cons(0, y0)))) A__TAIL(cons(z0, incr(head(nats)))) -> c10(MARK(incr(head(nats)))) A__TAIL(cons(z0, incr(head(pairs)))) -> c10(MARK(incr(head(pairs)))) A__TAIL(cons(z0, head(head(nats)))) -> c10(MARK(head(head(nats)))) A__TAIL(cons(z0, head(head(pairs)))) -> c10(MARK(head(head(pairs)))) A__TAIL(cons(z0, tail(head(nats)))) -> c10(MARK(tail(head(nats)))) A__TAIL(cons(z0, tail(head(pairs)))) -> c10(MARK(tail(head(pairs)))) A__TAIL(cons(z0, head(cons(nats, y0)))) -> c10(MARK(head(cons(nats, y0)))) A__TAIL(cons(z0, head(cons(incr(y0), y1)))) -> c10(MARK(head(cons(incr(y0), y1)))) A__TAIL(cons(z0, head(cons(pairs, y0)))) -> c10(MARK(head(cons(pairs, y0)))) A__TAIL(cons(z0, head(cons(odds, y0)))) -> c10(MARK(head(cons(odds, y0)))) A__TAIL(cons(z0, head(cons(head(y0), y1)))) -> c10(MARK(head(cons(head(y0), y1)))) A__TAIL(cons(z0, head(cons(tail(y0), y1)))) -> c10(MARK(head(cons(tail(y0), y1)))) A__TAIL(cons(z0, head(cons(cons(y0, y1), y2)))) -> c10(MARK(head(cons(cons(y0, y1), y2)))) A__TAIL(cons(z0, head(cons(s(y0), y1)))) -> c10(MARK(head(cons(s(y0), y1)))) A__TAIL(cons(z0, head(cons(y0, y1)))) -> c10(MARK(head(cons(y0, y1)))) A__TAIL(cons(z0, head(cons(0, y0)))) -> c10(MARK(head(cons(0, y0)))) ---------------------------------------- (180) Obligation: Complexity Dependency Tuples Problem Rules: a__nats -> cons(0, incr(nats)) a__nats -> nats a__pairs -> cons(0, incr(odds)) a__pairs -> pairs a__odds -> a__incr(a__pairs) a__odds -> odds a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) 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(nats) -> a__nats mark(incr(z0)) -> a__incr(mark(z0)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c17(MARK(s(z0))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(tail(incr(incr(z0)))) -> c17(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(odds))) -> c17(A__TAIL(a__incr(a__odds)), MARK(incr(odds))) MARK(tail(incr(head(z0)))) -> c17(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c17(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c17(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c17(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c17(MARK(incr(x0))) MARK(tail(head(incr(z0)))) -> c17(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(odds))) -> c17(A__TAIL(a__head(a__odds)), MARK(head(odds))) MARK(tail(head(head(z0)))) -> c17(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c17(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c17(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c17(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c17(MARK(head(x0))) MARK(tail(tail(nats))) -> c17(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(incr(z0)))) -> c17(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(pairs))) -> c17(A__TAIL(a__tail(a__pairs)), MARK(tail(pairs))) MARK(tail(tail(odds))) -> c17(A__TAIL(a__tail(a__odds)), MARK(tail(odds))) MARK(tail(tail(head(z0)))) -> c17(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c17(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c17(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c17(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c17(MARK(tail(x0))) MARK(tail(nats)) -> c17(A__TAIL(cons(0, incr(nats)))) MARK(tail(pairs)) -> c17(A__TAIL(cons(0, incr(odds)))) A__INCR(cons(cons(y0, y1), z1)) -> c6(MARK(cons(y0, y1))) A__INCR(cons(s(y0), z1)) -> c6(MARK(s(y0))) A__INCR(cons(incr(cons(y0, y1)), z1)) -> c6(MARK(incr(cons(y0, y1)))) A__INCR(cons(incr(s(y0)), z1)) -> c6(MARK(incr(s(y0)))) A__INCR(cons(head(cons(y0, y1)), z1)) -> c6(MARK(head(cons(y0, y1)))) A__INCR(cons(head(s(y0)), z1)) -> c6(MARK(head(s(y0)))) A__INCR(cons(tail(cons(y0, y1)), z1)) -> c6(MARK(tail(cons(y0, y1)))) A__INCR(cons(tail(s(y0)), z1)) -> c6(MARK(tail(s(y0)))) A__INCR(cons(incr(incr(nats)), z1)) -> c6(MARK(incr(incr(nats)))) A__INCR(cons(incr(incr(incr(y0))), z1)) -> c6(MARK(incr(incr(incr(y0))))) A__INCR(cons(incr(incr(pairs)), z1)) -> c6(MARK(incr(incr(pairs)))) A__INCR(cons(incr(incr(odds)), z1)) -> c6(MARK(incr(incr(odds)))) A__INCR(cons(incr(incr(head(y0))), z1)) -> c6(MARK(incr(incr(head(y0))))) A__INCR(cons(incr(incr(tail(y0))), z1)) -> c6(MARK(incr(incr(tail(y0))))) A__INCR(cons(incr(incr(cons(y0, y1))), z1)) -> c6(MARK(incr(incr(cons(y0, y1))))) A__INCR(cons(incr(incr(s(y0))), z1)) -> c6(MARK(incr(incr(s(y0))))) A__INCR(cons(incr(incr(y0)), z1)) -> c6(MARK(incr(incr(y0)))) A__INCR(cons(incr(odds), z1)) -> c6(MARK(incr(odds))) A__INCR(cons(incr(head(nats)), z1)) -> c6(MARK(incr(head(nats)))) A__INCR(cons(incr(head(incr(y0))), z1)) -> c6(MARK(incr(head(incr(y0))))) A__INCR(cons(incr(head(pairs)), z1)) -> c6(MARK(incr(head(pairs)))) A__INCR(cons(incr(head(odds)), z1)) -> c6(MARK(incr(head(odds)))) A__INCR(cons(incr(head(head(y0))), z1)) -> c6(MARK(incr(head(head(y0))))) A__INCR(cons(incr(head(tail(y0))), z1)) -> c6(MARK(incr(head(tail(y0))))) A__INCR(cons(incr(head(cons(y0, y1))), z1)) -> c6(MARK(incr(head(cons(y0, y1))))) A__INCR(cons(incr(head(s(y0))), z1)) -> c6(MARK(incr(head(s(y0))))) A__INCR(cons(incr(head(y0)), z1)) -> c6(MARK(incr(head(y0)))) A__INCR(cons(incr(tail(nats)), z1)) -> c6(MARK(incr(tail(nats)))) A__INCR(cons(incr(tail(incr(y0))), z1)) -> c6(MARK(incr(tail(incr(y0))))) A__INCR(cons(incr(tail(pairs)), z1)) -> c6(MARK(incr(tail(pairs)))) A__INCR(cons(incr(tail(odds)), z1)) -> c6(MARK(incr(tail(odds)))) A__INCR(cons(incr(tail(head(y0))), z1)) -> c6(MARK(incr(tail(head(y0))))) A__INCR(cons(incr(tail(tail(y0))), z1)) -> c6(MARK(incr(tail(tail(y0))))) A__INCR(cons(incr(tail(cons(y0, y1))), z1)) -> c6(MARK(incr(tail(cons(y0, y1))))) A__INCR(cons(incr(tail(s(y0))), z1)) -> c6(MARK(incr(tail(s(y0))))) A__INCR(cons(incr(tail(y0)), z1)) -> c6(MARK(incr(tail(y0)))) A__INCR(cons(head(incr(nats)), z1)) -> c6(MARK(head(incr(nats)))) A__INCR(cons(head(incr(incr(y0))), z1)) -> c6(MARK(head(incr(incr(y0))))) A__INCR(cons(head(incr(pairs)), z1)) -> c6(MARK(head(incr(pairs)))) A__INCR(cons(head(incr(odds)), z1)) -> c6(MARK(head(incr(odds)))) A__INCR(cons(head(incr(head(y0))), z1)) -> c6(MARK(head(incr(head(y0))))) A__INCR(cons(head(incr(tail(y0))), z1)) -> c6(MARK(head(incr(tail(y0))))) A__INCR(cons(head(incr(cons(y0, y1))), z1)) -> c6(MARK(head(incr(cons(y0, y1))))) A__INCR(cons(head(incr(s(y0))), z1)) -> c6(MARK(head(incr(s(y0))))) A__INCR(cons(head(incr(y0)), z1)) -> c6(MARK(head(incr(y0)))) A__INCR(cons(head(odds), z1)) -> c6(MARK(head(odds))) A__INCR(cons(head(head(nats)), z1)) -> c6(MARK(head(head(nats)))) A__INCR(cons(head(head(incr(y0))), z1)) -> c6(MARK(head(head(incr(y0))))) A__INCR(cons(head(head(pairs)), z1)) -> c6(MARK(head(head(pairs)))) A__INCR(cons(head(head(odds)), z1)) -> c6(MARK(head(head(odds)))) A__INCR(cons(head(head(head(y0))), z1)) -> c6(MARK(head(head(head(y0))))) A__INCR(cons(head(head(tail(y0))), z1)) -> c6(MARK(head(head(tail(y0))))) A__INCR(cons(head(head(cons(y0, y1))), z1)) -> c6(MARK(head(head(cons(y0, y1))))) A__INCR(cons(head(head(s(y0))), z1)) -> c6(MARK(head(head(s(y0))))) A__INCR(cons(head(head(y0)), z1)) -> c6(MARK(head(head(y0)))) A__INCR(cons(head(tail(nats)), z1)) -> c6(MARK(head(tail(nats)))) A__INCR(cons(head(tail(incr(y0))), z1)) -> c6(MARK(head(tail(incr(y0))))) A__INCR(cons(head(tail(pairs)), z1)) -> c6(MARK(head(tail(pairs)))) A__INCR(cons(head(tail(odds)), z1)) -> c6(MARK(head(tail(odds)))) A__INCR(cons(head(tail(head(y0))), z1)) -> c6(MARK(head(tail(head(y0))))) A__INCR(cons(head(tail(tail(y0))), z1)) -> c6(MARK(head(tail(tail(y0))))) A__INCR(cons(head(tail(cons(y0, y1))), z1)) -> c6(MARK(head(tail(cons(y0, y1))))) A__INCR(cons(head(tail(s(y0))), z1)) -> c6(MARK(head(tail(s(y0))))) A__INCR(cons(head(tail(y0)), z1)) -> c6(MARK(head(tail(y0)))) A__INCR(cons(tail(incr(nats)), z1)) -> c6(MARK(tail(incr(nats)))) A__INCR(cons(tail(incr(incr(y0))), z1)) -> c6(MARK(tail(incr(incr(y0))))) A__INCR(cons(tail(incr(pairs)), z1)) -> c6(MARK(tail(incr(pairs)))) A__INCR(cons(tail(incr(odds)), z1)) -> c6(MARK(tail(incr(odds)))) A__INCR(cons(tail(incr(head(y0))), z1)) -> c6(MARK(tail(incr(head(y0))))) A__INCR(cons(tail(incr(tail(y0))), z1)) -> c6(MARK(tail(incr(tail(y0))))) A__INCR(cons(tail(incr(cons(y0, y1))), z1)) -> c6(MARK(tail(incr(cons(y0, y1))))) A__INCR(cons(tail(incr(s(y0))), z1)) -> c6(MARK(tail(incr(s(y0))))) A__INCR(cons(tail(incr(y0)), z1)) -> c6(MARK(tail(incr(y0)))) A__INCR(cons(tail(odds), z1)) -> c6(MARK(tail(odds))) A__INCR(cons(tail(head(nats)), z1)) -> c6(MARK(tail(head(nats)))) A__INCR(cons(tail(head(incr(y0))), z1)) -> c6(MARK(tail(head(incr(y0))))) A__INCR(cons(tail(head(pairs)), z1)) -> c6(MARK(tail(head(pairs)))) A__INCR(cons(tail(head(odds)), z1)) -> c6(MARK(tail(head(odds)))) A__INCR(cons(tail(head(head(y0))), z1)) -> c6(MARK(tail(head(head(y0))))) A__INCR(cons(tail(head(tail(y0))), z1)) -> c6(MARK(tail(head(tail(y0))))) A__INCR(cons(tail(head(cons(y0, y1))), z1)) -> c6(MARK(tail(head(cons(y0, y1))))) A__INCR(cons(tail(head(s(y0))), z1)) -> c6(MARK(tail(head(s(y0))))) A__INCR(cons(tail(head(y0)), z1)) -> c6(MARK(tail(head(y0)))) A__INCR(cons(tail(tail(nats)), z1)) -> c6(MARK(tail(tail(nats)))) A__INCR(cons(tail(tail(incr(y0))), z1)) -> c6(MARK(tail(tail(incr(y0))))) A__INCR(cons(tail(tail(pairs)), z1)) -> c6(MARK(tail(tail(pairs)))) A__INCR(cons(tail(tail(odds)), z1)) -> c6(MARK(tail(tail(odds)))) A__INCR(cons(tail(tail(head(y0))), z1)) -> c6(MARK(tail(tail(head(y0))))) A__INCR(cons(tail(tail(tail(y0))), z1)) -> c6(MARK(tail(tail(tail(y0))))) A__INCR(cons(tail(tail(cons(y0, y1))), z1)) -> c6(MARK(tail(tail(cons(y0, y1))))) A__INCR(cons(tail(tail(s(y0))), z1)) -> c6(MARK(tail(tail(s(y0))))) A__INCR(cons(tail(tail(y0)), z1)) -> c6(MARK(tail(tail(y0)))) A__INCR(cons(tail(nats), z1)) -> c6(MARK(tail(nats))) A__INCR(cons(tail(pairs), z1)) -> c6(MARK(tail(pairs))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs))) MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs))) MARK(tail(incr(nats))) -> c17(A__TAIL(a__incr(a__nats))) MARK(tail(incr(pairs))) -> c17(A__TAIL(a__incr(a__pairs))) MARK(tail(odds)) -> c17(A__TAIL(a__incr(a__pairs))) MARK(incr(cons(nats, x1))) -> c13(A__INCR(cons(a__nats, x1)), MARK(cons(nats, x1))) MARK(incr(cons(incr(z0), x1))) -> c13(A__INCR(cons(a__incr(mark(z0)), x1)), MARK(cons(incr(z0), x1))) MARK(incr(cons(pairs, x1))) -> c13(A__INCR(cons(a__pairs, x1)), MARK(cons(pairs, x1))) MARK(incr(cons(odds, x1))) -> 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c10(MARK(incr(head(pairs)))) A__TAIL(cons(z0, head(head(nats)))) -> c10(MARK(head(head(nats)))) A__TAIL(cons(z0, head(head(pairs)))) -> c10(MARK(head(head(pairs)))) A__TAIL(cons(z0, tail(head(nats)))) -> c10(MARK(tail(head(nats)))) A__TAIL(cons(z0, tail(head(pairs)))) -> c10(MARK(tail(head(pairs)))) A__TAIL(cons(z0, head(cons(nats, y0)))) -> c10(MARK(head(cons(nats, y0)))) A__TAIL(cons(z0, head(cons(incr(y0), y1)))) -> c10(MARK(head(cons(incr(y0), y1)))) A__TAIL(cons(z0, head(cons(pairs, y0)))) -> c10(MARK(head(cons(pairs, y0)))) A__TAIL(cons(z0, head(cons(odds, y0)))) -> c10(MARK(head(cons(odds, y0)))) A__TAIL(cons(z0, head(cons(head(y0), y1)))) -> c10(MARK(head(cons(head(y0), y1)))) A__TAIL(cons(z0, head(cons(tail(y0), y1)))) -> c10(MARK(head(cons(tail(y0), y1)))) A__TAIL(cons(z0, head(cons(cons(y0, y1), y2)))) -> c10(MARK(head(cons(cons(y0, y1), y2)))) A__TAIL(cons(z0, head(cons(s(y0), y1)))) -> c10(MARK(head(cons(s(y0), y1)))) A__TAIL(cons(z0, head(cons(y0, y1)))) -> c10(MARK(head(cons(y0, y1)))) A__TAIL(cons(z0, head(cons(0, y0)))) -> c10(MARK(head(cons(0, y0)))) S tuples: MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c17(MARK(s(z0))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(tail(incr(incr(z0)))) -> c17(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(odds))) -> c17(A__TAIL(a__incr(a__odds)), MARK(incr(odds))) MARK(tail(incr(head(z0)))) -> c17(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c17(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c17(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c17(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c17(MARK(incr(x0))) MARK(tail(head(incr(z0)))) -> c17(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(odds))) -> c17(A__TAIL(a__head(a__odds)), MARK(head(odds))) MARK(tail(head(head(z0)))) -> c17(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c17(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c17(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c17(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c17(MARK(head(x0))) MARK(tail(tail(nats))) -> c17(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(incr(z0)))) -> c17(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(pairs))) -> c17(A__TAIL(a__tail(a__pairs)), MARK(tail(pairs))) MARK(tail(tail(odds))) -> c17(A__TAIL(a__tail(a__odds)), MARK(tail(odds))) MARK(tail(tail(head(z0)))) -> c17(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c17(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c17(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c17(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c17(MARK(tail(x0))) MARK(tail(nats)) -> c17(A__TAIL(cons(0, incr(nats)))) MARK(tail(pairs)) -> c17(A__TAIL(cons(0, incr(odds)))) A__INCR(cons(cons(y0, y1), z1)) -> c6(MARK(cons(y0, y1))) A__INCR(cons(s(y0), z1)) -> c6(MARK(s(y0))) A__INCR(cons(incr(cons(y0, y1)), z1)) -> c6(MARK(incr(cons(y0, y1)))) A__INCR(cons(incr(s(y0)), z1)) -> c6(MARK(incr(s(y0)))) A__INCR(cons(head(cons(y0, y1)), z1)) -> c6(MARK(head(cons(y0, y1)))) A__INCR(cons(head(s(y0)), z1)) -> c6(MARK(head(s(y0)))) A__INCR(cons(tail(cons(y0, y1)), z1)) -> c6(MARK(tail(cons(y0, y1)))) A__INCR(cons(tail(s(y0)), z1)) -> c6(MARK(tail(s(y0)))) A__INCR(cons(incr(incr(nats)), z1)) -> c6(MARK(incr(incr(nats)))) A__INCR(cons(incr(incr(incr(y0))), z1)) -> c6(MARK(incr(incr(incr(y0))))) A__INCR(cons(incr(incr(pairs)), z1)) -> c6(MARK(incr(incr(pairs)))) A__INCR(cons(incr(incr(odds)), z1)) -> c6(MARK(incr(incr(odds)))) A__INCR(cons(incr(incr(head(y0))), z1)) -> c6(MARK(incr(incr(head(y0))))) A__INCR(cons(incr(incr(tail(y0))), z1)) -> c6(MARK(incr(incr(tail(y0))))) A__INCR(cons(incr(incr(cons(y0, y1))), z1)) -> c6(MARK(incr(incr(cons(y0, y1))))) A__INCR(cons(incr(incr(s(y0))), z1)) -> c6(MARK(incr(incr(s(y0))))) A__INCR(cons(incr(incr(y0)), z1)) -> c6(MARK(incr(incr(y0)))) A__INCR(cons(incr(odds), z1)) -> c6(MARK(incr(odds))) A__INCR(cons(incr(head(nats)), z1)) -> c6(MARK(incr(head(nats)))) A__INCR(cons(incr(head(incr(y0))), z1)) -> c6(MARK(incr(head(incr(y0))))) A__INCR(cons(incr(head(pairs)), z1)) -> c6(MARK(incr(head(pairs)))) A__INCR(cons(incr(head(odds)), z1)) -> c6(MARK(incr(head(odds)))) A__INCR(cons(incr(head(head(y0))), z1)) -> c6(MARK(incr(head(head(y0))))) A__INCR(cons(incr(head(tail(y0))), z1)) -> c6(MARK(incr(head(tail(y0))))) A__INCR(cons(incr(head(cons(y0, y1))), z1)) -> c6(MARK(incr(head(cons(y0, y1))))) A__INCR(cons(incr(head(s(y0))), z1)) -> c6(MARK(incr(head(s(y0))))) A__INCR(cons(incr(head(y0)), z1)) -> c6(MARK(incr(head(y0)))) A__INCR(cons(incr(tail(nats)), z1)) -> c6(MARK(incr(tail(nats)))) A__INCR(cons(incr(tail(incr(y0))), z1)) -> c6(MARK(incr(tail(incr(y0))))) A__INCR(cons(incr(tail(pairs)), z1)) -> c6(MARK(incr(tail(pairs)))) A__INCR(cons(incr(tail(odds)), z1)) -> c6(MARK(incr(tail(odds)))) A__INCR(cons(incr(tail(head(y0))), z1)) -> c6(MARK(incr(tail(head(y0))))) A__INCR(cons(incr(tail(tail(y0))), z1)) -> c6(MARK(incr(tail(tail(y0))))) A__INCR(cons(incr(tail(cons(y0, y1))), z1)) -> c6(MARK(incr(tail(cons(y0, y1))))) A__INCR(cons(incr(tail(s(y0))), z1)) -> c6(MARK(incr(tail(s(y0))))) A__INCR(cons(incr(tail(y0)), z1)) -> c6(MARK(incr(tail(y0)))) A__INCR(cons(head(incr(nats)), z1)) -> c6(MARK(head(incr(nats)))) A__INCR(cons(head(incr(incr(y0))), z1)) -> c6(MARK(head(incr(incr(y0))))) A__INCR(cons(head(incr(pairs)), z1)) -> c6(MARK(head(incr(pairs)))) A__INCR(cons(head(incr(odds)), z1)) -> c6(MARK(head(incr(odds)))) A__INCR(cons(head(incr(head(y0))), z1)) -> c6(MARK(head(incr(head(y0))))) A__INCR(cons(head(incr(tail(y0))), z1)) -> c6(MARK(head(incr(tail(y0))))) A__INCR(cons(head(incr(cons(y0, y1))), z1)) -> c6(MARK(head(incr(cons(y0, y1))))) A__INCR(cons(head(incr(s(y0))), z1)) -> c6(MARK(head(incr(s(y0))))) A__INCR(cons(head(incr(y0)), z1)) -> c6(MARK(head(incr(y0)))) A__INCR(cons(head(odds), z1)) -> c6(MARK(head(odds))) A__INCR(cons(head(head(nats)), z1)) -> c6(MARK(head(head(nats)))) A__INCR(cons(head(head(incr(y0))), z1)) -> c6(MARK(head(head(incr(y0))))) A__INCR(cons(head(head(pairs)), z1)) -> c6(MARK(head(head(pairs)))) A__INCR(cons(head(head(odds)), z1)) -> c6(MARK(head(head(odds)))) A__INCR(cons(head(head(head(y0))), z1)) -> c6(MARK(head(head(head(y0))))) A__INCR(cons(head(head(tail(y0))), z1)) -> c6(MARK(head(head(tail(y0))))) A__INCR(cons(head(head(cons(y0, y1))), z1)) -> c6(MARK(head(head(cons(y0, y1))))) A__INCR(cons(head(head(s(y0))), z1)) -> c6(MARK(head(head(s(y0))))) A__INCR(cons(head(head(y0)), z1)) -> c6(MARK(head(head(y0)))) A__INCR(cons(head(tail(nats)), z1)) -> c6(MARK(head(tail(nats)))) A__INCR(cons(head(tail(incr(y0))), z1)) -> c6(MARK(head(tail(incr(y0))))) A__INCR(cons(head(tail(pairs)), z1)) -> c6(MARK(head(tail(pairs)))) A__INCR(cons(head(tail(odds)), z1)) -> c6(MARK(head(tail(odds)))) A__INCR(cons(head(tail(head(y0))), z1)) -> c6(MARK(head(tail(head(y0))))) A__INCR(cons(head(tail(tail(y0))), z1)) -> c6(MARK(head(tail(tail(y0))))) A__INCR(cons(head(tail(cons(y0, y1))), z1)) -> c6(MARK(head(tail(cons(y0, y1))))) A__INCR(cons(head(tail(s(y0))), z1)) -> c6(MARK(head(tail(s(y0))))) A__INCR(cons(head(tail(y0)), z1)) -> c6(MARK(head(tail(y0)))) A__INCR(cons(tail(incr(nats)), z1)) -> c6(MARK(tail(incr(nats)))) A__INCR(cons(tail(incr(incr(y0))), z1)) -> c6(MARK(tail(incr(incr(y0))))) A__INCR(cons(tail(incr(pairs)), z1)) -> c6(MARK(tail(incr(pairs)))) A__INCR(cons(tail(incr(odds)), z1)) -> c6(MARK(tail(incr(odds)))) 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c10(MARK(tail(head(odds)))) A__TAIL(cons(z0, tail(head(head(y0))))) -> c10(MARK(tail(head(head(y0))))) A__TAIL(cons(z0, tail(head(tail(y0))))) -> c10(MARK(tail(head(tail(y0))))) A__TAIL(cons(z0, tail(head(cons(y0, y1))))) -> c10(MARK(tail(head(cons(y0, y1))))) A__TAIL(cons(z0, tail(head(s(y0))))) -> c10(MARK(tail(head(s(y0))))) A__TAIL(cons(z0, tail(head(y0)))) -> c10(MARK(tail(head(y0)))) A__TAIL(cons(z0, tail(tail(nats)))) -> c10(MARK(tail(tail(nats)))) A__TAIL(cons(z0, tail(tail(incr(y0))))) -> c10(MARK(tail(tail(incr(y0))))) A__TAIL(cons(z0, tail(tail(pairs)))) -> c10(MARK(tail(tail(pairs)))) A__TAIL(cons(z0, tail(tail(odds)))) -> c10(MARK(tail(tail(odds)))) A__TAIL(cons(z0, tail(tail(head(y0))))) -> c10(MARK(tail(tail(head(y0))))) A__TAIL(cons(z0, tail(tail(tail(y0))))) -> c10(MARK(tail(tail(tail(y0))))) A__TAIL(cons(z0, tail(tail(cons(y0, y1))))) -> c10(MARK(tail(tail(cons(y0, y1))))) A__TAIL(cons(z0, tail(tail(s(y0))))) -> c10(MARK(tail(tail(s(y0))))) A__TAIL(cons(z0, tail(tail(y0)))) -> c10(MARK(tail(tail(y0)))) A__TAIL(cons(z0, tail(nats))) -> c10(MARK(tail(nats))) A__TAIL(cons(z0, tail(pairs))) -> c10(MARK(tail(pairs))) A__TAIL(cons(z0, incr(incr(nats)))) -> c10(MARK(incr(incr(nats)))) A__TAIL(cons(z0, incr(incr(pairs)))) -> c10(MARK(incr(incr(pairs)))) A__TAIL(cons(z0, incr(odds))) -> c10(MARK(incr(odds))) A__TAIL(cons(z0, head(incr(nats)))) -> c10(MARK(head(incr(nats)))) A__TAIL(cons(z0, head(incr(pairs)))) -> c10(MARK(head(incr(pairs)))) A__TAIL(cons(z0, head(odds))) -> c10(MARK(head(odds))) A__TAIL(cons(z0, tail(incr(nats)))) -> c10(MARK(tail(incr(nats)))) A__TAIL(cons(z0, tail(incr(pairs)))) -> c10(MARK(tail(incr(pairs)))) A__TAIL(cons(z0, tail(odds))) -> c10(MARK(tail(odds))) A__TAIL(cons(z0, incr(cons(nats, y0)))) -> c10(MARK(incr(cons(nats, y0)))) A__TAIL(cons(z0, incr(cons(incr(y0), y1)))) -> c10(MARK(incr(cons(incr(y0), y1)))) A__TAIL(cons(z0, incr(cons(pairs, y0)))) -> c10(MARK(incr(cons(pairs, y0)))) A__TAIL(cons(z0, incr(cons(odds, y0)))) -> c10(MARK(incr(cons(odds, y0)))) A__TAIL(cons(z0, incr(cons(head(y0), y1)))) -> c10(MARK(incr(cons(head(y0), y1)))) A__TAIL(cons(z0, incr(cons(tail(y0), y1)))) -> c10(MARK(incr(cons(tail(y0), y1)))) A__TAIL(cons(z0, incr(cons(cons(y0, y1), y2)))) -> c10(MARK(incr(cons(cons(y0, y1), y2)))) A__TAIL(cons(z0, incr(cons(s(y0), y1)))) -> c10(MARK(incr(cons(s(y0), y1)))) A__TAIL(cons(z0, incr(cons(y0, y1)))) -> c10(MARK(incr(cons(y0, y1)))) A__TAIL(cons(z0, incr(cons(0, y0)))) -> c10(MARK(incr(cons(0, y0)))) A__TAIL(cons(z0, incr(head(nats)))) -> c10(MARK(incr(head(nats)))) A__TAIL(cons(z0, incr(head(pairs)))) -> c10(MARK(incr(head(pairs)))) A__TAIL(cons(z0, head(head(nats)))) -> c10(MARK(head(head(nats)))) A__TAIL(cons(z0, head(head(pairs)))) -> c10(MARK(head(head(pairs)))) A__TAIL(cons(z0, tail(head(nats)))) -> c10(MARK(tail(head(nats)))) A__TAIL(cons(z0, tail(head(pairs)))) -> c10(MARK(tail(head(pairs)))) A__TAIL(cons(z0, head(cons(nats, y0)))) -> c10(MARK(head(cons(nats, y0)))) A__TAIL(cons(z0, head(cons(incr(y0), y1)))) -> c10(MARK(head(cons(incr(y0), y1)))) A__TAIL(cons(z0, head(cons(pairs, y0)))) -> c10(MARK(head(cons(pairs, y0)))) A__TAIL(cons(z0, head(cons(odds, y0)))) -> c10(MARK(head(cons(odds, y0)))) A__TAIL(cons(z0, head(cons(head(y0), y1)))) -> c10(MARK(head(cons(head(y0), y1)))) A__TAIL(cons(z0, head(cons(tail(y0), y1)))) -> c10(MARK(head(cons(tail(y0), y1)))) A__TAIL(cons(z0, head(cons(cons(y0, y1), y2)))) -> c10(MARK(head(cons(cons(y0, y1), y2)))) A__TAIL(cons(z0, head(cons(s(y0), y1)))) -> c10(MARK(head(cons(s(y0), y1)))) A__TAIL(cons(z0, head(cons(y0, y1)))) -> c10(MARK(head(cons(y0, y1)))) A__TAIL(cons(z0, head(cons(0, y0)))) -> c10(MARK(head(cons(0, y0)))) K tuples:none Defined Rule Symbols: a__nats, a__pairs, a__odds, a__incr_1, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: MARK_1, A__INCR_1, A__HEAD_1, A__TAIL_1 Compound Symbols: c18_1, c20_1, c13_1, c16_1, c17_2, c17_1, c13_2, c16_2, c6_1, c8_1, c10_1 ---------------------------------------- (181) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing nodes: MARK(tail(nats)) -> c17(A__TAIL(cons(0, incr(nats)))) A__HEAD(cons(tail(nats), z1)) -> c8(MARK(tail(nats))) A__INCR(cons(tail(nats), z1)) -> c6(MARK(tail(nats))) A__TAIL(cons(z0, tail(nats))) -> c10(MARK(tail(nats))) ---------------------------------------- (182) Obligation: Complexity Dependency Tuples Problem Rules: a__nats -> cons(0, incr(nats)) a__nats -> nats a__pairs -> cons(0, incr(odds)) a__pairs -> pairs a__odds -> a__incr(a__pairs) a__odds -> odds a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) 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(nats) -> a__nats mark(incr(z0)) -> a__incr(mark(z0)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c17(MARK(s(z0))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(tail(incr(incr(z0)))) -> c17(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(odds))) -> c17(A__TAIL(a__incr(a__odds)), MARK(incr(odds))) MARK(tail(incr(head(z0)))) -> c17(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c17(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c17(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c17(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c17(MARK(incr(x0))) MARK(tail(head(incr(z0)))) -> c17(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(odds))) -> c17(A__TAIL(a__head(a__odds)), MARK(head(odds))) MARK(tail(head(head(z0)))) -> c17(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c17(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c17(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c17(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c17(MARK(head(x0))) MARK(tail(tail(nats))) -> c17(A__TAIL(a__tail(a__nats)), MARK(tail(nats))) MARK(tail(tail(incr(z0)))) -> c17(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(pairs))) -> c17(A__TAIL(a__tail(a__pairs)), MARK(tail(pairs))) MARK(tail(tail(odds))) -> c17(A__TAIL(a__tail(a__odds)), MARK(tail(odds))) MARK(tail(tail(head(z0)))) -> c17(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c17(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c17(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c17(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c17(MARK(tail(x0))) MARK(tail(pairs)) -> c17(A__TAIL(cons(0, incr(odds)))) A__INCR(cons(cons(y0, y1), z1)) -> c6(MARK(cons(y0, y1))) A__INCR(cons(s(y0), z1)) -> c6(MARK(s(y0))) A__INCR(cons(incr(cons(y0, y1)), z1)) -> c6(MARK(incr(cons(y0, y1)))) A__INCR(cons(incr(s(y0)), z1)) -> c6(MARK(incr(s(y0)))) A__INCR(cons(head(cons(y0, y1)), z1)) -> c6(MARK(head(cons(y0, y1)))) A__INCR(cons(head(s(y0)), z1)) -> c6(MARK(head(s(y0)))) A__INCR(cons(tail(cons(y0, y1)), z1)) -> c6(MARK(tail(cons(y0, y1)))) A__INCR(cons(tail(s(y0)), z1)) -> c6(MARK(tail(s(y0)))) A__INCR(cons(incr(incr(nats)), z1)) -> c6(MARK(incr(incr(nats)))) A__INCR(cons(incr(incr(incr(y0))), z1)) -> c6(MARK(incr(incr(incr(y0))))) A__INCR(cons(incr(incr(pairs)), z1)) -> c6(MARK(incr(incr(pairs)))) A__INCR(cons(incr(incr(odds)), z1)) -> c6(MARK(incr(incr(odds)))) A__INCR(cons(incr(incr(head(y0))), z1)) -> c6(MARK(incr(incr(head(y0))))) A__INCR(cons(incr(incr(tail(y0))), z1)) -> c6(MARK(incr(incr(tail(y0))))) 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z1)) -> c6(MARK(incr(tail(incr(y0))))) A__INCR(cons(incr(tail(pairs)), z1)) -> c6(MARK(incr(tail(pairs)))) A__INCR(cons(incr(tail(odds)), z1)) -> c6(MARK(incr(tail(odds)))) A__INCR(cons(incr(tail(head(y0))), z1)) -> c6(MARK(incr(tail(head(y0))))) A__INCR(cons(incr(tail(tail(y0))), z1)) -> c6(MARK(incr(tail(tail(y0))))) A__INCR(cons(incr(tail(cons(y0, y1))), z1)) -> c6(MARK(incr(tail(cons(y0, y1))))) A__INCR(cons(incr(tail(s(y0))), z1)) -> c6(MARK(incr(tail(s(y0))))) A__INCR(cons(incr(tail(y0)), z1)) -> c6(MARK(incr(tail(y0)))) A__INCR(cons(head(incr(nats)), z1)) -> c6(MARK(head(incr(nats)))) A__INCR(cons(head(incr(incr(y0))), z1)) -> c6(MARK(head(incr(incr(y0))))) A__INCR(cons(head(incr(pairs)), z1)) -> c6(MARK(head(incr(pairs)))) A__INCR(cons(head(incr(odds)), z1)) -> c6(MARK(head(incr(odds)))) A__INCR(cons(head(incr(head(y0))), z1)) -> c6(MARK(head(incr(head(y0))))) A__INCR(cons(head(incr(tail(y0))), z1)) -> c6(MARK(head(incr(tail(y0))))) A__INCR(cons(head(incr(cons(y0, y1))), z1)) 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A__TAIL(cons(z0, tail(tail(nats)))) -> c10(MARK(tail(tail(nats)))) A__TAIL(cons(z0, tail(tail(incr(y0))))) -> c10(MARK(tail(tail(incr(y0))))) A__TAIL(cons(z0, tail(tail(pairs)))) -> c10(MARK(tail(tail(pairs)))) A__TAIL(cons(z0, tail(tail(odds)))) -> c10(MARK(tail(tail(odds)))) A__TAIL(cons(z0, tail(tail(head(y0))))) -> c10(MARK(tail(tail(head(y0))))) A__TAIL(cons(z0, tail(tail(tail(y0))))) -> c10(MARK(tail(tail(tail(y0))))) A__TAIL(cons(z0, tail(tail(cons(y0, y1))))) -> c10(MARK(tail(tail(cons(y0, y1))))) A__TAIL(cons(z0, tail(tail(s(y0))))) -> c10(MARK(tail(tail(s(y0))))) A__TAIL(cons(z0, tail(tail(y0)))) -> c10(MARK(tail(tail(y0)))) A__TAIL(cons(z0, tail(pairs))) -> c10(MARK(tail(pairs))) A__TAIL(cons(z0, incr(incr(nats)))) -> c10(MARK(incr(incr(nats)))) A__TAIL(cons(z0, incr(incr(pairs)))) -> c10(MARK(incr(incr(pairs)))) A__TAIL(cons(z0, incr(odds))) -> c10(MARK(incr(odds))) A__TAIL(cons(z0, head(incr(nats)))) -> c10(MARK(head(incr(nats)))) A__TAIL(cons(z0, head(incr(pairs)))) -> c10(MARK(head(incr(pairs)))) A__TAIL(cons(z0, head(odds))) -> c10(MARK(head(odds))) A__TAIL(cons(z0, tail(incr(nats)))) -> c10(MARK(tail(incr(nats)))) A__TAIL(cons(z0, tail(incr(pairs)))) -> c10(MARK(tail(incr(pairs)))) A__TAIL(cons(z0, tail(odds))) -> c10(MARK(tail(odds))) A__TAIL(cons(z0, incr(cons(nats, y0)))) -> c10(MARK(incr(cons(nats, y0)))) A__TAIL(cons(z0, incr(cons(incr(y0), y1)))) -> c10(MARK(incr(cons(incr(y0), y1)))) A__TAIL(cons(z0, incr(cons(pairs, y0)))) -> c10(MARK(incr(cons(pairs, y0)))) A__TAIL(cons(z0, incr(cons(odds, y0)))) -> c10(MARK(incr(cons(odds, y0)))) A__TAIL(cons(z0, incr(cons(head(y0), y1)))) -> c10(MARK(incr(cons(head(y0), y1)))) A__TAIL(cons(z0, incr(cons(tail(y0), y1)))) -> c10(MARK(incr(cons(tail(y0), y1)))) A__TAIL(cons(z0, incr(cons(cons(y0, y1), y2)))) -> c10(MARK(incr(cons(cons(y0, y1), y2)))) A__TAIL(cons(z0, incr(cons(s(y0), y1)))) -> c10(MARK(incr(cons(s(y0), y1)))) A__TAIL(cons(z0, incr(cons(y0, y1)))) -> c10(MARK(incr(cons(y0, y1)))) A__TAIL(cons(z0, incr(cons(0, y0)))) -> c10(MARK(incr(cons(0, y0)))) A__TAIL(cons(z0, incr(head(nats)))) -> c10(MARK(incr(head(nats)))) A__TAIL(cons(z0, incr(head(pairs)))) -> c10(MARK(incr(head(pairs)))) A__TAIL(cons(z0, head(head(nats)))) -> c10(MARK(head(head(nats)))) A__TAIL(cons(z0, head(head(pairs)))) -> c10(MARK(head(head(pairs)))) A__TAIL(cons(z0, tail(head(nats)))) -> c10(MARK(tail(head(nats)))) A__TAIL(cons(z0, tail(head(pairs)))) -> c10(MARK(tail(head(pairs)))) A__TAIL(cons(z0, head(cons(nats, y0)))) -> c10(MARK(head(cons(nats, y0)))) A__TAIL(cons(z0, head(cons(incr(y0), y1)))) -> c10(MARK(head(cons(incr(y0), y1)))) A__TAIL(cons(z0, head(cons(pairs, y0)))) -> c10(MARK(head(cons(pairs, y0)))) A__TAIL(cons(z0, head(cons(odds, y0)))) -> c10(MARK(head(cons(odds, y0)))) A__TAIL(cons(z0, head(cons(head(y0), y1)))) -> c10(MARK(head(cons(head(y0), y1)))) A__TAIL(cons(z0, head(cons(tail(y0), y1)))) -> c10(MARK(head(cons(tail(y0), y1)))) A__TAIL(cons(z0, head(cons(cons(y0, y1), y2)))) -> c10(MARK(head(cons(cons(y0, y1), y2)))) A__TAIL(cons(z0, head(cons(s(y0), y1)))) -> c10(MARK(head(cons(s(y0), y1)))) A__TAIL(cons(z0, head(cons(y0, y1)))) -> c10(MARK(head(cons(y0, y1)))) A__TAIL(cons(z0, head(cons(0, y0)))) -> c10(MARK(head(cons(0, y0)))) S tuples: MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c17(MARK(s(z0))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats)), MARK(tail(nats))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats)), MARK(tail(nats))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(tail(incr(incr(z0)))) -> c17(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(odds))) -> c17(A__TAIL(a__incr(a__odds)), MARK(incr(odds))) MARK(tail(incr(head(z0)))) -> c17(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c17(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c17(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c17(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c17(MARK(incr(x0))) MARK(tail(head(incr(z0)))) -> 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c8(MARK(incr(cons(head(y0), y1)))) A__HEAD(cons(incr(cons(tail(y0), y1)), z1)) -> c8(MARK(incr(cons(tail(y0), y1)))) A__HEAD(cons(incr(cons(cons(y0, y1), y2)), z1)) -> c8(MARK(incr(cons(cons(y0, y1), y2)))) A__HEAD(cons(incr(cons(s(y0), y1)), z1)) -> c8(MARK(incr(cons(s(y0), y1)))) A__HEAD(cons(incr(cons(y0, y1)), z1)) -> c8(MARK(incr(cons(y0, y1)))) A__HEAD(cons(incr(cons(0, y0)), z1)) -> c8(MARK(incr(cons(0, y0)))) MARK(incr(head(nats))) -> c13(A__INCR(a__head(a__nats))) MARK(incr(head(pairs))) -> c13(A__INCR(a__head(a__pairs))) MARK(head(head(nats))) -> c16(A__HEAD(a__head(a__nats))) MARK(head(head(pairs))) -> c16(A__HEAD(a__head(a__pairs))) MARK(tail(head(nats))) -> c17(A__TAIL(a__head(a__nats))) MARK(tail(head(pairs))) -> c17(A__TAIL(a__head(a__pairs))) MARK(head(cons(nats, x1))) -> c16(A__HEAD(cons(a__nats, x1)), MARK(cons(nats, x1))) MARK(head(cons(incr(z0), x1))) -> c16(A__HEAD(cons(a__incr(mark(z0)), x1)), MARK(cons(incr(z0), x1))) MARK(head(cons(pairs, x1))) -> c16(A__HEAD(cons(a__pairs, x1)), MARK(cons(pairs, x1))) MARK(head(cons(odds, x1))) -> c16(A__HEAD(cons(a__odds, x1)), MARK(cons(odds, x1))) MARK(head(cons(head(z0), x1))) -> c16(A__HEAD(cons(a__head(mark(z0)), x1)), MARK(cons(head(z0), x1))) MARK(head(cons(tail(z0), x1))) -> c16(A__HEAD(cons(a__tail(mark(z0)), x1)), MARK(cons(tail(z0), x1))) MARK(head(cons(cons(z0, z1), x1))) -> c16(A__HEAD(cons(cons(mark(z0), z1), x1)), MARK(cons(cons(z0, z1), x1))) MARK(head(cons(s(z0), x1))) -> c16(A__HEAD(cons(s(mark(z0)), x1)), MARK(cons(s(z0), x1))) MARK(head(cons(x0, x1))) -> c16(MARK(cons(x0, x1))) MARK(head(cons(0, x1))) -> c16(MARK(cons(0, x1))) A__TAIL(cons(z0, cons(y0, y1))) -> c10(MARK(cons(y0, y1))) A__TAIL(cons(z0, s(y0))) -> c10(MARK(s(y0))) A__TAIL(cons(z0, incr(s(y0)))) -> c10(MARK(incr(s(y0)))) A__TAIL(cons(z0, head(s(y0)))) -> c10(MARK(head(s(y0)))) A__TAIL(cons(z0, tail(cons(y0, y1)))) -> c10(MARK(tail(cons(y0, y1)))) A__TAIL(cons(z0, tail(s(y0)))) -> c10(MARK(tail(s(y0)))) A__TAIL(cons(z0, incr(incr(incr(y0))))) -> c10(MARK(incr(incr(incr(y0))))) A__TAIL(cons(z0, incr(incr(odds)))) -> c10(MARK(incr(incr(odds)))) A__TAIL(cons(z0, incr(incr(head(y0))))) -> c10(MARK(incr(incr(head(y0))))) A__TAIL(cons(z0, incr(incr(tail(y0))))) -> c10(MARK(incr(incr(tail(y0))))) A__TAIL(cons(z0, incr(incr(cons(y0, y1))))) -> c10(MARK(incr(incr(cons(y0, y1))))) A__TAIL(cons(z0, incr(incr(s(y0))))) -> c10(MARK(incr(incr(s(y0))))) A__TAIL(cons(z0, incr(incr(y0)))) -> c10(MARK(incr(incr(y0)))) A__TAIL(cons(z0, incr(head(incr(y0))))) -> c10(MARK(incr(head(incr(y0))))) A__TAIL(cons(z0, incr(head(odds)))) -> c10(MARK(incr(head(odds)))) A__TAIL(cons(z0, incr(head(head(y0))))) -> c10(MARK(incr(head(head(y0))))) A__TAIL(cons(z0, incr(head(tail(y0))))) -> c10(MARK(incr(head(tail(y0))))) A__TAIL(cons(z0, incr(head(cons(y0, y1))))) -> c10(MARK(incr(head(cons(y0, y1))))) A__TAIL(cons(z0, incr(head(s(y0))))) -> c10(MARK(incr(head(s(y0))))) A__TAIL(cons(z0, incr(head(y0)))) -> c10(MARK(incr(head(y0)))) A__TAIL(cons(z0, incr(tail(nats)))) -> c10(MARK(incr(tail(nats)))) A__TAIL(cons(z0, incr(tail(incr(y0))))) -> c10(MARK(incr(tail(incr(y0))))) A__TAIL(cons(z0, incr(tail(pairs)))) -> c10(MARK(incr(tail(pairs)))) A__TAIL(cons(z0, incr(tail(odds)))) -> c10(MARK(incr(tail(odds)))) A__TAIL(cons(z0, incr(tail(head(y0))))) -> c10(MARK(incr(tail(head(y0))))) A__TAIL(cons(z0, incr(tail(tail(y0))))) -> c10(MARK(incr(tail(tail(y0))))) A__TAIL(cons(z0, incr(tail(cons(y0, y1))))) -> c10(MARK(incr(tail(cons(y0, y1))))) A__TAIL(cons(z0, incr(tail(s(y0))))) -> c10(MARK(incr(tail(s(y0))))) A__TAIL(cons(z0, incr(tail(y0)))) -> c10(MARK(incr(tail(y0)))) A__TAIL(cons(z0, head(incr(incr(y0))))) -> c10(MARK(head(incr(incr(y0))))) A__TAIL(cons(z0, head(incr(odds)))) -> c10(MARK(head(incr(odds)))) A__TAIL(cons(z0, head(incr(head(y0))))) -> c10(MARK(head(incr(head(y0))))) A__TAIL(cons(z0, head(incr(tail(y0))))) -> c10(MARK(head(incr(tail(y0))))) A__TAIL(cons(z0, head(incr(cons(y0, y1))))) -> c10(MARK(head(incr(cons(y0, y1))))) A__TAIL(cons(z0, head(incr(s(y0))))) -> c10(MARK(head(incr(s(y0))))) A__TAIL(cons(z0, head(incr(y0)))) -> c10(MARK(head(incr(y0)))) A__TAIL(cons(z0, head(head(incr(y0))))) -> c10(MARK(head(head(incr(y0))))) A__TAIL(cons(z0, head(head(odds)))) -> c10(MARK(head(head(odds)))) A__TAIL(cons(z0, head(head(head(y0))))) -> c10(MARK(head(head(head(y0))))) A__TAIL(cons(z0, head(head(tail(y0))))) -> c10(MARK(head(head(tail(y0))))) A__TAIL(cons(z0, head(head(cons(y0, y1))))) -> c10(MARK(head(head(cons(y0, y1))))) A__TAIL(cons(z0, head(head(s(y0))))) -> c10(MARK(head(head(s(y0))))) A__TAIL(cons(z0, head(head(y0)))) -> c10(MARK(head(head(y0)))) A__TAIL(cons(z0, head(tail(nats)))) -> c10(MARK(head(tail(nats)))) A__TAIL(cons(z0, head(tail(incr(y0))))) -> c10(MARK(head(tail(incr(y0))))) A__TAIL(cons(z0, head(tail(pairs)))) -> c10(MARK(head(tail(pairs)))) A__TAIL(cons(z0, head(tail(odds)))) -> c10(MARK(head(tail(odds)))) A__TAIL(cons(z0, head(tail(head(y0))))) -> c10(MARK(head(tail(head(y0))))) A__TAIL(cons(z0, head(tail(tail(y0))))) -> c10(MARK(head(tail(tail(y0))))) A__TAIL(cons(z0, head(tail(cons(y0, y1))))) -> c10(MARK(head(tail(cons(y0, y1))))) A__TAIL(cons(z0, head(tail(s(y0))))) -> c10(MARK(head(tail(s(y0))))) A__TAIL(cons(z0, head(tail(y0)))) -> c10(MARK(head(tail(y0)))) A__TAIL(cons(z0, tail(incr(incr(y0))))) -> c10(MARK(tail(incr(incr(y0))))) A__TAIL(cons(z0, tail(incr(odds)))) -> c10(MARK(tail(incr(odds)))) A__TAIL(cons(z0, tail(incr(head(y0))))) -> c10(MARK(tail(incr(head(y0))))) A__TAIL(cons(z0, tail(incr(tail(y0))))) -> c10(MARK(tail(incr(tail(y0))))) A__TAIL(cons(z0, tail(incr(cons(y0, y1))))) -> c10(MARK(tail(incr(cons(y0, y1))))) A__TAIL(cons(z0, tail(incr(s(y0))))) -> c10(MARK(tail(incr(s(y0))))) A__TAIL(cons(z0, tail(incr(y0)))) -> c10(MARK(tail(incr(y0)))) A__TAIL(cons(z0, tail(head(incr(y0))))) -> c10(MARK(tail(head(incr(y0))))) A__TAIL(cons(z0, tail(head(odds)))) -> c10(MARK(tail(head(odds)))) A__TAIL(cons(z0, tail(head(head(y0))))) -> c10(MARK(tail(head(head(y0))))) A__TAIL(cons(z0, tail(head(tail(y0))))) -> c10(MARK(tail(head(tail(y0))))) A__TAIL(cons(z0, tail(head(cons(y0, y1))))) -> c10(MARK(tail(head(cons(y0, y1))))) A__TAIL(cons(z0, tail(head(s(y0))))) -> c10(MARK(tail(head(s(y0))))) A__TAIL(cons(z0, tail(head(y0)))) -> c10(MARK(tail(head(y0)))) A__TAIL(cons(z0, tail(tail(nats)))) -> c10(MARK(tail(tail(nats)))) A__TAIL(cons(z0, tail(tail(incr(y0))))) -> c10(MARK(tail(tail(incr(y0))))) A__TAIL(cons(z0, tail(tail(pairs)))) -> c10(MARK(tail(tail(pairs)))) A__TAIL(cons(z0, tail(tail(odds)))) -> c10(MARK(tail(tail(odds)))) A__TAIL(cons(z0, tail(tail(head(y0))))) -> c10(MARK(tail(tail(head(y0))))) A__TAIL(cons(z0, tail(tail(tail(y0))))) -> c10(MARK(tail(tail(tail(y0))))) A__TAIL(cons(z0, tail(tail(cons(y0, y1))))) -> c10(MARK(tail(tail(cons(y0, y1))))) A__TAIL(cons(z0, tail(tail(s(y0))))) -> c10(MARK(tail(tail(s(y0))))) A__TAIL(cons(z0, tail(tail(y0)))) -> c10(MARK(tail(tail(y0)))) A__TAIL(cons(z0, tail(pairs))) -> c10(MARK(tail(pairs))) A__TAIL(cons(z0, incr(incr(nats)))) -> c10(MARK(incr(incr(nats)))) A__TAIL(cons(z0, incr(incr(pairs)))) -> c10(MARK(incr(incr(pairs)))) A__TAIL(cons(z0, incr(odds))) -> c10(MARK(incr(odds))) A__TAIL(cons(z0, head(incr(nats)))) -> c10(MARK(head(incr(nats)))) A__TAIL(cons(z0, head(incr(pairs)))) -> c10(MARK(head(incr(pairs)))) A__TAIL(cons(z0, head(odds))) -> c10(MARK(head(odds))) A__TAIL(cons(z0, tail(incr(nats)))) -> c10(MARK(tail(incr(nats)))) A__TAIL(cons(z0, tail(incr(pairs)))) -> c10(MARK(tail(incr(pairs)))) A__TAIL(cons(z0, tail(odds))) -> c10(MARK(tail(odds))) A__TAIL(cons(z0, incr(cons(nats, y0)))) -> c10(MARK(incr(cons(nats, y0)))) A__TAIL(cons(z0, incr(cons(incr(y0), y1)))) -> c10(MARK(incr(cons(incr(y0), y1)))) A__TAIL(cons(z0, incr(cons(pairs, y0)))) -> c10(MARK(incr(cons(pairs, y0)))) A__TAIL(cons(z0, incr(cons(odds, y0)))) -> c10(MARK(incr(cons(odds, y0)))) A__TAIL(cons(z0, incr(cons(head(y0), y1)))) -> c10(MARK(incr(cons(head(y0), y1)))) A__TAIL(cons(z0, incr(cons(tail(y0), y1)))) -> c10(MARK(incr(cons(tail(y0), y1)))) A__TAIL(cons(z0, incr(cons(cons(y0, y1), y2)))) -> c10(MARK(incr(cons(cons(y0, y1), y2)))) A__TAIL(cons(z0, incr(cons(s(y0), y1)))) -> c10(MARK(incr(cons(s(y0), y1)))) A__TAIL(cons(z0, incr(cons(y0, y1)))) -> c10(MARK(incr(cons(y0, y1)))) A__TAIL(cons(z0, incr(cons(0, y0)))) -> c10(MARK(incr(cons(0, y0)))) A__TAIL(cons(z0, incr(head(nats)))) -> c10(MARK(incr(head(nats)))) A__TAIL(cons(z0, incr(head(pairs)))) -> c10(MARK(incr(head(pairs)))) A__TAIL(cons(z0, head(head(nats)))) -> c10(MARK(head(head(nats)))) A__TAIL(cons(z0, head(head(pairs)))) -> c10(MARK(head(head(pairs)))) A__TAIL(cons(z0, tail(head(nats)))) -> c10(MARK(tail(head(nats)))) A__TAIL(cons(z0, tail(head(pairs)))) -> c10(MARK(tail(head(pairs)))) A__TAIL(cons(z0, head(cons(nats, y0)))) -> c10(MARK(head(cons(nats, y0)))) A__TAIL(cons(z0, head(cons(incr(y0), y1)))) -> c10(MARK(head(cons(incr(y0), y1)))) A__TAIL(cons(z0, head(cons(pairs, y0)))) -> c10(MARK(head(cons(pairs, y0)))) A__TAIL(cons(z0, head(cons(odds, y0)))) -> c10(MARK(head(cons(odds, y0)))) A__TAIL(cons(z0, head(cons(head(y0), y1)))) -> c10(MARK(head(cons(head(y0), y1)))) A__TAIL(cons(z0, head(cons(tail(y0), y1)))) -> c10(MARK(head(cons(tail(y0), y1)))) A__TAIL(cons(z0, head(cons(cons(y0, y1), y2)))) -> c10(MARK(head(cons(cons(y0, y1), y2)))) A__TAIL(cons(z0, head(cons(s(y0), y1)))) -> c10(MARK(head(cons(s(y0), y1)))) A__TAIL(cons(z0, head(cons(y0, y1)))) -> c10(MARK(head(cons(y0, y1)))) A__TAIL(cons(z0, head(cons(0, y0)))) -> c10(MARK(head(cons(0, y0)))) K tuples:none Defined Rule Symbols: a__nats, a__pairs, a__odds, a__incr_1, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: MARK_1, A__INCR_1, A__HEAD_1, A__TAIL_1 Compound Symbols: c18_1, c20_1, c13_1, c16_1, c17_2, c17_1, c13_2, c16_2, c6_1, c8_1, c10_1 ---------------------------------------- (183) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (184) Obligation: Complexity Dependency Tuples Problem Rules: a__nats -> cons(0, incr(nats)) a__nats -> nats a__pairs -> cons(0, incr(odds)) a__pairs -> pairs a__odds -> a__incr(a__pairs) a__odds -> odds a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) 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(nats) -> a__nats mark(incr(z0)) -> a__incr(mark(z0)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c17(MARK(s(z0))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) 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A__TAIL(cons(z0, tail(head(pairs)))) -> c10(MARK(tail(head(pairs)))) A__TAIL(cons(z0, head(cons(nats, y0)))) -> c10(MARK(head(cons(nats, y0)))) A__TAIL(cons(z0, head(cons(incr(y0), y1)))) -> c10(MARK(head(cons(incr(y0), y1)))) A__TAIL(cons(z0, head(cons(pairs, y0)))) -> c10(MARK(head(cons(pairs, y0)))) A__TAIL(cons(z0, head(cons(odds, y0)))) -> c10(MARK(head(cons(odds, y0)))) A__TAIL(cons(z0, head(cons(head(y0), y1)))) -> c10(MARK(head(cons(head(y0), y1)))) A__TAIL(cons(z0, head(cons(tail(y0), y1)))) -> c10(MARK(head(cons(tail(y0), y1)))) A__TAIL(cons(z0, head(cons(cons(y0, y1), y2)))) -> c10(MARK(head(cons(cons(y0, y1), y2)))) A__TAIL(cons(z0, head(cons(s(y0), y1)))) -> c10(MARK(head(cons(s(y0), y1)))) A__TAIL(cons(z0, head(cons(y0, y1)))) -> c10(MARK(head(cons(y0, y1)))) A__TAIL(cons(z0, head(cons(0, y0)))) -> c10(MARK(head(cons(0, y0)))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats))) MARK(tail(tail(nats))) -> c17(A__TAIL(a__tail(a__nats))) S tuples: MARK(cons(z0, z1)) -> c18(MARK(z0)) MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c17(MARK(s(z0))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(tail(incr(incr(z0)))) -> c17(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(odds))) -> c17(A__TAIL(a__incr(a__odds)), MARK(incr(odds))) MARK(tail(incr(head(z0)))) -> c17(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c17(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c17(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c17(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c17(MARK(incr(x0))) MARK(tail(head(incr(z0)))) -> c17(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(odds))) -> c17(A__TAIL(a__head(a__odds)), MARK(head(odds))) MARK(tail(head(head(z0)))) -> c17(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c17(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c17(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c17(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c17(MARK(head(x0))) MARK(tail(tail(incr(z0)))) -> c17(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(pairs))) -> c17(A__TAIL(a__tail(a__pairs)), MARK(tail(pairs))) MARK(tail(tail(odds))) -> c17(A__TAIL(a__tail(a__odds)), MARK(tail(odds))) MARK(tail(tail(head(z0)))) -> c17(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c17(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c17(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c17(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c17(MARK(tail(x0))) MARK(tail(pairs)) -> c17(A__TAIL(cons(0, incr(odds)))) A__INCR(cons(cons(y0, y1), z1)) -> c6(MARK(cons(y0, y1))) A__INCR(cons(s(y0), z1)) -> c6(MARK(s(y0))) A__INCR(cons(incr(cons(y0, y1)), z1)) -> c6(MARK(incr(cons(y0, y1)))) A__INCR(cons(incr(s(y0)), z1)) -> c6(MARK(incr(s(y0)))) A__INCR(cons(head(cons(y0, y1)), z1)) -> c6(MARK(head(cons(y0, y1)))) A__INCR(cons(head(s(y0)), z1)) -> c6(MARK(head(s(y0)))) A__INCR(cons(tail(cons(y0, y1)), z1)) -> c6(MARK(tail(cons(y0, y1)))) A__INCR(cons(tail(s(y0)), z1)) -> c6(MARK(tail(s(y0)))) A__INCR(cons(incr(incr(nats)), z1)) -> c6(MARK(incr(incr(nats)))) A__INCR(cons(incr(incr(incr(y0))), z1)) -> c6(MARK(incr(incr(incr(y0))))) A__INCR(cons(incr(incr(pairs)), z1)) -> c6(MARK(incr(incr(pairs)))) A__INCR(cons(incr(incr(odds)), z1)) -> c6(MARK(incr(incr(odds)))) A__INCR(cons(incr(incr(head(y0))), z1)) -> c6(MARK(incr(incr(head(y0))))) A__INCR(cons(incr(incr(tail(y0))), z1)) -> c6(MARK(incr(incr(tail(y0))))) A__INCR(cons(incr(incr(cons(y0, y1))), z1)) -> c6(MARK(incr(incr(cons(y0, y1))))) A__INCR(cons(incr(incr(s(y0))), z1)) -> c6(MARK(incr(incr(s(y0))))) A__INCR(cons(incr(incr(y0)), z1)) -> c6(MARK(incr(incr(y0)))) A__INCR(cons(incr(odds), z1)) -> c6(MARK(incr(odds))) A__INCR(cons(incr(head(nats)), z1)) -> c6(MARK(incr(head(nats)))) A__INCR(cons(incr(head(incr(y0))), z1)) -> c6(MARK(incr(head(incr(y0))))) A__INCR(cons(incr(head(pairs)), z1)) -> c6(MARK(incr(head(pairs)))) A__INCR(cons(incr(head(odds)), z1)) -> c6(MARK(incr(head(odds)))) A__INCR(cons(incr(head(head(y0))), z1)) -> c6(MARK(incr(head(head(y0))))) A__INCR(cons(incr(head(tail(y0))), z1)) -> c6(MARK(incr(head(tail(y0))))) A__INCR(cons(incr(head(cons(y0, y1))), z1)) -> c6(MARK(incr(head(cons(y0, y1))))) A__INCR(cons(incr(head(s(y0))), z1)) -> c6(MARK(incr(head(s(y0))))) A__INCR(cons(incr(head(y0)), z1)) -> c6(MARK(incr(head(y0)))) A__INCR(cons(incr(tail(nats)), z1)) -> c6(MARK(incr(tail(nats)))) A__INCR(cons(incr(tail(incr(y0))), z1)) -> c6(MARK(incr(tail(incr(y0))))) A__INCR(cons(incr(tail(pairs)), z1)) -> c6(MARK(incr(tail(pairs)))) A__INCR(cons(incr(tail(odds)), z1)) -> c6(MARK(incr(tail(odds)))) A__INCR(cons(incr(tail(head(y0))), z1)) -> c6(MARK(incr(tail(head(y0))))) A__INCR(cons(incr(tail(tail(y0))), z1)) -> c6(MARK(incr(tail(tail(y0))))) A__INCR(cons(incr(tail(cons(y0, y1))), z1)) -> c6(MARK(incr(tail(cons(y0, y1))))) A__INCR(cons(incr(tail(s(y0))), z1)) -> c6(MARK(incr(tail(s(y0))))) A__INCR(cons(incr(tail(y0)), z1)) -> c6(MARK(incr(tail(y0)))) A__INCR(cons(head(incr(nats)), z1)) -> c6(MARK(head(incr(nats)))) A__INCR(cons(head(incr(incr(y0))), z1)) -> c6(MARK(head(incr(incr(y0))))) A__INCR(cons(head(incr(pairs)), z1)) -> c6(MARK(head(incr(pairs)))) A__INCR(cons(head(incr(odds)), z1)) -> c6(MARK(head(incr(odds)))) A__INCR(cons(head(incr(head(y0))), z1)) -> c6(MARK(head(incr(head(y0))))) A__INCR(cons(head(incr(tail(y0))), z1)) -> c6(MARK(head(incr(tail(y0))))) A__INCR(cons(head(incr(cons(y0, y1))), z1)) -> c6(MARK(head(incr(cons(y0, y1))))) A__INCR(cons(head(incr(s(y0))), z1)) -> c6(MARK(head(incr(s(y0))))) A__INCR(cons(head(incr(y0)), z1)) -> c6(MARK(head(incr(y0)))) A__INCR(cons(head(odds), z1)) -> c6(MARK(head(odds))) A__INCR(cons(head(head(nats)), z1)) -> c6(MARK(head(head(nats)))) A__INCR(cons(head(head(incr(y0))), z1)) -> c6(MARK(head(head(incr(y0))))) A__INCR(cons(head(head(pairs)), z1)) -> c6(MARK(head(head(pairs)))) A__INCR(cons(head(head(odds)), z1)) -> c6(MARK(head(head(odds)))) A__INCR(cons(head(head(head(y0))), z1)) -> c6(MARK(head(head(head(y0))))) A__INCR(cons(head(head(tail(y0))), z1)) -> c6(MARK(head(head(tail(y0))))) A__INCR(cons(head(head(cons(y0, y1))), z1)) -> c6(MARK(head(head(cons(y0, y1))))) A__INCR(cons(head(head(s(y0))), z1)) -> c6(MARK(head(head(s(y0))))) A__INCR(cons(head(head(y0)), z1)) -> c6(MARK(head(head(y0)))) A__INCR(cons(head(tail(nats)), z1)) -> c6(MARK(head(tail(nats)))) A__INCR(cons(head(tail(incr(y0))), z1)) -> c6(MARK(head(tail(incr(y0))))) A__INCR(cons(head(tail(pairs)), z1)) -> c6(MARK(head(tail(pairs)))) A__INCR(cons(head(tail(odds)), z1)) -> c6(MARK(head(tail(odds)))) A__INCR(cons(head(tail(head(y0))), z1)) -> c6(MARK(head(tail(head(y0))))) A__INCR(cons(head(tail(tail(y0))), z1)) -> c6(MARK(head(tail(tail(y0))))) A__INCR(cons(head(tail(cons(y0, y1))), z1)) -> c6(MARK(head(tail(cons(y0, y1))))) A__INCR(cons(head(tail(s(y0))), z1)) -> c6(MARK(head(tail(s(y0))))) A__INCR(cons(head(tail(y0)), z1)) -> c6(MARK(head(tail(y0)))) A__INCR(cons(tail(incr(nats)), z1)) -> c6(MARK(tail(incr(nats)))) A__INCR(cons(tail(incr(incr(y0))), z1)) -> c6(MARK(tail(incr(incr(y0))))) A__INCR(cons(tail(incr(pairs)), z1)) -> c6(MARK(tail(incr(pairs)))) A__INCR(cons(tail(incr(odds)), z1)) -> c6(MARK(tail(incr(odds)))) A__INCR(cons(tail(incr(head(y0))), z1)) -> c6(MARK(tail(incr(head(y0))))) A__INCR(cons(tail(incr(tail(y0))), z1)) -> c6(MARK(tail(incr(tail(y0))))) A__INCR(cons(tail(incr(cons(y0, y1))), z1)) -> c6(MARK(tail(incr(cons(y0, y1))))) A__INCR(cons(tail(incr(s(y0))), z1)) -> c6(MARK(tail(incr(s(y0))))) A__INCR(cons(tail(incr(y0)), z1)) -> c6(MARK(tail(incr(y0)))) A__INCR(cons(tail(odds), z1)) -> c6(MARK(tail(odds))) A__INCR(cons(tail(head(nats)), z1)) -> c6(MARK(tail(head(nats)))) A__INCR(cons(tail(head(incr(y0))), z1)) -> c6(MARK(tail(head(incr(y0))))) A__INCR(cons(tail(head(pairs)), z1)) -> c6(MARK(tail(head(pairs)))) A__INCR(cons(tail(head(odds)), z1)) -> c6(MARK(tail(head(odds)))) A__INCR(cons(tail(head(head(y0))), z1)) -> c6(MARK(tail(head(head(y0))))) A__INCR(cons(tail(head(tail(y0))), z1)) -> c6(MARK(tail(head(tail(y0))))) A__INCR(cons(tail(head(cons(y0, y1))), z1)) -> c6(MARK(tail(head(cons(y0, y1))))) A__INCR(cons(tail(head(s(y0))), z1)) -> c6(MARK(tail(head(s(y0))))) A__INCR(cons(tail(head(y0)), z1)) -> c6(MARK(tail(head(y0)))) A__INCR(cons(tail(tail(nats)), z1)) -> c6(MARK(tail(tail(nats)))) A__INCR(cons(tail(tail(incr(y0))), z1)) -> c6(MARK(tail(tail(incr(y0))))) A__INCR(cons(tail(tail(pairs)), z1)) -> c6(MARK(tail(tail(pairs)))) A__INCR(cons(tail(tail(odds)), z1)) -> c6(MARK(tail(tail(odds)))) A__INCR(cons(tail(tail(head(y0))), z1)) -> c6(MARK(tail(tail(head(y0))))) A__INCR(cons(tail(tail(tail(y0))), z1)) -> c6(MARK(tail(tail(tail(y0))))) A__INCR(cons(tail(tail(cons(y0, y1))), z1)) -> c6(MARK(tail(tail(cons(y0, y1))))) A__INCR(cons(tail(tail(s(y0))), z1)) -> c6(MARK(tail(tail(s(y0))))) A__INCR(cons(tail(tail(y0)), z1)) -> c6(MARK(tail(tail(y0)))) A__INCR(cons(tail(pairs), z1)) -> c6(MARK(tail(pairs))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs))) MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs))) MARK(tail(incr(nats))) -> c17(A__TAIL(a__incr(a__nats))) MARK(tail(incr(pairs))) -> c17(A__TAIL(a__incr(a__pairs))) MARK(tail(odds)) -> c17(A__TAIL(a__incr(a__pairs))) MARK(incr(cons(nats, x1))) -> c13(A__INCR(cons(a__nats, x1)), MARK(cons(nats, x1))) MARK(incr(cons(incr(z0), x1))) -> 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A__TAIL(cons(z0, tail(tail(pairs)))) -> c10(MARK(tail(tail(pairs)))) A__TAIL(cons(z0, tail(tail(odds)))) -> c10(MARK(tail(tail(odds)))) A__TAIL(cons(z0, tail(tail(head(y0))))) -> c10(MARK(tail(tail(head(y0))))) A__TAIL(cons(z0, tail(tail(tail(y0))))) -> c10(MARK(tail(tail(tail(y0))))) A__TAIL(cons(z0, tail(tail(cons(y0, y1))))) -> c10(MARK(tail(tail(cons(y0, y1))))) A__TAIL(cons(z0, tail(tail(s(y0))))) -> c10(MARK(tail(tail(s(y0))))) A__TAIL(cons(z0, tail(tail(y0)))) -> c10(MARK(tail(tail(y0)))) A__TAIL(cons(z0, tail(pairs))) -> c10(MARK(tail(pairs))) A__TAIL(cons(z0, incr(incr(nats)))) -> c10(MARK(incr(incr(nats)))) A__TAIL(cons(z0, incr(incr(pairs)))) -> c10(MARK(incr(incr(pairs)))) A__TAIL(cons(z0, incr(odds))) -> c10(MARK(incr(odds))) A__TAIL(cons(z0, head(incr(nats)))) -> c10(MARK(head(incr(nats)))) A__TAIL(cons(z0, head(incr(pairs)))) -> c10(MARK(head(incr(pairs)))) A__TAIL(cons(z0, head(odds))) -> c10(MARK(head(odds))) A__TAIL(cons(z0, tail(incr(nats)))) -> 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c10(MARK(incr(head(nats)))) A__TAIL(cons(z0, incr(head(pairs)))) -> c10(MARK(incr(head(pairs)))) A__TAIL(cons(z0, head(head(nats)))) -> c10(MARK(head(head(nats)))) A__TAIL(cons(z0, head(head(pairs)))) -> c10(MARK(head(head(pairs)))) A__TAIL(cons(z0, tail(head(nats)))) -> c10(MARK(tail(head(nats)))) A__TAIL(cons(z0, tail(head(pairs)))) -> c10(MARK(tail(head(pairs)))) A__TAIL(cons(z0, head(cons(nats, y0)))) -> c10(MARK(head(cons(nats, y0)))) A__TAIL(cons(z0, head(cons(incr(y0), y1)))) -> c10(MARK(head(cons(incr(y0), y1)))) A__TAIL(cons(z0, head(cons(pairs, y0)))) -> c10(MARK(head(cons(pairs, y0)))) A__TAIL(cons(z0, head(cons(odds, y0)))) -> c10(MARK(head(cons(odds, y0)))) A__TAIL(cons(z0, head(cons(head(y0), y1)))) -> c10(MARK(head(cons(head(y0), y1)))) A__TAIL(cons(z0, head(cons(tail(y0), y1)))) -> c10(MARK(head(cons(tail(y0), y1)))) A__TAIL(cons(z0, head(cons(cons(y0, y1), y2)))) -> c10(MARK(head(cons(cons(y0, y1), y2)))) A__TAIL(cons(z0, head(cons(s(y0), y1)))) -> c10(MARK(head(cons(s(y0), y1)))) A__TAIL(cons(z0, head(cons(y0, y1)))) -> c10(MARK(head(cons(y0, y1)))) A__TAIL(cons(z0, head(cons(0, y0)))) -> c10(MARK(head(cons(0, y0)))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats))) MARK(tail(tail(nats))) -> c17(A__TAIL(a__tail(a__nats))) K tuples:none Defined Rule Symbols: a__nats, a__pairs, a__odds, a__incr_1, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: MARK_1, A__INCR_1, A__HEAD_1, A__TAIL_1 Compound Symbols: c18_1, c20_1, c13_1, c16_1, c17_2, c17_1, c13_2, c16_2, c6_1, c8_1, c10_1 ---------------------------------------- (185) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace MARK(cons(z0, z1)) -> c18(MARK(z0)) by MARK(cons(cons(y0, y1), z1)) -> c18(MARK(cons(y0, y1))) MARK(cons(s(y0), z1)) -> c18(MARK(s(y0))) MARK(cons(incr(s(y0)), z1)) -> c18(MARK(incr(s(y0)))) MARK(cons(head(s(y0)), z1)) -> c18(MARK(head(s(y0)))) MARK(cons(tail(cons(y0, y1)), z1)) -> c18(MARK(tail(cons(y0, y1)))) MARK(cons(tail(s(y0)), z1)) -> c18(MARK(tail(s(y0)))) MARK(cons(incr(incr(incr(y0))), z1)) -> c18(MARK(incr(incr(incr(y0))))) MARK(cons(incr(incr(odds)), z1)) -> c18(MARK(incr(incr(odds)))) MARK(cons(incr(incr(head(y0))), z1)) -> c18(MARK(incr(incr(head(y0))))) MARK(cons(incr(incr(tail(y0))), z1)) -> c18(MARK(incr(incr(tail(y0))))) MARK(cons(incr(incr(cons(y0, y1))), z1)) -> c18(MARK(incr(incr(cons(y0, y1))))) MARK(cons(incr(incr(s(y0))), z1)) -> c18(MARK(incr(incr(s(y0))))) MARK(cons(incr(incr(y0)), z1)) -> c18(MARK(incr(incr(y0)))) MARK(cons(incr(head(incr(y0))), z1)) -> c18(MARK(incr(head(incr(y0))))) MARK(cons(incr(head(odds)), z1)) -> c18(MARK(incr(head(odds)))) MARK(cons(incr(head(head(y0))), z1)) -> c18(MARK(incr(head(head(y0))))) MARK(cons(incr(head(tail(y0))), z1)) -> c18(MARK(incr(head(tail(y0))))) MARK(cons(incr(head(cons(y0, y1))), z1)) -> c18(MARK(incr(head(cons(y0, y1))))) MARK(cons(incr(head(s(y0))), z1)) -> c18(MARK(incr(head(s(y0))))) MARK(cons(incr(head(y0)), z1)) -> c18(MARK(incr(head(y0)))) MARK(cons(incr(tail(incr(y0))), z1)) -> c18(MARK(incr(tail(incr(y0))))) MARK(cons(incr(tail(pairs)), z1)) -> c18(MARK(incr(tail(pairs)))) MARK(cons(incr(tail(odds)), z1)) -> c18(MARK(incr(tail(odds)))) MARK(cons(incr(tail(head(y0))), z1)) -> c18(MARK(incr(tail(head(y0))))) MARK(cons(incr(tail(tail(y0))), z1)) -> c18(MARK(incr(tail(tail(y0))))) MARK(cons(incr(tail(cons(y0, y1))), z1)) -> c18(MARK(incr(tail(cons(y0, y1))))) MARK(cons(incr(tail(s(y0))), z1)) -> c18(MARK(incr(tail(s(y0))))) MARK(cons(incr(tail(y0)), z1)) -> c18(MARK(incr(tail(y0)))) MARK(cons(head(incr(incr(y0))), z1)) -> c18(MARK(head(incr(incr(y0))))) MARK(cons(head(incr(odds)), z1)) -> c18(MARK(head(incr(odds)))) MARK(cons(head(incr(head(y0))), z1)) -> c18(MARK(head(incr(head(y0))))) MARK(cons(head(incr(tail(y0))), z1)) -> c18(MARK(head(incr(tail(y0))))) MARK(cons(head(incr(cons(y0, y1))), z1)) -> c18(MARK(head(incr(cons(y0, y1))))) MARK(cons(head(incr(s(y0))), z1)) -> c18(MARK(head(incr(s(y0))))) MARK(cons(head(incr(y0)), z1)) -> c18(MARK(head(incr(y0)))) MARK(cons(head(head(incr(y0))), z1)) -> c18(MARK(head(head(incr(y0))))) MARK(cons(head(head(odds)), z1)) -> c18(MARK(head(head(odds)))) MARK(cons(head(head(head(y0))), z1)) -> c18(MARK(head(head(head(y0))))) MARK(cons(head(head(tail(y0))), z1)) -> c18(MARK(head(head(tail(y0))))) MARK(cons(head(head(cons(y0, y1))), z1)) -> c18(MARK(head(head(cons(y0, y1))))) MARK(cons(head(head(s(y0))), z1)) -> c18(MARK(head(head(s(y0))))) MARK(cons(head(head(y0)), z1)) -> c18(MARK(head(head(y0)))) MARK(cons(head(tail(incr(y0))), z1)) -> c18(MARK(head(tail(incr(y0))))) MARK(cons(head(tail(pairs)), z1)) -> c18(MARK(head(tail(pairs)))) MARK(cons(head(tail(odds)), z1)) -> c18(MARK(head(tail(odds)))) MARK(cons(head(tail(head(y0))), z1)) -> c18(MARK(head(tail(head(y0))))) MARK(cons(head(tail(tail(y0))), z1)) -> c18(MARK(head(tail(tail(y0))))) MARK(cons(head(tail(cons(y0, y1))), z1)) -> c18(MARK(head(tail(cons(y0, y1))))) MARK(cons(head(tail(s(y0))), z1)) -> c18(MARK(head(tail(s(y0))))) MARK(cons(head(tail(y0)), z1)) -> c18(MARK(head(tail(y0)))) MARK(cons(tail(incr(incr(y0))), z1)) -> c18(MARK(tail(incr(incr(y0))))) MARK(cons(tail(incr(odds)), z1)) -> c18(MARK(tail(incr(odds)))) MARK(cons(tail(incr(head(y0))), z1)) -> c18(MARK(tail(incr(head(y0))))) MARK(cons(tail(incr(tail(y0))), z1)) -> c18(MARK(tail(incr(tail(y0))))) MARK(cons(tail(incr(cons(y0, y1))), z1)) -> c18(MARK(tail(incr(cons(y0, y1))))) MARK(cons(tail(incr(s(y0))), z1)) -> c18(MARK(tail(incr(s(y0))))) MARK(cons(tail(incr(y0)), z1)) -> c18(MARK(tail(incr(y0)))) MARK(cons(tail(head(incr(y0))), z1)) -> c18(MARK(tail(head(incr(y0))))) MARK(cons(tail(head(odds)), z1)) -> c18(MARK(tail(head(odds)))) MARK(cons(tail(head(head(y0))), z1)) -> c18(MARK(tail(head(head(y0))))) MARK(cons(tail(head(tail(y0))), z1)) -> c18(MARK(tail(head(tail(y0))))) MARK(cons(tail(head(cons(y0, y1))), z1)) -> c18(MARK(tail(head(cons(y0, y1))))) MARK(cons(tail(head(s(y0))), z1)) -> c18(MARK(tail(head(s(y0))))) MARK(cons(tail(head(y0)), z1)) -> c18(MARK(tail(head(y0)))) MARK(cons(tail(tail(incr(y0))), z1)) -> c18(MARK(tail(tail(incr(y0))))) MARK(cons(tail(tail(pairs)), z1)) -> c18(MARK(tail(tail(pairs)))) MARK(cons(tail(tail(odds)), z1)) -> c18(MARK(tail(tail(odds)))) MARK(cons(tail(tail(head(y0))), z1)) -> c18(MARK(tail(tail(head(y0))))) MARK(cons(tail(tail(tail(y0))), z1)) -> c18(MARK(tail(tail(tail(y0))))) MARK(cons(tail(tail(cons(y0, y1))), z1)) -> c18(MARK(tail(tail(cons(y0, y1))))) MARK(cons(tail(tail(s(y0))), z1)) -> c18(MARK(tail(tail(s(y0))))) MARK(cons(tail(tail(y0)), z1)) -> c18(MARK(tail(tail(y0)))) MARK(cons(tail(pairs), z1)) -> c18(MARK(tail(pairs))) MARK(cons(incr(incr(nats)), z1)) -> c18(MARK(incr(incr(nats)))) MARK(cons(incr(incr(pairs)), z1)) -> c18(MARK(incr(incr(pairs)))) MARK(cons(incr(odds), z1)) -> c18(MARK(incr(odds))) MARK(cons(head(incr(nats)), z1)) -> c18(MARK(head(incr(nats)))) MARK(cons(head(incr(pairs)), z1)) -> c18(MARK(head(incr(pairs)))) MARK(cons(head(odds), z1)) -> c18(MARK(head(odds))) MARK(cons(tail(incr(nats)), z1)) -> c18(MARK(tail(incr(nats)))) MARK(cons(tail(incr(pairs)), z1)) -> c18(MARK(tail(incr(pairs)))) MARK(cons(tail(odds), z1)) -> c18(MARK(tail(odds))) MARK(cons(incr(cons(nats, y0)), z1)) -> c18(MARK(incr(cons(nats, y0)))) MARK(cons(incr(cons(incr(y0), y1)), z1)) -> c18(MARK(incr(cons(incr(y0), y1)))) MARK(cons(incr(cons(pairs, y0)), z1)) -> c18(MARK(incr(cons(pairs, y0)))) MARK(cons(incr(cons(odds, y0)), z1)) -> c18(MARK(incr(cons(odds, y0)))) MARK(cons(incr(cons(head(y0), y1)), z1)) -> c18(MARK(incr(cons(head(y0), y1)))) MARK(cons(incr(cons(tail(y0), y1)), z1)) -> c18(MARK(incr(cons(tail(y0), y1)))) MARK(cons(incr(cons(cons(y0, y1), y2)), z1)) -> c18(MARK(incr(cons(cons(y0, y1), y2)))) MARK(cons(incr(cons(s(y0), y1)), z1)) -> c18(MARK(incr(cons(s(y0), y1)))) MARK(cons(incr(cons(y0, y1)), z1)) -> c18(MARK(incr(cons(y0, y1)))) MARK(cons(incr(cons(0, y0)), z1)) -> c18(MARK(incr(cons(0, y0)))) MARK(cons(incr(head(nats)), z1)) -> c18(MARK(incr(head(nats)))) MARK(cons(incr(head(pairs)), z1)) -> c18(MARK(incr(head(pairs)))) MARK(cons(head(head(nats)), z1)) -> c18(MARK(head(head(nats)))) MARK(cons(head(head(pairs)), z1)) -> c18(MARK(head(head(pairs)))) MARK(cons(tail(head(nats)), z1)) -> c18(MARK(tail(head(nats)))) MARK(cons(tail(head(pairs)), z1)) -> c18(MARK(tail(head(pairs)))) MARK(cons(head(cons(nats, y0)), z1)) -> c18(MARK(head(cons(nats, y0)))) MARK(cons(head(cons(incr(y0), y1)), z1)) -> c18(MARK(head(cons(incr(y0), y1)))) MARK(cons(head(cons(pairs, y0)), z1)) -> c18(MARK(head(cons(pairs, y0)))) MARK(cons(head(cons(odds, y0)), z1)) -> c18(MARK(head(cons(odds, y0)))) MARK(cons(head(cons(head(y0), y1)), z1)) -> c18(MARK(head(cons(head(y0), y1)))) MARK(cons(head(cons(tail(y0), y1)), z1)) -> c18(MARK(head(cons(tail(y0), y1)))) MARK(cons(head(cons(cons(y0, y1), y2)), z1)) -> c18(MARK(head(cons(cons(y0, y1), y2)))) MARK(cons(head(cons(s(y0), y1)), z1)) -> c18(MARK(head(cons(s(y0), y1)))) MARK(cons(head(cons(y0, y1)), z1)) -> c18(MARK(head(cons(y0, y1)))) MARK(cons(head(cons(0, y0)), z1)) -> c18(MARK(head(cons(0, y0)))) MARK(cons(incr(tail(nats)), z1)) -> c18(MARK(incr(tail(nats)))) MARK(cons(head(tail(nats)), z1)) -> c18(MARK(head(tail(nats)))) MARK(cons(tail(tail(nats)), z1)) -> c18(MARK(tail(tail(nats)))) ---------------------------------------- (186) Obligation: Complexity Dependency Tuples Problem Rules: a__nats -> cons(0, incr(nats)) a__nats -> nats a__pairs -> cons(0, incr(odds)) a__pairs -> pairs a__odds -> a__incr(a__pairs) a__odds -> odds a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) 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(nats) -> a__nats mark(incr(z0)) -> a__incr(mark(z0)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c17(MARK(s(z0))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(tail(incr(incr(z0)))) -> c17(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(odds))) -> c17(A__TAIL(a__incr(a__odds)), MARK(incr(odds))) MARK(tail(incr(head(z0)))) -> c17(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c17(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c17(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c17(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c17(MARK(incr(x0))) MARK(tail(head(incr(z0)))) -> c17(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(odds))) -> c17(A__TAIL(a__head(a__odds)), MARK(head(odds))) MARK(tail(head(head(z0)))) -> c17(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> 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MARK(cons(head(head(tail(y0))), z1)) -> c18(MARK(head(head(tail(y0))))) MARK(cons(head(head(cons(y0, y1))), z1)) -> c18(MARK(head(head(cons(y0, y1))))) MARK(cons(head(head(s(y0))), z1)) -> c18(MARK(head(head(s(y0))))) MARK(cons(head(head(y0)), z1)) -> c18(MARK(head(head(y0)))) MARK(cons(head(tail(incr(y0))), z1)) -> c18(MARK(head(tail(incr(y0))))) MARK(cons(head(tail(pairs)), z1)) -> c18(MARK(head(tail(pairs)))) MARK(cons(head(tail(odds)), z1)) -> c18(MARK(head(tail(odds)))) MARK(cons(head(tail(head(y0))), z1)) -> c18(MARK(head(tail(head(y0))))) MARK(cons(head(tail(tail(y0))), z1)) -> c18(MARK(head(tail(tail(y0))))) MARK(cons(head(tail(cons(y0, y1))), z1)) -> c18(MARK(head(tail(cons(y0, y1))))) MARK(cons(head(tail(s(y0))), z1)) -> c18(MARK(head(tail(s(y0))))) MARK(cons(head(tail(y0)), z1)) -> c18(MARK(head(tail(y0)))) MARK(cons(tail(incr(incr(y0))), z1)) -> c18(MARK(tail(incr(incr(y0))))) MARK(cons(tail(incr(odds)), z1)) -> c18(MARK(tail(incr(odds)))) MARK(cons(tail(incr(head(y0))), z1)) -> c18(MARK(tail(incr(head(y0))))) MARK(cons(tail(incr(tail(y0))), z1)) -> c18(MARK(tail(incr(tail(y0))))) MARK(cons(tail(incr(cons(y0, y1))), z1)) -> c18(MARK(tail(incr(cons(y0, y1))))) MARK(cons(tail(incr(s(y0))), z1)) -> c18(MARK(tail(incr(s(y0))))) MARK(cons(tail(incr(y0)), z1)) -> c18(MARK(tail(incr(y0)))) MARK(cons(tail(head(incr(y0))), z1)) -> c18(MARK(tail(head(incr(y0))))) MARK(cons(tail(head(odds)), z1)) -> c18(MARK(tail(head(odds)))) MARK(cons(tail(head(head(y0))), z1)) -> c18(MARK(tail(head(head(y0))))) MARK(cons(tail(head(tail(y0))), z1)) -> c18(MARK(tail(head(tail(y0))))) MARK(cons(tail(head(cons(y0, y1))), z1)) -> c18(MARK(tail(head(cons(y0, y1))))) MARK(cons(tail(head(s(y0))), z1)) -> c18(MARK(tail(head(s(y0))))) MARK(cons(tail(head(y0)), z1)) -> c18(MARK(tail(head(y0)))) MARK(cons(tail(tail(incr(y0))), z1)) -> c18(MARK(tail(tail(incr(y0))))) MARK(cons(tail(tail(pairs)), z1)) -> c18(MARK(tail(tail(pairs)))) MARK(cons(tail(tail(odds)), z1)) -> c18(MARK(tail(tail(odds)))) MARK(cons(tail(tail(head(y0))), z1)) -> c18(MARK(tail(tail(head(y0))))) MARK(cons(tail(tail(tail(y0))), z1)) -> c18(MARK(tail(tail(tail(y0))))) MARK(cons(tail(tail(cons(y0, y1))), z1)) -> c18(MARK(tail(tail(cons(y0, y1))))) MARK(cons(tail(tail(s(y0))), z1)) -> c18(MARK(tail(tail(s(y0))))) MARK(cons(tail(tail(y0)), z1)) -> c18(MARK(tail(tail(y0)))) MARK(cons(tail(pairs), z1)) -> c18(MARK(tail(pairs))) MARK(cons(incr(incr(nats)), z1)) -> c18(MARK(incr(incr(nats)))) MARK(cons(incr(incr(pairs)), z1)) -> c18(MARK(incr(incr(pairs)))) MARK(cons(incr(odds), z1)) -> c18(MARK(incr(odds))) MARK(cons(head(incr(nats)), z1)) -> c18(MARK(head(incr(nats)))) MARK(cons(head(incr(pairs)), z1)) -> c18(MARK(head(incr(pairs)))) MARK(cons(head(odds), z1)) -> c18(MARK(head(odds))) MARK(cons(tail(incr(nats)), z1)) -> c18(MARK(tail(incr(nats)))) MARK(cons(tail(incr(pairs)), z1)) -> c18(MARK(tail(incr(pairs)))) MARK(cons(tail(odds), z1)) -> c18(MARK(tail(odds))) MARK(cons(incr(cons(nats, y0)), z1)) -> c18(MARK(incr(cons(nats, y0)))) MARK(cons(incr(cons(incr(y0), y1)), z1)) -> c18(MARK(incr(cons(incr(y0), y1)))) MARK(cons(incr(cons(pairs, y0)), z1)) -> c18(MARK(incr(cons(pairs, y0)))) MARK(cons(incr(cons(odds, y0)), z1)) -> c18(MARK(incr(cons(odds, y0)))) MARK(cons(incr(cons(head(y0), y1)), z1)) -> c18(MARK(incr(cons(head(y0), y1)))) MARK(cons(incr(cons(tail(y0), y1)), z1)) -> c18(MARK(incr(cons(tail(y0), y1)))) MARK(cons(incr(cons(cons(y0, y1), y2)), z1)) -> c18(MARK(incr(cons(cons(y0, y1), y2)))) MARK(cons(incr(cons(s(y0), y1)), z1)) -> c18(MARK(incr(cons(s(y0), y1)))) MARK(cons(incr(cons(y0, y1)), z1)) -> c18(MARK(incr(cons(y0, y1)))) MARK(cons(incr(cons(0, y0)), z1)) -> c18(MARK(incr(cons(0, y0)))) MARK(cons(incr(head(nats)), z1)) -> c18(MARK(incr(head(nats)))) MARK(cons(incr(head(pairs)), z1)) -> c18(MARK(incr(head(pairs)))) MARK(cons(head(head(nats)), z1)) -> c18(MARK(head(head(nats)))) MARK(cons(head(head(pairs)), z1)) -> c18(MARK(head(head(pairs)))) MARK(cons(tail(head(nats)), z1)) -> c18(MARK(tail(head(nats)))) MARK(cons(tail(head(pairs)), z1)) -> c18(MARK(tail(head(pairs)))) MARK(cons(head(cons(nats, y0)), z1)) -> c18(MARK(head(cons(nats, y0)))) MARK(cons(head(cons(incr(y0), y1)), z1)) -> c18(MARK(head(cons(incr(y0), y1)))) MARK(cons(head(cons(pairs, y0)), z1)) -> c18(MARK(head(cons(pairs, y0)))) MARK(cons(head(cons(odds, y0)), z1)) -> c18(MARK(head(cons(odds, y0)))) MARK(cons(head(cons(head(y0), y1)), z1)) -> c18(MARK(head(cons(head(y0), y1)))) MARK(cons(head(cons(tail(y0), y1)), z1)) -> c18(MARK(head(cons(tail(y0), y1)))) MARK(cons(head(cons(cons(y0, y1), y2)), z1)) -> c18(MARK(head(cons(cons(y0, y1), y2)))) MARK(cons(head(cons(s(y0), y1)), z1)) -> c18(MARK(head(cons(s(y0), y1)))) MARK(cons(head(cons(y0, y1)), z1)) -> c18(MARK(head(cons(y0, y1)))) MARK(cons(head(cons(0, y0)), z1)) -> c18(MARK(head(cons(0, y0)))) MARK(cons(incr(tail(nats)), z1)) -> c18(MARK(incr(tail(nats)))) MARK(cons(head(tail(nats)), z1)) -> c18(MARK(head(tail(nats)))) MARK(cons(tail(tail(nats)), z1)) -> c18(MARK(tail(tail(nats)))) S tuples: MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c17(MARK(s(z0))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(tail(incr(incr(z0)))) -> c17(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(odds))) -> c17(A__TAIL(a__incr(a__odds)), MARK(incr(odds))) MARK(tail(incr(head(z0)))) -> c17(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c17(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c17(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c17(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c17(MARK(incr(x0))) MARK(tail(head(incr(z0)))) -> c17(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(odds))) -> c17(A__TAIL(a__head(a__odds)), MARK(head(odds))) MARK(tail(head(head(z0)))) -> c17(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c17(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c17(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c17(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c17(MARK(head(x0))) MARK(tail(tail(incr(z0)))) -> c17(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(pairs))) -> c17(A__TAIL(a__tail(a__pairs)), MARK(tail(pairs))) MARK(tail(tail(odds))) -> c17(A__TAIL(a__tail(a__odds)), MARK(tail(odds))) MARK(tail(tail(head(z0)))) -> c17(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c17(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c17(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c17(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c17(MARK(tail(x0))) MARK(tail(pairs)) -> c17(A__TAIL(cons(0, incr(odds)))) A__INCR(cons(cons(y0, y1), z1)) -> c6(MARK(cons(y0, y1))) A__INCR(cons(s(y0), z1)) -> c6(MARK(s(y0))) A__INCR(cons(incr(cons(y0, y1)), z1)) -> c6(MARK(incr(cons(y0, y1)))) A__INCR(cons(incr(s(y0)), z1)) -> c6(MARK(incr(s(y0)))) A__INCR(cons(head(cons(y0, y1)), z1)) -> c6(MARK(head(cons(y0, y1)))) A__INCR(cons(head(s(y0)), z1)) -> c6(MARK(head(s(y0)))) A__INCR(cons(tail(cons(y0, y1)), z1)) -> c6(MARK(tail(cons(y0, y1)))) A__INCR(cons(tail(s(y0)), z1)) -> c6(MARK(tail(s(y0)))) A__INCR(cons(incr(incr(nats)), z1)) -> c6(MARK(incr(incr(nats)))) A__INCR(cons(incr(incr(incr(y0))), z1)) -> c6(MARK(incr(incr(incr(y0))))) A__INCR(cons(incr(incr(pairs)), z1)) -> c6(MARK(incr(incr(pairs)))) A__INCR(cons(incr(incr(odds)), z1)) -> c6(MARK(incr(incr(odds)))) A__INCR(cons(incr(incr(head(y0))), z1)) -> c6(MARK(incr(incr(head(y0))))) A__INCR(cons(incr(incr(tail(y0))), z1)) -> c6(MARK(incr(incr(tail(y0))))) A__INCR(cons(incr(incr(cons(y0, y1))), z1)) -> c6(MARK(incr(incr(cons(y0, y1))))) A__INCR(cons(incr(incr(s(y0))), z1)) -> c6(MARK(incr(incr(s(y0))))) A__INCR(cons(incr(incr(y0)), z1)) -> c6(MARK(incr(incr(y0)))) A__INCR(cons(incr(odds), z1)) -> c6(MARK(incr(odds))) A__INCR(cons(incr(head(nats)), z1)) -> c6(MARK(incr(head(nats)))) A__INCR(cons(incr(head(incr(y0))), z1)) -> c6(MARK(incr(head(incr(y0))))) A__INCR(cons(incr(head(pairs)), z1)) -> c6(MARK(incr(head(pairs)))) A__INCR(cons(incr(head(odds)), z1)) -> c6(MARK(incr(head(odds)))) A__INCR(cons(incr(head(head(y0))), z1)) -> c6(MARK(incr(head(head(y0))))) A__INCR(cons(incr(head(tail(y0))), z1)) -> c6(MARK(incr(head(tail(y0))))) A__INCR(cons(incr(head(cons(y0, y1))), z1)) -> c6(MARK(incr(head(cons(y0, y1))))) A__INCR(cons(incr(head(s(y0))), z1)) -> c6(MARK(incr(head(s(y0))))) A__INCR(cons(incr(head(y0)), z1)) -> c6(MARK(incr(head(y0)))) A__INCR(cons(incr(tail(nats)), z1)) -> c6(MARK(incr(tail(nats)))) A__INCR(cons(incr(tail(incr(y0))), z1)) -> c6(MARK(incr(tail(incr(y0))))) A__INCR(cons(incr(tail(pairs)), z1)) -> 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MARK(cons(cons(z0, z1), x1))) MARK(head(cons(s(z0), x1))) -> c16(A__HEAD(cons(s(mark(z0)), x1)), MARK(cons(s(z0), x1))) MARK(head(cons(x0, x1))) -> c16(MARK(cons(x0, x1))) MARK(head(cons(0, x1))) -> c16(MARK(cons(0, x1))) A__TAIL(cons(z0, cons(y0, y1))) -> c10(MARK(cons(y0, y1))) A__TAIL(cons(z0, s(y0))) -> c10(MARK(s(y0))) A__TAIL(cons(z0, incr(s(y0)))) -> c10(MARK(incr(s(y0)))) A__TAIL(cons(z0, head(s(y0)))) -> c10(MARK(head(s(y0)))) A__TAIL(cons(z0, tail(cons(y0, y1)))) -> c10(MARK(tail(cons(y0, y1)))) A__TAIL(cons(z0, tail(s(y0)))) -> c10(MARK(tail(s(y0)))) A__TAIL(cons(z0, incr(incr(incr(y0))))) -> c10(MARK(incr(incr(incr(y0))))) A__TAIL(cons(z0, incr(incr(odds)))) -> c10(MARK(incr(incr(odds)))) A__TAIL(cons(z0, incr(incr(head(y0))))) -> c10(MARK(incr(incr(head(y0))))) A__TAIL(cons(z0, incr(incr(tail(y0))))) -> c10(MARK(incr(incr(tail(y0))))) A__TAIL(cons(z0, incr(incr(cons(y0, y1))))) -> c10(MARK(incr(incr(cons(y0, y1))))) A__TAIL(cons(z0, incr(incr(s(y0))))) -> c10(MARK(incr(incr(s(y0))))) A__TAIL(cons(z0, incr(incr(y0)))) -> c10(MARK(incr(incr(y0)))) A__TAIL(cons(z0, incr(head(incr(y0))))) -> c10(MARK(incr(head(incr(y0))))) A__TAIL(cons(z0, incr(head(odds)))) -> c10(MARK(incr(head(odds)))) A__TAIL(cons(z0, incr(head(head(y0))))) -> c10(MARK(incr(head(head(y0))))) A__TAIL(cons(z0, incr(head(tail(y0))))) -> c10(MARK(incr(head(tail(y0))))) A__TAIL(cons(z0, incr(head(cons(y0, y1))))) -> c10(MARK(incr(head(cons(y0, y1))))) A__TAIL(cons(z0, incr(head(s(y0))))) -> c10(MARK(incr(head(s(y0))))) A__TAIL(cons(z0, incr(head(y0)))) -> c10(MARK(incr(head(y0)))) A__TAIL(cons(z0, incr(tail(nats)))) -> c10(MARK(incr(tail(nats)))) A__TAIL(cons(z0, incr(tail(incr(y0))))) -> c10(MARK(incr(tail(incr(y0))))) A__TAIL(cons(z0, incr(tail(pairs)))) -> c10(MARK(incr(tail(pairs)))) A__TAIL(cons(z0, incr(tail(odds)))) -> c10(MARK(incr(tail(odds)))) A__TAIL(cons(z0, incr(tail(head(y0))))) -> c10(MARK(incr(tail(head(y0))))) A__TAIL(cons(z0, incr(tail(tail(y0))))) -> c10(MARK(incr(tail(tail(y0))))) A__TAIL(cons(z0, incr(tail(cons(y0, y1))))) -> c10(MARK(incr(tail(cons(y0, y1))))) A__TAIL(cons(z0, incr(tail(s(y0))))) -> c10(MARK(incr(tail(s(y0))))) A__TAIL(cons(z0, incr(tail(y0)))) -> c10(MARK(incr(tail(y0)))) A__TAIL(cons(z0, head(incr(incr(y0))))) -> c10(MARK(head(incr(incr(y0))))) A__TAIL(cons(z0, head(incr(odds)))) -> c10(MARK(head(incr(odds)))) A__TAIL(cons(z0, head(incr(head(y0))))) -> c10(MARK(head(incr(head(y0))))) A__TAIL(cons(z0, head(incr(tail(y0))))) -> c10(MARK(head(incr(tail(y0))))) A__TAIL(cons(z0, head(incr(cons(y0, y1))))) -> c10(MARK(head(incr(cons(y0, y1))))) A__TAIL(cons(z0, head(incr(s(y0))))) -> c10(MARK(head(incr(s(y0))))) A__TAIL(cons(z0, head(incr(y0)))) -> c10(MARK(head(incr(y0)))) A__TAIL(cons(z0, head(head(incr(y0))))) -> c10(MARK(head(head(incr(y0))))) A__TAIL(cons(z0, head(head(odds)))) -> c10(MARK(head(head(odds)))) A__TAIL(cons(z0, head(head(head(y0))))) -> c10(MARK(head(head(head(y0))))) A__TAIL(cons(z0, head(head(tail(y0))))) -> c10(MARK(head(head(tail(y0))))) A__TAIL(cons(z0, head(head(cons(y0, y1))))) -> c10(MARK(head(head(cons(y0, y1))))) A__TAIL(cons(z0, head(head(s(y0))))) -> c10(MARK(head(head(s(y0))))) A__TAIL(cons(z0, head(head(y0)))) -> c10(MARK(head(head(y0)))) A__TAIL(cons(z0, head(tail(nats)))) -> c10(MARK(head(tail(nats)))) A__TAIL(cons(z0, head(tail(incr(y0))))) -> c10(MARK(head(tail(incr(y0))))) A__TAIL(cons(z0, head(tail(pairs)))) -> c10(MARK(head(tail(pairs)))) A__TAIL(cons(z0, head(tail(odds)))) -> c10(MARK(head(tail(odds)))) A__TAIL(cons(z0, head(tail(head(y0))))) -> c10(MARK(head(tail(head(y0))))) A__TAIL(cons(z0, head(tail(tail(y0))))) -> c10(MARK(head(tail(tail(y0))))) A__TAIL(cons(z0, head(tail(cons(y0, y1))))) -> c10(MARK(head(tail(cons(y0, y1))))) A__TAIL(cons(z0, head(tail(s(y0))))) -> c10(MARK(head(tail(s(y0))))) A__TAIL(cons(z0, head(tail(y0)))) -> c10(MARK(head(tail(y0)))) A__TAIL(cons(z0, tail(incr(incr(y0))))) -> c10(MARK(tail(incr(incr(y0))))) A__TAIL(cons(z0, tail(incr(odds)))) -> c10(MARK(tail(incr(odds)))) A__TAIL(cons(z0, tail(incr(head(y0))))) -> c10(MARK(tail(incr(head(y0))))) A__TAIL(cons(z0, tail(incr(tail(y0))))) -> c10(MARK(tail(incr(tail(y0))))) A__TAIL(cons(z0, tail(incr(cons(y0, y1))))) -> c10(MARK(tail(incr(cons(y0, y1))))) A__TAIL(cons(z0, tail(incr(s(y0))))) -> c10(MARK(tail(incr(s(y0))))) A__TAIL(cons(z0, tail(incr(y0)))) -> c10(MARK(tail(incr(y0)))) A__TAIL(cons(z0, tail(head(incr(y0))))) -> c10(MARK(tail(head(incr(y0))))) A__TAIL(cons(z0, tail(head(odds)))) -> c10(MARK(tail(head(odds)))) A__TAIL(cons(z0, tail(head(head(y0))))) -> c10(MARK(tail(head(head(y0))))) A__TAIL(cons(z0, tail(head(tail(y0))))) -> c10(MARK(tail(head(tail(y0))))) A__TAIL(cons(z0, tail(head(cons(y0, y1))))) -> c10(MARK(tail(head(cons(y0, y1))))) A__TAIL(cons(z0, tail(head(s(y0))))) -> c10(MARK(tail(head(s(y0))))) A__TAIL(cons(z0, tail(head(y0)))) -> c10(MARK(tail(head(y0)))) A__TAIL(cons(z0, tail(tail(nats)))) -> c10(MARK(tail(tail(nats)))) A__TAIL(cons(z0, tail(tail(incr(y0))))) -> c10(MARK(tail(tail(incr(y0))))) A__TAIL(cons(z0, tail(tail(pairs)))) -> c10(MARK(tail(tail(pairs)))) A__TAIL(cons(z0, tail(tail(odds)))) -> c10(MARK(tail(tail(odds)))) A__TAIL(cons(z0, tail(tail(head(y0))))) -> c10(MARK(tail(tail(head(y0))))) A__TAIL(cons(z0, tail(tail(tail(y0))))) -> c10(MARK(tail(tail(tail(y0))))) A__TAIL(cons(z0, tail(tail(cons(y0, y1))))) -> c10(MARK(tail(tail(cons(y0, y1))))) A__TAIL(cons(z0, tail(tail(s(y0))))) -> c10(MARK(tail(tail(s(y0))))) A__TAIL(cons(z0, tail(tail(y0)))) -> c10(MARK(tail(tail(y0)))) A__TAIL(cons(z0, tail(pairs))) -> c10(MARK(tail(pairs))) A__TAIL(cons(z0, incr(incr(nats)))) -> c10(MARK(incr(incr(nats)))) A__TAIL(cons(z0, incr(incr(pairs)))) -> c10(MARK(incr(incr(pairs)))) A__TAIL(cons(z0, incr(odds))) -> c10(MARK(incr(odds))) A__TAIL(cons(z0, head(incr(nats)))) -> c10(MARK(head(incr(nats)))) A__TAIL(cons(z0, head(incr(pairs)))) -> c10(MARK(head(incr(pairs)))) A__TAIL(cons(z0, head(odds))) -> c10(MARK(head(odds))) A__TAIL(cons(z0, tail(incr(nats)))) -> c10(MARK(tail(incr(nats)))) A__TAIL(cons(z0, tail(incr(pairs)))) -> c10(MARK(tail(incr(pairs)))) A__TAIL(cons(z0, tail(odds))) -> c10(MARK(tail(odds))) A__TAIL(cons(z0, incr(cons(nats, y0)))) -> c10(MARK(incr(cons(nats, y0)))) A__TAIL(cons(z0, incr(cons(incr(y0), y1)))) -> c10(MARK(incr(cons(incr(y0), y1)))) A__TAIL(cons(z0, incr(cons(pairs, y0)))) -> c10(MARK(incr(cons(pairs, y0)))) A__TAIL(cons(z0, incr(cons(odds, y0)))) -> c10(MARK(incr(cons(odds, y0)))) A__TAIL(cons(z0, incr(cons(head(y0), y1)))) -> c10(MARK(incr(cons(head(y0), y1)))) A__TAIL(cons(z0, incr(cons(tail(y0), y1)))) -> c10(MARK(incr(cons(tail(y0), y1)))) A__TAIL(cons(z0, incr(cons(cons(y0, y1), y2)))) -> c10(MARK(incr(cons(cons(y0, y1), y2)))) A__TAIL(cons(z0, incr(cons(s(y0), y1)))) -> c10(MARK(incr(cons(s(y0), y1)))) A__TAIL(cons(z0, incr(cons(y0, y1)))) -> c10(MARK(incr(cons(y0, y1)))) A__TAIL(cons(z0, incr(cons(0, y0)))) -> c10(MARK(incr(cons(0, y0)))) A__TAIL(cons(z0, incr(head(nats)))) -> c10(MARK(incr(head(nats)))) A__TAIL(cons(z0, incr(head(pairs)))) -> c10(MARK(incr(head(pairs)))) A__TAIL(cons(z0, head(head(nats)))) -> c10(MARK(head(head(nats)))) A__TAIL(cons(z0, head(head(pairs)))) -> c10(MARK(head(head(pairs)))) A__TAIL(cons(z0, tail(head(nats)))) -> c10(MARK(tail(head(nats)))) A__TAIL(cons(z0, tail(head(pairs)))) -> c10(MARK(tail(head(pairs)))) A__TAIL(cons(z0, head(cons(nats, y0)))) -> c10(MARK(head(cons(nats, y0)))) A__TAIL(cons(z0, head(cons(incr(y0), y1)))) -> c10(MARK(head(cons(incr(y0), y1)))) A__TAIL(cons(z0, head(cons(pairs, y0)))) -> c10(MARK(head(cons(pairs, y0)))) A__TAIL(cons(z0, head(cons(odds, y0)))) -> c10(MARK(head(cons(odds, y0)))) A__TAIL(cons(z0, head(cons(head(y0), y1)))) -> c10(MARK(head(cons(head(y0), y1)))) A__TAIL(cons(z0, head(cons(tail(y0), y1)))) -> c10(MARK(head(cons(tail(y0), y1)))) A__TAIL(cons(z0, head(cons(cons(y0, y1), y2)))) -> c10(MARK(head(cons(cons(y0, y1), y2)))) A__TAIL(cons(z0, head(cons(s(y0), y1)))) -> c10(MARK(head(cons(s(y0), y1)))) A__TAIL(cons(z0, head(cons(y0, y1)))) -> c10(MARK(head(cons(y0, y1)))) A__TAIL(cons(z0, head(cons(0, y0)))) -> c10(MARK(head(cons(0, y0)))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats))) MARK(tail(tail(nats))) -> c17(A__TAIL(a__tail(a__nats))) MARK(cons(cons(y0, y1), z1)) -> c18(MARK(cons(y0, y1))) MARK(cons(s(y0), z1)) -> c18(MARK(s(y0))) MARK(cons(incr(s(y0)), z1)) -> c18(MARK(incr(s(y0)))) MARK(cons(head(s(y0)), z1)) -> c18(MARK(head(s(y0)))) MARK(cons(tail(cons(y0, y1)), z1)) -> c18(MARK(tail(cons(y0, y1)))) MARK(cons(tail(s(y0)), z1)) -> c18(MARK(tail(s(y0)))) MARK(cons(incr(incr(incr(y0))), z1)) -> c18(MARK(incr(incr(incr(y0))))) MARK(cons(incr(incr(odds)), z1)) -> c18(MARK(incr(incr(odds)))) MARK(cons(incr(incr(head(y0))), z1)) -> c18(MARK(incr(incr(head(y0))))) MARK(cons(incr(incr(tail(y0))), z1)) -> c18(MARK(incr(incr(tail(y0))))) MARK(cons(incr(incr(cons(y0, y1))), z1)) -> c18(MARK(incr(incr(cons(y0, y1))))) MARK(cons(incr(incr(s(y0))), z1)) -> c18(MARK(incr(incr(s(y0))))) MARK(cons(incr(incr(y0)), z1)) -> c18(MARK(incr(incr(y0)))) MARK(cons(incr(head(incr(y0))), z1)) -> c18(MARK(incr(head(incr(y0))))) MARK(cons(incr(head(odds)), z1)) -> c18(MARK(incr(head(odds)))) MARK(cons(incr(head(head(y0))), z1)) -> c18(MARK(incr(head(head(y0))))) MARK(cons(incr(head(tail(y0))), z1)) -> c18(MARK(incr(head(tail(y0))))) MARK(cons(incr(head(cons(y0, y1))), z1)) -> c18(MARK(incr(head(cons(y0, y1))))) MARK(cons(incr(head(s(y0))), z1)) -> c18(MARK(incr(head(s(y0))))) MARK(cons(incr(head(y0)), z1)) -> c18(MARK(incr(head(y0)))) MARK(cons(incr(tail(incr(y0))), z1)) -> c18(MARK(incr(tail(incr(y0))))) MARK(cons(incr(tail(pairs)), z1)) -> c18(MARK(incr(tail(pairs)))) MARK(cons(incr(tail(odds)), z1)) -> c18(MARK(incr(tail(odds)))) MARK(cons(incr(tail(head(y0))), z1)) -> c18(MARK(incr(tail(head(y0))))) MARK(cons(incr(tail(tail(y0))), z1)) -> c18(MARK(incr(tail(tail(y0))))) MARK(cons(incr(tail(cons(y0, y1))), z1)) -> c18(MARK(incr(tail(cons(y0, y1))))) MARK(cons(incr(tail(s(y0))), z1)) -> c18(MARK(incr(tail(s(y0))))) MARK(cons(incr(tail(y0)), z1)) -> c18(MARK(incr(tail(y0)))) MARK(cons(head(incr(incr(y0))), z1)) -> c18(MARK(head(incr(incr(y0))))) MARK(cons(head(incr(odds)), z1)) -> c18(MARK(head(incr(odds)))) MARK(cons(head(incr(head(y0))), z1)) -> c18(MARK(head(incr(head(y0))))) MARK(cons(head(incr(tail(y0))), z1)) -> c18(MARK(head(incr(tail(y0))))) MARK(cons(head(incr(cons(y0, y1))), z1)) -> c18(MARK(head(incr(cons(y0, y1))))) MARK(cons(head(incr(s(y0))), z1)) -> c18(MARK(head(incr(s(y0))))) MARK(cons(head(incr(y0)), z1)) -> c18(MARK(head(incr(y0)))) MARK(cons(head(head(incr(y0))), z1)) -> c18(MARK(head(head(incr(y0))))) MARK(cons(head(head(odds)), z1)) -> c18(MARK(head(head(odds)))) MARK(cons(head(head(head(y0))), z1)) -> c18(MARK(head(head(head(y0))))) MARK(cons(head(head(tail(y0))), z1)) -> c18(MARK(head(head(tail(y0))))) MARK(cons(head(head(cons(y0, y1))), z1)) -> c18(MARK(head(head(cons(y0, y1))))) MARK(cons(head(head(s(y0))), z1)) -> c18(MARK(head(head(s(y0))))) MARK(cons(head(head(y0)), z1)) -> c18(MARK(head(head(y0)))) MARK(cons(head(tail(incr(y0))), z1)) -> c18(MARK(head(tail(incr(y0))))) MARK(cons(head(tail(pairs)), z1)) -> c18(MARK(head(tail(pairs)))) MARK(cons(head(tail(odds)), z1)) -> c18(MARK(head(tail(odds)))) MARK(cons(head(tail(head(y0))), z1)) -> c18(MARK(head(tail(head(y0))))) MARK(cons(head(tail(tail(y0))), z1)) -> c18(MARK(head(tail(tail(y0))))) MARK(cons(head(tail(cons(y0, y1))), z1)) -> c18(MARK(head(tail(cons(y0, y1))))) MARK(cons(head(tail(s(y0))), z1)) -> c18(MARK(head(tail(s(y0))))) MARK(cons(head(tail(y0)), z1)) -> c18(MARK(head(tail(y0)))) MARK(cons(tail(incr(incr(y0))), z1)) -> c18(MARK(tail(incr(incr(y0))))) MARK(cons(tail(incr(odds)), z1)) -> c18(MARK(tail(incr(odds)))) MARK(cons(tail(incr(head(y0))), z1)) -> c18(MARK(tail(incr(head(y0))))) MARK(cons(tail(incr(tail(y0))), z1)) -> c18(MARK(tail(incr(tail(y0))))) MARK(cons(tail(incr(cons(y0, y1))), z1)) -> c18(MARK(tail(incr(cons(y0, y1))))) MARK(cons(tail(incr(s(y0))), z1)) -> c18(MARK(tail(incr(s(y0))))) MARK(cons(tail(incr(y0)), z1)) -> c18(MARK(tail(incr(y0)))) MARK(cons(tail(head(incr(y0))), z1)) -> c18(MARK(tail(head(incr(y0))))) MARK(cons(tail(head(odds)), z1)) -> c18(MARK(tail(head(odds)))) MARK(cons(tail(head(head(y0))), z1)) -> c18(MARK(tail(head(head(y0))))) MARK(cons(tail(head(tail(y0))), z1)) -> c18(MARK(tail(head(tail(y0))))) MARK(cons(tail(head(cons(y0, y1))), z1)) -> c18(MARK(tail(head(cons(y0, y1))))) MARK(cons(tail(head(s(y0))), z1)) -> c18(MARK(tail(head(s(y0))))) MARK(cons(tail(head(y0)), z1)) -> c18(MARK(tail(head(y0)))) MARK(cons(tail(tail(incr(y0))), z1)) -> c18(MARK(tail(tail(incr(y0))))) MARK(cons(tail(tail(pairs)), z1)) -> c18(MARK(tail(tail(pairs)))) MARK(cons(tail(tail(odds)), z1)) -> c18(MARK(tail(tail(odds)))) MARK(cons(tail(tail(head(y0))), z1)) -> c18(MARK(tail(tail(head(y0))))) MARK(cons(tail(tail(tail(y0))), z1)) -> c18(MARK(tail(tail(tail(y0))))) MARK(cons(tail(tail(cons(y0, y1))), z1)) -> c18(MARK(tail(tail(cons(y0, y1))))) MARK(cons(tail(tail(s(y0))), z1)) -> c18(MARK(tail(tail(s(y0))))) MARK(cons(tail(tail(y0)), z1)) -> c18(MARK(tail(tail(y0)))) MARK(cons(tail(pairs), z1)) -> c18(MARK(tail(pairs))) MARK(cons(incr(incr(nats)), z1)) -> c18(MARK(incr(incr(nats)))) MARK(cons(incr(incr(pairs)), z1)) -> c18(MARK(incr(incr(pairs)))) MARK(cons(incr(odds), z1)) -> c18(MARK(incr(odds))) MARK(cons(head(incr(nats)), z1)) -> c18(MARK(head(incr(nats)))) MARK(cons(head(incr(pairs)), z1)) -> c18(MARK(head(incr(pairs)))) MARK(cons(head(odds), z1)) -> c18(MARK(head(odds))) MARK(cons(tail(incr(nats)), z1)) -> c18(MARK(tail(incr(nats)))) MARK(cons(tail(incr(pairs)), z1)) -> c18(MARK(tail(incr(pairs)))) MARK(cons(tail(odds), z1)) -> c18(MARK(tail(odds))) MARK(cons(incr(cons(nats, y0)), z1)) -> c18(MARK(incr(cons(nats, y0)))) MARK(cons(incr(cons(incr(y0), y1)), z1)) -> c18(MARK(incr(cons(incr(y0), y1)))) MARK(cons(incr(cons(pairs, y0)), z1)) -> c18(MARK(incr(cons(pairs, y0)))) MARK(cons(incr(cons(odds, y0)), z1)) -> c18(MARK(incr(cons(odds, y0)))) MARK(cons(incr(cons(head(y0), y1)), z1)) -> c18(MARK(incr(cons(head(y0), y1)))) MARK(cons(incr(cons(tail(y0), y1)), z1)) -> c18(MARK(incr(cons(tail(y0), y1)))) MARK(cons(incr(cons(cons(y0, y1), y2)), z1)) -> c18(MARK(incr(cons(cons(y0, y1), y2)))) MARK(cons(incr(cons(s(y0), y1)), z1)) -> c18(MARK(incr(cons(s(y0), y1)))) MARK(cons(incr(cons(y0, y1)), z1)) -> c18(MARK(incr(cons(y0, y1)))) MARK(cons(incr(cons(0, y0)), z1)) -> c18(MARK(incr(cons(0, y0)))) MARK(cons(incr(head(nats)), z1)) -> c18(MARK(incr(head(nats)))) MARK(cons(incr(head(pairs)), z1)) -> c18(MARK(incr(head(pairs)))) MARK(cons(head(head(nats)), z1)) -> c18(MARK(head(head(nats)))) MARK(cons(head(head(pairs)), z1)) -> c18(MARK(head(head(pairs)))) MARK(cons(tail(head(nats)), z1)) -> c18(MARK(tail(head(nats)))) MARK(cons(tail(head(pairs)), z1)) -> c18(MARK(tail(head(pairs)))) MARK(cons(head(cons(nats, y0)), z1)) -> c18(MARK(head(cons(nats, y0)))) MARK(cons(head(cons(incr(y0), y1)), z1)) -> c18(MARK(head(cons(incr(y0), y1)))) MARK(cons(head(cons(pairs, y0)), z1)) -> c18(MARK(head(cons(pairs, y0)))) MARK(cons(head(cons(odds, y0)), z1)) -> c18(MARK(head(cons(odds, y0)))) MARK(cons(head(cons(head(y0), y1)), z1)) -> c18(MARK(head(cons(head(y0), y1)))) MARK(cons(head(cons(tail(y0), y1)), z1)) -> c18(MARK(head(cons(tail(y0), y1)))) MARK(cons(head(cons(cons(y0, y1), y2)), z1)) -> c18(MARK(head(cons(cons(y0, y1), y2)))) MARK(cons(head(cons(s(y0), y1)), z1)) -> c18(MARK(head(cons(s(y0), y1)))) MARK(cons(head(cons(y0, y1)), z1)) -> c18(MARK(head(cons(y0, y1)))) MARK(cons(head(cons(0, y0)), z1)) -> c18(MARK(head(cons(0, y0)))) MARK(cons(incr(tail(nats)), z1)) -> c18(MARK(incr(tail(nats)))) MARK(cons(head(tail(nats)), z1)) -> c18(MARK(head(tail(nats)))) MARK(cons(tail(tail(nats)), z1)) -> c18(MARK(tail(tail(nats)))) K tuples:none Defined Rule Symbols: a__nats, a__pairs, a__odds, a__incr_1, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: MARK_1, A__INCR_1, A__HEAD_1, A__TAIL_1 Compound Symbols: c20_1, c13_1, c16_1, c17_2, c17_1, c13_2, c16_2, c6_1, c8_1, c10_1, c18_1 ---------------------------------------- (187) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing nodes: MARK(head(cons(0, x1))) -> c16(MARK(cons(0, x1))) MARK(incr(cons(0, x1))) -> c13(MARK(cons(0, x1))) ---------------------------------------- (188) Obligation: Complexity Dependency Tuples Problem Rules: a__nats -> cons(0, incr(nats)) a__nats -> nats a__pairs -> cons(0, incr(odds)) a__pairs -> pairs a__odds -> a__incr(a__pairs) a__odds -> odds a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) 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(nats) -> a__nats mark(incr(z0)) -> a__incr(mark(z0)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c17(MARK(s(z0))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(tail(incr(incr(z0)))) -> c17(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(odds))) -> c17(A__TAIL(a__incr(a__odds)), MARK(incr(odds))) MARK(tail(incr(head(z0)))) -> c17(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c17(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c17(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c17(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c17(MARK(incr(x0))) MARK(tail(head(incr(z0)))) -> c17(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(odds))) -> c17(A__TAIL(a__head(a__odds)), MARK(head(odds))) MARK(tail(head(head(z0)))) -> c17(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c17(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c17(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c17(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c17(MARK(head(x0))) MARK(tail(tail(incr(z0)))) -> c17(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(pairs))) -> c17(A__TAIL(a__tail(a__pairs)), MARK(tail(pairs))) MARK(tail(tail(odds))) -> c17(A__TAIL(a__tail(a__odds)), MARK(tail(odds))) MARK(tail(tail(head(z0)))) -> c17(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c17(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c17(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c17(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c17(MARK(tail(x0))) MARK(tail(pairs)) -> c17(A__TAIL(cons(0, incr(odds)))) A__INCR(cons(cons(y0, y1), z1)) -> c6(MARK(cons(y0, y1))) A__INCR(cons(s(y0), z1)) -> c6(MARK(s(y0))) A__INCR(cons(incr(cons(y0, y1)), z1)) -> c6(MARK(incr(cons(y0, y1)))) A__INCR(cons(incr(s(y0)), z1)) -> c6(MARK(incr(s(y0)))) A__INCR(cons(head(cons(y0, y1)), z1)) -> c6(MARK(head(cons(y0, y1)))) A__INCR(cons(head(s(y0)), z1)) -> c6(MARK(head(s(y0)))) A__INCR(cons(tail(cons(y0, y1)), z1)) -> c6(MARK(tail(cons(y0, y1)))) A__INCR(cons(tail(s(y0)), z1)) -> c6(MARK(tail(s(y0)))) A__INCR(cons(incr(incr(nats)), z1)) -> c6(MARK(incr(incr(nats)))) A__INCR(cons(incr(incr(incr(y0))), z1)) -> c6(MARK(incr(incr(incr(y0))))) A__INCR(cons(incr(incr(pairs)), z1)) -> c6(MARK(incr(incr(pairs)))) A__INCR(cons(incr(incr(odds)), z1)) -> c6(MARK(incr(incr(odds)))) A__INCR(cons(incr(incr(head(y0))), z1)) -> c6(MARK(incr(incr(head(y0))))) A__INCR(cons(incr(incr(tail(y0))), z1)) -> c6(MARK(incr(incr(tail(y0))))) A__INCR(cons(incr(incr(cons(y0, y1))), z1)) -> c6(MARK(incr(incr(cons(y0, y1))))) A__INCR(cons(incr(incr(s(y0))), z1)) -> c6(MARK(incr(incr(s(y0))))) A__INCR(cons(incr(incr(y0)), z1)) -> c6(MARK(incr(incr(y0)))) A__INCR(cons(incr(odds), z1)) -> c6(MARK(incr(odds))) A__INCR(cons(incr(head(nats)), z1)) -> c6(MARK(incr(head(nats)))) A__INCR(cons(incr(head(incr(y0))), z1)) -> c6(MARK(incr(head(incr(y0))))) A__INCR(cons(incr(head(pairs)), z1)) -> c6(MARK(incr(head(pairs)))) A__INCR(cons(incr(head(odds)), z1)) -> c6(MARK(incr(head(odds)))) A__INCR(cons(incr(head(head(y0))), z1)) -> c6(MARK(incr(head(head(y0))))) A__INCR(cons(incr(head(tail(y0))), z1)) -> c6(MARK(incr(head(tail(y0))))) A__INCR(cons(incr(head(cons(y0, y1))), z1)) -> c6(MARK(incr(head(cons(y0, y1))))) A__INCR(cons(incr(head(s(y0))), z1)) -> c6(MARK(incr(head(s(y0))))) A__INCR(cons(incr(head(y0)), z1)) -> c6(MARK(incr(head(y0)))) A__INCR(cons(incr(tail(nats)), z1)) -> c6(MARK(incr(tail(nats)))) A__INCR(cons(incr(tail(incr(y0))), z1)) -> c6(MARK(incr(tail(incr(y0))))) A__INCR(cons(incr(tail(pairs)), z1)) -> c6(MARK(incr(tail(pairs)))) A__INCR(cons(incr(tail(odds)), z1)) -> c6(MARK(incr(tail(odds)))) A__INCR(cons(incr(tail(head(y0))), z1)) -> c6(MARK(incr(tail(head(y0))))) A__INCR(cons(incr(tail(tail(y0))), z1)) -> c6(MARK(incr(tail(tail(y0))))) A__INCR(cons(incr(tail(cons(y0, y1))), z1)) -> c6(MARK(incr(tail(cons(y0, y1))))) A__INCR(cons(incr(tail(s(y0))), z1)) -> c6(MARK(incr(tail(s(y0))))) A__INCR(cons(incr(tail(y0)), z1)) -> c6(MARK(incr(tail(y0)))) A__INCR(cons(head(incr(nats)), z1)) -> c6(MARK(head(incr(nats)))) A__INCR(cons(head(incr(incr(y0))), z1)) -> c6(MARK(head(incr(incr(y0))))) A__INCR(cons(head(incr(pairs)), z1)) -> c6(MARK(head(incr(pairs)))) A__INCR(cons(head(incr(odds)), z1)) -> c6(MARK(head(incr(odds)))) A__INCR(cons(head(incr(head(y0))), z1)) -> c6(MARK(head(incr(head(y0))))) A__INCR(cons(head(incr(tail(y0))), z1)) -> c6(MARK(head(incr(tail(y0))))) A__INCR(cons(head(incr(cons(y0, y1))), z1)) -> c6(MARK(head(incr(cons(y0, y1))))) A__INCR(cons(head(incr(s(y0))), z1)) -> c6(MARK(head(incr(s(y0))))) A__INCR(cons(head(incr(y0)), z1)) -> c6(MARK(head(incr(y0)))) A__INCR(cons(head(odds), z1)) -> c6(MARK(head(odds))) A__INCR(cons(head(head(nats)), z1)) -> c6(MARK(head(head(nats)))) A__INCR(cons(head(head(incr(y0))), z1)) -> c6(MARK(head(head(incr(y0))))) A__INCR(cons(head(head(pairs)), z1)) -> c6(MARK(head(head(pairs)))) A__INCR(cons(head(head(odds)), z1)) -> c6(MARK(head(head(odds)))) A__INCR(cons(head(head(head(y0))), z1)) -> c6(MARK(head(head(head(y0))))) A__INCR(cons(head(head(tail(y0))), z1)) -> c6(MARK(head(head(tail(y0))))) A__INCR(cons(head(head(cons(y0, y1))), z1)) -> c6(MARK(head(head(cons(y0, y1))))) A__INCR(cons(head(head(s(y0))), z1)) -> c6(MARK(head(head(s(y0))))) A__INCR(cons(head(head(y0)), z1)) -> c6(MARK(head(head(y0)))) A__INCR(cons(head(tail(nats)), z1)) -> c6(MARK(head(tail(nats)))) A__INCR(cons(head(tail(incr(y0))), z1)) -> c6(MARK(head(tail(incr(y0))))) A__INCR(cons(head(tail(pairs)), z1)) -> c6(MARK(head(tail(pairs)))) A__INCR(cons(head(tail(odds)), z1)) -> c6(MARK(head(tail(odds)))) A__INCR(cons(head(tail(head(y0))), z1)) -> c6(MARK(head(tail(head(y0))))) A__INCR(cons(head(tail(tail(y0))), z1)) -> c6(MARK(head(tail(tail(y0))))) A__INCR(cons(head(tail(cons(y0, y1))), z1)) -> c6(MARK(head(tail(cons(y0, y1))))) A__INCR(cons(head(tail(s(y0))), z1)) -> c6(MARK(head(tail(s(y0))))) A__INCR(cons(head(tail(y0)), z1)) -> c6(MARK(head(tail(y0)))) A__INCR(cons(tail(incr(nats)), z1)) -> c6(MARK(tail(incr(nats)))) A__INCR(cons(tail(incr(incr(y0))), z1)) -> c6(MARK(tail(incr(incr(y0))))) A__INCR(cons(tail(incr(pairs)), z1)) -> c6(MARK(tail(incr(pairs)))) A__INCR(cons(tail(incr(odds)), z1)) -> c6(MARK(tail(incr(odds)))) A__INCR(cons(tail(incr(head(y0))), z1)) -> c6(MARK(tail(incr(head(y0))))) A__INCR(cons(tail(incr(tail(y0))), z1)) -> c6(MARK(tail(incr(tail(y0))))) A__INCR(cons(tail(incr(cons(y0, y1))), z1)) -> c6(MARK(tail(incr(cons(y0, y1))))) A__INCR(cons(tail(incr(s(y0))), z1)) -> c6(MARK(tail(incr(s(y0))))) A__INCR(cons(tail(incr(y0)), z1)) -> c6(MARK(tail(incr(y0)))) A__INCR(cons(tail(odds), z1)) -> c6(MARK(tail(odds))) A__INCR(cons(tail(head(nats)), z1)) -> c6(MARK(tail(head(nats)))) A__INCR(cons(tail(head(incr(y0))), z1)) -> c6(MARK(tail(head(incr(y0))))) A__INCR(cons(tail(head(pairs)), z1)) -> c6(MARK(tail(head(pairs)))) A__INCR(cons(tail(head(odds)), z1)) -> c6(MARK(tail(head(odds)))) A__INCR(cons(tail(head(head(y0))), z1)) -> c6(MARK(tail(head(head(y0))))) A__INCR(cons(tail(head(tail(y0))), z1)) -> c6(MARK(tail(head(tail(y0))))) A__INCR(cons(tail(head(cons(y0, y1))), z1)) -> c6(MARK(tail(head(cons(y0, y1))))) A__INCR(cons(tail(head(s(y0))), z1)) -> c6(MARK(tail(head(s(y0))))) A__INCR(cons(tail(head(y0)), z1)) -> c6(MARK(tail(head(y0)))) 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c10(MARK(head(incr(nats)))) A__TAIL(cons(z0, head(incr(pairs)))) -> c10(MARK(head(incr(pairs)))) A__TAIL(cons(z0, head(odds))) -> c10(MARK(head(odds))) A__TAIL(cons(z0, tail(incr(nats)))) -> c10(MARK(tail(incr(nats)))) A__TAIL(cons(z0, tail(incr(pairs)))) -> c10(MARK(tail(incr(pairs)))) A__TAIL(cons(z0, tail(odds))) -> c10(MARK(tail(odds))) A__TAIL(cons(z0, incr(cons(nats, y0)))) -> c10(MARK(incr(cons(nats, y0)))) A__TAIL(cons(z0, incr(cons(incr(y0), y1)))) -> c10(MARK(incr(cons(incr(y0), y1)))) A__TAIL(cons(z0, incr(cons(pairs, y0)))) -> c10(MARK(incr(cons(pairs, y0)))) A__TAIL(cons(z0, incr(cons(odds, y0)))) -> c10(MARK(incr(cons(odds, y0)))) A__TAIL(cons(z0, incr(cons(head(y0), y1)))) -> c10(MARK(incr(cons(head(y0), y1)))) A__TAIL(cons(z0, incr(cons(tail(y0), y1)))) -> c10(MARK(incr(cons(tail(y0), y1)))) A__TAIL(cons(z0, incr(cons(cons(y0, y1), y2)))) -> c10(MARK(incr(cons(cons(y0, y1), y2)))) A__TAIL(cons(z0, incr(cons(s(y0), y1)))) -> c10(MARK(incr(cons(s(y0), y1)))) A__TAIL(cons(z0, incr(cons(y0, y1)))) -> c10(MARK(incr(cons(y0, y1)))) A__TAIL(cons(z0, incr(cons(0, y0)))) -> c10(MARK(incr(cons(0, y0)))) A__TAIL(cons(z0, incr(head(nats)))) -> c10(MARK(incr(head(nats)))) A__TAIL(cons(z0, incr(head(pairs)))) -> c10(MARK(incr(head(pairs)))) A__TAIL(cons(z0, head(head(nats)))) -> c10(MARK(head(head(nats)))) A__TAIL(cons(z0, head(head(pairs)))) -> c10(MARK(head(head(pairs)))) A__TAIL(cons(z0, tail(head(nats)))) -> c10(MARK(tail(head(nats)))) A__TAIL(cons(z0, tail(head(pairs)))) -> c10(MARK(tail(head(pairs)))) A__TAIL(cons(z0, head(cons(nats, y0)))) -> c10(MARK(head(cons(nats, y0)))) A__TAIL(cons(z0, head(cons(incr(y0), y1)))) -> c10(MARK(head(cons(incr(y0), y1)))) A__TAIL(cons(z0, head(cons(pairs, y0)))) -> c10(MARK(head(cons(pairs, y0)))) A__TAIL(cons(z0, head(cons(odds, y0)))) -> c10(MARK(head(cons(odds, y0)))) A__TAIL(cons(z0, head(cons(head(y0), y1)))) -> c10(MARK(head(cons(head(y0), y1)))) A__TAIL(cons(z0, head(cons(tail(y0), y1)))) -> c10(MARK(head(cons(tail(y0), y1)))) A__TAIL(cons(z0, head(cons(cons(y0, y1), y2)))) -> c10(MARK(head(cons(cons(y0, y1), y2)))) A__TAIL(cons(z0, head(cons(s(y0), y1)))) -> c10(MARK(head(cons(s(y0), y1)))) A__TAIL(cons(z0, head(cons(y0, y1)))) -> c10(MARK(head(cons(y0, y1)))) A__TAIL(cons(z0, head(cons(0, y0)))) -> c10(MARK(head(cons(0, y0)))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats))) MARK(tail(tail(nats))) -> c17(A__TAIL(a__tail(a__nats))) MARK(cons(cons(y0, y1), z1)) -> c18(MARK(cons(y0, y1))) MARK(cons(s(y0), z1)) -> c18(MARK(s(y0))) MARK(cons(incr(s(y0)), z1)) -> c18(MARK(incr(s(y0)))) MARK(cons(head(s(y0)), z1)) -> c18(MARK(head(s(y0)))) MARK(cons(tail(cons(y0, y1)), z1)) -> c18(MARK(tail(cons(y0, y1)))) MARK(cons(tail(s(y0)), z1)) -> c18(MARK(tail(s(y0)))) MARK(cons(incr(incr(incr(y0))), z1)) -> c18(MARK(incr(incr(incr(y0))))) MARK(cons(incr(incr(odds)), z1)) -> c18(MARK(incr(incr(odds)))) MARK(cons(incr(incr(head(y0))), z1)) -> c18(MARK(incr(incr(head(y0))))) MARK(cons(incr(incr(tail(y0))), z1)) -> c18(MARK(incr(incr(tail(y0))))) MARK(cons(incr(incr(cons(y0, y1))), z1)) -> c18(MARK(incr(incr(cons(y0, y1))))) MARK(cons(incr(incr(s(y0))), z1)) -> c18(MARK(incr(incr(s(y0))))) MARK(cons(incr(incr(y0)), z1)) -> c18(MARK(incr(incr(y0)))) MARK(cons(incr(head(incr(y0))), z1)) -> c18(MARK(incr(head(incr(y0))))) MARK(cons(incr(head(odds)), z1)) -> c18(MARK(incr(head(odds)))) MARK(cons(incr(head(head(y0))), z1)) -> c18(MARK(incr(head(head(y0))))) MARK(cons(incr(head(tail(y0))), z1)) -> c18(MARK(incr(head(tail(y0))))) MARK(cons(incr(head(cons(y0, y1))), z1)) -> c18(MARK(incr(head(cons(y0, y1))))) MARK(cons(incr(head(s(y0))), z1)) -> c18(MARK(incr(head(s(y0))))) MARK(cons(incr(head(y0)), z1)) -> c18(MARK(incr(head(y0)))) MARK(cons(incr(tail(incr(y0))), z1)) -> c18(MARK(incr(tail(incr(y0))))) MARK(cons(incr(tail(pairs)), z1)) -> c18(MARK(incr(tail(pairs)))) MARK(cons(incr(tail(odds)), z1)) -> c18(MARK(incr(tail(odds)))) MARK(cons(incr(tail(head(y0))), z1)) -> c18(MARK(incr(tail(head(y0))))) MARK(cons(incr(tail(tail(y0))), z1)) -> c18(MARK(incr(tail(tail(y0))))) MARK(cons(incr(tail(cons(y0, y1))), z1)) -> c18(MARK(incr(tail(cons(y0, y1))))) MARK(cons(incr(tail(s(y0))), z1)) -> c18(MARK(incr(tail(s(y0))))) MARK(cons(incr(tail(y0)), z1)) -> c18(MARK(incr(tail(y0)))) MARK(cons(head(incr(incr(y0))), z1)) -> c18(MARK(head(incr(incr(y0))))) MARK(cons(head(incr(odds)), z1)) -> c18(MARK(head(incr(odds)))) MARK(cons(head(incr(head(y0))), z1)) -> c18(MARK(head(incr(head(y0))))) MARK(cons(head(incr(tail(y0))), z1)) -> c18(MARK(head(incr(tail(y0))))) MARK(cons(head(incr(cons(y0, y1))), z1)) -> c18(MARK(head(incr(cons(y0, y1))))) MARK(cons(head(incr(s(y0))), z1)) -> c18(MARK(head(incr(s(y0))))) MARK(cons(head(incr(y0)), z1)) -> c18(MARK(head(incr(y0)))) MARK(cons(head(head(incr(y0))), z1)) -> c18(MARK(head(head(incr(y0))))) MARK(cons(head(head(odds)), z1)) -> c18(MARK(head(head(odds)))) MARK(cons(head(head(head(y0))), z1)) -> c18(MARK(head(head(head(y0))))) MARK(cons(head(head(tail(y0))), z1)) -> c18(MARK(head(head(tail(y0))))) MARK(cons(head(head(cons(y0, y1))), z1)) -> c18(MARK(head(head(cons(y0, y1))))) MARK(cons(head(head(s(y0))), z1)) -> c18(MARK(head(head(s(y0))))) MARK(cons(head(head(y0)), z1)) -> c18(MARK(head(head(y0)))) MARK(cons(head(tail(incr(y0))), z1)) -> c18(MARK(head(tail(incr(y0))))) MARK(cons(head(tail(pairs)), z1)) -> c18(MARK(head(tail(pairs)))) MARK(cons(head(tail(odds)), z1)) -> c18(MARK(head(tail(odds)))) MARK(cons(head(tail(head(y0))), z1)) -> c18(MARK(head(tail(head(y0))))) MARK(cons(head(tail(tail(y0))), z1)) -> c18(MARK(head(tail(tail(y0))))) MARK(cons(head(tail(cons(y0, y1))), z1)) -> c18(MARK(head(tail(cons(y0, y1))))) MARK(cons(head(tail(s(y0))), z1)) -> c18(MARK(head(tail(s(y0))))) MARK(cons(head(tail(y0)), z1)) -> c18(MARK(head(tail(y0)))) MARK(cons(tail(incr(incr(y0))), z1)) -> c18(MARK(tail(incr(incr(y0))))) MARK(cons(tail(incr(odds)), z1)) -> c18(MARK(tail(incr(odds)))) MARK(cons(tail(incr(head(y0))), z1)) -> c18(MARK(tail(incr(head(y0))))) MARK(cons(tail(incr(tail(y0))), z1)) -> c18(MARK(tail(incr(tail(y0))))) MARK(cons(tail(incr(cons(y0, y1))), z1)) -> c18(MARK(tail(incr(cons(y0, y1))))) MARK(cons(tail(incr(s(y0))), z1)) -> c18(MARK(tail(incr(s(y0))))) MARK(cons(tail(incr(y0)), z1)) -> c18(MARK(tail(incr(y0)))) MARK(cons(tail(head(incr(y0))), z1)) -> c18(MARK(tail(head(incr(y0))))) MARK(cons(tail(head(odds)), z1)) -> c18(MARK(tail(head(odds)))) MARK(cons(tail(head(head(y0))), z1)) -> c18(MARK(tail(head(head(y0))))) MARK(cons(tail(head(tail(y0))), z1)) -> c18(MARK(tail(head(tail(y0))))) MARK(cons(tail(head(cons(y0, y1))), z1)) -> c18(MARK(tail(head(cons(y0, y1))))) MARK(cons(tail(head(s(y0))), z1)) -> c18(MARK(tail(head(s(y0))))) MARK(cons(tail(head(y0)), z1)) -> c18(MARK(tail(head(y0)))) MARK(cons(tail(tail(incr(y0))), z1)) -> c18(MARK(tail(tail(incr(y0))))) MARK(cons(tail(tail(pairs)), z1)) -> c18(MARK(tail(tail(pairs)))) MARK(cons(tail(tail(odds)), z1)) -> c18(MARK(tail(tail(odds)))) MARK(cons(tail(tail(head(y0))), z1)) -> c18(MARK(tail(tail(head(y0))))) MARK(cons(tail(tail(tail(y0))), z1)) -> c18(MARK(tail(tail(tail(y0))))) MARK(cons(tail(tail(cons(y0, y1))), z1)) -> c18(MARK(tail(tail(cons(y0, y1))))) MARK(cons(tail(tail(s(y0))), z1)) -> c18(MARK(tail(tail(s(y0))))) MARK(cons(tail(tail(y0)), z1)) -> c18(MARK(tail(tail(y0)))) MARK(cons(tail(pairs), z1)) -> c18(MARK(tail(pairs))) MARK(cons(incr(incr(nats)), z1)) -> c18(MARK(incr(incr(nats)))) MARK(cons(incr(incr(pairs)), z1)) -> c18(MARK(incr(incr(pairs)))) MARK(cons(incr(odds), z1)) -> c18(MARK(incr(odds))) MARK(cons(head(incr(nats)), z1)) -> c18(MARK(head(incr(nats)))) MARK(cons(head(incr(pairs)), z1)) -> c18(MARK(head(incr(pairs)))) MARK(cons(head(odds), z1)) -> c18(MARK(head(odds))) MARK(cons(tail(incr(nats)), z1)) -> c18(MARK(tail(incr(nats)))) MARK(cons(tail(incr(pairs)), z1)) -> c18(MARK(tail(incr(pairs)))) MARK(cons(tail(odds), z1)) -> c18(MARK(tail(odds))) MARK(cons(incr(cons(nats, y0)), z1)) -> c18(MARK(incr(cons(nats, y0)))) MARK(cons(incr(cons(incr(y0), y1)), z1)) -> c18(MARK(incr(cons(incr(y0), y1)))) MARK(cons(incr(cons(pairs, y0)), z1)) -> c18(MARK(incr(cons(pairs, y0)))) MARK(cons(incr(cons(odds, y0)), z1)) -> c18(MARK(incr(cons(odds, y0)))) MARK(cons(incr(cons(head(y0), y1)), z1)) -> c18(MARK(incr(cons(head(y0), y1)))) MARK(cons(incr(cons(tail(y0), y1)), z1)) -> c18(MARK(incr(cons(tail(y0), y1)))) MARK(cons(incr(cons(cons(y0, y1), y2)), z1)) -> c18(MARK(incr(cons(cons(y0, y1), y2)))) MARK(cons(incr(cons(s(y0), y1)), z1)) -> c18(MARK(incr(cons(s(y0), y1)))) MARK(cons(incr(cons(y0, y1)), z1)) -> c18(MARK(incr(cons(y0, y1)))) MARK(cons(incr(cons(0, y0)), z1)) -> c18(MARK(incr(cons(0, y0)))) MARK(cons(incr(head(nats)), z1)) -> c18(MARK(incr(head(nats)))) MARK(cons(incr(head(pairs)), z1)) -> c18(MARK(incr(head(pairs)))) MARK(cons(head(head(nats)), z1)) -> c18(MARK(head(head(nats)))) MARK(cons(head(head(pairs)), z1)) -> c18(MARK(head(head(pairs)))) MARK(cons(tail(head(nats)), z1)) -> c18(MARK(tail(head(nats)))) MARK(cons(tail(head(pairs)), z1)) -> c18(MARK(tail(head(pairs)))) MARK(cons(head(cons(nats, y0)), z1)) -> c18(MARK(head(cons(nats, y0)))) MARK(cons(head(cons(incr(y0), y1)), z1)) -> c18(MARK(head(cons(incr(y0), y1)))) MARK(cons(head(cons(pairs, y0)), z1)) -> c18(MARK(head(cons(pairs, y0)))) MARK(cons(head(cons(odds, y0)), z1)) -> c18(MARK(head(cons(odds, y0)))) MARK(cons(head(cons(head(y0), y1)), z1)) -> c18(MARK(head(cons(head(y0), y1)))) MARK(cons(head(cons(tail(y0), y1)), z1)) -> c18(MARK(head(cons(tail(y0), y1)))) MARK(cons(head(cons(cons(y0, y1), y2)), z1)) -> c18(MARK(head(cons(cons(y0, y1), y2)))) MARK(cons(head(cons(s(y0), y1)), z1)) -> c18(MARK(head(cons(s(y0), y1)))) MARK(cons(head(cons(y0, y1)), z1)) -> c18(MARK(head(cons(y0, y1)))) MARK(cons(head(cons(0, y0)), z1)) -> c18(MARK(head(cons(0, y0)))) MARK(cons(incr(tail(nats)), z1)) -> c18(MARK(incr(tail(nats)))) MARK(cons(head(tail(nats)), z1)) -> c18(MARK(head(tail(nats)))) MARK(cons(tail(tail(nats)), z1)) -> c18(MARK(tail(tail(nats)))) S tuples: MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c17(MARK(s(z0))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), 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y1))), z1)) -> c18(MARK(head(head(cons(y0, y1))))) MARK(cons(head(head(s(y0))), z1)) -> c18(MARK(head(head(s(y0))))) MARK(cons(head(head(y0)), z1)) -> c18(MARK(head(head(y0)))) MARK(cons(head(tail(incr(y0))), z1)) -> c18(MARK(head(tail(incr(y0))))) MARK(cons(head(tail(pairs)), z1)) -> c18(MARK(head(tail(pairs)))) MARK(cons(head(tail(odds)), z1)) -> c18(MARK(head(tail(odds)))) MARK(cons(head(tail(head(y0))), z1)) -> c18(MARK(head(tail(head(y0))))) MARK(cons(head(tail(tail(y0))), z1)) -> c18(MARK(head(tail(tail(y0))))) MARK(cons(head(tail(cons(y0, y1))), z1)) -> c18(MARK(head(tail(cons(y0, y1))))) MARK(cons(head(tail(s(y0))), z1)) -> c18(MARK(head(tail(s(y0))))) MARK(cons(head(tail(y0)), z1)) -> c18(MARK(head(tail(y0)))) MARK(cons(tail(incr(incr(y0))), z1)) -> c18(MARK(tail(incr(incr(y0))))) MARK(cons(tail(incr(odds)), z1)) -> c18(MARK(tail(incr(odds)))) MARK(cons(tail(incr(head(y0))), z1)) -> c18(MARK(tail(incr(head(y0))))) MARK(cons(tail(incr(tail(y0))), z1)) -> c18(MARK(tail(incr(tail(y0))))) MARK(cons(tail(incr(cons(y0, y1))), z1)) -> c18(MARK(tail(incr(cons(y0, y1))))) MARK(cons(tail(incr(s(y0))), z1)) -> c18(MARK(tail(incr(s(y0))))) MARK(cons(tail(incr(y0)), z1)) -> c18(MARK(tail(incr(y0)))) MARK(cons(tail(head(incr(y0))), z1)) -> c18(MARK(tail(head(incr(y0))))) MARK(cons(tail(head(odds)), z1)) -> c18(MARK(tail(head(odds)))) MARK(cons(tail(head(head(y0))), z1)) -> c18(MARK(tail(head(head(y0))))) MARK(cons(tail(head(tail(y0))), z1)) -> c18(MARK(tail(head(tail(y0))))) MARK(cons(tail(head(cons(y0, y1))), z1)) -> c18(MARK(tail(head(cons(y0, y1))))) MARK(cons(tail(head(s(y0))), z1)) -> c18(MARK(tail(head(s(y0))))) MARK(cons(tail(head(y0)), z1)) -> c18(MARK(tail(head(y0)))) MARK(cons(tail(tail(incr(y0))), z1)) -> c18(MARK(tail(tail(incr(y0))))) MARK(cons(tail(tail(pairs)), z1)) -> c18(MARK(tail(tail(pairs)))) MARK(cons(tail(tail(odds)), z1)) -> c18(MARK(tail(tail(odds)))) MARK(cons(tail(tail(head(y0))), z1)) -> c18(MARK(tail(tail(head(y0))))) MARK(cons(tail(tail(tail(y0))), z1)) -> c18(MARK(tail(tail(tail(y0))))) MARK(cons(tail(tail(cons(y0, y1))), z1)) -> c18(MARK(tail(tail(cons(y0, y1))))) MARK(cons(tail(tail(s(y0))), z1)) -> c18(MARK(tail(tail(s(y0))))) MARK(cons(tail(tail(y0)), z1)) -> c18(MARK(tail(tail(y0)))) MARK(cons(tail(pairs), z1)) -> c18(MARK(tail(pairs))) MARK(cons(incr(incr(nats)), z1)) -> c18(MARK(incr(incr(nats)))) MARK(cons(incr(incr(pairs)), z1)) -> c18(MARK(incr(incr(pairs)))) MARK(cons(incr(odds), z1)) -> c18(MARK(incr(odds))) MARK(cons(head(incr(nats)), z1)) -> c18(MARK(head(incr(nats)))) MARK(cons(head(incr(pairs)), z1)) -> c18(MARK(head(incr(pairs)))) MARK(cons(head(odds), z1)) -> c18(MARK(head(odds))) MARK(cons(tail(incr(nats)), z1)) -> c18(MARK(tail(incr(nats)))) MARK(cons(tail(incr(pairs)), z1)) -> c18(MARK(tail(incr(pairs)))) MARK(cons(tail(odds), z1)) -> c18(MARK(tail(odds))) MARK(cons(incr(cons(nats, y0)), z1)) -> c18(MARK(incr(cons(nats, y0)))) MARK(cons(incr(cons(incr(y0), y1)), z1)) -> c18(MARK(incr(cons(incr(y0), y1)))) MARK(cons(incr(cons(pairs, y0)), z1)) -> c18(MARK(incr(cons(pairs, y0)))) MARK(cons(incr(cons(odds, y0)), z1)) -> c18(MARK(incr(cons(odds, y0)))) MARK(cons(incr(cons(head(y0), y1)), z1)) -> c18(MARK(incr(cons(head(y0), y1)))) MARK(cons(incr(cons(tail(y0), y1)), z1)) -> c18(MARK(incr(cons(tail(y0), y1)))) MARK(cons(incr(cons(cons(y0, y1), y2)), z1)) -> c18(MARK(incr(cons(cons(y0, y1), y2)))) MARK(cons(incr(cons(s(y0), y1)), z1)) -> c18(MARK(incr(cons(s(y0), y1)))) MARK(cons(incr(cons(y0, y1)), z1)) -> c18(MARK(incr(cons(y0, y1)))) MARK(cons(incr(cons(0, y0)), z1)) -> c18(MARK(incr(cons(0, y0)))) MARK(cons(incr(head(nats)), z1)) -> c18(MARK(incr(head(nats)))) MARK(cons(incr(head(pairs)), z1)) -> c18(MARK(incr(head(pairs)))) MARK(cons(head(head(nats)), z1)) -> c18(MARK(head(head(nats)))) MARK(cons(head(head(pairs)), z1)) -> c18(MARK(head(head(pairs)))) MARK(cons(tail(head(nats)), z1)) -> c18(MARK(tail(head(nats)))) MARK(cons(tail(head(pairs)), z1)) -> c18(MARK(tail(head(pairs)))) MARK(cons(head(cons(nats, y0)), z1)) -> c18(MARK(head(cons(nats, y0)))) MARK(cons(head(cons(incr(y0), y1)), z1)) -> c18(MARK(head(cons(incr(y0), y1)))) MARK(cons(head(cons(pairs, y0)), z1)) -> c18(MARK(head(cons(pairs, y0)))) MARK(cons(head(cons(odds, y0)), z1)) -> c18(MARK(head(cons(odds, y0)))) MARK(cons(head(cons(head(y0), y1)), z1)) -> c18(MARK(head(cons(head(y0), y1)))) MARK(cons(head(cons(tail(y0), y1)), z1)) -> c18(MARK(head(cons(tail(y0), y1)))) MARK(cons(head(cons(cons(y0, y1), y2)), z1)) -> c18(MARK(head(cons(cons(y0, y1), y2)))) MARK(cons(head(cons(s(y0), y1)), z1)) -> c18(MARK(head(cons(s(y0), y1)))) MARK(cons(head(cons(y0, y1)), z1)) -> c18(MARK(head(cons(y0, y1)))) MARK(cons(head(cons(0, y0)), z1)) -> c18(MARK(head(cons(0, y0)))) MARK(cons(incr(tail(nats)), z1)) -> c18(MARK(incr(tail(nats)))) MARK(cons(head(tail(nats)), z1)) -> c18(MARK(head(tail(nats)))) MARK(cons(tail(tail(nats)), z1)) -> c18(MARK(tail(tail(nats)))) K tuples:none Defined Rule Symbols: a__nats, a__pairs, a__odds, a__incr_1, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: MARK_1, A__INCR_1, A__HEAD_1, A__TAIL_1 Compound Symbols: c20_1, c13_1, c16_1, c17_2, c17_1, c13_2, c16_2, c6_1, c8_1, c10_1, c18_1 ---------------------------------------- (189) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 6 trailing tuple parts ---------------------------------------- (190) Obligation: Complexity Dependency Tuples Problem Rules: a__nats -> cons(0, incr(nats)) a__nats -> nats a__pairs -> cons(0, incr(odds)) a__pairs -> pairs a__odds -> a__incr(a__pairs) a__odds -> odds a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) 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(nats) -> a__nats mark(incr(z0)) -> a__incr(mark(z0)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c17(MARK(s(z0))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(tail(incr(incr(z0)))) -> c17(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(odds))) -> c17(A__TAIL(a__incr(a__odds)), MARK(incr(odds))) MARK(tail(incr(head(z0)))) -> c17(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c17(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c17(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c17(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c17(MARK(incr(x0))) MARK(tail(head(incr(z0)))) -> c17(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(odds))) -> c17(A__TAIL(a__head(a__odds)), MARK(head(odds))) MARK(tail(head(head(z0)))) -> c17(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c17(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c17(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c17(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c17(MARK(head(x0))) MARK(tail(tail(incr(z0)))) -> c17(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(pairs))) -> c17(A__TAIL(a__tail(a__pairs)), MARK(tail(pairs))) MARK(tail(tail(odds))) -> c17(A__TAIL(a__tail(a__odds)), MARK(tail(odds))) MARK(tail(tail(head(z0)))) -> c17(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c17(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c17(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c17(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c17(MARK(tail(x0))) MARK(tail(pairs)) -> c17(A__TAIL(cons(0, incr(odds)))) A__INCR(cons(cons(y0, y1), z1)) -> c6(MARK(cons(y0, y1))) A__INCR(cons(s(y0), z1)) -> c6(MARK(s(y0))) A__INCR(cons(incr(cons(y0, y1)), z1)) -> c6(MARK(incr(cons(y0, y1)))) A__INCR(cons(incr(s(y0)), z1)) -> c6(MARK(incr(s(y0)))) A__INCR(cons(head(cons(y0, y1)), z1)) -> c6(MARK(head(cons(y0, y1)))) A__INCR(cons(head(s(y0)), z1)) -> c6(MARK(head(s(y0)))) A__INCR(cons(tail(cons(y0, y1)), z1)) -> c6(MARK(tail(cons(y0, y1)))) A__INCR(cons(tail(s(y0)), z1)) -> c6(MARK(tail(s(y0)))) A__INCR(cons(incr(incr(nats)), z1)) -> c6(MARK(incr(incr(nats)))) A__INCR(cons(incr(incr(incr(y0))), z1)) -> c6(MARK(incr(incr(incr(y0))))) A__INCR(cons(incr(incr(pairs)), z1)) -> c6(MARK(incr(incr(pairs)))) A__INCR(cons(incr(incr(odds)), z1)) -> c6(MARK(incr(incr(odds)))) A__INCR(cons(incr(incr(head(y0))), z1)) -> c6(MARK(incr(incr(head(y0))))) A__INCR(cons(incr(incr(tail(y0))), z1)) -> c6(MARK(incr(incr(tail(y0))))) A__INCR(cons(incr(incr(cons(y0, y1))), z1)) -> c6(MARK(incr(incr(cons(y0, y1))))) A__INCR(cons(incr(incr(s(y0))), z1)) -> c6(MARK(incr(incr(s(y0))))) A__INCR(cons(incr(incr(y0)), z1)) -> c6(MARK(incr(incr(y0)))) A__INCR(cons(incr(odds), z1)) -> c6(MARK(incr(odds))) A__INCR(cons(incr(head(nats)), z1)) -> c6(MARK(incr(head(nats)))) A__INCR(cons(incr(head(incr(y0))), z1)) -> c6(MARK(incr(head(incr(y0))))) A__INCR(cons(incr(head(pairs)), z1)) -> c6(MARK(incr(head(pairs)))) A__INCR(cons(incr(head(odds)), z1)) -> c6(MARK(incr(head(odds)))) A__INCR(cons(incr(head(head(y0))), z1)) -> c6(MARK(incr(head(head(y0))))) A__INCR(cons(incr(head(tail(y0))), z1)) -> c6(MARK(incr(head(tail(y0))))) A__INCR(cons(incr(head(cons(y0, y1))), z1)) -> c6(MARK(incr(head(cons(y0, y1))))) A__INCR(cons(incr(head(s(y0))), z1)) -> c6(MARK(incr(head(s(y0))))) A__INCR(cons(incr(head(y0)), z1)) -> c6(MARK(incr(head(y0)))) A__INCR(cons(incr(tail(nats)), z1)) -> c6(MARK(incr(tail(nats)))) A__INCR(cons(incr(tail(incr(y0))), z1)) -> c6(MARK(incr(tail(incr(y0))))) A__INCR(cons(incr(tail(pairs)), z1)) -> c6(MARK(incr(tail(pairs)))) A__INCR(cons(incr(tail(odds)), z1)) -> c6(MARK(incr(tail(odds)))) A__INCR(cons(incr(tail(head(y0))), z1)) -> c6(MARK(incr(tail(head(y0))))) A__INCR(cons(incr(tail(tail(y0))), z1)) -> c6(MARK(incr(tail(tail(y0))))) A__INCR(cons(incr(tail(cons(y0, y1))), z1)) -> c6(MARK(incr(tail(cons(y0, y1))))) A__INCR(cons(incr(tail(s(y0))), z1)) -> c6(MARK(incr(tail(s(y0))))) A__INCR(cons(incr(tail(y0)), z1)) -> c6(MARK(incr(tail(y0)))) A__INCR(cons(head(incr(nats)), z1)) -> c6(MARK(head(incr(nats)))) A__INCR(cons(head(incr(incr(y0))), z1)) -> c6(MARK(head(incr(incr(y0))))) A__INCR(cons(head(incr(pairs)), z1)) -> c6(MARK(head(incr(pairs)))) A__INCR(cons(head(incr(odds)), z1)) -> c6(MARK(head(incr(odds)))) A__INCR(cons(head(incr(head(y0))), z1)) -> c6(MARK(head(incr(head(y0))))) A__INCR(cons(head(incr(tail(y0))), z1)) -> c6(MARK(head(incr(tail(y0))))) A__INCR(cons(head(incr(cons(y0, y1))), z1)) -> c6(MARK(head(incr(cons(y0, y1))))) A__INCR(cons(head(incr(s(y0))), z1)) -> c6(MARK(head(incr(s(y0))))) A__INCR(cons(head(incr(y0)), z1)) -> c6(MARK(head(incr(y0)))) A__INCR(cons(head(odds), z1)) -> 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z1)) -> c18(MARK(head(cons(s(y0), y1)))) MARK(cons(head(cons(y0, y1)), z1)) -> c18(MARK(head(cons(y0, y1)))) MARK(cons(head(cons(0, y0)), z1)) -> c18(MARK(head(cons(0, y0)))) MARK(cons(incr(tail(nats)), z1)) -> c18(MARK(incr(tail(nats)))) MARK(cons(head(tail(nats)), z1)) -> c18(MARK(head(tail(nats)))) MARK(cons(tail(tail(nats)), z1)) -> c18(MARK(tail(tail(nats)))) MARK(incr(cons(nats, x1))) -> c13(A__INCR(cons(a__nats, x1))) MARK(incr(cons(pairs, x1))) -> c13(A__INCR(cons(a__pairs, x1))) MARK(incr(cons(odds, x1))) -> c13(A__INCR(cons(a__odds, x1))) MARK(head(cons(nats, x1))) -> c16(A__HEAD(cons(a__nats, x1))) MARK(head(cons(pairs, x1))) -> c16(A__HEAD(cons(a__pairs, x1))) MARK(head(cons(odds, x1))) -> c16(A__HEAD(cons(a__odds, x1))) S tuples: MARK(s(z0)) -> c20(MARK(z0)) MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c17(MARK(s(z0))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(tail(incr(incr(z0)))) -> c17(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(odds))) -> c17(A__TAIL(a__incr(a__odds)), MARK(incr(odds))) MARK(tail(incr(head(z0)))) -> c17(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c17(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c17(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c17(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c17(MARK(incr(x0))) MARK(tail(head(incr(z0)))) -> c17(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(odds))) -> c17(A__TAIL(a__head(a__odds)), MARK(head(odds))) MARK(tail(head(head(z0)))) -> c17(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c17(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c17(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c17(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c17(MARK(head(x0))) MARK(tail(tail(incr(z0)))) -> c17(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(pairs))) -> c17(A__TAIL(a__tail(a__pairs)), MARK(tail(pairs))) MARK(tail(tail(odds))) -> c17(A__TAIL(a__tail(a__odds)), MARK(tail(odds))) MARK(tail(tail(head(z0)))) -> c17(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c17(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c17(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c17(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c17(MARK(tail(x0))) MARK(tail(pairs)) -> c17(A__TAIL(cons(0, incr(odds)))) A__INCR(cons(cons(y0, y1), z1)) -> c6(MARK(cons(y0, y1))) A__INCR(cons(s(y0), z1)) -> c6(MARK(s(y0))) A__INCR(cons(incr(cons(y0, y1)), z1)) -> c6(MARK(incr(cons(y0, y1)))) A__INCR(cons(incr(s(y0)), z1)) -> c6(MARK(incr(s(y0)))) A__INCR(cons(head(cons(y0, y1)), z1)) -> c6(MARK(head(cons(y0, y1)))) A__INCR(cons(head(s(y0)), z1)) -> c6(MARK(head(s(y0)))) A__INCR(cons(tail(cons(y0, y1)), z1)) -> c6(MARK(tail(cons(y0, y1)))) A__INCR(cons(tail(s(y0)), z1)) -> c6(MARK(tail(s(y0)))) A__INCR(cons(incr(incr(nats)), z1)) -> c6(MARK(incr(incr(nats)))) A__INCR(cons(incr(incr(incr(y0))), z1)) -> c6(MARK(incr(incr(incr(y0))))) A__INCR(cons(incr(incr(pairs)), z1)) -> c6(MARK(incr(incr(pairs)))) A__INCR(cons(incr(incr(odds)), z1)) -> c6(MARK(incr(incr(odds)))) A__INCR(cons(incr(incr(head(y0))), z1)) -> c6(MARK(incr(incr(head(y0))))) A__INCR(cons(incr(incr(tail(y0))), z1)) -> c6(MARK(incr(incr(tail(y0))))) A__INCR(cons(incr(incr(cons(y0, y1))), z1)) -> c6(MARK(incr(incr(cons(y0, y1))))) A__INCR(cons(incr(incr(s(y0))), z1)) -> c6(MARK(incr(incr(s(y0))))) A__INCR(cons(incr(incr(y0)), z1)) -> c6(MARK(incr(incr(y0)))) A__INCR(cons(incr(odds), z1)) -> c6(MARK(incr(odds))) A__INCR(cons(incr(head(nats)), z1)) -> c6(MARK(incr(head(nats)))) A__INCR(cons(incr(head(incr(y0))), z1)) -> c6(MARK(incr(head(incr(y0))))) A__INCR(cons(incr(head(pairs)), z1)) -> c6(MARK(incr(head(pairs)))) A__INCR(cons(incr(head(odds)), z1)) -> c6(MARK(incr(head(odds)))) A__INCR(cons(incr(head(head(y0))), z1)) -> c6(MARK(incr(head(head(y0))))) A__INCR(cons(incr(head(tail(y0))), z1)) -> c6(MARK(incr(head(tail(y0))))) A__INCR(cons(incr(head(cons(y0, y1))), z1)) -> c6(MARK(incr(head(cons(y0, y1))))) A__INCR(cons(incr(head(s(y0))), z1)) -> c6(MARK(incr(head(s(y0))))) A__INCR(cons(incr(head(y0)), z1)) -> c6(MARK(incr(head(y0)))) A__INCR(cons(incr(tail(nats)), z1)) -> c6(MARK(incr(tail(nats)))) A__INCR(cons(incr(tail(incr(y0))), z1)) -> c6(MARK(incr(tail(incr(y0))))) A__INCR(cons(incr(tail(pairs)), z1)) -> c6(MARK(incr(tail(pairs)))) A__INCR(cons(incr(tail(odds)), z1)) -> c6(MARK(incr(tail(odds)))) A__INCR(cons(incr(tail(head(y0))), z1)) -> c6(MARK(incr(tail(head(y0))))) A__INCR(cons(incr(tail(tail(y0))), z1)) -> c6(MARK(incr(tail(tail(y0))))) A__INCR(cons(incr(tail(cons(y0, y1))), z1)) -> c6(MARK(incr(tail(cons(y0, y1))))) A__INCR(cons(incr(tail(s(y0))), z1)) -> c6(MARK(incr(tail(s(y0))))) A__INCR(cons(incr(tail(y0)), z1)) -> c6(MARK(incr(tail(y0)))) A__INCR(cons(head(incr(nats)), z1)) -> c6(MARK(head(incr(nats)))) A__INCR(cons(head(incr(incr(y0))), z1)) -> c6(MARK(head(incr(incr(y0))))) A__INCR(cons(head(incr(pairs)), z1)) -> c6(MARK(head(incr(pairs)))) A__INCR(cons(head(incr(odds)), z1)) -> c6(MARK(head(incr(odds)))) A__INCR(cons(head(incr(head(y0))), z1)) -> c6(MARK(head(incr(head(y0))))) A__INCR(cons(head(incr(tail(y0))), z1)) -> c6(MARK(head(incr(tail(y0))))) A__INCR(cons(head(incr(cons(y0, y1))), z1)) -> c6(MARK(head(incr(cons(y0, y1))))) A__INCR(cons(head(incr(s(y0))), z1)) -> c6(MARK(head(incr(s(y0))))) A__INCR(cons(head(incr(y0)), z1)) -> c6(MARK(head(incr(y0)))) A__INCR(cons(head(odds), z1)) -> c6(MARK(head(odds))) A__INCR(cons(head(head(nats)), z1)) -> c6(MARK(head(head(nats)))) A__INCR(cons(head(head(incr(y0))), z1)) -> c6(MARK(head(head(incr(y0))))) A__INCR(cons(head(head(pairs)), z1)) -> c6(MARK(head(head(pairs)))) A__INCR(cons(head(head(odds)), z1)) -> c6(MARK(head(head(odds)))) A__INCR(cons(head(head(head(y0))), z1)) -> c6(MARK(head(head(head(y0))))) A__INCR(cons(head(head(tail(y0))), z1)) -> c6(MARK(head(head(tail(y0))))) A__INCR(cons(head(head(cons(y0, y1))), z1)) -> c6(MARK(head(head(cons(y0, y1))))) A__INCR(cons(head(head(s(y0))), z1)) -> c6(MARK(head(head(s(y0))))) A__INCR(cons(head(head(y0)), z1)) -> c6(MARK(head(head(y0)))) A__INCR(cons(head(tail(nats)), z1)) -> c6(MARK(head(tail(nats)))) A__INCR(cons(head(tail(incr(y0))), z1)) -> c6(MARK(head(tail(incr(y0))))) A__INCR(cons(head(tail(pairs)), z1)) -> c6(MARK(head(tail(pairs)))) A__INCR(cons(head(tail(odds)), z1)) -> c6(MARK(head(tail(odds)))) A__INCR(cons(head(tail(head(y0))), z1)) -> c6(MARK(head(tail(head(y0))))) A__INCR(cons(head(tail(tail(y0))), z1)) -> c6(MARK(head(tail(tail(y0))))) A__INCR(cons(head(tail(cons(y0, y1))), z1)) -> c6(MARK(head(tail(cons(y0, y1))))) A__INCR(cons(head(tail(s(y0))), z1)) -> c6(MARK(head(tail(s(y0))))) A__INCR(cons(head(tail(y0)), z1)) -> c6(MARK(head(tail(y0)))) A__INCR(cons(tail(incr(nats)), z1)) -> c6(MARK(tail(incr(nats)))) A__INCR(cons(tail(incr(incr(y0))), z1)) -> c6(MARK(tail(incr(incr(y0))))) A__INCR(cons(tail(incr(pairs)), z1)) -> c6(MARK(tail(incr(pairs)))) A__INCR(cons(tail(incr(odds)), z1)) -> c6(MARK(tail(incr(odds)))) A__INCR(cons(tail(incr(head(y0))), z1)) -> c6(MARK(tail(incr(head(y0))))) A__INCR(cons(tail(incr(tail(y0))), z1)) -> c6(MARK(tail(incr(tail(y0))))) A__INCR(cons(tail(incr(cons(y0, y1))), z1)) -> c6(MARK(tail(incr(cons(y0, y1))))) A__INCR(cons(tail(incr(s(y0))), z1)) -> c6(MARK(tail(incr(s(y0))))) A__INCR(cons(tail(incr(y0)), z1)) -> c6(MARK(tail(incr(y0)))) A__INCR(cons(tail(odds), z1)) -> c6(MARK(tail(odds))) A__INCR(cons(tail(head(nats)), z1)) -> c6(MARK(tail(head(nats)))) A__INCR(cons(tail(head(incr(y0))), z1)) -> c6(MARK(tail(head(incr(y0))))) A__INCR(cons(tail(head(pairs)), z1)) -> c6(MARK(tail(head(pairs)))) A__INCR(cons(tail(head(odds)), z1)) -> c6(MARK(tail(head(odds)))) 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tail(incr(nats)))) -> c10(MARK(tail(incr(nats)))) A__TAIL(cons(z0, tail(incr(pairs)))) -> c10(MARK(tail(incr(pairs)))) A__TAIL(cons(z0, tail(odds))) -> c10(MARK(tail(odds))) A__TAIL(cons(z0, incr(cons(nats, y0)))) -> c10(MARK(incr(cons(nats, y0)))) A__TAIL(cons(z0, incr(cons(incr(y0), y1)))) -> c10(MARK(incr(cons(incr(y0), y1)))) A__TAIL(cons(z0, incr(cons(pairs, y0)))) -> c10(MARK(incr(cons(pairs, y0)))) A__TAIL(cons(z0, incr(cons(odds, y0)))) -> c10(MARK(incr(cons(odds, y0)))) A__TAIL(cons(z0, incr(cons(head(y0), y1)))) -> c10(MARK(incr(cons(head(y0), y1)))) A__TAIL(cons(z0, incr(cons(tail(y0), y1)))) -> c10(MARK(incr(cons(tail(y0), y1)))) A__TAIL(cons(z0, incr(cons(cons(y0, y1), y2)))) -> c10(MARK(incr(cons(cons(y0, y1), y2)))) A__TAIL(cons(z0, incr(cons(s(y0), y1)))) -> c10(MARK(incr(cons(s(y0), y1)))) A__TAIL(cons(z0, incr(cons(y0, y1)))) -> c10(MARK(incr(cons(y0, y1)))) A__TAIL(cons(z0, incr(cons(0, y0)))) -> c10(MARK(incr(cons(0, y0)))) A__TAIL(cons(z0, incr(head(nats)))) -> c10(MARK(incr(head(nats)))) A__TAIL(cons(z0, incr(head(pairs)))) -> c10(MARK(incr(head(pairs)))) A__TAIL(cons(z0, head(head(nats)))) -> c10(MARK(head(head(nats)))) A__TAIL(cons(z0, head(head(pairs)))) -> c10(MARK(head(head(pairs)))) A__TAIL(cons(z0, tail(head(nats)))) -> c10(MARK(tail(head(nats)))) A__TAIL(cons(z0, tail(head(pairs)))) -> c10(MARK(tail(head(pairs)))) A__TAIL(cons(z0, head(cons(nats, y0)))) -> c10(MARK(head(cons(nats, y0)))) A__TAIL(cons(z0, head(cons(incr(y0), y1)))) -> c10(MARK(head(cons(incr(y0), y1)))) A__TAIL(cons(z0, head(cons(pairs, y0)))) -> c10(MARK(head(cons(pairs, y0)))) A__TAIL(cons(z0, head(cons(odds, y0)))) -> c10(MARK(head(cons(odds, y0)))) A__TAIL(cons(z0, head(cons(head(y0), y1)))) -> c10(MARK(head(cons(head(y0), y1)))) A__TAIL(cons(z0, head(cons(tail(y0), y1)))) -> c10(MARK(head(cons(tail(y0), y1)))) A__TAIL(cons(z0, head(cons(cons(y0, y1), y2)))) -> c10(MARK(head(cons(cons(y0, y1), y2)))) A__TAIL(cons(z0, head(cons(s(y0), y1)))) -> c10(MARK(head(cons(s(y0), y1)))) A__TAIL(cons(z0, head(cons(y0, y1)))) -> c10(MARK(head(cons(y0, y1)))) A__TAIL(cons(z0, head(cons(0, y0)))) -> c10(MARK(head(cons(0, y0)))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats))) MARK(tail(tail(nats))) -> c17(A__TAIL(a__tail(a__nats))) MARK(cons(cons(y0, y1), z1)) -> c18(MARK(cons(y0, y1))) MARK(cons(s(y0), z1)) -> c18(MARK(s(y0))) MARK(cons(incr(s(y0)), z1)) -> c18(MARK(incr(s(y0)))) MARK(cons(head(s(y0)), z1)) -> c18(MARK(head(s(y0)))) MARK(cons(tail(cons(y0, y1)), z1)) -> c18(MARK(tail(cons(y0, y1)))) MARK(cons(tail(s(y0)), z1)) -> c18(MARK(tail(s(y0)))) MARK(cons(incr(incr(incr(y0))), z1)) -> c18(MARK(incr(incr(incr(y0))))) MARK(cons(incr(incr(odds)), z1)) -> c18(MARK(incr(incr(odds)))) MARK(cons(incr(incr(head(y0))), z1)) -> c18(MARK(incr(incr(head(y0))))) MARK(cons(incr(incr(tail(y0))), z1)) -> c18(MARK(incr(incr(tail(y0))))) MARK(cons(incr(incr(cons(y0, y1))), z1)) -> c18(MARK(incr(incr(cons(y0, y1))))) MARK(cons(incr(incr(s(y0))), z1)) -> c18(MARK(incr(incr(s(y0))))) MARK(cons(incr(incr(y0)), z1)) -> c18(MARK(incr(incr(y0)))) MARK(cons(incr(head(incr(y0))), z1)) -> c18(MARK(incr(head(incr(y0))))) MARK(cons(incr(head(odds)), z1)) -> c18(MARK(incr(head(odds)))) MARK(cons(incr(head(head(y0))), z1)) -> c18(MARK(incr(head(head(y0))))) MARK(cons(incr(head(tail(y0))), z1)) -> c18(MARK(incr(head(tail(y0))))) MARK(cons(incr(head(cons(y0, y1))), z1)) -> c18(MARK(incr(head(cons(y0, y1))))) MARK(cons(incr(head(s(y0))), z1)) -> c18(MARK(incr(head(s(y0))))) MARK(cons(incr(head(y0)), z1)) -> c18(MARK(incr(head(y0)))) MARK(cons(incr(tail(incr(y0))), z1)) -> c18(MARK(incr(tail(incr(y0))))) MARK(cons(incr(tail(pairs)), z1)) -> c18(MARK(incr(tail(pairs)))) MARK(cons(incr(tail(odds)), z1)) -> c18(MARK(incr(tail(odds)))) MARK(cons(incr(tail(head(y0))), z1)) -> c18(MARK(incr(tail(head(y0))))) MARK(cons(incr(tail(tail(y0))), z1)) -> c18(MARK(incr(tail(tail(y0))))) MARK(cons(incr(tail(cons(y0, y1))), z1)) -> c18(MARK(incr(tail(cons(y0, y1))))) MARK(cons(incr(tail(s(y0))), z1)) -> c18(MARK(incr(tail(s(y0))))) MARK(cons(incr(tail(y0)), z1)) -> c18(MARK(incr(tail(y0)))) MARK(cons(head(incr(incr(y0))), z1)) -> c18(MARK(head(incr(incr(y0))))) MARK(cons(head(incr(odds)), z1)) -> c18(MARK(head(incr(odds)))) MARK(cons(head(incr(head(y0))), z1)) -> c18(MARK(head(incr(head(y0))))) MARK(cons(head(incr(tail(y0))), z1)) -> c18(MARK(head(incr(tail(y0))))) MARK(cons(head(incr(cons(y0, y1))), z1)) -> c18(MARK(head(incr(cons(y0, y1))))) MARK(cons(head(incr(s(y0))), z1)) -> c18(MARK(head(incr(s(y0))))) MARK(cons(head(incr(y0)), z1)) -> c18(MARK(head(incr(y0)))) MARK(cons(head(head(incr(y0))), z1)) -> c18(MARK(head(head(incr(y0))))) MARK(cons(head(head(odds)), z1)) -> c18(MARK(head(head(odds)))) MARK(cons(head(head(head(y0))), z1)) -> c18(MARK(head(head(head(y0))))) MARK(cons(head(head(tail(y0))), z1)) -> c18(MARK(head(head(tail(y0))))) MARK(cons(head(head(cons(y0, y1))), z1)) -> c18(MARK(head(head(cons(y0, y1))))) MARK(cons(head(head(s(y0))), z1)) -> c18(MARK(head(head(s(y0))))) MARK(cons(head(head(y0)), z1)) -> c18(MARK(head(head(y0)))) MARK(cons(head(tail(incr(y0))), z1)) -> c18(MARK(head(tail(incr(y0))))) MARK(cons(head(tail(pairs)), z1)) -> c18(MARK(head(tail(pairs)))) MARK(cons(head(tail(odds)), z1)) -> c18(MARK(head(tail(odds)))) MARK(cons(head(tail(head(y0))), z1)) -> c18(MARK(head(tail(head(y0))))) MARK(cons(head(tail(tail(y0))), z1)) -> c18(MARK(head(tail(tail(y0))))) MARK(cons(head(tail(cons(y0, y1))), z1)) -> c18(MARK(head(tail(cons(y0, y1))))) MARK(cons(head(tail(s(y0))), z1)) -> c18(MARK(head(tail(s(y0))))) MARK(cons(head(tail(y0)), z1)) -> c18(MARK(head(tail(y0)))) MARK(cons(tail(incr(incr(y0))), z1)) -> c18(MARK(tail(incr(incr(y0))))) MARK(cons(tail(incr(odds)), z1)) -> c18(MARK(tail(incr(odds)))) MARK(cons(tail(incr(head(y0))), z1)) -> c18(MARK(tail(incr(head(y0))))) MARK(cons(tail(incr(tail(y0))), z1)) -> c18(MARK(tail(incr(tail(y0))))) MARK(cons(tail(incr(cons(y0, y1))), z1)) -> c18(MARK(tail(incr(cons(y0, y1))))) MARK(cons(tail(incr(s(y0))), z1)) -> c18(MARK(tail(incr(s(y0))))) MARK(cons(tail(incr(y0)), z1)) -> c18(MARK(tail(incr(y0)))) MARK(cons(tail(head(incr(y0))), z1)) -> c18(MARK(tail(head(incr(y0))))) MARK(cons(tail(head(odds)), z1)) -> c18(MARK(tail(head(odds)))) MARK(cons(tail(head(head(y0))), z1)) -> c18(MARK(tail(head(head(y0))))) MARK(cons(tail(head(tail(y0))), z1)) -> c18(MARK(tail(head(tail(y0))))) MARK(cons(tail(head(cons(y0, y1))), z1)) -> c18(MARK(tail(head(cons(y0, y1))))) MARK(cons(tail(head(s(y0))), z1)) -> c18(MARK(tail(head(s(y0))))) MARK(cons(tail(head(y0)), z1)) -> c18(MARK(tail(head(y0)))) MARK(cons(tail(tail(incr(y0))), z1)) -> c18(MARK(tail(tail(incr(y0))))) MARK(cons(tail(tail(pairs)), z1)) -> c18(MARK(tail(tail(pairs)))) MARK(cons(tail(tail(odds)), z1)) -> c18(MARK(tail(tail(odds)))) MARK(cons(tail(tail(head(y0))), z1)) -> c18(MARK(tail(tail(head(y0))))) MARK(cons(tail(tail(tail(y0))), z1)) -> c18(MARK(tail(tail(tail(y0))))) MARK(cons(tail(tail(cons(y0, y1))), z1)) -> c18(MARK(tail(tail(cons(y0, y1))))) MARK(cons(tail(tail(s(y0))), z1)) -> c18(MARK(tail(tail(s(y0))))) MARK(cons(tail(tail(y0)), z1)) -> c18(MARK(tail(tail(y0)))) MARK(cons(tail(pairs), z1)) -> c18(MARK(tail(pairs))) MARK(cons(incr(incr(nats)), z1)) -> c18(MARK(incr(incr(nats)))) MARK(cons(incr(incr(pairs)), z1)) -> c18(MARK(incr(incr(pairs)))) MARK(cons(incr(odds), z1)) -> c18(MARK(incr(odds))) MARK(cons(head(incr(nats)), z1)) -> c18(MARK(head(incr(nats)))) MARK(cons(head(incr(pairs)), z1)) -> c18(MARK(head(incr(pairs)))) MARK(cons(head(odds), z1)) -> c18(MARK(head(odds))) MARK(cons(tail(incr(nats)), z1)) -> c18(MARK(tail(incr(nats)))) MARK(cons(tail(incr(pairs)), z1)) -> c18(MARK(tail(incr(pairs)))) MARK(cons(tail(odds), z1)) -> c18(MARK(tail(odds))) MARK(cons(incr(cons(nats, y0)), z1)) -> c18(MARK(incr(cons(nats, y0)))) MARK(cons(incr(cons(incr(y0), y1)), z1)) -> c18(MARK(incr(cons(incr(y0), y1)))) MARK(cons(incr(cons(pairs, y0)), z1)) -> c18(MARK(incr(cons(pairs, y0)))) MARK(cons(incr(cons(odds, y0)), z1)) -> c18(MARK(incr(cons(odds, y0)))) MARK(cons(incr(cons(head(y0), y1)), z1)) -> c18(MARK(incr(cons(head(y0), y1)))) MARK(cons(incr(cons(tail(y0), y1)), z1)) -> c18(MARK(incr(cons(tail(y0), y1)))) MARK(cons(incr(cons(cons(y0, y1), y2)), z1)) -> c18(MARK(incr(cons(cons(y0, y1), y2)))) MARK(cons(incr(cons(s(y0), y1)), z1)) -> c18(MARK(incr(cons(s(y0), y1)))) MARK(cons(incr(cons(y0, y1)), z1)) -> c18(MARK(incr(cons(y0, y1)))) MARK(cons(incr(cons(0, y0)), z1)) -> c18(MARK(incr(cons(0, y0)))) MARK(cons(incr(head(nats)), z1)) -> c18(MARK(incr(head(nats)))) MARK(cons(incr(head(pairs)), z1)) -> c18(MARK(incr(head(pairs)))) MARK(cons(head(head(nats)), z1)) -> c18(MARK(head(head(nats)))) MARK(cons(head(head(pairs)), z1)) -> c18(MARK(head(head(pairs)))) MARK(cons(tail(head(nats)), z1)) -> c18(MARK(tail(head(nats)))) MARK(cons(tail(head(pairs)), z1)) -> c18(MARK(tail(head(pairs)))) MARK(cons(head(cons(nats, y0)), z1)) -> c18(MARK(head(cons(nats, y0)))) MARK(cons(head(cons(incr(y0), y1)), z1)) -> c18(MARK(head(cons(incr(y0), y1)))) MARK(cons(head(cons(pairs, y0)), z1)) -> c18(MARK(head(cons(pairs, y0)))) MARK(cons(head(cons(odds, y0)), z1)) -> c18(MARK(head(cons(odds, y0)))) MARK(cons(head(cons(head(y0), y1)), z1)) -> c18(MARK(head(cons(head(y0), y1)))) MARK(cons(head(cons(tail(y0), y1)), z1)) -> c18(MARK(head(cons(tail(y0), y1)))) MARK(cons(head(cons(cons(y0, y1), y2)), z1)) -> c18(MARK(head(cons(cons(y0, y1), y2)))) MARK(cons(head(cons(s(y0), y1)), z1)) -> c18(MARK(head(cons(s(y0), y1)))) MARK(cons(head(cons(y0, y1)), z1)) -> c18(MARK(head(cons(y0, y1)))) MARK(cons(head(cons(0, y0)), z1)) -> c18(MARK(head(cons(0, y0)))) MARK(cons(incr(tail(nats)), z1)) -> c18(MARK(incr(tail(nats)))) MARK(cons(head(tail(nats)), z1)) -> c18(MARK(head(tail(nats)))) MARK(cons(tail(tail(nats)), z1)) -> c18(MARK(tail(tail(nats)))) MARK(incr(cons(nats, x1))) -> c13(A__INCR(cons(a__nats, x1))) MARK(incr(cons(pairs, x1))) -> c13(A__INCR(cons(a__pairs, x1))) MARK(incr(cons(odds, x1))) -> c13(A__INCR(cons(a__odds, x1))) MARK(head(cons(nats, x1))) -> c16(A__HEAD(cons(a__nats, x1))) MARK(head(cons(pairs, x1))) -> c16(A__HEAD(cons(a__pairs, x1))) MARK(head(cons(odds, x1))) -> c16(A__HEAD(cons(a__odds, x1))) K tuples:none Defined Rule Symbols: a__nats, a__pairs, a__odds, a__incr_1, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: MARK_1, A__INCR_1, A__HEAD_1, A__TAIL_1 Compound Symbols: c20_1, c13_1, c16_1, c17_2, c17_1, c13_2, c16_2, c6_1, c8_1, c10_1, c18_1 ---------------------------------------- (191) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace MARK(s(z0)) -> c20(MARK(z0)) by MARK(s(s(y0))) -> c20(MARK(s(y0))) MARK(s(incr(s(y0)))) -> c20(MARK(incr(s(y0)))) MARK(s(head(s(y0)))) -> c20(MARK(head(s(y0)))) MARK(s(tail(cons(y0, y1)))) -> c20(MARK(tail(cons(y0, y1)))) MARK(s(tail(s(y0)))) -> c20(MARK(tail(s(y0)))) MARK(s(incr(incr(incr(y0))))) -> c20(MARK(incr(incr(incr(y0))))) MARK(s(incr(incr(odds)))) -> c20(MARK(incr(incr(odds)))) MARK(s(incr(incr(head(y0))))) -> c20(MARK(incr(incr(head(y0))))) MARK(s(incr(incr(tail(y0))))) -> c20(MARK(incr(incr(tail(y0))))) MARK(s(incr(incr(cons(y0, y1))))) -> c20(MARK(incr(incr(cons(y0, y1))))) MARK(s(incr(incr(s(y0))))) -> c20(MARK(incr(incr(s(y0))))) MARK(s(incr(incr(y0)))) -> c20(MARK(incr(incr(y0)))) MARK(s(incr(head(incr(y0))))) -> c20(MARK(incr(head(incr(y0))))) MARK(s(incr(head(odds)))) -> c20(MARK(incr(head(odds)))) MARK(s(incr(head(head(y0))))) -> c20(MARK(incr(head(head(y0))))) MARK(s(incr(head(tail(y0))))) -> c20(MARK(incr(head(tail(y0))))) MARK(s(incr(head(cons(y0, y1))))) -> c20(MARK(incr(head(cons(y0, y1))))) MARK(s(incr(head(s(y0))))) -> c20(MARK(incr(head(s(y0))))) MARK(s(incr(head(y0)))) -> c20(MARK(incr(head(y0)))) MARK(s(incr(tail(incr(y0))))) -> c20(MARK(incr(tail(incr(y0))))) MARK(s(incr(tail(pairs)))) -> c20(MARK(incr(tail(pairs)))) MARK(s(incr(tail(odds)))) -> c20(MARK(incr(tail(odds)))) MARK(s(incr(tail(head(y0))))) -> c20(MARK(incr(tail(head(y0))))) MARK(s(incr(tail(tail(y0))))) -> c20(MARK(incr(tail(tail(y0))))) MARK(s(incr(tail(cons(y0, y1))))) -> c20(MARK(incr(tail(cons(y0, y1))))) MARK(s(incr(tail(s(y0))))) -> c20(MARK(incr(tail(s(y0))))) MARK(s(incr(tail(y0)))) -> c20(MARK(incr(tail(y0)))) MARK(s(head(incr(incr(y0))))) -> c20(MARK(head(incr(incr(y0))))) MARK(s(head(incr(odds)))) -> c20(MARK(head(incr(odds)))) MARK(s(head(incr(head(y0))))) -> c20(MARK(head(incr(head(y0))))) MARK(s(head(incr(tail(y0))))) -> c20(MARK(head(incr(tail(y0))))) MARK(s(head(incr(cons(y0, y1))))) -> c20(MARK(head(incr(cons(y0, y1))))) MARK(s(head(incr(s(y0))))) -> c20(MARK(head(incr(s(y0))))) MARK(s(head(incr(y0)))) -> c20(MARK(head(incr(y0)))) MARK(s(head(head(incr(y0))))) -> c20(MARK(head(head(incr(y0))))) MARK(s(head(head(odds)))) -> c20(MARK(head(head(odds)))) MARK(s(head(head(head(y0))))) -> c20(MARK(head(head(head(y0))))) MARK(s(head(head(tail(y0))))) -> c20(MARK(head(head(tail(y0))))) MARK(s(head(head(cons(y0, y1))))) -> c20(MARK(head(head(cons(y0, y1))))) MARK(s(head(head(s(y0))))) -> c20(MARK(head(head(s(y0))))) MARK(s(head(head(y0)))) -> c20(MARK(head(head(y0)))) MARK(s(head(tail(incr(y0))))) -> c20(MARK(head(tail(incr(y0))))) MARK(s(head(tail(pairs)))) -> c20(MARK(head(tail(pairs)))) MARK(s(head(tail(odds)))) -> c20(MARK(head(tail(odds)))) MARK(s(head(tail(head(y0))))) -> c20(MARK(head(tail(head(y0))))) MARK(s(head(tail(tail(y0))))) -> c20(MARK(head(tail(tail(y0))))) MARK(s(head(tail(cons(y0, y1))))) -> c20(MARK(head(tail(cons(y0, y1))))) MARK(s(head(tail(s(y0))))) -> c20(MARK(head(tail(s(y0))))) MARK(s(head(tail(y0)))) -> c20(MARK(head(tail(y0)))) MARK(s(tail(incr(incr(y0))))) -> c20(MARK(tail(incr(incr(y0))))) MARK(s(tail(incr(odds)))) -> c20(MARK(tail(incr(odds)))) MARK(s(tail(incr(head(y0))))) -> c20(MARK(tail(incr(head(y0))))) MARK(s(tail(incr(tail(y0))))) -> c20(MARK(tail(incr(tail(y0))))) MARK(s(tail(incr(cons(y0, y1))))) -> c20(MARK(tail(incr(cons(y0, y1))))) MARK(s(tail(incr(s(y0))))) -> c20(MARK(tail(incr(s(y0))))) MARK(s(tail(incr(y0)))) -> c20(MARK(tail(incr(y0)))) MARK(s(tail(head(incr(y0))))) -> c20(MARK(tail(head(incr(y0))))) MARK(s(tail(head(odds)))) -> c20(MARK(tail(head(odds)))) MARK(s(tail(head(head(y0))))) -> c20(MARK(tail(head(head(y0))))) MARK(s(tail(head(tail(y0))))) -> c20(MARK(tail(head(tail(y0))))) MARK(s(tail(head(cons(y0, y1))))) -> c20(MARK(tail(head(cons(y0, y1))))) MARK(s(tail(head(s(y0))))) -> c20(MARK(tail(head(s(y0))))) MARK(s(tail(head(y0)))) -> c20(MARK(tail(head(y0)))) MARK(s(tail(tail(incr(y0))))) -> c20(MARK(tail(tail(incr(y0))))) MARK(s(tail(tail(pairs)))) -> c20(MARK(tail(tail(pairs)))) MARK(s(tail(tail(odds)))) -> c20(MARK(tail(tail(odds)))) MARK(s(tail(tail(head(y0))))) -> c20(MARK(tail(tail(head(y0))))) MARK(s(tail(tail(tail(y0))))) -> c20(MARK(tail(tail(tail(y0))))) MARK(s(tail(tail(cons(y0, y1))))) -> c20(MARK(tail(tail(cons(y0, y1))))) MARK(s(tail(tail(s(y0))))) -> c20(MARK(tail(tail(s(y0))))) MARK(s(tail(tail(y0)))) -> c20(MARK(tail(tail(y0)))) MARK(s(tail(pairs))) -> c20(MARK(tail(pairs))) MARK(s(incr(incr(nats)))) -> c20(MARK(incr(incr(nats)))) MARK(s(incr(incr(pairs)))) -> c20(MARK(incr(incr(pairs)))) MARK(s(incr(odds))) -> c20(MARK(incr(odds))) MARK(s(head(incr(nats)))) -> c20(MARK(head(incr(nats)))) MARK(s(head(incr(pairs)))) -> c20(MARK(head(incr(pairs)))) MARK(s(head(odds))) -> c20(MARK(head(odds))) MARK(s(tail(incr(nats)))) -> c20(MARK(tail(incr(nats)))) MARK(s(tail(incr(pairs)))) -> c20(MARK(tail(incr(pairs)))) MARK(s(tail(odds))) -> c20(MARK(tail(odds))) MARK(s(incr(cons(incr(y0), y1)))) -> c20(MARK(incr(cons(incr(y0), y1)))) MARK(s(incr(cons(head(y0), y1)))) -> c20(MARK(incr(cons(head(y0), y1)))) MARK(s(incr(cons(tail(y0), y1)))) -> c20(MARK(incr(cons(tail(y0), y1)))) MARK(s(incr(cons(cons(y0, y1), y2)))) -> c20(MARK(incr(cons(cons(y0, y1), y2)))) MARK(s(incr(cons(s(y0), y1)))) -> c20(MARK(incr(cons(s(y0), y1)))) MARK(s(incr(cons(y0, y1)))) -> c20(MARK(incr(cons(y0, y1)))) MARK(s(incr(head(nats)))) -> c20(MARK(incr(head(nats)))) MARK(s(incr(head(pairs)))) -> c20(MARK(incr(head(pairs)))) MARK(s(head(head(nats)))) -> c20(MARK(head(head(nats)))) MARK(s(head(head(pairs)))) -> c20(MARK(head(head(pairs)))) MARK(s(tail(head(nats)))) -> c20(MARK(tail(head(nats)))) MARK(s(tail(head(pairs)))) -> c20(MARK(tail(head(pairs)))) MARK(s(head(cons(incr(y0), y1)))) -> c20(MARK(head(cons(incr(y0), y1)))) MARK(s(head(cons(head(y0), y1)))) -> c20(MARK(head(cons(head(y0), y1)))) MARK(s(head(cons(tail(y0), y1)))) -> c20(MARK(head(cons(tail(y0), y1)))) MARK(s(head(cons(cons(y0, y1), y2)))) -> c20(MARK(head(cons(cons(y0, y1), y2)))) MARK(s(head(cons(s(y0), y1)))) -> c20(MARK(head(cons(s(y0), y1)))) MARK(s(head(cons(y0, y1)))) -> c20(MARK(head(cons(y0, y1)))) MARK(s(incr(tail(nats)))) -> c20(MARK(incr(tail(nats)))) MARK(s(head(tail(nats)))) -> c20(MARK(head(tail(nats)))) MARK(s(tail(tail(nats)))) -> c20(MARK(tail(tail(nats)))) MARK(s(cons(cons(y0, y1), y2))) -> c20(MARK(cons(cons(y0, y1), y2))) MARK(s(cons(s(y0), y1))) -> c20(MARK(cons(s(y0), y1))) MARK(s(cons(incr(s(y0)), y1))) -> c20(MARK(cons(incr(s(y0)), y1))) MARK(s(cons(head(s(y0)), y1))) -> c20(MARK(cons(head(s(y0)), y1))) MARK(s(cons(tail(cons(y0, y1)), y2))) -> c20(MARK(cons(tail(cons(y0, y1)), y2))) MARK(s(cons(tail(s(y0)), y1))) -> c20(MARK(cons(tail(s(y0)), y1))) MARK(s(cons(incr(incr(incr(y0))), y1))) -> c20(MARK(cons(incr(incr(incr(y0))), y1))) MARK(s(cons(incr(incr(odds)), y0))) -> c20(MARK(cons(incr(incr(odds)), y0))) MARK(s(cons(incr(incr(head(y0))), y1))) -> c20(MARK(cons(incr(incr(head(y0))), y1))) MARK(s(cons(incr(incr(tail(y0))), y1))) -> c20(MARK(cons(incr(incr(tail(y0))), y1))) MARK(s(cons(incr(incr(cons(y0, y1))), y2))) -> c20(MARK(cons(incr(incr(cons(y0, y1))), y2))) MARK(s(cons(incr(incr(s(y0))), y1))) -> c20(MARK(cons(incr(incr(s(y0))), y1))) MARK(s(cons(incr(incr(y0)), y1))) -> c20(MARK(cons(incr(incr(y0)), y1))) MARK(s(cons(incr(head(incr(y0))), y1))) -> c20(MARK(cons(incr(head(incr(y0))), y1))) MARK(s(cons(incr(head(odds)), y0))) -> c20(MARK(cons(incr(head(odds)), y0))) MARK(s(cons(incr(head(head(y0))), y1))) -> c20(MARK(cons(incr(head(head(y0))), y1))) MARK(s(cons(incr(head(tail(y0))), y1))) -> c20(MARK(cons(incr(head(tail(y0))), y1))) MARK(s(cons(incr(head(cons(y0, y1))), y2))) -> c20(MARK(cons(incr(head(cons(y0, y1))), y2))) MARK(s(cons(incr(head(s(y0))), y1))) -> c20(MARK(cons(incr(head(s(y0))), y1))) MARK(s(cons(incr(head(y0)), y1))) -> c20(MARK(cons(incr(head(y0)), y1))) MARK(s(cons(incr(tail(incr(y0))), y1))) -> c20(MARK(cons(incr(tail(incr(y0))), y1))) MARK(s(cons(incr(tail(pairs)), y0))) -> c20(MARK(cons(incr(tail(pairs)), y0))) MARK(s(cons(incr(tail(odds)), y0))) -> c20(MARK(cons(incr(tail(odds)), y0))) MARK(s(cons(incr(tail(head(y0))), y1))) -> c20(MARK(cons(incr(tail(head(y0))), y1))) MARK(s(cons(incr(tail(tail(y0))), y1))) -> c20(MARK(cons(incr(tail(tail(y0))), y1))) MARK(s(cons(incr(tail(cons(y0, y1))), y2))) -> c20(MARK(cons(incr(tail(cons(y0, y1))), y2))) MARK(s(cons(incr(tail(s(y0))), y1))) -> c20(MARK(cons(incr(tail(s(y0))), y1))) MARK(s(cons(incr(tail(y0)), y1))) -> c20(MARK(cons(incr(tail(y0)), y1))) MARK(s(cons(head(incr(incr(y0))), y1))) -> c20(MARK(cons(head(incr(incr(y0))), y1))) MARK(s(cons(head(incr(odds)), y0))) -> c20(MARK(cons(head(incr(odds)), y0))) MARK(s(cons(head(incr(head(y0))), y1))) -> c20(MARK(cons(head(incr(head(y0))), y1))) MARK(s(cons(head(incr(tail(y0))), y1))) -> c20(MARK(cons(head(incr(tail(y0))), y1))) MARK(s(cons(head(incr(cons(y0, y1))), y2))) -> c20(MARK(cons(head(incr(cons(y0, y1))), y2))) MARK(s(cons(head(incr(s(y0))), y1))) -> c20(MARK(cons(head(incr(s(y0))), y1))) MARK(s(cons(head(incr(y0)), y1))) -> c20(MARK(cons(head(incr(y0)), y1))) MARK(s(cons(head(head(incr(y0))), y1))) -> c20(MARK(cons(head(head(incr(y0))), y1))) MARK(s(cons(head(head(odds)), y0))) -> c20(MARK(cons(head(head(odds)), y0))) MARK(s(cons(head(head(head(y0))), y1))) -> c20(MARK(cons(head(head(head(y0))), y1))) MARK(s(cons(head(head(tail(y0))), y1))) -> c20(MARK(cons(head(head(tail(y0))), y1))) MARK(s(cons(head(head(cons(y0, y1))), y2))) -> c20(MARK(cons(head(head(cons(y0, y1))), y2))) MARK(s(cons(head(head(s(y0))), y1))) -> c20(MARK(cons(head(head(s(y0))), y1))) MARK(s(cons(head(head(y0)), y1))) -> c20(MARK(cons(head(head(y0)), y1))) MARK(s(cons(head(tail(incr(y0))), y1))) -> c20(MARK(cons(head(tail(incr(y0))), y1))) MARK(s(cons(head(tail(pairs)), y0))) -> c20(MARK(cons(head(tail(pairs)), y0))) MARK(s(cons(head(tail(odds)), y0))) -> c20(MARK(cons(head(tail(odds)), y0))) MARK(s(cons(head(tail(head(y0))), y1))) -> c20(MARK(cons(head(tail(head(y0))), y1))) MARK(s(cons(head(tail(tail(y0))), y1))) -> c20(MARK(cons(head(tail(tail(y0))), y1))) MARK(s(cons(head(tail(cons(y0, y1))), y2))) -> c20(MARK(cons(head(tail(cons(y0, y1))), y2))) MARK(s(cons(head(tail(s(y0))), y1))) -> c20(MARK(cons(head(tail(s(y0))), y1))) MARK(s(cons(head(tail(y0)), y1))) -> c20(MARK(cons(head(tail(y0)), y1))) MARK(s(cons(tail(incr(incr(y0))), y1))) -> c20(MARK(cons(tail(incr(incr(y0))), y1))) MARK(s(cons(tail(incr(odds)), y0))) -> c20(MARK(cons(tail(incr(odds)), y0))) MARK(s(cons(tail(incr(head(y0))), y1))) -> c20(MARK(cons(tail(incr(head(y0))), y1))) MARK(s(cons(tail(incr(tail(y0))), y1))) -> c20(MARK(cons(tail(incr(tail(y0))), y1))) MARK(s(cons(tail(incr(cons(y0, y1))), y2))) -> c20(MARK(cons(tail(incr(cons(y0, y1))), y2))) MARK(s(cons(tail(incr(s(y0))), y1))) -> c20(MARK(cons(tail(incr(s(y0))), y1))) MARK(s(cons(tail(incr(y0)), y1))) -> c20(MARK(cons(tail(incr(y0)), y1))) MARK(s(cons(tail(head(incr(y0))), y1))) -> c20(MARK(cons(tail(head(incr(y0))), y1))) MARK(s(cons(tail(head(odds)), y0))) -> c20(MARK(cons(tail(head(odds)), y0))) MARK(s(cons(tail(head(head(y0))), y1))) -> c20(MARK(cons(tail(head(head(y0))), y1))) MARK(s(cons(tail(head(tail(y0))), y1))) -> c20(MARK(cons(tail(head(tail(y0))), y1))) MARK(s(cons(tail(head(cons(y0, y1))), y2))) -> c20(MARK(cons(tail(head(cons(y0, y1))), y2))) MARK(s(cons(tail(head(s(y0))), y1))) -> c20(MARK(cons(tail(head(s(y0))), y1))) MARK(s(cons(tail(head(y0)), y1))) -> c20(MARK(cons(tail(head(y0)), y1))) MARK(s(cons(tail(tail(incr(y0))), y1))) -> c20(MARK(cons(tail(tail(incr(y0))), y1))) MARK(s(cons(tail(tail(pairs)), y0))) -> c20(MARK(cons(tail(tail(pairs)), y0))) MARK(s(cons(tail(tail(odds)), y0))) -> c20(MARK(cons(tail(tail(odds)), y0))) MARK(s(cons(tail(tail(head(y0))), y1))) -> c20(MARK(cons(tail(tail(head(y0))), y1))) MARK(s(cons(tail(tail(tail(y0))), y1))) -> c20(MARK(cons(tail(tail(tail(y0))), y1))) MARK(s(cons(tail(tail(cons(y0, y1))), y2))) -> c20(MARK(cons(tail(tail(cons(y0, y1))), y2))) MARK(s(cons(tail(tail(s(y0))), y1))) -> c20(MARK(cons(tail(tail(s(y0))), y1))) MARK(s(cons(tail(tail(y0)), y1))) -> c20(MARK(cons(tail(tail(y0)), y1))) MARK(s(cons(tail(pairs), y0))) -> c20(MARK(cons(tail(pairs), y0))) MARK(s(cons(incr(incr(nats)), y0))) -> c20(MARK(cons(incr(incr(nats)), y0))) MARK(s(cons(incr(incr(pairs)), y0))) -> c20(MARK(cons(incr(incr(pairs)), y0))) MARK(s(cons(incr(odds), y0))) -> c20(MARK(cons(incr(odds), y0))) MARK(s(cons(head(incr(nats)), y0))) -> c20(MARK(cons(head(incr(nats)), y0))) MARK(s(cons(head(incr(pairs)), y0))) -> c20(MARK(cons(head(incr(pairs)), y0))) MARK(s(cons(head(odds), y0))) -> c20(MARK(cons(head(odds), y0))) MARK(s(cons(tail(incr(nats)), y0))) -> c20(MARK(cons(tail(incr(nats)), y0))) MARK(s(cons(tail(incr(pairs)), y0))) -> c20(MARK(cons(tail(incr(pairs)), y0))) MARK(s(cons(tail(odds), y0))) -> c20(MARK(cons(tail(odds), y0))) MARK(s(cons(incr(cons(nats, y0)), y1))) -> c20(MARK(cons(incr(cons(nats, y0)), y1))) MARK(s(cons(incr(cons(incr(y0), y1)), y2))) -> c20(MARK(cons(incr(cons(incr(y0), y1)), y2))) MARK(s(cons(incr(cons(pairs, y0)), y1))) -> c20(MARK(cons(incr(cons(pairs, y0)), y1))) MARK(s(cons(incr(cons(odds, y0)), y1))) -> c20(MARK(cons(incr(cons(odds, y0)), y1))) MARK(s(cons(incr(cons(head(y0), y1)), y2))) -> c20(MARK(cons(incr(cons(head(y0), y1)), y2))) MARK(s(cons(incr(cons(tail(y0), y1)), y2))) -> c20(MARK(cons(incr(cons(tail(y0), y1)), y2))) MARK(s(cons(incr(cons(cons(y0, y1), y2)), y3))) -> c20(MARK(cons(incr(cons(cons(y0, y1), y2)), y3))) MARK(s(cons(incr(cons(s(y0), y1)), y2))) -> c20(MARK(cons(incr(cons(s(y0), y1)), y2))) MARK(s(cons(incr(cons(y0, y1)), y2))) -> c20(MARK(cons(incr(cons(y0, y1)), y2))) MARK(s(cons(incr(cons(0, y0)), y1))) -> c20(MARK(cons(incr(cons(0, y0)), y1))) MARK(s(cons(incr(head(nats)), y0))) -> c20(MARK(cons(incr(head(nats)), y0))) MARK(s(cons(incr(head(pairs)), y0))) -> c20(MARK(cons(incr(head(pairs)), y0))) MARK(s(cons(head(head(nats)), y0))) -> c20(MARK(cons(head(head(nats)), y0))) MARK(s(cons(head(head(pairs)), y0))) -> c20(MARK(cons(head(head(pairs)), y0))) MARK(s(cons(tail(head(nats)), y0))) -> c20(MARK(cons(tail(head(nats)), y0))) MARK(s(cons(tail(head(pairs)), y0))) -> c20(MARK(cons(tail(head(pairs)), y0))) MARK(s(cons(head(cons(nats, y0)), y1))) -> c20(MARK(cons(head(cons(nats, y0)), y1))) MARK(s(cons(head(cons(incr(y0), y1)), y2))) -> c20(MARK(cons(head(cons(incr(y0), y1)), y2))) MARK(s(cons(head(cons(pairs, y0)), y1))) -> c20(MARK(cons(head(cons(pairs, y0)), y1))) MARK(s(cons(head(cons(odds, y0)), y1))) -> c20(MARK(cons(head(cons(odds, y0)), y1))) MARK(s(cons(head(cons(head(y0), y1)), y2))) -> c20(MARK(cons(head(cons(head(y0), y1)), y2))) MARK(s(cons(head(cons(tail(y0), y1)), y2))) -> c20(MARK(cons(head(cons(tail(y0), y1)), y2))) MARK(s(cons(head(cons(cons(y0, y1), y2)), y3))) -> c20(MARK(cons(head(cons(cons(y0, y1), y2)), y3))) MARK(s(cons(head(cons(s(y0), y1)), y2))) -> c20(MARK(cons(head(cons(s(y0), y1)), y2))) MARK(s(cons(head(cons(y0, y1)), y2))) -> c20(MARK(cons(head(cons(y0, y1)), y2))) MARK(s(cons(head(cons(0, y0)), y1))) -> c20(MARK(cons(head(cons(0, y0)), y1))) MARK(s(cons(incr(tail(nats)), y0))) -> c20(MARK(cons(incr(tail(nats)), y0))) MARK(s(cons(head(tail(nats)), y0))) -> c20(MARK(cons(head(tail(nats)), y0))) MARK(s(cons(tail(tail(nats)), y0))) -> c20(MARK(cons(tail(tail(nats)), y0))) MARK(s(incr(cons(nats, y0)))) -> c20(MARK(incr(cons(nats, y0)))) MARK(s(incr(cons(pairs, y0)))) -> c20(MARK(incr(cons(pairs, y0)))) MARK(s(incr(cons(odds, y0)))) -> c20(MARK(incr(cons(odds, y0)))) MARK(s(head(cons(nats, y0)))) -> c20(MARK(head(cons(nats, y0)))) MARK(s(head(cons(pairs, y0)))) -> c20(MARK(head(cons(pairs, y0)))) MARK(s(head(cons(odds, y0)))) -> c20(MARK(head(cons(odds, y0)))) ---------------------------------------- (192) Obligation: Complexity Dependency Tuples Problem Rules: a__nats -> cons(0, incr(nats)) a__nats -> nats a__pairs -> cons(0, incr(odds)) a__pairs -> pairs a__odds -> a__incr(a__pairs) a__odds -> odds a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) 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(nats) -> a__nats mark(incr(z0)) -> a__incr(mark(z0)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c17(MARK(s(z0))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(head(incr(z0)))) -> 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MARK(cons(incr(head(head(y0))), z1)) -> c18(MARK(incr(head(head(y0))))) MARK(cons(incr(head(tail(y0))), z1)) -> c18(MARK(incr(head(tail(y0))))) MARK(cons(incr(head(cons(y0, y1))), z1)) -> c18(MARK(incr(head(cons(y0, y1))))) MARK(cons(incr(head(s(y0))), z1)) -> c18(MARK(incr(head(s(y0))))) MARK(cons(incr(head(y0)), z1)) -> c18(MARK(incr(head(y0)))) MARK(cons(incr(tail(incr(y0))), z1)) -> c18(MARK(incr(tail(incr(y0))))) MARK(cons(incr(tail(pairs)), z1)) -> c18(MARK(incr(tail(pairs)))) MARK(cons(incr(tail(odds)), z1)) -> c18(MARK(incr(tail(odds)))) MARK(cons(incr(tail(head(y0))), z1)) -> c18(MARK(incr(tail(head(y0))))) MARK(cons(incr(tail(tail(y0))), z1)) -> c18(MARK(incr(tail(tail(y0))))) MARK(cons(incr(tail(cons(y0, y1))), z1)) -> c18(MARK(incr(tail(cons(y0, y1))))) MARK(cons(incr(tail(s(y0))), z1)) -> c18(MARK(incr(tail(s(y0))))) MARK(cons(incr(tail(y0)), z1)) -> c18(MARK(incr(tail(y0)))) MARK(cons(head(incr(incr(y0))), z1)) -> c18(MARK(head(incr(incr(y0))))) MARK(cons(head(incr(odds)), z1)) -> c18(MARK(head(incr(odds)))) MARK(cons(head(incr(head(y0))), z1)) -> c18(MARK(head(incr(head(y0))))) MARK(cons(head(incr(tail(y0))), z1)) -> c18(MARK(head(incr(tail(y0))))) MARK(cons(head(incr(cons(y0, y1))), z1)) -> c18(MARK(head(incr(cons(y0, y1))))) MARK(cons(head(incr(s(y0))), z1)) -> c18(MARK(head(incr(s(y0))))) MARK(cons(head(incr(y0)), z1)) -> c18(MARK(head(incr(y0)))) MARK(cons(head(head(incr(y0))), z1)) -> c18(MARK(head(head(incr(y0))))) MARK(cons(head(head(odds)), z1)) -> c18(MARK(head(head(odds)))) MARK(cons(head(head(head(y0))), z1)) -> c18(MARK(head(head(head(y0))))) MARK(cons(head(head(tail(y0))), z1)) -> c18(MARK(head(head(tail(y0))))) MARK(cons(head(head(cons(y0, y1))), z1)) -> c18(MARK(head(head(cons(y0, y1))))) MARK(cons(head(head(s(y0))), z1)) -> c18(MARK(head(head(s(y0))))) MARK(cons(head(head(y0)), z1)) -> c18(MARK(head(head(y0)))) MARK(cons(head(tail(incr(y0))), z1)) -> c18(MARK(head(tail(incr(y0))))) MARK(cons(head(tail(pairs)), 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c18(MARK(tail(head(incr(y0))))) MARK(cons(tail(head(odds)), z1)) -> c18(MARK(tail(head(odds)))) MARK(cons(tail(head(head(y0))), z1)) -> c18(MARK(tail(head(head(y0))))) MARK(cons(tail(head(tail(y0))), z1)) -> c18(MARK(tail(head(tail(y0))))) MARK(cons(tail(head(cons(y0, y1))), z1)) -> c18(MARK(tail(head(cons(y0, y1))))) MARK(cons(tail(head(s(y0))), z1)) -> c18(MARK(tail(head(s(y0))))) MARK(cons(tail(head(y0)), z1)) -> c18(MARK(tail(head(y0)))) MARK(cons(tail(tail(incr(y0))), z1)) -> c18(MARK(tail(tail(incr(y0))))) MARK(cons(tail(tail(pairs)), z1)) -> c18(MARK(tail(tail(pairs)))) MARK(cons(tail(tail(odds)), z1)) -> c18(MARK(tail(tail(odds)))) MARK(cons(tail(tail(head(y0))), z1)) -> c18(MARK(tail(tail(head(y0))))) MARK(cons(tail(tail(tail(y0))), z1)) -> c18(MARK(tail(tail(tail(y0))))) MARK(cons(tail(tail(cons(y0, y1))), z1)) -> c18(MARK(tail(tail(cons(y0, y1))))) MARK(cons(tail(tail(s(y0))), z1)) -> c18(MARK(tail(tail(s(y0))))) MARK(cons(tail(tail(y0)), z1)) -> c18(MARK(tail(tail(y0)))) MARK(cons(tail(pairs), z1)) -> c18(MARK(tail(pairs))) MARK(cons(incr(incr(nats)), z1)) -> c18(MARK(incr(incr(nats)))) MARK(cons(incr(incr(pairs)), z1)) -> c18(MARK(incr(incr(pairs)))) MARK(cons(incr(odds), z1)) -> c18(MARK(incr(odds))) MARK(cons(head(incr(nats)), z1)) -> c18(MARK(head(incr(nats)))) MARK(cons(head(incr(pairs)), z1)) -> c18(MARK(head(incr(pairs)))) MARK(cons(head(odds), z1)) -> c18(MARK(head(odds))) MARK(cons(tail(incr(nats)), z1)) -> c18(MARK(tail(incr(nats)))) MARK(cons(tail(incr(pairs)), z1)) -> c18(MARK(tail(incr(pairs)))) MARK(cons(tail(odds), z1)) -> c18(MARK(tail(odds))) MARK(cons(incr(cons(nats, y0)), z1)) -> c18(MARK(incr(cons(nats, y0)))) MARK(cons(incr(cons(incr(y0), y1)), z1)) -> c18(MARK(incr(cons(incr(y0), y1)))) MARK(cons(incr(cons(pairs, y0)), z1)) -> c18(MARK(incr(cons(pairs, y0)))) MARK(cons(incr(cons(odds, y0)), z1)) -> c18(MARK(incr(cons(odds, y0)))) MARK(cons(incr(cons(head(y0), y1)), z1)) -> c18(MARK(incr(cons(head(y0), y1)))) MARK(cons(incr(cons(tail(y0), y1)), z1)) -> c18(MARK(incr(cons(tail(y0), y1)))) MARK(cons(incr(cons(cons(y0, y1), y2)), z1)) -> c18(MARK(incr(cons(cons(y0, y1), y2)))) MARK(cons(incr(cons(s(y0), y1)), z1)) -> c18(MARK(incr(cons(s(y0), y1)))) MARK(cons(incr(cons(y0, y1)), z1)) -> c18(MARK(incr(cons(y0, y1)))) MARK(cons(incr(cons(0, y0)), z1)) -> c18(MARK(incr(cons(0, y0)))) MARK(cons(incr(head(nats)), z1)) -> c18(MARK(incr(head(nats)))) MARK(cons(incr(head(pairs)), z1)) -> c18(MARK(incr(head(pairs)))) MARK(cons(head(head(nats)), z1)) -> c18(MARK(head(head(nats)))) MARK(cons(head(head(pairs)), z1)) -> c18(MARK(head(head(pairs)))) MARK(cons(tail(head(nats)), z1)) -> c18(MARK(tail(head(nats)))) MARK(cons(tail(head(pairs)), z1)) -> c18(MARK(tail(head(pairs)))) MARK(cons(head(cons(nats, y0)), z1)) -> c18(MARK(head(cons(nats, y0)))) MARK(cons(head(cons(incr(y0), y1)), z1)) -> c18(MARK(head(cons(incr(y0), y1)))) MARK(cons(head(cons(pairs, y0)), z1)) -> c18(MARK(head(cons(pairs, y0)))) MARK(cons(head(cons(odds, y0)), z1)) -> c18(MARK(head(cons(odds, y0)))) MARK(cons(head(cons(head(y0), y1)), z1)) -> c18(MARK(head(cons(head(y0), y1)))) MARK(cons(head(cons(tail(y0), y1)), z1)) -> c18(MARK(head(cons(tail(y0), y1)))) MARK(cons(head(cons(cons(y0, y1), y2)), z1)) -> c18(MARK(head(cons(cons(y0, y1), y2)))) MARK(cons(head(cons(s(y0), y1)), z1)) -> c18(MARK(head(cons(s(y0), y1)))) MARK(cons(head(cons(y0, y1)), z1)) -> c18(MARK(head(cons(y0, y1)))) MARK(cons(head(cons(0, y0)), z1)) -> c18(MARK(head(cons(0, y0)))) MARK(cons(incr(tail(nats)), z1)) -> c18(MARK(incr(tail(nats)))) MARK(cons(head(tail(nats)), z1)) -> c18(MARK(head(tail(nats)))) MARK(cons(tail(tail(nats)), z1)) -> c18(MARK(tail(tail(nats)))) MARK(incr(cons(nats, x1))) -> c13(A__INCR(cons(a__nats, x1))) MARK(incr(cons(pairs, x1))) -> c13(A__INCR(cons(a__pairs, x1))) MARK(incr(cons(odds, x1))) -> c13(A__INCR(cons(a__odds, x1))) MARK(head(cons(nats, x1))) -> c16(A__HEAD(cons(a__nats, x1))) MARK(head(cons(pairs, x1))) -> c16(A__HEAD(cons(a__pairs, x1))) MARK(head(cons(odds, x1))) -> c16(A__HEAD(cons(a__odds, x1))) MARK(s(s(y0))) -> c20(MARK(s(y0))) MARK(s(incr(s(y0)))) -> c20(MARK(incr(s(y0)))) MARK(s(head(s(y0)))) -> c20(MARK(head(s(y0)))) MARK(s(tail(cons(y0, y1)))) -> c20(MARK(tail(cons(y0, y1)))) MARK(s(tail(s(y0)))) -> c20(MARK(tail(s(y0)))) MARK(s(incr(incr(incr(y0))))) -> c20(MARK(incr(incr(incr(y0))))) MARK(s(incr(incr(odds)))) -> c20(MARK(incr(incr(odds)))) MARK(s(incr(incr(head(y0))))) -> c20(MARK(incr(incr(head(y0))))) MARK(s(incr(incr(tail(y0))))) -> c20(MARK(incr(incr(tail(y0))))) MARK(s(incr(incr(cons(y0, y1))))) -> c20(MARK(incr(incr(cons(y0, y1))))) MARK(s(incr(incr(s(y0))))) -> c20(MARK(incr(incr(s(y0))))) MARK(s(incr(incr(y0)))) -> c20(MARK(incr(incr(y0)))) MARK(s(incr(head(incr(y0))))) -> c20(MARK(incr(head(incr(y0))))) MARK(s(incr(head(odds)))) -> c20(MARK(incr(head(odds)))) MARK(s(incr(head(head(y0))))) -> c20(MARK(incr(head(head(y0))))) MARK(s(incr(head(tail(y0))))) -> c20(MARK(incr(head(tail(y0))))) MARK(s(incr(head(cons(y0, y1))))) -> c20(MARK(incr(head(cons(y0, y1))))) MARK(s(incr(head(s(y0))))) -> c20(MARK(incr(head(s(y0))))) MARK(s(incr(head(y0)))) -> c20(MARK(incr(head(y0)))) MARK(s(incr(tail(incr(y0))))) -> c20(MARK(incr(tail(incr(y0))))) MARK(s(incr(tail(pairs)))) -> c20(MARK(incr(tail(pairs)))) MARK(s(incr(tail(odds)))) -> c20(MARK(incr(tail(odds)))) MARK(s(incr(tail(head(y0))))) -> c20(MARK(incr(tail(head(y0))))) MARK(s(incr(tail(tail(y0))))) -> c20(MARK(incr(tail(tail(y0))))) MARK(s(incr(tail(cons(y0, y1))))) -> c20(MARK(incr(tail(cons(y0, y1))))) MARK(s(incr(tail(s(y0))))) -> c20(MARK(incr(tail(s(y0))))) MARK(s(incr(tail(y0)))) -> c20(MARK(incr(tail(y0)))) MARK(s(head(incr(incr(y0))))) -> c20(MARK(head(incr(incr(y0))))) MARK(s(head(incr(odds)))) -> c20(MARK(head(incr(odds)))) MARK(s(head(incr(head(y0))))) -> c20(MARK(head(incr(head(y0))))) MARK(s(head(incr(tail(y0))))) -> c20(MARK(head(incr(tail(y0))))) MARK(s(head(incr(cons(y0, y1))))) -> c20(MARK(head(incr(cons(y0, y1))))) MARK(s(head(incr(s(y0))))) -> c20(MARK(head(incr(s(y0))))) MARK(s(head(incr(y0)))) -> c20(MARK(head(incr(y0)))) MARK(s(head(head(incr(y0))))) -> c20(MARK(head(head(incr(y0))))) MARK(s(head(head(odds)))) -> c20(MARK(head(head(odds)))) MARK(s(head(head(head(y0))))) -> c20(MARK(head(head(head(y0))))) MARK(s(head(head(tail(y0))))) -> c20(MARK(head(head(tail(y0))))) MARK(s(head(head(cons(y0, y1))))) -> c20(MARK(head(head(cons(y0, y1))))) MARK(s(head(head(s(y0))))) -> c20(MARK(head(head(s(y0))))) MARK(s(head(head(y0)))) -> c20(MARK(head(head(y0)))) MARK(s(head(tail(incr(y0))))) -> c20(MARK(head(tail(incr(y0))))) MARK(s(head(tail(pairs)))) -> c20(MARK(head(tail(pairs)))) MARK(s(head(tail(odds)))) -> c20(MARK(head(tail(odds)))) MARK(s(head(tail(head(y0))))) -> c20(MARK(head(tail(head(y0))))) MARK(s(head(tail(tail(y0))))) -> c20(MARK(head(tail(tail(y0))))) MARK(s(head(tail(cons(y0, y1))))) -> c20(MARK(head(tail(cons(y0, y1))))) MARK(s(head(tail(s(y0))))) -> c20(MARK(head(tail(s(y0))))) MARK(s(head(tail(y0)))) -> c20(MARK(head(tail(y0)))) MARK(s(tail(incr(incr(y0))))) -> c20(MARK(tail(incr(incr(y0))))) MARK(s(tail(incr(odds)))) -> c20(MARK(tail(incr(odds)))) MARK(s(tail(incr(head(y0))))) -> c20(MARK(tail(incr(head(y0))))) MARK(s(tail(incr(tail(y0))))) -> c20(MARK(tail(incr(tail(y0))))) MARK(s(tail(incr(cons(y0, y1))))) -> c20(MARK(tail(incr(cons(y0, y1))))) MARK(s(tail(incr(s(y0))))) -> c20(MARK(tail(incr(s(y0))))) MARK(s(tail(incr(y0)))) -> c20(MARK(tail(incr(y0)))) MARK(s(tail(head(incr(y0))))) -> c20(MARK(tail(head(incr(y0))))) MARK(s(tail(head(odds)))) -> c20(MARK(tail(head(odds)))) MARK(s(tail(head(head(y0))))) -> c20(MARK(tail(head(head(y0))))) MARK(s(tail(head(tail(y0))))) -> c20(MARK(tail(head(tail(y0))))) MARK(s(tail(head(cons(y0, y1))))) -> c20(MARK(tail(head(cons(y0, y1))))) MARK(s(tail(head(s(y0))))) -> c20(MARK(tail(head(s(y0))))) MARK(s(tail(head(y0)))) -> c20(MARK(tail(head(y0)))) MARK(s(tail(tail(incr(y0))))) -> c20(MARK(tail(tail(incr(y0))))) MARK(s(tail(tail(pairs)))) -> c20(MARK(tail(tail(pairs)))) MARK(s(tail(tail(odds)))) -> c20(MARK(tail(tail(odds)))) MARK(s(tail(tail(head(y0))))) -> c20(MARK(tail(tail(head(y0))))) MARK(s(tail(tail(tail(y0))))) -> c20(MARK(tail(tail(tail(y0))))) MARK(s(tail(tail(cons(y0, y1))))) -> c20(MARK(tail(tail(cons(y0, y1))))) MARK(s(tail(tail(s(y0))))) -> c20(MARK(tail(tail(s(y0))))) MARK(s(tail(tail(y0)))) -> c20(MARK(tail(tail(y0)))) MARK(s(tail(pairs))) -> c20(MARK(tail(pairs))) MARK(s(incr(incr(nats)))) -> c20(MARK(incr(incr(nats)))) MARK(s(incr(incr(pairs)))) -> c20(MARK(incr(incr(pairs)))) MARK(s(incr(odds))) -> c20(MARK(incr(odds))) MARK(s(head(incr(nats)))) -> c20(MARK(head(incr(nats)))) MARK(s(head(incr(pairs)))) -> c20(MARK(head(incr(pairs)))) MARK(s(head(odds))) -> c20(MARK(head(odds))) MARK(s(tail(incr(nats)))) -> c20(MARK(tail(incr(nats)))) MARK(s(tail(incr(pairs)))) -> c20(MARK(tail(incr(pairs)))) MARK(s(tail(odds))) -> c20(MARK(tail(odds))) MARK(s(incr(cons(incr(y0), y1)))) -> c20(MARK(incr(cons(incr(y0), y1)))) MARK(s(incr(cons(head(y0), y1)))) -> c20(MARK(incr(cons(head(y0), y1)))) MARK(s(incr(cons(tail(y0), y1)))) -> c20(MARK(incr(cons(tail(y0), y1)))) MARK(s(incr(cons(cons(y0, y1), y2)))) -> c20(MARK(incr(cons(cons(y0, y1), y2)))) MARK(s(incr(cons(s(y0), y1)))) -> c20(MARK(incr(cons(s(y0), y1)))) MARK(s(incr(cons(y0, y1)))) -> c20(MARK(incr(cons(y0, y1)))) MARK(s(incr(head(nats)))) -> c20(MARK(incr(head(nats)))) MARK(s(incr(head(pairs)))) -> c20(MARK(incr(head(pairs)))) MARK(s(head(head(nats)))) -> c20(MARK(head(head(nats)))) MARK(s(head(head(pairs)))) -> c20(MARK(head(head(pairs)))) MARK(s(tail(head(nats)))) -> c20(MARK(tail(head(nats)))) MARK(s(tail(head(pairs)))) -> c20(MARK(tail(head(pairs)))) MARK(s(head(cons(incr(y0), y1)))) -> c20(MARK(head(cons(incr(y0), y1)))) MARK(s(head(cons(head(y0), y1)))) -> c20(MARK(head(cons(head(y0), y1)))) MARK(s(head(cons(tail(y0), y1)))) -> c20(MARK(head(cons(tail(y0), y1)))) MARK(s(head(cons(cons(y0, y1), y2)))) -> c20(MARK(head(cons(cons(y0, y1), y2)))) MARK(s(head(cons(s(y0), y1)))) -> c20(MARK(head(cons(s(y0), y1)))) MARK(s(head(cons(y0, y1)))) -> c20(MARK(head(cons(y0, y1)))) MARK(s(incr(tail(nats)))) -> c20(MARK(incr(tail(nats)))) MARK(s(head(tail(nats)))) -> c20(MARK(head(tail(nats)))) MARK(s(tail(tail(nats)))) -> c20(MARK(tail(tail(nats)))) MARK(s(cons(cons(y0, y1), y2))) -> c20(MARK(cons(cons(y0, y1), y2))) MARK(s(cons(s(y0), y1))) -> c20(MARK(cons(s(y0), y1))) MARK(s(cons(incr(s(y0)), y1))) -> c20(MARK(cons(incr(s(y0)), y1))) MARK(s(cons(head(s(y0)), y1))) -> c20(MARK(cons(head(s(y0)), y1))) MARK(s(cons(tail(cons(y0, y1)), y2))) -> c20(MARK(cons(tail(cons(y0, y1)), y2))) MARK(s(cons(tail(s(y0)), y1))) -> c20(MARK(cons(tail(s(y0)), y1))) MARK(s(cons(incr(incr(incr(y0))), y1))) -> c20(MARK(cons(incr(incr(incr(y0))), y1))) MARK(s(cons(incr(incr(odds)), y0))) -> c20(MARK(cons(incr(incr(odds)), y0))) MARK(s(cons(incr(incr(head(y0))), y1))) -> c20(MARK(cons(incr(incr(head(y0))), y1))) MARK(s(cons(incr(incr(tail(y0))), y1))) -> c20(MARK(cons(incr(incr(tail(y0))), y1))) MARK(s(cons(incr(incr(cons(y0, y1))), y2))) -> c20(MARK(cons(incr(incr(cons(y0, y1))), y2))) MARK(s(cons(incr(incr(s(y0))), y1))) -> c20(MARK(cons(incr(incr(s(y0))), y1))) MARK(s(cons(incr(incr(y0)), y1))) -> c20(MARK(cons(incr(incr(y0)), y1))) MARK(s(cons(incr(head(incr(y0))), y1))) -> c20(MARK(cons(incr(head(incr(y0))), y1))) MARK(s(cons(incr(head(odds)), y0))) -> c20(MARK(cons(incr(head(odds)), y0))) MARK(s(cons(incr(head(head(y0))), y1))) -> c20(MARK(cons(incr(head(head(y0))), y1))) MARK(s(cons(incr(head(tail(y0))), y1))) -> c20(MARK(cons(incr(head(tail(y0))), y1))) MARK(s(cons(incr(head(cons(y0, y1))), y2))) -> c20(MARK(cons(incr(head(cons(y0, y1))), y2))) MARK(s(cons(incr(head(s(y0))), y1))) -> c20(MARK(cons(incr(head(s(y0))), y1))) MARK(s(cons(incr(head(y0)), y1))) -> c20(MARK(cons(incr(head(y0)), y1))) MARK(s(cons(incr(tail(incr(y0))), y1))) -> c20(MARK(cons(incr(tail(incr(y0))), y1))) MARK(s(cons(incr(tail(pairs)), y0))) -> c20(MARK(cons(incr(tail(pairs)), y0))) MARK(s(cons(incr(tail(odds)), y0))) -> c20(MARK(cons(incr(tail(odds)), y0))) MARK(s(cons(incr(tail(head(y0))), y1))) -> c20(MARK(cons(incr(tail(head(y0))), y1))) MARK(s(cons(incr(tail(tail(y0))), y1))) -> c20(MARK(cons(incr(tail(tail(y0))), y1))) MARK(s(cons(incr(tail(cons(y0, y1))), y2))) -> c20(MARK(cons(incr(tail(cons(y0, y1))), y2))) MARK(s(cons(incr(tail(s(y0))), y1))) -> c20(MARK(cons(incr(tail(s(y0))), y1))) MARK(s(cons(incr(tail(y0)), y1))) -> c20(MARK(cons(incr(tail(y0)), y1))) MARK(s(cons(head(incr(incr(y0))), y1))) -> c20(MARK(cons(head(incr(incr(y0))), y1))) MARK(s(cons(head(incr(odds)), y0))) -> c20(MARK(cons(head(incr(odds)), y0))) MARK(s(cons(head(incr(head(y0))), y1))) -> c20(MARK(cons(head(incr(head(y0))), y1))) MARK(s(cons(head(incr(tail(y0))), y1))) -> c20(MARK(cons(head(incr(tail(y0))), y1))) MARK(s(cons(head(incr(cons(y0, y1))), y2))) -> c20(MARK(cons(head(incr(cons(y0, y1))), y2))) MARK(s(cons(head(incr(s(y0))), y1))) -> c20(MARK(cons(head(incr(s(y0))), y1))) MARK(s(cons(head(incr(y0)), y1))) -> c20(MARK(cons(head(incr(y0)), y1))) MARK(s(cons(head(head(incr(y0))), y1))) -> c20(MARK(cons(head(head(incr(y0))), y1))) MARK(s(cons(head(head(odds)), y0))) -> c20(MARK(cons(head(head(odds)), y0))) MARK(s(cons(head(head(head(y0))), y1))) -> c20(MARK(cons(head(head(head(y0))), y1))) MARK(s(cons(head(head(tail(y0))), y1))) -> c20(MARK(cons(head(head(tail(y0))), y1))) MARK(s(cons(head(head(cons(y0, y1))), y2))) -> c20(MARK(cons(head(head(cons(y0, y1))), y2))) MARK(s(cons(head(head(s(y0))), y1))) -> c20(MARK(cons(head(head(s(y0))), y1))) MARK(s(cons(head(head(y0)), y1))) -> c20(MARK(cons(head(head(y0)), y1))) MARK(s(cons(head(tail(incr(y0))), y1))) -> c20(MARK(cons(head(tail(incr(y0))), y1))) MARK(s(cons(head(tail(pairs)), y0))) -> c20(MARK(cons(head(tail(pairs)), y0))) MARK(s(cons(head(tail(odds)), y0))) -> c20(MARK(cons(head(tail(odds)), y0))) MARK(s(cons(head(tail(head(y0))), y1))) -> c20(MARK(cons(head(tail(head(y0))), y1))) MARK(s(cons(head(tail(tail(y0))), y1))) -> c20(MARK(cons(head(tail(tail(y0))), y1))) MARK(s(cons(head(tail(cons(y0, y1))), y2))) -> c20(MARK(cons(head(tail(cons(y0, y1))), y2))) MARK(s(cons(head(tail(s(y0))), y1))) -> c20(MARK(cons(head(tail(s(y0))), y1))) MARK(s(cons(head(tail(y0)), y1))) -> c20(MARK(cons(head(tail(y0)), y1))) MARK(s(cons(tail(incr(incr(y0))), y1))) -> c20(MARK(cons(tail(incr(incr(y0))), y1))) MARK(s(cons(tail(incr(odds)), y0))) -> c20(MARK(cons(tail(incr(odds)), y0))) MARK(s(cons(tail(incr(head(y0))), y1))) -> c20(MARK(cons(tail(incr(head(y0))), y1))) MARK(s(cons(tail(incr(tail(y0))), y1))) -> c20(MARK(cons(tail(incr(tail(y0))), y1))) MARK(s(cons(tail(incr(cons(y0, y1))), y2))) -> c20(MARK(cons(tail(incr(cons(y0, y1))), y2))) MARK(s(cons(tail(incr(s(y0))), y1))) -> c20(MARK(cons(tail(incr(s(y0))), y1))) MARK(s(cons(tail(incr(y0)), y1))) -> c20(MARK(cons(tail(incr(y0)), y1))) MARK(s(cons(tail(head(incr(y0))), y1))) -> c20(MARK(cons(tail(head(incr(y0))), y1))) MARK(s(cons(tail(head(odds)), y0))) -> c20(MARK(cons(tail(head(odds)), y0))) MARK(s(cons(tail(head(head(y0))), y1))) -> c20(MARK(cons(tail(head(head(y0))), y1))) MARK(s(cons(tail(head(tail(y0))), y1))) -> c20(MARK(cons(tail(head(tail(y0))), y1))) MARK(s(cons(tail(head(cons(y0, y1))), y2))) -> c20(MARK(cons(tail(head(cons(y0, y1))), y2))) MARK(s(cons(tail(head(s(y0))), y1))) -> c20(MARK(cons(tail(head(s(y0))), y1))) MARK(s(cons(tail(head(y0)), y1))) -> c20(MARK(cons(tail(head(y0)), y1))) MARK(s(cons(tail(tail(incr(y0))), y1))) -> c20(MARK(cons(tail(tail(incr(y0))), y1))) MARK(s(cons(tail(tail(pairs)), y0))) -> c20(MARK(cons(tail(tail(pairs)), y0))) MARK(s(cons(tail(tail(odds)), y0))) -> c20(MARK(cons(tail(tail(odds)), y0))) MARK(s(cons(tail(tail(head(y0))), y1))) -> c20(MARK(cons(tail(tail(head(y0))), y1))) MARK(s(cons(tail(tail(tail(y0))), y1))) -> c20(MARK(cons(tail(tail(tail(y0))), y1))) MARK(s(cons(tail(tail(cons(y0, y1))), y2))) -> c20(MARK(cons(tail(tail(cons(y0, y1))), y2))) MARK(s(cons(tail(tail(s(y0))), y1))) -> c20(MARK(cons(tail(tail(s(y0))), y1))) MARK(s(cons(tail(tail(y0)), y1))) -> c20(MARK(cons(tail(tail(y0)), y1))) MARK(s(cons(tail(pairs), y0))) -> c20(MARK(cons(tail(pairs), y0))) MARK(s(cons(incr(incr(nats)), y0))) -> c20(MARK(cons(incr(incr(nats)), y0))) MARK(s(cons(incr(incr(pairs)), y0))) -> c20(MARK(cons(incr(incr(pairs)), y0))) MARK(s(cons(incr(odds), y0))) -> c20(MARK(cons(incr(odds), y0))) MARK(s(cons(head(incr(nats)), y0))) -> c20(MARK(cons(head(incr(nats)), y0))) MARK(s(cons(head(incr(pairs)), y0))) -> c20(MARK(cons(head(incr(pairs)), y0))) MARK(s(cons(head(odds), y0))) -> c20(MARK(cons(head(odds), y0))) MARK(s(cons(tail(incr(nats)), y0))) -> c20(MARK(cons(tail(incr(nats)), y0))) MARK(s(cons(tail(incr(pairs)), y0))) -> c20(MARK(cons(tail(incr(pairs)), y0))) MARK(s(cons(tail(odds), y0))) -> c20(MARK(cons(tail(odds), y0))) MARK(s(cons(incr(cons(nats, y0)), y1))) -> c20(MARK(cons(incr(cons(nats, y0)), y1))) MARK(s(cons(incr(cons(incr(y0), y1)), y2))) -> c20(MARK(cons(incr(cons(incr(y0), y1)), y2))) MARK(s(cons(incr(cons(pairs, y0)), y1))) -> c20(MARK(cons(incr(cons(pairs, y0)), y1))) MARK(s(cons(incr(cons(odds, y0)), y1))) -> c20(MARK(cons(incr(cons(odds, y0)), y1))) MARK(s(cons(incr(cons(head(y0), y1)), y2))) -> c20(MARK(cons(incr(cons(head(y0), y1)), y2))) MARK(s(cons(incr(cons(tail(y0), y1)), y2))) -> c20(MARK(cons(incr(cons(tail(y0), y1)), y2))) MARK(s(cons(incr(cons(cons(y0, y1), y2)), y3))) -> c20(MARK(cons(incr(cons(cons(y0, y1), y2)), y3))) MARK(s(cons(incr(cons(s(y0), y1)), y2))) -> c20(MARK(cons(incr(cons(s(y0), y1)), y2))) MARK(s(cons(incr(cons(y0, y1)), y2))) -> c20(MARK(cons(incr(cons(y0, y1)), y2))) MARK(s(cons(incr(cons(0, y0)), y1))) -> c20(MARK(cons(incr(cons(0, y0)), y1))) MARK(s(cons(incr(head(nats)), y0))) -> c20(MARK(cons(incr(head(nats)), y0))) MARK(s(cons(incr(head(pairs)), y0))) -> c20(MARK(cons(incr(head(pairs)), y0))) MARK(s(cons(head(head(nats)), y0))) -> c20(MARK(cons(head(head(nats)), y0))) MARK(s(cons(head(head(pairs)), y0))) -> c20(MARK(cons(head(head(pairs)), y0))) MARK(s(cons(tail(head(nats)), y0))) -> c20(MARK(cons(tail(head(nats)), y0))) MARK(s(cons(tail(head(pairs)), y0))) -> c20(MARK(cons(tail(head(pairs)), y0))) MARK(s(cons(head(cons(nats, y0)), y1))) -> c20(MARK(cons(head(cons(nats, y0)), y1))) MARK(s(cons(head(cons(incr(y0), y1)), y2))) -> c20(MARK(cons(head(cons(incr(y0), y1)), y2))) MARK(s(cons(head(cons(pairs, y0)), y1))) -> c20(MARK(cons(head(cons(pairs, y0)), y1))) MARK(s(cons(head(cons(odds, y0)), y1))) -> c20(MARK(cons(head(cons(odds, y0)), y1))) MARK(s(cons(head(cons(head(y0), y1)), y2))) -> c20(MARK(cons(head(cons(head(y0), y1)), y2))) MARK(s(cons(head(cons(tail(y0), y1)), y2))) -> c20(MARK(cons(head(cons(tail(y0), y1)), y2))) MARK(s(cons(head(cons(cons(y0, y1), y2)), y3))) -> c20(MARK(cons(head(cons(cons(y0, y1), y2)), y3))) MARK(s(cons(head(cons(s(y0), y1)), y2))) -> c20(MARK(cons(head(cons(s(y0), y1)), y2))) MARK(s(cons(head(cons(y0, y1)), y2))) -> c20(MARK(cons(head(cons(y0, y1)), y2))) MARK(s(cons(head(cons(0, y0)), y1))) -> c20(MARK(cons(head(cons(0, y0)), y1))) MARK(s(cons(incr(tail(nats)), y0))) -> c20(MARK(cons(incr(tail(nats)), y0))) MARK(s(cons(head(tail(nats)), y0))) -> c20(MARK(cons(head(tail(nats)), y0))) MARK(s(cons(tail(tail(nats)), y0))) -> c20(MARK(cons(tail(tail(nats)), y0))) MARK(s(incr(cons(nats, y0)))) -> c20(MARK(incr(cons(nats, y0)))) MARK(s(incr(cons(pairs, y0)))) -> c20(MARK(incr(cons(pairs, y0)))) MARK(s(incr(cons(odds, y0)))) -> c20(MARK(incr(cons(odds, y0)))) MARK(s(head(cons(nats, y0)))) -> c20(MARK(head(cons(nats, y0)))) MARK(s(head(cons(pairs, y0)))) -> c20(MARK(head(cons(pairs, y0)))) MARK(s(head(cons(odds, y0)))) -> c20(MARK(head(cons(odds, y0)))) S tuples: MARK(incr(s(z0))) -> c13(MARK(s(z0))) MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c17(MARK(s(z0))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(tail(incr(incr(z0)))) -> c17(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(odds))) -> c17(A__TAIL(a__incr(a__odds)), MARK(incr(odds))) MARK(tail(incr(head(z0)))) -> c17(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c17(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c17(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c17(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c17(MARK(incr(x0))) MARK(tail(head(incr(z0)))) -> c17(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(odds))) -> c17(A__TAIL(a__head(a__odds)), MARK(head(odds))) MARK(tail(head(head(z0)))) -> c17(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c17(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c17(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c17(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) 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c18(MARK(head(odds))) MARK(cons(tail(incr(nats)), z1)) -> c18(MARK(tail(incr(nats)))) MARK(cons(tail(incr(pairs)), z1)) -> c18(MARK(tail(incr(pairs)))) MARK(cons(tail(odds), z1)) -> c18(MARK(tail(odds))) MARK(cons(incr(cons(nats, y0)), z1)) -> c18(MARK(incr(cons(nats, y0)))) MARK(cons(incr(cons(incr(y0), y1)), z1)) -> c18(MARK(incr(cons(incr(y0), y1)))) MARK(cons(incr(cons(pairs, y0)), z1)) -> c18(MARK(incr(cons(pairs, y0)))) MARK(cons(incr(cons(odds, y0)), z1)) -> c18(MARK(incr(cons(odds, y0)))) MARK(cons(incr(cons(head(y0), y1)), z1)) -> c18(MARK(incr(cons(head(y0), y1)))) MARK(cons(incr(cons(tail(y0), y1)), z1)) -> c18(MARK(incr(cons(tail(y0), y1)))) MARK(cons(incr(cons(cons(y0, y1), y2)), z1)) -> c18(MARK(incr(cons(cons(y0, y1), y2)))) MARK(cons(incr(cons(s(y0), y1)), z1)) -> c18(MARK(incr(cons(s(y0), y1)))) MARK(cons(incr(cons(y0, y1)), z1)) -> c18(MARK(incr(cons(y0, y1)))) MARK(cons(incr(cons(0, y0)), z1)) -> c18(MARK(incr(cons(0, y0)))) MARK(cons(incr(head(nats)), z1)) -> 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MARK(s(incr(incr(odds)))) -> c20(MARK(incr(incr(odds)))) MARK(s(incr(incr(head(y0))))) -> c20(MARK(incr(incr(head(y0))))) MARK(s(incr(incr(tail(y0))))) -> c20(MARK(incr(incr(tail(y0))))) MARK(s(incr(incr(cons(y0, y1))))) -> c20(MARK(incr(incr(cons(y0, y1))))) MARK(s(incr(incr(s(y0))))) -> c20(MARK(incr(incr(s(y0))))) MARK(s(incr(incr(y0)))) -> c20(MARK(incr(incr(y0)))) MARK(s(incr(head(incr(y0))))) -> c20(MARK(incr(head(incr(y0))))) MARK(s(incr(head(odds)))) -> c20(MARK(incr(head(odds)))) MARK(s(incr(head(head(y0))))) -> c20(MARK(incr(head(head(y0))))) MARK(s(incr(head(tail(y0))))) -> c20(MARK(incr(head(tail(y0))))) MARK(s(incr(head(cons(y0, y1))))) -> c20(MARK(incr(head(cons(y0, y1))))) MARK(s(incr(head(s(y0))))) -> c20(MARK(incr(head(s(y0))))) MARK(s(incr(head(y0)))) -> c20(MARK(incr(head(y0)))) MARK(s(incr(tail(incr(y0))))) -> c20(MARK(incr(tail(incr(y0))))) MARK(s(incr(tail(pairs)))) -> c20(MARK(incr(tail(pairs)))) MARK(s(incr(tail(odds)))) -> c20(MARK(incr(tail(odds)))) MARK(s(incr(tail(head(y0))))) -> c20(MARK(incr(tail(head(y0))))) MARK(s(incr(tail(tail(y0))))) -> c20(MARK(incr(tail(tail(y0))))) MARK(s(incr(tail(cons(y0, y1))))) -> c20(MARK(incr(tail(cons(y0, y1))))) MARK(s(incr(tail(s(y0))))) -> c20(MARK(incr(tail(s(y0))))) MARK(s(incr(tail(y0)))) -> c20(MARK(incr(tail(y0)))) MARK(s(head(incr(incr(y0))))) -> c20(MARK(head(incr(incr(y0))))) MARK(s(head(incr(odds)))) -> c20(MARK(head(incr(odds)))) MARK(s(head(incr(head(y0))))) -> c20(MARK(head(incr(head(y0))))) MARK(s(head(incr(tail(y0))))) -> c20(MARK(head(incr(tail(y0))))) MARK(s(head(incr(cons(y0, y1))))) -> c20(MARK(head(incr(cons(y0, y1))))) MARK(s(head(incr(s(y0))))) -> c20(MARK(head(incr(s(y0))))) MARK(s(head(incr(y0)))) -> c20(MARK(head(incr(y0)))) MARK(s(head(head(incr(y0))))) -> c20(MARK(head(head(incr(y0))))) MARK(s(head(head(odds)))) -> c20(MARK(head(head(odds)))) MARK(s(head(head(head(y0))))) -> c20(MARK(head(head(head(y0))))) MARK(s(head(head(tail(y0))))) -> c20(MARK(head(head(tail(y0))))) MARK(s(head(head(cons(y0, y1))))) -> c20(MARK(head(head(cons(y0, y1))))) MARK(s(head(head(s(y0))))) -> c20(MARK(head(head(s(y0))))) MARK(s(head(head(y0)))) -> c20(MARK(head(head(y0)))) MARK(s(head(tail(incr(y0))))) -> c20(MARK(head(tail(incr(y0))))) MARK(s(head(tail(pairs)))) -> c20(MARK(head(tail(pairs)))) MARK(s(head(tail(odds)))) -> c20(MARK(head(tail(odds)))) MARK(s(head(tail(head(y0))))) -> c20(MARK(head(tail(head(y0))))) MARK(s(head(tail(tail(y0))))) -> c20(MARK(head(tail(tail(y0))))) MARK(s(head(tail(cons(y0, y1))))) -> c20(MARK(head(tail(cons(y0, y1))))) MARK(s(head(tail(s(y0))))) -> c20(MARK(head(tail(s(y0))))) MARK(s(head(tail(y0)))) -> c20(MARK(head(tail(y0)))) MARK(s(tail(incr(incr(y0))))) -> c20(MARK(tail(incr(incr(y0))))) MARK(s(tail(incr(odds)))) -> c20(MARK(tail(incr(odds)))) MARK(s(tail(incr(head(y0))))) -> c20(MARK(tail(incr(head(y0))))) MARK(s(tail(incr(tail(y0))))) -> c20(MARK(tail(incr(tail(y0))))) MARK(s(tail(incr(cons(y0, y1))))) -> c20(MARK(tail(incr(cons(y0, y1))))) MARK(s(tail(incr(s(y0))))) -> c20(MARK(tail(incr(s(y0))))) MARK(s(tail(incr(y0)))) -> c20(MARK(tail(incr(y0)))) MARK(s(tail(head(incr(y0))))) -> c20(MARK(tail(head(incr(y0))))) MARK(s(tail(head(odds)))) -> c20(MARK(tail(head(odds)))) MARK(s(tail(head(head(y0))))) -> c20(MARK(tail(head(head(y0))))) MARK(s(tail(head(tail(y0))))) -> c20(MARK(tail(head(tail(y0))))) MARK(s(tail(head(cons(y0, y1))))) -> c20(MARK(tail(head(cons(y0, y1))))) MARK(s(tail(head(s(y0))))) -> c20(MARK(tail(head(s(y0))))) MARK(s(tail(head(y0)))) -> c20(MARK(tail(head(y0)))) MARK(s(tail(tail(incr(y0))))) -> c20(MARK(tail(tail(incr(y0))))) MARK(s(tail(tail(pairs)))) -> c20(MARK(tail(tail(pairs)))) MARK(s(tail(tail(odds)))) -> c20(MARK(tail(tail(odds)))) MARK(s(tail(tail(head(y0))))) -> c20(MARK(tail(tail(head(y0))))) MARK(s(tail(tail(tail(y0))))) -> c20(MARK(tail(tail(tail(y0))))) MARK(s(tail(tail(cons(y0, y1))))) -> c20(MARK(tail(tail(cons(y0, y1))))) MARK(s(tail(tail(s(y0))))) -> c20(MARK(tail(tail(s(y0))))) MARK(s(tail(tail(y0)))) -> c20(MARK(tail(tail(y0)))) MARK(s(tail(pairs))) -> c20(MARK(tail(pairs))) MARK(s(incr(incr(nats)))) -> c20(MARK(incr(incr(nats)))) MARK(s(incr(incr(pairs)))) -> c20(MARK(incr(incr(pairs)))) MARK(s(incr(odds))) -> c20(MARK(incr(odds))) MARK(s(head(incr(nats)))) -> c20(MARK(head(incr(nats)))) MARK(s(head(incr(pairs)))) -> c20(MARK(head(incr(pairs)))) MARK(s(head(odds))) -> c20(MARK(head(odds))) MARK(s(tail(incr(nats)))) -> c20(MARK(tail(incr(nats)))) MARK(s(tail(incr(pairs)))) -> c20(MARK(tail(incr(pairs)))) MARK(s(tail(odds))) -> c20(MARK(tail(odds))) MARK(s(incr(cons(incr(y0), y1)))) -> c20(MARK(incr(cons(incr(y0), y1)))) MARK(s(incr(cons(head(y0), y1)))) -> c20(MARK(incr(cons(head(y0), y1)))) MARK(s(incr(cons(tail(y0), y1)))) -> c20(MARK(incr(cons(tail(y0), y1)))) MARK(s(incr(cons(cons(y0, y1), y2)))) -> c20(MARK(incr(cons(cons(y0, y1), y2)))) MARK(s(incr(cons(s(y0), y1)))) -> c20(MARK(incr(cons(s(y0), y1)))) MARK(s(incr(cons(y0, y1)))) -> c20(MARK(incr(cons(y0, y1)))) MARK(s(incr(head(nats)))) -> c20(MARK(incr(head(nats)))) MARK(s(incr(head(pairs)))) -> c20(MARK(incr(head(pairs)))) MARK(s(head(head(nats)))) -> c20(MARK(head(head(nats)))) MARK(s(head(head(pairs)))) -> c20(MARK(head(head(pairs)))) MARK(s(tail(head(nats)))) -> c20(MARK(tail(head(nats)))) MARK(s(tail(head(pairs)))) -> c20(MARK(tail(head(pairs)))) MARK(s(head(cons(incr(y0), y1)))) -> c20(MARK(head(cons(incr(y0), y1)))) MARK(s(head(cons(head(y0), y1)))) -> c20(MARK(head(cons(head(y0), y1)))) MARK(s(head(cons(tail(y0), y1)))) -> c20(MARK(head(cons(tail(y0), y1)))) MARK(s(head(cons(cons(y0, y1), y2)))) -> c20(MARK(head(cons(cons(y0, y1), y2)))) MARK(s(head(cons(s(y0), y1)))) -> c20(MARK(head(cons(s(y0), y1)))) MARK(s(head(cons(y0, y1)))) -> c20(MARK(head(cons(y0, y1)))) MARK(s(incr(tail(nats)))) -> c20(MARK(incr(tail(nats)))) MARK(s(head(tail(nats)))) -> c20(MARK(head(tail(nats)))) MARK(s(tail(tail(nats)))) -> c20(MARK(tail(tail(nats)))) MARK(s(cons(cons(y0, y1), y2))) -> c20(MARK(cons(cons(y0, y1), y2))) MARK(s(cons(s(y0), y1))) -> c20(MARK(cons(s(y0), y1))) MARK(s(cons(incr(s(y0)), y1))) -> c20(MARK(cons(incr(s(y0)), y1))) MARK(s(cons(head(s(y0)), y1))) -> c20(MARK(cons(head(s(y0)), y1))) MARK(s(cons(tail(cons(y0, y1)), y2))) -> c20(MARK(cons(tail(cons(y0, y1)), y2))) MARK(s(cons(tail(s(y0)), y1))) -> c20(MARK(cons(tail(s(y0)), y1))) MARK(s(cons(incr(incr(incr(y0))), y1))) -> c20(MARK(cons(incr(incr(incr(y0))), y1))) MARK(s(cons(incr(incr(odds)), y0))) -> c20(MARK(cons(incr(incr(odds)), y0))) MARK(s(cons(incr(incr(head(y0))), y1))) -> c20(MARK(cons(incr(incr(head(y0))), y1))) MARK(s(cons(incr(incr(tail(y0))), y1))) -> c20(MARK(cons(incr(incr(tail(y0))), y1))) MARK(s(cons(incr(incr(cons(y0, y1))), y2))) -> c20(MARK(cons(incr(incr(cons(y0, y1))), y2))) MARK(s(cons(incr(incr(s(y0))), y1))) -> c20(MARK(cons(incr(incr(s(y0))), y1))) MARK(s(cons(incr(incr(y0)), y1))) -> c20(MARK(cons(incr(incr(y0)), y1))) MARK(s(cons(incr(head(incr(y0))), y1))) -> c20(MARK(cons(incr(head(incr(y0))), y1))) MARK(s(cons(incr(head(odds)), y0))) -> c20(MARK(cons(incr(head(odds)), y0))) MARK(s(cons(incr(head(head(y0))), y1))) -> c20(MARK(cons(incr(head(head(y0))), y1))) MARK(s(cons(incr(head(tail(y0))), y1))) -> c20(MARK(cons(incr(head(tail(y0))), y1))) MARK(s(cons(incr(head(cons(y0, y1))), y2))) -> c20(MARK(cons(incr(head(cons(y0, y1))), y2))) MARK(s(cons(incr(head(s(y0))), y1))) -> c20(MARK(cons(incr(head(s(y0))), y1))) MARK(s(cons(incr(head(y0)), y1))) -> c20(MARK(cons(incr(head(y0)), y1))) MARK(s(cons(incr(tail(incr(y0))), y1))) -> c20(MARK(cons(incr(tail(incr(y0))), y1))) MARK(s(cons(incr(tail(pairs)), y0))) -> c20(MARK(cons(incr(tail(pairs)), y0))) MARK(s(cons(incr(tail(odds)), y0))) -> c20(MARK(cons(incr(tail(odds)), y0))) MARK(s(cons(incr(tail(head(y0))), y1))) -> c20(MARK(cons(incr(tail(head(y0))), y1))) MARK(s(cons(incr(tail(tail(y0))), y1))) -> c20(MARK(cons(incr(tail(tail(y0))), y1))) MARK(s(cons(incr(tail(cons(y0, y1))), y2))) -> c20(MARK(cons(incr(tail(cons(y0, y1))), y2))) MARK(s(cons(incr(tail(s(y0))), y1))) -> c20(MARK(cons(incr(tail(s(y0))), y1))) MARK(s(cons(incr(tail(y0)), y1))) -> c20(MARK(cons(incr(tail(y0)), y1))) MARK(s(cons(head(incr(incr(y0))), y1))) -> c20(MARK(cons(head(incr(incr(y0))), y1))) MARK(s(cons(head(incr(odds)), y0))) -> c20(MARK(cons(head(incr(odds)), y0))) MARK(s(cons(head(incr(head(y0))), y1))) -> c20(MARK(cons(head(incr(head(y0))), y1))) MARK(s(cons(head(incr(tail(y0))), y1))) -> c20(MARK(cons(head(incr(tail(y0))), y1))) MARK(s(cons(head(incr(cons(y0, y1))), y2))) -> c20(MARK(cons(head(incr(cons(y0, y1))), y2))) MARK(s(cons(head(incr(s(y0))), y1))) -> c20(MARK(cons(head(incr(s(y0))), y1))) MARK(s(cons(head(incr(y0)), y1))) -> c20(MARK(cons(head(incr(y0)), y1))) MARK(s(cons(head(head(incr(y0))), y1))) -> c20(MARK(cons(head(head(incr(y0))), y1))) MARK(s(cons(head(head(odds)), y0))) -> c20(MARK(cons(head(head(odds)), y0))) MARK(s(cons(head(head(head(y0))), y1))) -> c20(MARK(cons(head(head(head(y0))), y1))) MARK(s(cons(head(head(tail(y0))), y1))) -> c20(MARK(cons(head(head(tail(y0))), y1))) MARK(s(cons(head(head(cons(y0, y1))), y2))) -> c20(MARK(cons(head(head(cons(y0, y1))), y2))) MARK(s(cons(head(head(s(y0))), y1))) -> c20(MARK(cons(head(head(s(y0))), y1))) MARK(s(cons(head(head(y0)), y1))) -> c20(MARK(cons(head(head(y0)), y1))) MARK(s(cons(head(tail(incr(y0))), y1))) -> c20(MARK(cons(head(tail(incr(y0))), y1))) MARK(s(cons(head(tail(pairs)), y0))) -> c20(MARK(cons(head(tail(pairs)), y0))) MARK(s(cons(head(tail(odds)), y0))) -> c20(MARK(cons(head(tail(odds)), y0))) MARK(s(cons(head(tail(head(y0))), y1))) -> c20(MARK(cons(head(tail(head(y0))), y1))) MARK(s(cons(head(tail(tail(y0))), y1))) -> c20(MARK(cons(head(tail(tail(y0))), y1))) MARK(s(cons(head(tail(cons(y0, y1))), y2))) -> c20(MARK(cons(head(tail(cons(y0, y1))), y2))) MARK(s(cons(head(tail(s(y0))), y1))) -> c20(MARK(cons(head(tail(s(y0))), y1))) MARK(s(cons(head(tail(y0)), y1))) -> c20(MARK(cons(head(tail(y0)), y1))) MARK(s(cons(tail(incr(incr(y0))), y1))) -> c20(MARK(cons(tail(incr(incr(y0))), y1))) MARK(s(cons(tail(incr(odds)), y0))) -> c20(MARK(cons(tail(incr(odds)), y0))) MARK(s(cons(tail(incr(head(y0))), y1))) -> c20(MARK(cons(tail(incr(head(y0))), y1))) MARK(s(cons(tail(incr(tail(y0))), y1))) -> c20(MARK(cons(tail(incr(tail(y0))), y1))) MARK(s(cons(tail(incr(cons(y0, y1))), y2))) -> c20(MARK(cons(tail(incr(cons(y0, y1))), y2))) MARK(s(cons(tail(incr(s(y0))), y1))) -> c20(MARK(cons(tail(incr(s(y0))), y1))) MARK(s(cons(tail(incr(y0)), y1))) -> c20(MARK(cons(tail(incr(y0)), y1))) MARK(s(cons(tail(head(incr(y0))), y1))) -> c20(MARK(cons(tail(head(incr(y0))), y1))) MARK(s(cons(tail(head(odds)), y0))) -> c20(MARK(cons(tail(head(odds)), y0))) MARK(s(cons(tail(head(head(y0))), y1))) -> c20(MARK(cons(tail(head(head(y0))), y1))) MARK(s(cons(tail(head(tail(y0))), y1))) -> c20(MARK(cons(tail(head(tail(y0))), y1))) MARK(s(cons(tail(head(cons(y0, y1))), y2))) -> c20(MARK(cons(tail(head(cons(y0, y1))), y2))) MARK(s(cons(tail(head(s(y0))), y1))) -> c20(MARK(cons(tail(head(s(y0))), y1))) MARK(s(cons(tail(head(y0)), y1))) -> c20(MARK(cons(tail(head(y0)), y1))) MARK(s(cons(tail(tail(incr(y0))), y1))) -> c20(MARK(cons(tail(tail(incr(y0))), y1))) MARK(s(cons(tail(tail(pairs)), y0))) -> c20(MARK(cons(tail(tail(pairs)), y0))) MARK(s(cons(tail(tail(odds)), y0))) -> c20(MARK(cons(tail(tail(odds)), y0))) MARK(s(cons(tail(tail(head(y0))), y1))) -> c20(MARK(cons(tail(tail(head(y0))), y1))) MARK(s(cons(tail(tail(tail(y0))), y1))) -> c20(MARK(cons(tail(tail(tail(y0))), y1))) MARK(s(cons(tail(tail(cons(y0, y1))), y2))) -> c20(MARK(cons(tail(tail(cons(y0, y1))), y2))) MARK(s(cons(tail(tail(s(y0))), y1))) -> c20(MARK(cons(tail(tail(s(y0))), y1))) MARK(s(cons(tail(tail(y0)), y1))) -> c20(MARK(cons(tail(tail(y0)), y1))) MARK(s(cons(tail(pairs), y0))) -> c20(MARK(cons(tail(pairs), y0))) MARK(s(cons(incr(incr(nats)), y0))) -> c20(MARK(cons(incr(incr(nats)), y0))) MARK(s(cons(incr(incr(pairs)), y0))) -> c20(MARK(cons(incr(incr(pairs)), y0))) MARK(s(cons(incr(odds), y0))) -> c20(MARK(cons(incr(odds), y0))) MARK(s(cons(head(incr(nats)), y0))) -> c20(MARK(cons(head(incr(nats)), y0))) MARK(s(cons(head(incr(pairs)), y0))) -> c20(MARK(cons(head(incr(pairs)), y0))) MARK(s(cons(head(odds), y0))) -> c20(MARK(cons(head(odds), y0))) MARK(s(cons(tail(incr(nats)), y0))) -> c20(MARK(cons(tail(incr(nats)), y0))) MARK(s(cons(tail(incr(pairs)), y0))) -> c20(MARK(cons(tail(incr(pairs)), y0))) MARK(s(cons(tail(odds), y0))) -> c20(MARK(cons(tail(odds), y0))) MARK(s(cons(incr(cons(nats, y0)), y1))) -> c20(MARK(cons(incr(cons(nats, y0)), y1))) MARK(s(cons(incr(cons(incr(y0), y1)), y2))) -> c20(MARK(cons(incr(cons(incr(y0), y1)), y2))) MARK(s(cons(incr(cons(pairs, y0)), y1))) -> c20(MARK(cons(incr(cons(pairs, y0)), y1))) MARK(s(cons(incr(cons(odds, y0)), y1))) -> c20(MARK(cons(incr(cons(odds, y0)), y1))) MARK(s(cons(incr(cons(head(y0), y1)), y2))) -> c20(MARK(cons(incr(cons(head(y0), y1)), y2))) MARK(s(cons(incr(cons(tail(y0), y1)), y2))) -> c20(MARK(cons(incr(cons(tail(y0), y1)), y2))) MARK(s(cons(incr(cons(cons(y0, y1), y2)), y3))) -> c20(MARK(cons(incr(cons(cons(y0, y1), y2)), y3))) MARK(s(cons(incr(cons(s(y0), y1)), y2))) -> c20(MARK(cons(incr(cons(s(y0), y1)), y2))) MARK(s(cons(incr(cons(y0, y1)), y2))) -> c20(MARK(cons(incr(cons(y0, y1)), y2))) MARK(s(cons(incr(cons(0, y0)), y1))) -> c20(MARK(cons(incr(cons(0, y0)), y1))) MARK(s(cons(incr(head(nats)), y0))) -> c20(MARK(cons(incr(head(nats)), y0))) MARK(s(cons(incr(head(pairs)), y0))) -> c20(MARK(cons(incr(head(pairs)), y0))) MARK(s(cons(head(head(nats)), y0))) -> c20(MARK(cons(head(head(nats)), y0))) MARK(s(cons(head(head(pairs)), y0))) -> c20(MARK(cons(head(head(pairs)), y0))) MARK(s(cons(tail(head(nats)), y0))) -> c20(MARK(cons(tail(head(nats)), y0))) MARK(s(cons(tail(head(pairs)), y0))) -> c20(MARK(cons(tail(head(pairs)), y0))) MARK(s(cons(head(cons(nats, y0)), y1))) -> c20(MARK(cons(head(cons(nats, y0)), y1))) MARK(s(cons(head(cons(incr(y0), y1)), y2))) -> c20(MARK(cons(head(cons(incr(y0), y1)), y2))) MARK(s(cons(head(cons(pairs, y0)), y1))) -> c20(MARK(cons(head(cons(pairs, y0)), y1))) MARK(s(cons(head(cons(odds, y0)), y1))) -> c20(MARK(cons(head(cons(odds, y0)), y1))) MARK(s(cons(head(cons(head(y0), y1)), y2))) -> c20(MARK(cons(head(cons(head(y0), y1)), y2))) MARK(s(cons(head(cons(tail(y0), y1)), y2))) -> c20(MARK(cons(head(cons(tail(y0), y1)), y2))) MARK(s(cons(head(cons(cons(y0, y1), y2)), y3))) -> c20(MARK(cons(head(cons(cons(y0, y1), y2)), y3))) MARK(s(cons(head(cons(s(y0), y1)), y2))) -> c20(MARK(cons(head(cons(s(y0), y1)), y2))) MARK(s(cons(head(cons(y0, y1)), y2))) -> c20(MARK(cons(head(cons(y0, y1)), y2))) MARK(s(cons(head(cons(0, y0)), y1))) -> c20(MARK(cons(head(cons(0, y0)), y1))) MARK(s(cons(incr(tail(nats)), y0))) -> c20(MARK(cons(incr(tail(nats)), y0))) MARK(s(cons(head(tail(nats)), y0))) -> c20(MARK(cons(head(tail(nats)), y0))) MARK(s(cons(tail(tail(nats)), y0))) -> c20(MARK(cons(tail(tail(nats)), y0))) MARK(s(incr(cons(nats, y0)))) -> c20(MARK(incr(cons(nats, y0)))) MARK(s(incr(cons(pairs, y0)))) -> c20(MARK(incr(cons(pairs, y0)))) MARK(s(incr(cons(odds, y0)))) -> c20(MARK(incr(cons(odds, y0)))) MARK(s(head(cons(nats, y0)))) -> c20(MARK(head(cons(nats, y0)))) MARK(s(head(cons(pairs, y0)))) -> c20(MARK(head(cons(pairs, y0)))) MARK(s(head(cons(odds, y0)))) -> c20(MARK(head(cons(odds, y0)))) K tuples:none Defined Rule Symbols: a__nats, a__pairs, a__odds, a__incr_1, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: MARK_1, A__INCR_1, A__HEAD_1, A__TAIL_1 Compound Symbols: c13_1, c16_1, c17_2, c17_1, c13_2, c16_2, c6_1, c8_1, c10_1, c18_1, c20_1 ---------------------------------------- (193) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace MARK(incr(s(z0))) -> c13(MARK(s(z0))) by MARK(incr(s(s(y0)))) -> c13(MARK(s(s(y0)))) MARK(incr(s(incr(s(y0))))) -> c13(MARK(s(incr(s(y0))))) MARK(incr(s(head(s(y0))))) -> c13(MARK(s(head(s(y0))))) MARK(incr(s(tail(cons(y0, y1))))) -> c13(MARK(s(tail(cons(y0, y1))))) MARK(incr(s(tail(s(y0))))) -> c13(MARK(s(tail(s(y0))))) MARK(incr(s(incr(incr(incr(y0)))))) -> c13(MARK(s(incr(incr(incr(y0)))))) MARK(incr(s(incr(incr(odds))))) -> c13(MARK(s(incr(incr(odds))))) MARK(incr(s(incr(incr(head(y0)))))) -> c13(MARK(s(incr(incr(head(y0)))))) MARK(incr(s(incr(incr(tail(y0)))))) -> c13(MARK(s(incr(incr(tail(y0)))))) MARK(incr(s(incr(incr(cons(y0, y1)))))) -> c13(MARK(s(incr(incr(cons(y0, y1)))))) MARK(incr(s(incr(incr(s(y0)))))) -> c13(MARK(s(incr(incr(s(y0)))))) MARK(incr(s(incr(incr(y0))))) -> c13(MARK(s(incr(incr(y0))))) MARK(incr(s(incr(head(incr(y0)))))) -> c13(MARK(s(incr(head(incr(y0)))))) MARK(incr(s(incr(head(odds))))) -> c13(MARK(s(incr(head(odds))))) MARK(incr(s(incr(head(head(y0)))))) -> c13(MARK(s(incr(head(head(y0)))))) MARK(incr(s(incr(head(tail(y0)))))) -> c13(MARK(s(incr(head(tail(y0)))))) MARK(incr(s(incr(head(cons(y0, y1)))))) -> c13(MARK(s(incr(head(cons(y0, y1)))))) MARK(incr(s(incr(head(s(y0)))))) -> c13(MARK(s(incr(head(s(y0)))))) MARK(incr(s(incr(head(y0))))) -> c13(MARK(s(incr(head(y0))))) MARK(incr(s(incr(tail(incr(y0)))))) -> c13(MARK(s(incr(tail(incr(y0)))))) MARK(incr(s(incr(tail(pairs))))) -> c13(MARK(s(incr(tail(pairs))))) MARK(incr(s(incr(tail(odds))))) -> c13(MARK(s(incr(tail(odds))))) MARK(incr(s(incr(tail(head(y0)))))) -> c13(MARK(s(incr(tail(head(y0)))))) MARK(incr(s(incr(tail(tail(y0)))))) -> c13(MARK(s(incr(tail(tail(y0)))))) MARK(incr(s(incr(tail(cons(y0, y1)))))) -> c13(MARK(s(incr(tail(cons(y0, y1)))))) MARK(incr(s(incr(tail(s(y0)))))) -> c13(MARK(s(incr(tail(s(y0)))))) MARK(incr(s(incr(tail(y0))))) -> c13(MARK(s(incr(tail(y0))))) MARK(incr(s(head(incr(incr(y0)))))) -> c13(MARK(s(head(incr(incr(y0)))))) MARK(incr(s(head(incr(odds))))) -> c13(MARK(s(head(incr(odds))))) MARK(incr(s(head(incr(head(y0)))))) -> c13(MARK(s(head(incr(head(y0)))))) MARK(incr(s(head(incr(tail(y0)))))) -> c13(MARK(s(head(incr(tail(y0)))))) MARK(incr(s(head(incr(cons(y0, y1)))))) -> c13(MARK(s(head(incr(cons(y0, y1)))))) MARK(incr(s(head(incr(s(y0)))))) -> c13(MARK(s(head(incr(s(y0)))))) MARK(incr(s(head(incr(y0))))) -> c13(MARK(s(head(incr(y0))))) MARK(incr(s(head(head(incr(y0)))))) -> c13(MARK(s(head(head(incr(y0)))))) MARK(incr(s(head(head(odds))))) -> c13(MARK(s(head(head(odds))))) MARK(incr(s(head(head(head(y0)))))) -> c13(MARK(s(head(head(head(y0)))))) MARK(incr(s(head(head(tail(y0)))))) -> c13(MARK(s(head(head(tail(y0)))))) MARK(incr(s(head(head(cons(y0, y1)))))) -> c13(MARK(s(head(head(cons(y0, y1)))))) MARK(incr(s(head(head(s(y0)))))) -> c13(MARK(s(head(head(s(y0)))))) MARK(incr(s(head(head(y0))))) -> c13(MARK(s(head(head(y0))))) MARK(incr(s(head(tail(incr(y0)))))) -> c13(MARK(s(head(tail(incr(y0)))))) MARK(incr(s(head(tail(pairs))))) -> c13(MARK(s(head(tail(pairs))))) MARK(incr(s(head(tail(odds))))) -> c13(MARK(s(head(tail(odds))))) MARK(incr(s(head(tail(head(y0)))))) -> c13(MARK(s(head(tail(head(y0)))))) MARK(incr(s(head(tail(tail(y0)))))) -> c13(MARK(s(head(tail(tail(y0)))))) MARK(incr(s(head(tail(cons(y0, y1)))))) -> c13(MARK(s(head(tail(cons(y0, y1)))))) MARK(incr(s(head(tail(s(y0)))))) -> c13(MARK(s(head(tail(s(y0)))))) MARK(incr(s(head(tail(y0))))) -> c13(MARK(s(head(tail(y0))))) MARK(incr(s(tail(incr(incr(y0)))))) -> c13(MARK(s(tail(incr(incr(y0)))))) MARK(incr(s(tail(incr(odds))))) -> c13(MARK(s(tail(incr(odds))))) MARK(incr(s(tail(incr(head(y0)))))) -> c13(MARK(s(tail(incr(head(y0)))))) MARK(incr(s(tail(incr(tail(y0)))))) -> c13(MARK(s(tail(incr(tail(y0)))))) MARK(incr(s(tail(incr(cons(y0, y1)))))) -> c13(MARK(s(tail(incr(cons(y0, y1)))))) MARK(incr(s(tail(incr(s(y0)))))) -> c13(MARK(s(tail(incr(s(y0)))))) MARK(incr(s(tail(incr(y0))))) -> c13(MARK(s(tail(incr(y0))))) MARK(incr(s(tail(head(incr(y0)))))) -> c13(MARK(s(tail(head(incr(y0)))))) MARK(incr(s(tail(head(odds))))) -> c13(MARK(s(tail(head(odds))))) MARK(incr(s(tail(head(head(y0)))))) -> c13(MARK(s(tail(head(head(y0)))))) MARK(incr(s(tail(head(tail(y0)))))) -> c13(MARK(s(tail(head(tail(y0)))))) MARK(incr(s(tail(head(cons(y0, y1)))))) -> c13(MARK(s(tail(head(cons(y0, y1)))))) MARK(incr(s(tail(head(s(y0)))))) -> c13(MARK(s(tail(head(s(y0)))))) MARK(incr(s(tail(head(y0))))) -> c13(MARK(s(tail(head(y0))))) MARK(incr(s(tail(tail(incr(y0)))))) -> c13(MARK(s(tail(tail(incr(y0)))))) MARK(incr(s(tail(tail(pairs))))) -> c13(MARK(s(tail(tail(pairs))))) MARK(incr(s(tail(tail(odds))))) -> c13(MARK(s(tail(tail(odds))))) MARK(incr(s(tail(tail(head(y0)))))) -> c13(MARK(s(tail(tail(head(y0)))))) MARK(incr(s(tail(tail(tail(y0)))))) -> c13(MARK(s(tail(tail(tail(y0)))))) MARK(incr(s(tail(tail(cons(y0, y1)))))) -> c13(MARK(s(tail(tail(cons(y0, y1)))))) MARK(incr(s(tail(tail(s(y0)))))) -> c13(MARK(s(tail(tail(s(y0)))))) MARK(incr(s(tail(tail(y0))))) -> c13(MARK(s(tail(tail(y0))))) MARK(incr(s(tail(pairs)))) -> c13(MARK(s(tail(pairs)))) MARK(incr(s(incr(incr(nats))))) -> c13(MARK(s(incr(incr(nats))))) MARK(incr(s(incr(incr(pairs))))) -> c13(MARK(s(incr(incr(pairs))))) MARK(incr(s(incr(odds)))) -> c13(MARK(s(incr(odds)))) MARK(incr(s(head(incr(nats))))) -> c13(MARK(s(head(incr(nats))))) MARK(incr(s(head(incr(pairs))))) -> c13(MARK(s(head(incr(pairs))))) MARK(incr(s(head(odds)))) -> c13(MARK(s(head(odds)))) MARK(incr(s(tail(incr(nats))))) -> c13(MARK(s(tail(incr(nats))))) MARK(incr(s(tail(incr(pairs))))) -> c13(MARK(s(tail(incr(pairs))))) MARK(incr(s(tail(odds)))) -> c13(MARK(s(tail(odds)))) MARK(incr(s(incr(cons(incr(y0), y1))))) -> c13(MARK(s(incr(cons(incr(y0), y1))))) MARK(incr(s(incr(cons(head(y0), y1))))) -> c13(MARK(s(incr(cons(head(y0), y1))))) MARK(incr(s(incr(cons(tail(y0), y1))))) -> c13(MARK(s(incr(cons(tail(y0), y1))))) MARK(incr(s(incr(cons(cons(y0, y1), y2))))) -> c13(MARK(s(incr(cons(cons(y0, y1), y2))))) MARK(incr(s(incr(cons(s(y0), y1))))) -> c13(MARK(s(incr(cons(s(y0), y1))))) MARK(incr(s(incr(cons(y0, y1))))) -> c13(MARK(s(incr(cons(y0, y1))))) MARK(incr(s(incr(head(nats))))) -> c13(MARK(s(incr(head(nats))))) MARK(incr(s(incr(head(pairs))))) -> c13(MARK(s(incr(head(pairs))))) MARK(incr(s(head(head(nats))))) -> c13(MARK(s(head(head(nats))))) MARK(incr(s(head(head(pairs))))) -> c13(MARK(s(head(head(pairs))))) MARK(incr(s(tail(head(nats))))) -> c13(MARK(s(tail(head(nats))))) MARK(incr(s(tail(head(pairs))))) -> c13(MARK(s(tail(head(pairs))))) MARK(incr(s(head(cons(incr(y0), y1))))) -> c13(MARK(s(head(cons(incr(y0), y1))))) MARK(incr(s(head(cons(head(y0), y1))))) -> c13(MARK(s(head(cons(head(y0), y1))))) MARK(incr(s(head(cons(tail(y0), y1))))) -> c13(MARK(s(head(cons(tail(y0), y1))))) MARK(incr(s(head(cons(cons(y0, y1), y2))))) -> c13(MARK(s(head(cons(cons(y0, y1), y2))))) MARK(incr(s(head(cons(s(y0), y1))))) -> c13(MARK(s(head(cons(s(y0), y1))))) MARK(incr(s(head(cons(y0, y1))))) -> c13(MARK(s(head(cons(y0, y1))))) MARK(incr(s(incr(tail(nats))))) -> c13(MARK(s(incr(tail(nats))))) MARK(incr(s(head(tail(nats))))) -> c13(MARK(s(head(tail(nats))))) MARK(incr(s(tail(tail(nats))))) -> c13(MARK(s(tail(tail(nats))))) MARK(incr(s(cons(cons(y0, y1), y2)))) -> c13(MARK(s(cons(cons(y0, y1), y2)))) MARK(incr(s(cons(s(y0), y1)))) -> c13(MARK(s(cons(s(y0), y1)))) MARK(incr(s(cons(incr(s(y0)), y1)))) -> c13(MARK(s(cons(incr(s(y0)), y1)))) MARK(incr(s(cons(head(s(y0)), y1)))) -> c13(MARK(s(cons(head(s(y0)), y1)))) MARK(incr(s(cons(tail(cons(y0, y1)), y2)))) -> c13(MARK(s(cons(tail(cons(y0, y1)), y2)))) MARK(incr(s(cons(tail(s(y0)), y1)))) -> c13(MARK(s(cons(tail(s(y0)), y1)))) MARK(incr(s(cons(incr(incr(incr(y0))), y1)))) -> c13(MARK(s(cons(incr(incr(incr(y0))), y1)))) MARK(incr(s(cons(incr(incr(odds)), y0)))) -> c13(MARK(s(cons(incr(incr(odds)), y0)))) MARK(incr(s(cons(incr(incr(head(y0))), y1)))) -> c13(MARK(s(cons(incr(incr(head(y0))), y1)))) MARK(incr(s(cons(incr(incr(tail(y0))), y1)))) -> c13(MARK(s(cons(incr(incr(tail(y0))), y1)))) MARK(incr(s(cons(incr(incr(cons(y0, y1))), y2)))) -> c13(MARK(s(cons(incr(incr(cons(y0, y1))), y2)))) MARK(incr(s(cons(incr(incr(s(y0))), y1)))) -> c13(MARK(s(cons(incr(incr(s(y0))), y1)))) MARK(incr(s(cons(incr(incr(y0)), y1)))) -> c13(MARK(s(cons(incr(incr(y0)), y1)))) MARK(incr(s(cons(incr(head(incr(y0))), y1)))) -> c13(MARK(s(cons(incr(head(incr(y0))), y1)))) MARK(incr(s(cons(incr(head(odds)), y0)))) -> c13(MARK(s(cons(incr(head(odds)), y0)))) MARK(incr(s(cons(incr(head(head(y0))), y1)))) -> c13(MARK(s(cons(incr(head(head(y0))), y1)))) MARK(incr(s(cons(incr(head(tail(y0))), y1)))) -> c13(MARK(s(cons(incr(head(tail(y0))), y1)))) MARK(incr(s(cons(incr(head(cons(y0, y1))), y2)))) -> c13(MARK(s(cons(incr(head(cons(y0, y1))), y2)))) MARK(incr(s(cons(incr(head(s(y0))), y1)))) -> c13(MARK(s(cons(incr(head(s(y0))), y1)))) MARK(incr(s(cons(incr(head(y0)), y1)))) -> c13(MARK(s(cons(incr(head(y0)), y1)))) MARK(incr(s(cons(incr(tail(incr(y0))), y1)))) -> c13(MARK(s(cons(incr(tail(incr(y0))), y1)))) MARK(incr(s(cons(incr(tail(pairs)), y0)))) -> c13(MARK(s(cons(incr(tail(pairs)), y0)))) MARK(incr(s(cons(incr(tail(odds)), y0)))) -> c13(MARK(s(cons(incr(tail(odds)), y0)))) MARK(incr(s(cons(incr(tail(head(y0))), y1)))) -> c13(MARK(s(cons(incr(tail(head(y0))), y1)))) MARK(incr(s(cons(incr(tail(tail(y0))), y1)))) -> c13(MARK(s(cons(incr(tail(tail(y0))), y1)))) MARK(incr(s(cons(incr(tail(cons(y0, y1))), y2)))) -> c13(MARK(s(cons(incr(tail(cons(y0, y1))), y2)))) MARK(incr(s(cons(incr(tail(s(y0))), y1)))) -> c13(MARK(s(cons(incr(tail(s(y0))), y1)))) MARK(incr(s(cons(incr(tail(y0)), y1)))) -> c13(MARK(s(cons(incr(tail(y0)), y1)))) MARK(incr(s(cons(head(incr(incr(y0))), y1)))) -> c13(MARK(s(cons(head(incr(incr(y0))), y1)))) MARK(incr(s(cons(head(incr(odds)), y0)))) -> c13(MARK(s(cons(head(incr(odds)), y0)))) MARK(incr(s(cons(head(incr(head(y0))), y1)))) -> c13(MARK(s(cons(head(incr(head(y0))), y1)))) MARK(incr(s(cons(head(incr(tail(y0))), y1)))) -> c13(MARK(s(cons(head(incr(tail(y0))), y1)))) MARK(incr(s(cons(head(incr(cons(y0, y1))), y2)))) -> c13(MARK(s(cons(head(incr(cons(y0, y1))), y2)))) MARK(incr(s(cons(head(incr(s(y0))), y1)))) -> c13(MARK(s(cons(head(incr(s(y0))), y1)))) MARK(incr(s(cons(head(incr(y0)), y1)))) -> c13(MARK(s(cons(head(incr(y0)), y1)))) MARK(incr(s(cons(head(head(incr(y0))), y1)))) -> c13(MARK(s(cons(head(head(incr(y0))), y1)))) MARK(incr(s(cons(head(head(odds)), y0)))) -> c13(MARK(s(cons(head(head(odds)), y0)))) MARK(incr(s(cons(head(head(head(y0))), y1)))) -> c13(MARK(s(cons(head(head(head(y0))), y1)))) MARK(incr(s(cons(head(head(tail(y0))), y1)))) -> c13(MARK(s(cons(head(head(tail(y0))), y1)))) MARK(incr(s(cons(head(head(cons(y0, y1))), y2)))) -> c13(MARK(s(cons(head(head(cons(y0, y1))), y2)))) MARK(incr(s(cons(head(head(s(y0))), y1)))) -> c13(MARK(s(cons(head(head(s(y0))), y1)))) MARK(incr(s(cons(head(head(y0)), y1)))) -> c13(MARK(s(cons(head(head(y0)), y1)))) MARK(incr(s(cons(head(tail(incr(y0))), y1)))) -> c13(MARK(s(cons(head(tail(incr(y0))), y1)))) MARK(incr(s(cons(head(tail(pairs)), y0)))) -> c13(MARK(s(cons(head(tail(pairs)), y0)))) MARK(incr(s(cons(head(tail(odds)), y0)))) -> c13(MARK(s(cons(head(tail(odds)), y0)))) MARK(incr(s(cons(head(tail(head(y0))), y1)))) -> c13(MARK(s(cons(head(tail(head(y0))), y1)))) MARK(incr(s(cons(head(tail(tail(y0))), y1)))) -> c13(MARK(s(cons(head(tail(tail(y0))), y1)))) MARK(incr(s(cons(head(tail(cons(y0, y1))), y2)))) -> c13(MARK(s(cons(head(tail(cons(y0, y1))), y2)))) MARK(incr(s(cons(head(tail(s(y0))), y1)))) -> c13(MARK(s(cons(head(tail(s(y0))), y1)))) MARK(incr(s(cons(head(tail(y0)), y1)))) -> c13(MARK(s(cons(head(tail(y0)), y1)))) MARK(incr(s(cons(tail(incr(incr(y0))), y1)))) -> c13(MARK(s(cons(tail(incr(incr(y0))), y1)))) MARK(incr(s(cons(tail(incr(odds)), y0)))) -> c13(MARK(s(cons(tail(incr(odds)), y0)))) MARK(incr(s(cons(tail(incr(head(y0))), y1)))) -> c13(MARK(s(cons(tail(incr(head(y0))), y1)))) MARK(incr(s(cons(tail(incr(tail(y0))), y1)))) -> c13(MARK(s(cons(tail(incr(tail(y0))), y1)))) MARK(incr(s(cons(tail(incr(cons(y0, y1))), y2)))) -> c13(MARK(s(cons(tail(incr(cons(y0, y1))), y2)))) MARK(incr(s(cons(tail(incr(s(y0))), y1)))) -> c13(MARK(s(cons(tail(incr(s(y0))), y1)))) MARK(incr(s(cons(tail(incr(y0)), y1)))) -> c13(MARK(s(cons(tail(incr(y0)), y1)))) MARK(incr(s(cons(tail(head(incr(y0))), y1)))) -> c13(MARK(s(cons(tail(head(incr(y0))), y1)))) MARK(incr(s(cons(tail(head(odds)), y0)))) -> c13(MARK(s(cons(tail(head(odds)), y0)))) MARK(incr(s(cons(tail(head(head(y0))), y1)))) -> c13(MARK(s(cons(tail(head(head(y0))), y1)))) MARK(incr(s(cons(tail(head(tail(y0))), y1)))) -> c13(MARK(s(cons(tail(head(tail(y0))), y1)))) MARK(incr(s(cons(tail(head(cons(y0, y1))), y2)))) -> c13(MARK(s(cons(tail(head(cons(y0, y1))), y2)))) MARK(incr(s(cons(tail(head(s(y0))), y1)))) -> c13(MARK(s(cons(tail(head(s(y0))), y1)))) MARK(incr(s(cons(tail(head(y0)), y1)))) -> c13(MARK(s(cons(tail(head(y0)), y1)))) MARK(incr(s(cons(tail(tail(incr(y0))), y1)))) -> c13(MARK(s(cons(tail(tail(incr(y0))), y1)))) MARK(incr(s(cons(tail(tail(pairs)), y0)))) -> c13(MARK(s(cons(tail(tail(pairs)), y0)))) MARK(incr(s(cons(tail(tail(odds)), y0)))) -> c13(MARK(s(cons(tail(tail(odds)), y0)))) MARK(incr(s(cons(tail(tail(head(y0))), y1)))) -> c13(MARK(s(cons(tail(tail(head(y0))), y1)))) MARK(incr(s(cons(tail(tail(tail(y0))), y1)))) -> c13(MARK(s(cons(tail(tail(tail(y0))), y1)))) MARK(incr(s(cons(tail(tail(cons(y0, y1))), y2)))) -> c13(MARK(s(cons(tail(tail(cons(y0, y1))), y2)))) MARK(incr(s(cons(tail(tail(s(y0))), y1)))) -> c13(MARK(s(cons(tail(tail(s(y0))), y1)))) MARK(incr(s(cons(tail(tail(y0)), y1)))) -> c13(MARK(s(cons(tail(tail(y0)), y1)))) MARK(incr(s(cons(tail(pairs), y0)))) -> c13(MARK(s(cons(tail(pairs), y0)))) MARK(incr(s(cons(incr(incr(nats)), y0)))) -> c13(MARK(s(cons(incr(incr(nats)), y0)))) MARK(incr(s(cons(incr(incr(pairs)), y0)))) -> c13(MARK(s(cons(incr(incr(pairs)), y0)))) MARK(incr(s(cons(incr(odds), y0)))) -> c13(MARK(s(cons(incr(odds), y0)))) MARK(incr(s(cons(head(incr(nats)), y0)))) -> c13(MARK(s(cons(head(incr(nats)), y0)))) MARK(incr(s(cons(head(incr(pairs)), y0)))) -> c13(MARK(s(cons(head(incr(pairs)), y0)))) MARK(incr(s(cons(head(odds), y0)))) -> c13(MARK(s(cons(head(odds), y0)))) MARK(incr(s(cons(tail(incr(nats)), y0)))) -> c13(MARK(s(cons(tail(incr(nats)), y0)))) MARK(incr(s(cons(tail(incr(pairs)), y0)))) -> c13(MARK(s(cons(tail(incr(pairs)), y0)))) MARK(incr(s(cons(tail(odds), y0)))) -> c13(MARK(s(cons(tail(odds), y0)))) MARK(incr(s(cons(incr(cons(nats, y0)), y1)))) -> c13(MARK(s(cons(incr(cons(nats, y0)), y1)))) MARK(incr(s(cons(incr(cons(incr(y0), y1)), y2)))) -> c13(MARK(s(cons(incr(cons(incr(y0), y1)), y2)))) MARK(incr(s(cons(incr(cons(pairs, y0)), y1)))) -> c13(MARK(s(cons(incr(cons(pairs, y0)), y1)))) MARK(incr(s(cons(incr(cons(odds, y0)), y1)))) -> c13(MARK(s(cons(incr(cons(odds, y0)), y1)))) MARK(incr(s(cons(incr(cons(head(y0), y1)), y2)))) -> c13(MARK(s(cons(incr(cons(head(y0), y1)), y2)))) MARK(incr(s(cons(incr(cons(tail(y0), y1)), y2)))) -> c13(MARK(s(cons(incr(cons(tail(y0), y1)), y2)))) MARK(incr(s(cons(incr(cons(cons(y0, y1), y2)), y3)))) -> c13(MARK(s(cons(incr(cons(cons(y0, y1), y2)), y3)))) MARK(incr(s(cons(incr(cons(s(y0), y1)), y2)))) -> c13(MARK(s(cons(incr(cons(s(y0), y1)), y2)))) MARK(incr(s(cons(incr(cons(y0, y1)), y2)))) -> c13(MARK(s(cons(incr(cons(y0, y1)), y2)))) MARK(incr(s(cons(incr(cons(0, y0)), y1)))) -> c13(MARK(s(cons(incr(cons(0, y0)), y1)))) MARK(incr(s(cons(incr(head(nats)), y0)))) -> c13(MARK(s(cons(incr(head(nats)), y0)))) MARK(incr(s(cons(incr(head(pairs)), y0)))) -> c13(MARK(s(cons(incr(head(pairs)), y0)))) MARK(incr(s(cons(head(head(nats)), y0)))) -> c13(MARK(s(cons(head(head(nats)), y0)))) MARK(incr(s(cons(head(head(pairs)), y0)))) -> c13(MARK(s(cons(head(head(pairs)), y0)))) MARK(incr(s(cons(tail(head(nats)), y0)))) -> c13(MARK(s(cons(tail(head(nats)), y0)))) MARK(incr(s(cons(tail(head(pairs)), y0)))) -> c13(MARK(s(cons(tail(head(pairs)), y0)))) MARK(incr(s(cons(head(cons(nats, y0)), y1)))) -> c13(MARK(s(cons(head(cons(nats, y0)), y1)))) MARK(incr(s(cons(head(cons(incr(y0), y1)), y2)))) -> c13(MARK(s(cons(head(cons(incr(y0), y1)), y2)))) MARK(incr(s(cons(head(cons(pairs, y0)), y1)))) -> c13(MARK(s(cons(head(cons(pairs, y0)), y1)))) MARK(incr(s(cons(head(cons(odds, y0)), y1)))) -> c13(MARK(s(cons(head(cons(odds, y0)), y1)))) MARK(incr(s(cons(head(cons(head(y0), y1)), y2)))) -> c13(MARK(s(cons(head(cons(head(y0), y1)), y2)))) MARK(incr(s(cons(head(cons(tail(y0), y1)), y2)))) -> c13(MARK(s(cons(head(cons(tail(y0), y1)), y2)))) MARK(incr(s(cons(head(cons(cons(y0, y1), y2)), y3)))) -> c13(MARK(s(cons(head(cons(cons(y0, y1), y2)), y3)))) MARK(incr(s(cons(head(cons(s(y0), y1)), y2)))) -> c13(MARK(s(cons(head(cons(s(y0), y1)), y2)))) MARK(incr(s(cons(head(cons(y0, y1)), y2)))) -> c13(MARK(s(cons(head(cons(y0, y1)), y2)))) MARK(incr(s(cons(head(cons(0, y0)), y1)))) -> c13(MARK(s(cons(head(cons(0, y0)), y1)))) MARK(incr(s(cons(incr(tail(nats)), y0)))) -> c13(MARK(s(cons(incr(tail(nats)), y0)))) MARK(incr(s(cons(head(tail(nats)), y0)))) -> c13(MARK(s(cons(head(tail(nats)), y0)))) MARK(incr(s(cons(tail(tail(nats)), y0)))) -> c13(MARK(s(cons(tail(tail(nats)), y0)))) MARK(incr(s(incr(cons(nats, y0))))) -> c13(MARK(s(incr(cons(nats, y0))))) MARK(incr(s(incr(cons(pairs, y0))))) -> c13(MARK(s(incr(cons(pairs, y0))))) MARK(incr(s(incr(cons(odds, y0))))) -> c13(MARK(s(incr(cons(odds, y0))))) MARK(incr(s(head(cons(nats, y0))))) -> c13(MARK(s(head(cons(nats, y0))))) MARK(incr(s(head(cons(pairs, y0))))) -> c13(MARK(s(head(cons(pairs, y0))))) MARK(incr(s(head(cons(odds, y0))))) -> c13(MARK(s(head(cons(odds, y0))))) ---------------------------------------- (194) Obligation: Complexity Dependency Tuples Problem Rules: a__nats -> cons(0, incr(nats)) a__nats -> nats a__pairs -> cons(0, incr(odds)) a__pairs -> pairs a__odds -> a__incr(a__pairs) a__odds -> odds a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) 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(nats) -> a__nats mark(incr(z0)) -> a__incr(mark(z0)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c17(MARK(s(z0))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(tail(incr(incr(z0)))) -> c17(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(odds))) -> c17(A__TAIL(a__incr(a__odds)), MARK(incr(odds))) MARK(tail(incr(head(z0)))) -> c17(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c17(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c17(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c17(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c17(MARK(incr(x0))) MARK(tail(head(incr(z0)))) -> c17(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(odds))) -> c17(A__TAIL(a__head(a__odds)), MARK(head(odds))) MARK(tail(head(head(z0)))) -> c17(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c17(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c17(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c17(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c17(MARK(head(x0))) MARK(tail(tail(incr(z0)))) -> c17(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(pairs))) -> c17(A__TAIL(a__tail(a__pairs)), MARK(tail(pairs))) MARK(tail(tail(odds))) -> c17(A__TAIL(a__tail(a__odds)), MARK(tail(odds))) MARK(tail(tail(head(z0)))) -> c17(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> 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MARK(cons(tail(tail(s(y0))), z1)) -> c18(MARK(tail(tail(s(y0))))) MARK(cons(tail(tail(y0)), z1)) -> c18(MARK(tail(tail(y0)))) MARK(cons(tail(pairs), z1)) -> c18(MARK(tail(pairs))) MARK(cons(incr(incr(nats)), z1)) -> c18(MARK(incr(incr(nats)))) MARK(cons(incr(incr(pairs)), z1)) -> c18(MARK(incr(incr(pairs)))) MARK(cons(incr(odds), z1)) -> c18(MARK(incr(odds))) MARK(cons(head(incr(nats)), z1)) -> c18(MARK(head(incr(nats)))) MARK(cons(head(incr(pairs)), z1)) -> c18(MARK(head(incr(pairs)))) MARK(cons(head(odds), z1)) -> c18(MARK(head(odds))) MARK(cons(tail(incr(nats)), z1)) -> c18(MARK(tail(incr(nats)))) MARK(cons(tail(incr(pairs)), z1)) -> c18(MARK(tail(incr(pairs)))) MARK(cons(tail(odds), z1)) -> c18(MARK(tail(odds))) MARK(cons(incr(cons(nats, y0)), z1)) -> c18(MARK(incr(cons(nats, y0)))) MARK(cons(incr(cons(incr(y0), y1)), z1)) -> c18(MARK(incr(cons(incr(y0), y1)))) MARK(cons(incr(cons(pairs, y0)), z1)) -> c18(MARK(incr(cons(pairs, y0)))) MARK(cons(incr(cons(odds, y0)), z1)) -> c18(MARK(incr(cons(odds, y0)))) MARK(cons(incr(cons(head(y0), y1)), z1)) -> c18(MARK(incr(cons(head(y0), y1)))) MARK(cons(incr(cons(tail(y0), y1)), z1)) -> c18(MARK(incr(cons(tail(y0), y1)))) MARK(cons(incr(cons(cons(y0, y1), y2)), z1)) -> c18(MARK(incr(cons(cons(y0, y1), y2)))) MARK(cons(incr(cons(s(y0), y1)), z1)) -> c18(MARK(incr(cons(s(y0), y1)))) MARK(cons(incr(cons(y0, y1)), z1)) -> c18(MARK(incr(cons(y0, y1)))) MARK(cons(incr(cons(0, y0)), z1)) -> c18(MARK(incr(cons(0, y0)))) MARK(cons(incr(head(nats)), z1)) -> c18(MARK(incr(head(nats)))) MARK(cons(incr(head(pairs)), z1)) -> c18(MARK(incr(head(pairs)))) MARK(cons(head(head(nats)), z1)) -> c18(MARK(head(head(nats)))) MARK(cons(head(head(pairs)), z1)) -> c18(MARK(head(head(pairs)))) MARK(cons(tail(head(nats)), z1)) -> c18(MARK(tail(head(nats)))) MARK(cons(tail(head(pairs)), z1)) -> c18(MARK(tail(head(pairs)))) MARK(cons(head(cons(nats, y0)), z1)) -> c18(MARK(head(cons(nats, y0)))) MARK(cons(head(cons(incr(y0), y1)), z1)) -> c18(MARK(head(cons(incr(y0), y1)))) MARK(cons(head(cons(pairs, y0)), z1)) -> c18(MARK(head(cons(pairs, y0)))) MARK(cons(head(cons(odds, y0)), z1)) -> c18(MARK(head(cons(odds, y0)))) MARK(cons(head(cons(head(y0), y1)), z1)) -> c18(MARK(head(cons(head(y0), y1)))) MARK(cons(head(cons(tail(y0), y1)), z1)) -> c18(MARK(head(cons(tail(y0), y1)))) MARK(cons(head(cons(cons(y0, y1), y2)), z1)) -> c18(MARK(head(cons(cons(y0, y1), y2)))) MARK(cons(head(cons(s(y0), y1)), z1)) -> c18(MARK(head(cons(s(y0), y1)))) MARK(cons(head(cons(y0, y1)), z1)) -> c18(MARK(head(cons(y0, y1)))) MARK(cons(head(cons(0, y0)), z1)) -> c18(MARK(head(cons(0, y0)))) MARK(cons(incr(tail(nats)), z1)) -> c18(MARK(incr(tail(nats)))) MARK(cons(head(tail(nats)), z1)) -> c18(MARK(head(tail(nats)))) MARK(cons(tail(tail(nats)), z1)) -> c18(MARK(tail(tail(nats)))) MARK(incr(cons(nats, x1))) -> c13(A__INCR(cons(a__nats, x1))) MARK(incr(cons(pairs, x1))) -> c13(A__INCR(cons(a__pairs, x1))) MARK(incr(cons(odds, x1))) -> 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c20(MARK(incr(head(odds)))) MARK(s(incr(head(head(y0))))) -> c20(MARK(incr(head(head(y0))))) MARK(s(incr(head(tail(y0))))) -> c20(MARK(incr(head(tail(y0))))) MARK(s(incr(head(cons(y0, y1))))) -> c20(MARK(incr(head(cons(y0, y1))))) MARK(s(incr(head(s(y0))))) -> c20(MARK(incr(head(s(y0))))) MARK(s(incr(head(y0)))) -> c20(MARK(incr(head(y0)))) MARK(s(incr(tail(incr(y0))))) -> c20(MARK(incr(tail(incr(y0))))) MARK(s(incr(tail(pairs)))) -> c20(MARK(incr(tail(pairs)))) MARK(s(incr(tail(odds)))) -> c20(MARK(incr(tail(odds)))) MARK(s(incr(tail(head(y0))))) -> c20(MARK(incr(tail(head(y0))))) MARK(s(incr(tail(tail(y0))))) -> c20(MARK(incr(tail(tail(y0))))) MARK(s(incr(tail(cons(y0, y1))))) -> c20(MARK(incr(tail(cons(y0, y1))))) MARK(s(incr(tail(s(y0))))) -> c20(MARK(incr(tail(s(y0))))) MARK(s(incr(tail(y0)))) -> c20(MARK(incr(tail(y0)))) MARK(s(head(incr(incr(y0))))) -> c20(MARK(head(incr(incr(y0))))) MARK(s(head(incr(odds)))) -> c20(MARK(head(incr(odds)))) MARK(s(head(incr(head(y0))))) -> c20(MARK(head(incr(head(y0))))) MARK(s(head(incr(tail(y0))))) -> c20(MARK(head(incr(tail(y0))))) MARK(s(head(incr(cons(y0, y1))))) -> c20(MARK(head(incr(cons(y0, y1))))) MARK(s(head(incr(s(y0))))) -> c20(MARK(head(incr(s(y0))))) MARK(s(head(incr(y0)))) -> c20(MARK(head(incr(y0)))) MARK(s(head(head(incr(y0))))) -> c20(MARK(head(head(incr(y0))))) MARK(s(head(head(odds)))) -> c20(MARK(head(head(odds)))) MARK(s(head(head(head(y0))))) -> c20(MARK(head(head(head(y0))))) MARK(s(head(head(tail(y0))))) -> c20(MARK(head(head(tail(y0))))) MARK(s(head(head(cons(y0, y1))))) -> c20(MARK(head(head(cons(y0, y1))))) MARK(s(head(head(s(y0))))) -> c20(MARK(head(head(s(y0))))) MARK(s(head(head(y0)))) -> c20(MARK(head(head(y0)))) MARK(s(head(tail(incr(y0))))) -> c20(MARK(head(tail(incr(y0))))) MARK(s(head(tail(pairs)))) -> c20(MARK(head(tail(pairs)))) MARK(s(head(tail(odds)))) -> c20(MARK(head(tail(odds)))) MARK(s(head(tail(head(y0))))) -> c20(MARK(head(tail(head(y0))))) MARK(s(head(tail(tail(y0))))) -> c20(MARK(head(tail(tail(y0))))) MARK(s(head(tail(cons(y0, y1))))) -> c20(MARK(head(tail(cons(y0, y1))))) MARK(s(head(tail(s(y0))))) -> c20(MARK(head(tail(s(y0))))) MARK(s(head(tail(y0)))) -> c20(MARK(head(tail(y0)))) MARK(s(tail(incr(incr(y0))))) -> c20(MARK(tail(incr(incr(y0))))) MARK(s(tail(incr(odds)))) -> c20(MARK(tail(incr(odds)))) MARK(s(tail(incr(head(y0))))) -> c20(MARK(tail(incr(head(y0))))) MARK(s(tail(incr(tail(y0))))) -> c20(MARK(tail(incr(tail(y0))))) MARK(s(tail(incr(cons(y0, y1))))) -> c20(MARK(tail(incr(cons(y0, y1))))) MARK(s(tail(incr(s(y0))))) -> c20(MARK(tail(incr(s(y0))))) MARK(s(tail(incr(y0)))) -> c20(MARK(tail(incr(y0)))) MARK(s(tail(head(incr(y0))))) -> c20(MARK(tail(head(incr(y0))))) MARK(s(tail(head(odds)))) -> c20(MARK(tail(head(odds)))) MARK(s(tail(head(head(y0))))) -> c20(MARK(tail(head(head(y0))))) MARK(s(tail(head(tail(y0))))) -> c20(MARK(tail(head(tail(y0))))) MARK(s(tail(head(cons(y0, y1))))) -> c20(MARK(tail(head(cons(y0, y1))))) MARK(s(tail(head(s(y0))))) -> c20(MARK(tail(head(s(y0))))) MARK(s(tail(head(y0)))) -> c20(MARK(tail(head(y0)))) MARK(s(tail(tail(incr(y0))))) -> c20(MARK(tail(tail(incr(y0))))) MARK(s(tail(tail(pairs)))) -> c20(MARK(tail(tail(pairs)))) MARK(s(tail(tail(odds)))) -> c20(MARK(tail(tail(odds)))) MARK(s(tail(tail(head(y0))))) -> c20(MARK(tail(tail(head(y0))))) MARK(s(tail(tail(tail(y0))))) -> c20(MARK(tail(tail(tail(y0))))) MARK(s(tail(tail(cons(y0, y1))))) -> c20(MARK(tail(tail(cons(y0, y1))))) MARK(s(tail(tail(s(y0))))) -> c20(MARK(tail(tail(s(y0))))) MARK(s(tail(tail(y0)))) -> c20(MARK(tail(tail(y0)))) MARK(s(tail(pairs))) -> c20(MARK(tail(pairs))) MARK(s(incr(incr(nats)))) -> c20(MARK(incr(incr(nats)))) MARK(s(incr(incr(pairs)))) -> c20(MARK(incr(incr(pairs)))) MARK(s(incr(odds))) -> c20(MARK(incr(odds))) MARK(s(head(incr(nats)))) -> c20(MARK(head(incr(nats)))) MARK(s(head(incr(pairs)))) -> c20(MARK(head(incr(pairs)))) MARK(s(head(odds))) -> c20(MARK(head(odds))) MARK(s(tail(incr(nats)))) -> c20(MARK(tail(incr(nats)))) MARK(s(tail(incr(pairs)))) -> c20(MARK(tail(incr(pairs)))) MARK(s(tail(odds))) -> c20(MARK(tail(odds))) MARK(s(incr(cons(incr(y0), y1)))) -> c20(MARK(incr(cons(incr(y0), y1)))) MARK(s(incr(cons(head(y0), y1)))) -> c20(MARK(incr(cons(head(y0), y1)))) MARK(s(incr(cons(tail(y0), y1)))) -> c20(MARK(incr(cons(tail(y0), y1)))) MARK(s(incr(cons(cons(y0, y1), y2)))) -> c20(MARK(incr(cons(cons(y0, y1), y2)))) MARK(s(incr(cons(s(y0), y1)))) -> c20(MARK(incr(cons(s(y0), y1)))) MARK(s(incr(cons(y0, y1)))) -> c20(MARK(incr(cons(y0, y1)))) MARK(s(incr(head(nats)))) -> c20(MARK(incr(head(nats)))) MARK(s(incr(head(pairs)))) -> c20(MARK(incr(head(pairs)))) MARK(s(head(head(nats)))) -> c20(MARK(head(head(nats)))) MARK(s(head(head(pairs)))) -> c20(MARK(head(head(pairs)))) MARK(s(tail(head(nats)))) -> c20(MARK(tail(head(nats)))) MARK(s(tail(head(pairs)))) -> c20(MARK(tail(head(pairs)))) MARK(s(head(cons(incr(y0), y1)))) -> c20(MARK(head(cons(incr(y0), y1)))) MARK(s(head(cons(head(y0), y1)))) -> c20(MARK(head(cons(head(y0), y1)))) MARK(s(head(cons(tail(y0), y1)))) -> c20(MARK(head(cons(tail(y0), y1)))) MARK(s(head(cons(cons(y0, y1), y2)))) -> c20(MARK(head(cons(cons(y0, y1), y2)))) MARK(s(head(cons(s(y0), y1)))) -> c20(MARK(head(cons(s(y0), y1)))) MARK(s(head(cons(y0, y1)))) -> c20(MARK(head(cons(y0, y1)))) MARK(s(incr(tail(nats)))) -> c20(MARK(incr(tail(nats)))) MARK(s(head(tail(nats)))) -> c20(MARK(head(tail(nats)))) MARK(s(tail(tail(nats)))) -> c20(MARK(tail(tail(nats)))) MARK(s(cons(cons(y0, y1), y2))) -> c20(MARK(cons(cons(y0, y1), y2))) MARK(s(cons(s(y0), y1))) -> c20(MARK(cons(s(y0), y1))) MARK(s(cons(incr(s(y0)), y1))) -> c20(MARK(cons(incr(s(y0)), y1))) MARK(s(cons(head(s(y0)), y1))) -> c20(MARK(cons(head(s(y0)), y1))) MARK(s(cons(tail(cons(y0, y1)), y2))) -> c20(MARK(cons(tail(cons(y0, y1)), y2))) MARK(s(cons(tail(s(y0)), y1))) -> c20(MARK(cons(tail(s(y0)), y1))) MARK(s(cons(incr(incr(incr(y0))), y1))) -> c20(MARK(cons(incr(incr(incr(y0))), y1))) MARK(s(cons(incr(incr(odds)), y0))) -> c20(MARK(cons(incr(incr(odds)), y0))) MARK(s(cons(incr(incr(head(y0))), y1))) -> c20(MARK(cons(incr(incr(head(y0))), y1))) MARK(s(cons(incr(incr(tail(y0))), y1))) -> c20(MARK(cons(incr(incr(tail(y0))), y1))) MARK(s(cons(incr(incr(cons(y0, y1))), y2))) -> c20(MARK(cons(incr(incr(cons(y0, y1))), y2))) MARK(s(cons(incr(incr(s(y0))), y1))) -> c20(MARK(cons(incr(incr(s(y0))), y1))) MARK(s(cons(incr(incr(y0)), y1))) -> c20(MARK(cons(incr(incr(y0)), y1))) MARK(s(cons(incr(head(incr(y0))), y1))) -> c20(MARK(cons(incr(head(incr(y0))), y1))) MARK(s(cons(incr(head(odds)), y0))) -> c20(MARK(cons(incr(head(odds)), y0))) MARK(s(cons(incr(head(head(y0))), y1))) -> c20(MARK(cons(incr(head(head(y0))), y1))) MARK(s(cons(incr(head(tail(y0))), y1))) -> c20(MARK(cons(incr(head(tail(y0))), y1))) MARK(s(cons(incr(head(cons(y0, y1))), y2))) -> c20(MARK(cons(incr(head(cons(y0, y1))), y2))) MARK(s(cons(incr(head(s(y0))), y1))) -> c20(MARK(cons(incr(head(s(y0))), y1))) MARK(s(cons(incr(head(y0)), y1))) -> c20(MARK(cons(incr(head(y0)), y1))) MARK(s(cons(incr(tail(incr(y0))), y1))) -> c20(MARK(cons(incr(tail(incr(y0))), y1))) MARK(s(cons(incr(tail(pairs)), y0))) -> c20(MARK(cons(incr(tail(pairs)), y0))) MARK(s(cons(incr(tail(odds)), y0))) -> c20(MARK(cons(incr(tail(odds)), y0))) MARK(s(cons(incr(tail(head(y0))), y1))) -> c20(MARK(cons(incr(tail(head(y0))), y1))) MARK(s(cons(incr(tail(tail(y0))), y1))) -> c20(MARK(cons(incr(tail(tail(y0))), y1))) MARK(s(cons(incr(tail(cons(y0, y1))), y2))) -> c20(MARK(cons(incr(tail(cons(y0, y1))), y2))) MARK(s(cons(incr(tail(s(y0))), y1))) -> c20(MARK(cons(incr(tail(s(y0))), y1))) MARK(s(cons(incr(tail(y0)), y1))) -> c20(MARK(cons(incr(tail(y0)), y1))) MARK(s(cons(head(incr(incr(y0))), y1))) -> c20(MARK(cons(head(incr(incr(y0))), y1))) MARK(s(cons(head(incr(odds)), y0))) -> c20(MARK(cons(head(incr(odds)), y0))) MARK(s(cons(head(incr(head(y0))), y1))) -> c20(MARK(cons(head(incr(head(y0))), y1))) MARK(s(cons(head(incr(tail(y0))), y1))) -> c20(MARK(cons(head(incr(tail(y0))), y1))) MARK(s(cons(head(incr(cons(y0, y1))), y2))) -> c20(MARK(cons(head(incr(cons(y0, y1))), y2))) MARK(s(cons(head(incr(s(y0))), y1))) -> c20(MARK(cons(head(incr(s(y0))), y1))) MARK(s(cons(head(incr(y0)), y1))) -> c20(MARK(cons(head(incr(y0)), y1))) MARK(s(cons(head(head(incr(y0))), y1))) -> c20(MARK(cons(head(head(incr(y0))), y1))) MARK(s(cons(head(head(odds)), y0))) -> c20(MARK(cons(head(head(odds)), y0))) MARK(s(cons(head(head(head(y0))), y1))) -> c20(MARK(cons(head(head(head(y0))), y1))) MARK(s(cons(head(head(tail(y0))), y1))) -> c20(MARK(cons(head(head(tail(y0))), y1))) MARK(s(cons(head(head(cons(y0, y1))), y2))) -> c20(MARK(cons(head(head(cons(y0, y1))), y2))) MARK(s(cons(head(head(s(y0))), y1))) -> c20(MARK(cons(head(head(s(y0))), y1))) MARK(s(cons(head(head(y0)), y1))) -> c20(MARK(cons(head(head(y0)), y1))) MARK(s(cons(head(tail(incr(y0))), y1))) -> c20(MARK(cons(head(tail(incr(y0))), y1))) MARK(s(cons(head(tail(pairs)), y0))) -> c20(MARK(cons(head(tail(pairs)), y0))) MARK(s(cons(head(tail(odds)), y0))) -> c20(MARK(cons(head(tail(odds)), y0))) MARK(s(cons(head(tail(head(y0))), y1))) -> c20(MARK(cons(head(tail(head(y0))), y1))) MARK(s(cons(head(tail(tail(y0))), y1))) -> c20(MARK(cons(head(tail(tail(y0))), y1))) MARK(s(cons(head(tail(cons(y0, y1))), y2))) -> c20(MARK(cons(head(tail(cons(y0, y1))), y2))) MARK(s(cons(head(tail(s(y0))), y1))) -> c20(MARK(cons(head(tail(s(y0))), y1))) MARK(s(cons(head(tail(y0)), y1))) -> c20(MARK(cons(head(tail(y0)), y1))) MARK(s(cons(tail(incr(incr(y0))), y1))) -> c20(MARK(cons(tail(incr(incr(y0))), y1))) MARK(s(cons(tail(incr(odds)), y0))) -> c20(MARK(cons(tail(incr(odds)), y0))) MARK(s(cons(tail(incr(head(y0))), y1))) -> c20(MARK(cons(tail(incr(head(y0))), y1))) MARK(s(cons(tail(incr(tail(y0))), y1))) -> c20(MARK(cons(tail(incr(tail(y0))), y1))) MARK(s(cons(tail(incr(cons(y0, y1))), y2))) -> c20(MARK(cons(tail(incr(cons(y0, y1))), y2))) MARK(s(cons(tail(incr(s(y0))), y1))) -> c20(MARK(cons(tail(incr(s(y0))), y1))) MARK(s(cons(tail(incr(y0)), y1))) -> c20(MARK(cons(tail(incr(y0)), y1))) MARK(s(cons(tail(head(incr(y0))), y1))) -> c20(MARK(cons(tail(head(incr(y0))), y1))) MARK(s(cons(tail(head(odds)), y0))) -> c20(MARK(cons(tail(head(odds)), y0))) MARK(s(cons(tail(head(head(y0))), y1))) -> c20(MARK(cons(tail(head(head(y0))), y1))) MARK(s(cons(tail(head(tail(y0))), y1))) -> c20(MARK(cons(tail(head(tail(y0))), y1))) MARK(s(cons(tail(head(cons(y0, y1))), y2))) -> c20(MARK(cons(tail(head(cons(y0, y1))), y2))) MARK(s(cons(tail(head(s(y0))), y1))) -> c20(MARK(cons(tail(head(s(y0))), y1))) MARK(s(cons(tail(head(y0)), y1))) -> c20(MARK(cons(tail(head(y0)), y1))) MARK(s(cons(tail(tail(incr(y0))), y1))) -> c20(MARK(cons(tail(tail(incr(y0))), y1))) MARK(s(cons(tail(tail(pairs)), y0))) -> c20(MARK(cons(tail(tail(pairs)), y0))) MARK(s(cons(tail(tail(odds)), y0))) -> c20(MARK(cons(tail(tail(odds)), y0))) MARK(s(cons(tail(tail(head(y0))), y1))) -> c20(MARK(cons(tail(tail(head(y0))), y1))) MARK(s(cons(tail(tail(tail(y0))), y1))) -> c20(MARK(cons(tail(tail(tail(y0))), y1))) MARK(s(cons(tail(tail(cons(y0, y1))), y2))) -> c20(MARK(cons(tail(tail(cons(y0, y1))), y2))) MARK(s(cons(tail(tail(s(y0))), y1))) -> c20(MARK(cons(tail(tail(s(y0))), y1))) MARK(s(cons(tail(tail(y0)), y1))) -> c20(MARK(cons(tail(tail(y0)), y1))) MARK(s(cons(tail(pairs), y0))) -> c20(MARK(cons(tail(pairs), y0))) MARK(s(cons(incr(incr(nats)), y0))) -> c20(MARK(cons(incr(incr(nats)), y0))) MARK(s(cons(incr(incr(pairs)), y0))) -> c20(MARK(cons(incr(incr(pairs)), y0))) MARK(s(cons(incr(odds), y0))) -> c20(MARK(cons(incr(odds), y0))) MARK(s(cons(head(incr(nats)), y0))) -> c20(MARK(cons(head(incr(nats)), y0))) MARK(s(cons(head(incr(pairs)), y0))) -> c20(MARK(cons(head(incr(pairs)), y0))) MARK(s(cons(head(odds), y0))) -> c20(MARK(cons(head(odds), y0))) MARK(s(cons(tail(incr(nats)), y0))) -> c20(MARK(cons(tail(incr(nats)), y0))) MARK(s(cons(tail(incr(pairs)), y0))) -> c20(MARK(cons(tail(incr(pairs)), y0))) MARK(s(cons(tail(odds), y0))) -> c20(MARK(cons(tail(odds), y0))) MARK(s(cons(incr(cons(nats, y0)), y1))) -> c20(MARK(cons(incr(cons(nats, y0)), y1))) MARK(s(cons(incr(cons(incr(y0), y1)), y2))) -> c20(MARK(cons(incr(cons(incr(y0), y1)), y2))) MARK(s(cons(incr(cons(pairs, y0)), y1))) -> c20(MARK(cons(incr(cons(pairs, y0)), y1))) MARK(s(cons(incr(cons(odds, y0)), y1))) -> c20(MARK(cons(incr(cons(odds, y0)), y1))) MARK(s(cons(incr(cons(head(y0), y1)), y2))) -> c20(MARK(cons(incr(cons(head(y0), y1)), y2))) MARK(s(cons(incr(cons(tail(y0), y1)), y2))) -> c20(MARK(cons(incr(cons(tail(y0), y1)), y2))) MARK(s(cons(incr(cons(cons(y0, y1), y2)), y3))) -> c20(MARK(cons(incr(cons(cons(y0, y1), y2)), y3))) MARK(s(cons(incr(cons(s(y0), y1)), y2))) -> c20(MARK(cons(incr(cons(s(y0), y1)), y2))) MARK(s(cons(incr(cons(y0, y1)), y2))) -> c20(MARK(cons(incr(cons(y0, y1)), y2))) MARK(s(cons(incr(cons(0, y0)), y1))) -> c20(MARK(cons(incr(cons(0, y0)), y1))) MARK(s(cons(incr(head(nats)), y0))) -> c20(MARK(cons(incr(head(nats)), y0))) MARK(s(cons(incr(head(pairs)), y0))) -> c20(MARK(cons(incr(head(pairs)), y0))) MARK(s(cons(head(head(nats)), y0))) -> c20(MARK(cons(head(head(nats)), y0))) MARK(s(cons(head(head(pairs)), y0))) -> c20(MARK(cons(head(head(pairs)), y0))) MARK(s(cons(tail(head(nats)), y0))) -> c20(MARK(cons(tail(head(nats)), y0))) MARK(s(cons(tail(head(pairs)), y0))) -> c20(MARK(cons(tail(head(pairs)), y0))) MARK(s(cons(head(cons(nats, y0)), y1))) -> c20(MARK(cons(head(cons(nats, y0)), y1))) MARK(s(cons(head(cons(incr(y0), y1)), y2))) -> c20(MARK(cons(head(cons(incr(y0), y1)), y2))) MARK(s(cons(head(cons(pairs, y0)), y1))) -> c20(MARK(cons(head(cons(pairs, y0)), y1))) MARK(s(cons(head(cons(odds, y0)), y1))) -> c20(MARK(cons(head(cons(odds, y0)), y1))) MARK(s(cons(head(cons(head(y0), y1)), y2))) -> c20(MARK(cons(head(cons(head(y0), y1)), y2))) MARK(s(cons(head(cons(tail(y0), y1)), y2))) -> c20(MARK(cons(head(cons(tail(y0), y1)), y2))) MARK(s(cons(head(cons(cons(y0, y1), y2)), y3))) -> c20(MARK(cons(head(cons(cons(y0, y1), y2)), y3))) MARK(s(cons(head(cons(s(y0), y1)), y2))) -> c20(MARK(cons(head(cons(s(y0), y1)), y2))) MARK(s(cons(head(cons(y0, y1)), y2))) -> c20(MARK(cons(head(cons(y0, y1)), y2))) MARK(s(cons(head(cons(0, y0)), y1))) -> c20(MARK(cons(head(cons(0, y0)), y1))) MARK(s(cons(incr(tail(nats)), y0))) -> c20(MARK(cons(incr(tail(nats)), y0))) MARK(s(cons(head(tail(nats)), y0))) -> c20(MARK(cons(head(tail(nats)), y0))) MARK(s(cons(tail(tail(nats)), y0))) -> c20(MARK(cons(tail(tail(nats)), y0))) MARK(s(incr(cons(nats, y0)))) -> c20(MARK(incr(cons(nats, y0)))) MARK(s(incr(cons(pairs, y0)))) -> c20(MARK(incr(cons(pairs, y0)))) MARK(s(incr(cons(odds, y0)))) -> c20(MARK(incr(cons(odds, y0)))) MARK(s(head(cons(nats, y0)))) -> c20(MARK(head(cons(nats, y0)))) MARK(s(head(cons(pairs, y0)))) -> c20(MARK(head(cons(pairs, y0)))) MARK(s(head(cons(odds, y0)))) -> c20(MARK(head(cons(odds, y0)))) MARK(incr(s(s(y0)))) -> c13(MARK(s(s(y0)))) MARK(incr(s(incr(s(y0))))) -> c13(MARK(s(incr(s(y0))))) MARK(incr(s(head(s(y0))))) -> c13(MARK(s(head(s(y0))))) MARK(incr(s(tail(cons(y0, y1))))) -> c13(MARK(s(tail(cons(y0, y1))))) MARK(incr(s(tail(s(y0))))) -> c13(MARK(s(tail(s(y0))))) MARK(incr(s(incr(incr(incr(y0)))))) -> c13(MARK(s(incr(incr(incr(y0)))))) MARK(incr(s(incr(incr(odds))))) -> c13(MARK(s(incr(incr(odds))))) MARK(incr(s(incr(incr(head(y0)))))) -> c13(MARK(s(incr(incr(head(y0)))))) MARK(incr(s(incr(incr(tail(y0)))))) -> c13(MARK(s(incr(incr(tail(y0)))))) MARK(incr(s(incr(incr(cons(y0, y1)))))) -> c13(MARK(s(incr(incr(cons(y0, y1)))))) MARK(incr(s(incr(incr(s(y0)))))) -> c13(MARK(s(incr(incr(s(y0)))))) MARK(incr(s(incr(incr(y0))))) -> c13(MARK(s(incr(incr(y0))))) MARK(incr(s(incr(head(incr(y0)))))) -> c13(MARK(s(incr(head(incr(y0)))))) MARK(incr(s(incr(head(odds))))) -> c13(MARK(s(incr(head(odds))))) MARK(incr(s(incr(head(head(y0)))))) -> c13(MARK(s(incr(head(head(y0)))))) MARK(incr(s(incr(head(tail(y0)))))) -> c13(MARK(s(incr(head(tail(y0)))))) MARK(incr(s(incr(head(cons(y0, y1)))))) -> c13(MARK(s(incr(head(cons(y0, y1)))))) MARK(incr(s(incr(head(s(y0)))))) -> c13(MARK(s(incr(head(s(y0)))))) MARK(incr(s(incr(head(y0))))) -> c13(MARK(s(incr(head(y0))))) MARK(incr(s(incr(tail(incr(y0)))))) -> c13(MARK(s(incr(tail(incr(y0)))))) MARK(incr(s(incr(tail(pairs))))) -> c13(MARK(s(incr(tail(pairs))))) MARK(incr(s(incr(tail(odds))))) -> c13(MARK(s(incr(tail(odds))))) MARK(incr(s(incr(tail(head(y0)))))) -> c13(MARK(s(incr(tail(head(y0)))))) MARK(incr(s(incr(tail(tail(y0)))))) -> c13(MARK(s(incr(tail(tail(y0)))))) MARK(incr(s(incr(tail(cons(y0, y1)))))) -> c13(MARK(s(incr(tail(cons(y0, y1)))))) MARK(incr(s(incr(tail(s(y0)))))) -> c13(MARK(s(incr(tail(s(y0)))))) MARK(incr(s(incr(tail(y0))))) -> c13(MARK(s(incr(tail(y0))))) MARK(incr(s(head(incr(incr(y0)))))) -> c13(MARK(s(head(incr(incr(y0)))))) MARK(incr(s(head(incr(odds))))) -> c13(MARK(s(head(incr(odds))))) MARK(incr(s(head(incr(head(y0)))))) -> c13(MARK(s(head(incr(head(y0)))))) MARK(incr(s(head(incr(tail(y0)))))) -> c13(MARK(s(head(incr(tail(y0)))))) MARK(incr(s(head(incr(cons(y0, y1)))))) -> c13(MARK(s(head(incr(cons(y0, y1)))))) MARK(incr(s(head(incr(s(y0)))))) -> c13(MARK(s(head(incr(s(y0)))))) MARK(incr(s(head(incr(y0))))) -> c13(MARK(s(head(incr(y0))))) MARK(incr(s(head(head(incr(y0)))))) -> c13(MARK(s(head(head(incr(y0)))))) MARK(incr(s(head(head(odds))))) -> c13(MARK(s(head(head(odds))))) MARK(incr(s(head(head(head(y0)))))) -> c13(MARK(s(head(head(head(y0)))))) MARK(incr(s(head(head(tail(y0)))))) -> c13(MARK(s(head(head(tail(y0)))))) MARK(incr(s(head(head(cons(y0, y1)))))) -> c13(MARK(s(head(head(cons(y0, y1)))))) MARK(incr(s(head(head(s(y0)))))) -> c13(MARK(s(head(head(s(y0)))))) MARK(incr(s(head(head(y0))))) -> c13(MARK(s(head(head(y0))))) MARK(incr(s(head(tail(incr(y0)))))) -> c13(MARK(s(head(tail(incr(y0)))))) MARK(incr(s(head(tail(pairs))))) -> c13(MARK(s(head(tail(pairs))))) MARK(incr(s(head(tail(odds))))) -> c13(MARK(s(head(tail(odds))))) MARK(incr(s(head(tail(head(y0)))))) -> c13(MARK(s(head(tail(head(y0)))))) MARK(incr(s(head(tail(tail(y0)))))) -> c13(MARK(s(head(tail(tail(y0)))))) MARK(incr(s(head(tail(cons(y0, y1)))))) -> c13(MARK(s(head(tail(cons(y0, y1)))))) MARK(incr(s(head(tail(s(y0)))))) -> c13(MARK(s(head(tail(s(y0)))))) MARK(incr(s(head(tail(y0))))) -> c13(MARK(s(head(tail(y0))))) MARK(incr(s(tail(incr(incr(y0)))))) -> c13(MARK(s(tail(incr(incr(y0)))))) MARK(incr(s(tail(incr(odds))))) -> c13(MARK(s(tail(incr(odds))))) MARK(incr(s(tail(incr(head(y0)))))) -> c13(MARK(s(tail(incr(head(y0)))))) MARK(incr(s(tail(incr(tail(y0)))))) -> c13(MARK(s(tail(incr(tail(y0)))))) MARK(incr(s(tail(incr(cons(y0, y1)))))) -> c13(MARK(s(tail(incr(cons(y0, y1)))))) MARK(incr(s(tail(incr(s(y0)))))) -> c13(MARK(s(tail(incr(s(y0)))))) MARK(incr(s(tail(incr(y0))))) -> c13(MARK(s(tail(incr(y0))))) MARK(incr(s(tail(head(incr(y0)))))) -> c13(MARK(s(tail(head(incr(y0)))))) MARK(incr(s(tail(head(odds))))) -> c13(MARK(s(tail(head(odds))))) MARK(incr(s(tail(head(head(y0)))))) -> c13(MARK(s(tail(head(head(y0)))))) MARK(incr(s(tail(head(tail(y0)))))) -> c13(MARK(s(tail(head(tail(y0)))))) MARK(incr(s(tail(head(cons(y0, y1)))))) -> c13(MARK(s(tail(head(cons(y0, y1)))))) MARK(incr(s(tail(head(s(y0)))))) -> c13(MARK(s(tail(head(s(y0)))))) MARK(incr(s(tail(head(y0))))) -> c13(MARK(s(tail(head(y0))))) MARK(incr(s(tail(tail(incr(y0)))))) -> c13(MARK(s(tail(tail(incr(y0)))))) MARK(incr(s(tail(tail(pairs))))) -> c13(MARK(s(tail(tail(pairs))))) MARK(incr(s(tail(tail(odds))))) -> c13(MARK(s(tail(tail(odds))))) MARK(incr(s(tail(tail(head(y0)))))) -> c13(MARK(s(tail(tail(head(y0)))))) MARK(incr(s(tail(tail(tail(y0)))))) -> c13(MARK(s(tail(tail(tail(y0)))))) MARK(incr(s(tail(tail(cons(y0, y1)))))) -> c13(MARK(s(tail(tail(cons(y0, y1)))))) MARK(incr(s(tail(tail(s(y0)))))) -> c13(MARK(s(tail(tail(s(y0)))))) MARK(incr(s(tail(tail(y0))))) -> c13(MARK(s(tail(tail(y0))))) MARK(incr(s(tail(pairs)))) -> c13(MARK(s(tail(pairs)))) MARK(incr(s(incr(incr(nats))))) -> c13(MARK(s(incr(incr(nats))))) MARK(incr(s(incr(incr(pairs))))) -> c13(MARK(s(incr(incr(pairs))))) MARK(incr(s(incr(odds)))) -> c13(MARK(s(incr(odds)))) MARK(incr(s(head(incr(nats))))) -> c13(MARK(s(head(incr(nats))))) MARK(incr(s(head(incr(pairs))))) -> c13(MARK(s(head(incr(pairs))))) MARK(incr(s(head(odds)))) -> c13(MARK(s(head(odds)))) MARK(incr(s(tail(incr(nats))))) -> c13(MARK(s(tail(incr(nats))))) MARK(incr(s(tail(incr(pairs))))) -> c13(MARK(s(tail(incr(pairs))))) MARK(incr(s(tail(odds)))) -> c13(MARK(s(tail(odds)))) MARK(incr(s(incr(cons(incr(y0), y1))))) -> c13(MARK(s(incr(cons(incr(y0), y1))))) MARK(incr(s(incr(cons(head(y0), y1))))) -> c13(MARK(s(incr(cons(head(y0), y1))))) MARK(incr(s(incr(cons(tail(y0), y1))))) -> c13(MARK(s(incr(cons(tail(y0), y1))))) MARK(incr(s(incr(cons(cons(y0, y1), y2))))) -> c13(MARK(s(incr(cons(cons(y0, y1), y2))))) MARK(incr(s(incr(cons(s(y0), y1))))) -> c13(MARK(s(incr(cons(s(y0), y1))))) MARK(incr(s(incr(cons(y0, y1))))) -> c13(MARK(s(incr(cons(y0, y1))))) MARK(incr(s(incr(head(nats))))) -> c13(MARK(s(incr(head(nats))))) MARK(incr(s(incr(head(pairs))))) -> c13(MARK(s(incr(head(pairs))))) MARK(incr(s(head(head(nats))))) -> c13(MARK(s(head(head(nats))))) MARK(incr(s(head(head(pairs))))) -> c13(MARK(s(head(head(pairs))))) MARK(incr(s(tail(head(nats))))) -> c13(MARK(s(tail(head(nats))))) MARK(incr(s(tail(head(pairs))))) -> c13(MARK(s(tail(head(pairs))))) MARK(incr(s(head(cons(incr(y0), y1))))) -> c13(MARK(s(head(cons(incr(y0), y1))))) MARK(incr(s(head(cons(head(y0), y1))))) -> c13(MARK(s(head(cons(head(y0), y1))))) MARK(incr(s(head(cons(tail(y0), y1))))) -> c13(MARK(s(head(cons(tail(y0), y1))))) MARK(incr(s(head(cons(cons(y0, y1), y2))))) -> c13(MARK(s(head(cons(cons(y0, y1), y2))))) MARK(incr(s(head(cons(s(y0), y1))))) -> c13(MARK(s(head(cons(s(y0), y1))))) MARK(incr(s(head(cons(y0, y1))))) -> c13(MARK(s(head(cons(y0, y1))))) MARK(incr(s(incr(tail(nats))))) -> c13(MARK(s(incr(tail(nats))))) MARK(incr(s(head(tail(nats))))) -> c13(MARK(s(head(tail(nats))))) MARK(incr(s(tail(tail(nats))))) -> c13(MARK(s(tail(tail(nats))))) MARK(incr(s(cons(cons(y0, y1), y2)))) -> c13(MARK(s(cons(cons(y0, y1), y2)))) MARK(incr(s(cons(s(y0), y1)))) -> c13(MARK(s(cons(s(y0), y1)))) MARK(incr(s(cons(incr(s(y0)), y1)))) -> c13(MARK(s(cons(incr(s(y0)), y1)))) MARK(incr(s(cons(head(s(y0)), y1)))) -> c13(MARK(s(cons(head(s(y0)), y1)))) MARK(incr(s(cons(tail(cons(y0, y1)), y2)))) -> c13(MARK(s(cons(tail(cons(y0, y1)), y2)))) MARK(incr(s(cons(tail(s(y0)), y1)))) -> c13(MARK(s(cons(tail(s(y0)), y1)))) MARK(incr(s(cons(incr(incr(incr(y0))), y1)))) -> c13(MARK(s(cons(incr(incr(incr(y0))), y1)))) MARK(incr(s(cons(incr(incr(odds)), y0)))) -> c13(MARK(s(cons(incr(incr(odds)), y0)))) MARK(incr(s(cons(incr(incr(head(y0))), y1)))) -> c13(MARK(s(cons(incr(incr(head(y0))), y1)))) MARK(incr(s(cons(incr(incr(tail(y0))), y1)))) -> c13(MARK(s(cons(incr(incr(tail(y0))), y1)))) MARK(incr(s(cons(incr(incr(cons(y0, y1))), y2)))) -> c13(MARK(s(cons(incr(incr(cons(y0, y1))), y2)))) MARK(incr(s(cons(incr(incr(s(y0))), y1)))) -> c13(MARK(s(cons(incr(incr(s(y0))), y1)))) MARK(incr(s(cons(incr(incr(y0)), y1)))) -> c13(MARK(s(cons(incr(incr(y0)), y1)))) MARK(incr(s(cons(incr(head(incr(y0))), y1)))) -> c13(MARK(s(cons(incr(head(incr(y0))), y1)))) MARK(incr(s(cons(incr(head(odds)), y0)))) -> c13(MARK(s(cons(incr(head(odds)), y0)))) MARK(incr(s(cons(incr(head(head(y0))), y1)))) -> c13(MARK(s(cons(incr(head(head(y0))), y1)))) MARK(incr(s(cons(incr(head(tail(y0))), y1)))) -> c13(MARK(s(cons(incr(head(tail(y0))), y1)))) MARK(incr(s(cons(incr(head(cons(y0, y1))), y2)))) -> c13(MARK(s(cons(incr(head(cons(y0, y1))), y2)))) MARK(incr(s(cons(incr(head(s(y0))), y1)))) -> c13(MARK(s(cons(incr(head(s(y0))), y1)))) MARK(incr(s(cons(incr(head(y0)), y1)))) -> c13(MARK(s(cons(incr(head(y0)), y1)))) MARK(incr(s(cons(incr(tail(incr(y0))), y1)))) -> c13(MARK(s(cons(incr(tail(incr(y0))), y1)))) MARK(incr(s(cons(incr(tail(pairs)), y0)))) -> c13(MARK(s(cons(incr(tail(pairs)), y0)))) MARK(incr(s(cons(incr(tail(odds)), y0)))) -> c13(MARK(s(cons(incr(tail(odds)), y0)))) MARK(incr(s(cons(incr(tail(head(y0))), y1)))) -> c13(MARK(s(cons(incr(tail(head(y0))), y1)))) MARK(incr(s(cons(incr(tail(tail(y0))), y1)))) -> c13(MARK(s(cons(incr(tail(tail(y0))), y1)))) MARK(incr(s(cons(incr(tail(cons(y0, y1))), y2)))) -> c13(MARK(s(cons(incr(tail(cons(y0, y1))), y2)))) MARK(incr(s(cons(incr(tail(s(y0))), y1)))) -> c13(MARK(s(cons(incr(tail(s(y0))), y1)))) MARK(incr(s(cons(incr(tail(y0)), y1)))) -> c13(MARK(s(cons(incr(tail(y0)), y1)))) MARK(incr(s(cons(head(incr(incr(y0))), y1)))) -> c13(MARK(s(cons(head(incr(incr(y0))), y1)))) MARK(incr(s(cons(head(incr(odds)), y0)))) -> c13(MARK(s(cons(head(incr(odds)), y0)))) MARK(incr(s(cons(head(incr(head(y0))), y1)))) -> c13(MARK(s(cons(head(incr(head(y0))), y1)))) MARK(incr(s(cons(head(incr(tail(y0))), y1)))) -> c13(MARK(s(cons(head(incr(tail(y0))), y1)))) MARK(incr(s(cons(head(incr(cons(y0, y1))), y2)))) -> c13(MARK(s(cons(head(incr(cons(y0, y1))), y2)))) MARK(incr(s(cons(head(incr(s(y0))), y1)))) -> c13(MARK(s(cons(head(incr(s(y0))), y1)))) MARK(incr(s(cons(head(incr(y0)), y1)))) -> c13(MARK(s(cons(head(incr(y0)), y1)))) MARK(incr(s(cons(head(head(incr(y0))), y1)))) -> c13(MARK(s(cons(head(head(incr(y0))), y1)))) MARK(incr(s(cons(head(head(odds)), y0)))) -> c13(MARK(s(cons(head(head(odds)), y0)))) MARK(incr(s(cons(head(head(head(y0))), y1)))) -> c13(MARK(s(cons(head(head(head(y0))), y1)))) MARK(incr(s(cons(head(head(tail(y0))), y1)))) -> c13(MARK(s(cons(head(head(tail(y0))), y1)))) MARK(incr(s(cons(head(head(cons(y0, y1))), y2)))) -> c13(MARK(s(cons(head(head(cons(y0, y1))), y2)))) MARK(incr(s(cons(head(head(s(y0))), y1)))) -> c13(MARK(s(cons(head(head(s(y0))), y1)))) MARK(incr(s(cons(head(head(y0)), y1)))) -> c13(MARK(s(cons(head(head(y0)), y1)))) MARK(incr(s(cons(head(tail(incr(y0))), y1)))) -> c13(MARK(s(cons(head(tail(incr(y0))), y1)))) MARK(incr(s(cons(head(tail(pairs)), y0)))) -> c13(MARK(s(cons(head(tail(pairs)), y0)))) MARK(incr(s(cons(head(tail(odds)), y0)))) -> c13(MARK(s(cons(head(tail(odds)), y0)))) MARK(incr(s(cons(head(tail(head(y0))), y1)))) -> c13(MARK(s(cons(head(tail(head(y0))), y1)))) MARK(incr(s(cons(head(tail(tail(y0))), y1)))) -> c13(MARK(s(cons(head(tail(tail(y0))), y1)))) MARK(incr(s(cons(head(tail(cons(y0, y1))), y2)))) -> c13(MARK(s(cons(head(tail(cons(y0, y1))), y2)))) MARK(incr(s(cons(head(tail(s(y0))), y1)))) -> c13(MARK(s(cons(head(tail(s(y0))), y1)))) MARK(incr(s(cons(head(tail(y0)), y1)))) -> c13(MARK(s(cons(head(tail(y0)), y1)))) MARK(incr(s(cons(tail(incr(incr(y0))), y1)))) -> c13(MARK(s(cons(tail(incr(incr(y0))), y1)))) MARK(incr(s(cons(tail(incr(odds)), y0)))) -> c13(MARK(s(cons(tail(incr(odds)), y0)))) MARK(incr(s(cons(tail(incr(head(y0))), y1)))) -> c13(MARK(s(cons(tail(incr(head(y0))), y1)))) MARK(incr(s(cons(tail(incr(tail(y0))), y1)))) -> c13(MARK(s(cons(tail(incr(tail(y0))), y1)))) MARK(incr(s(cons(tail(incr(cons(y0, y1))), y2)))) -> c13(MARK(s(cons(tail(incr(cons(y0, y1))), y2)))) MARK(incr(s(cons(tail(incr(s(y0))), y1)))) -> c13(MARK(s(cons(tail(incr(s(y0))), y1)))) MARK(incr(s(cons(tail(incr(y0)), y1)))) -> c13(MARK(s(cons(tail(incr(y0)), y1)))) MARK(incr(s(cons(tail(head(incr(y0))), y1)))) -> c13(MARK(s(cons(tail(head(incr(y0))), y1)))) MARK(incr(s(cons(tail(head(odds)), y0)))) -> c13(MARK(s(cons(tail(head(odds)), y0)))) MARK(incr(s(cons(tail(head(head(y0))), y1)))) -> c13(MARK(s(cons(tail(head(head(y0))), y1)))) MARK(incr(s(cons(tail(head(tail(y0))), y1)))) -> c13(MARK(s(cons(tail(head(tail(y0))), y1)))) MARK(incr(s(cons(tail(head(cons(y0, y1))), y2)))) -> c13(MARK(s(cons(tail(head(cons(y0, y1))), y2)))) MARK(incr(s(cons(tail(head(s(y0))), y1)))) -> c13(MARK(s(cons(tail(head(s(y0))), y1)))) MARK(incr(s(cons(tail(head(y0)), y1)))) -> c13(MARK(s(cons(tail(head(y0)), y1)))) MARK(incr(s(cons(tail(tail(incr(y0))), y1)))) -> c13(MARK(s(cons(tail(tail(incr(y0))), y1)))) MARK(incr(s(cons(tail(tail(pairs)), y0)))) -> c13(MARK(s(cons(tail(tail(pairs)), y0)))) MARK(incr(s(cons(tail(tail(odds)), y0)))) -> c13(MARK(s(cons(tail(tail(odds)), y0)))) MARK(incr(s(cons(tail(tail(head(y0))), y1)))) -> c13(MARK(s(cons(tail(tail(head(y0))), y1)))) MARK(incr(s(cons(tail(tail(tail(y0))), y1)))) -> c13(MARK(s(cons(tail(tail(tail(y0))), y1)))) MARK(incr(s(cons(tail(tail(cons(y0, y1))), y2)))) -> c13(MARK(s(cons(tail(tail(cons(y0, y1))), y2)))) MARK(incr(s(cons(tail(tail(s(y0))), y1)))) -> c13(MARK(s(cons(tail(tail(s(y0))), y1)))) MARK(incr(s(cons(tail(tail(y0)), y1)))) -> c13(MARK(s(cons(tail(tail(y0)), y1)))) MARK(incr(s(cons(tail(pairs), y0)))) -> c13(MARK(s(cons(tail(pairs), y0)))) MARK(incr(s(cons(incr(incr(nats)), y0)))) -> c13(MARK(s(cons(incr(incr(nats)), y0)))) MARK(incr(s(cons(incr(incr(pairs)), y0)))) -> c13(MARK(s(cons(incr(incr(pairs)), y0)))) MARK(incr(s(cons(incr(odds), y0)))) -> c13(MARK(s(cons(incr(odds), y0)))) MARK(incr(s(cons(head(incr(nats)), y0)))) -> c13(MARK(s(cons(head(incr(nats)), y0)))) MARK(incr(s(cons(head(incr(pairs)), y0)))) -> c13(MARK(s(cons(head(incr(pairs)), y0)))) MARK(incr(s(cons(head(odds), y0)))) -> c13(MARK(s(cons(head(odds), y0)))) MARK(incr(s(cons(tail(incr(nats)), y0)))) -> c13(MARK(s(cons(tail(incr(nats)), y0)))) MARK(incr(s(cons(tail(incr(pairs)), y0)))) -> c13(MARK(s(cons(tail(incr(pairs)), y0)))) MARK(incr(s(cons(tail(odds), y0)))) -> c13(MARK(s(cons(tail(odds), y0)))) MARK(incr(s(cons(incr(cons(nats, y0)), y1)))) -> c13(MARK(s(cons(incr(cons(nats, y0)), y1)))) MARK(incr(s(cons(incr(cons(incr(y0), y1)), y2)))) -> c13(MARK(s(cons(incr(cons(incr(y0), y1)), y2)))) MARK(incr(s(cons(incr(cons(pairs, y0)), y1)))) -> c13(MARK(s(cons(incr(cons(pairs, y0)), y1)))) MARK(incr(s(cons(incr(cons(odds, y0)), y1)))) -> c13(MARK(s(cons(incr(cons(odds, y0)), y1)))) MARK(incr(s(cons(incr(cons(head(y0), y1)), y2)))) -> c13(MARK(s(cons(incr(cons(head(y0), y1)), y2)))) MARK(incr(s(cons(incr(cons(tail(y0), y1)), y2)))) -> c13(MARK(s(cons(incr(cons(tail(y0), y1)), y2)))) MARK(incr(s(cons(incr(cons(cons(y0, y1), y2)), y3)))) -> c13(MARK(s(cons(incr(cons(cons(y0, y1), y2)), y3)))) MARK(incr(s(cons(incr(cons(s(y0), y1)), y2)))) -> c13(MARK(s(cons(incr(cons(s(y0), y1)), y2)))) MARK(incr(s(cons(incr(cons(y0, y1)), y2)))) -> c13(MARK(s(cons(incr(cons(y0, y1)), y2)))) MARK(incr(s(cons(incr(cons(0, y0)), y1)))) -> c13(MARK(s(cons(incr(cons(0, y0)), y1)))) MARK(incr(s(cons(incr(head(nats)), y0)))) -> c13(MARK(s(cons(incr(head(nats)), y0)))) MARK(incr(s(cons(incr(head(pairs)), y0)))) -> c13(MARK(s(cons(incr(head(pairs)), y0)))) MARK(incr(s(cons(head(head(nats)), y0)))) -> c13(MARK(s(cons(head(head(nats)), y0)))) MARK(incr(s(cons(head(head(pairs)), y0)))) -> c13(MARK(s(cons(head(head(pairs)), y0)))) MARK(incr(s(cons(tail(head(nats)), y0)))) -> c13(MARK(s(cons(tail(head(nats)), y0)))) MARK(incr(s(cons(tail(head(pairs)), y0)))) -> c13(MARK(s(cons(tail(head(pairs)), y0)))) MARK(incr(s(cons(head(cons(nats, y0)), y1)))) -> c13(MARK(s(cons(head(cons(nats, y0)), y1)))) MARK(incr(s(cons(head(cons(incr(y0), y1)), y2)))) -> c13(MARK(s(cons(head(cons(incr(y0), y1)), y2)))) MARK(incr(s(cons(head(cons(pairs, y0)), y1)))) -> c13(MARK(s(cons(head(cons(pairs, y0)), y1)))) MARK(incr(s(cons(head(cons(odds, y0)), y1)))) -> c13(MARK(s(cons(head(cons(odds, y0)), y1)))) MARK(incr(s(cons(head(cons(head(y0), y1)), y2)))) -> c13(MARK(s(cons(head(cons(head(y0), y1)), y2)))) MARK(incr(s(cons(head(cons(tail(y0), y1)), y2)))) -> c13(MARK(s(cons(head(cons(tail(y0), y1)), y2)))) MARK(incr(s(cons(head(cons(cons(y0, y1), y2)), y3)))) -> c13(MARK(s(cons(head(cons(cons(y0, y1), y2)), y3)))) MARK(incr(s(cons(head(cons(s(y0), y1)), y2)))) -> c13(MARK(s(cons(head(cons(s(y0), y1)), y2)))) MARK(incr(s(cons(head(cons(y0, y1)), y2)))) -> c13(MARK(s(cons(head(cons(y0, y1)), y2)))) MARK(incr(s(cons(head(cons(0, y0)), y1)))) -> c13(MARK(s(cons(head(cons(0, y0)), y1)))) MARK(incr(s(cons(incr(tail(nats)), y0)))) -> c13(MARK(s(cons(incr(tail(nats)), y0)))) MARK(incr(s(cons(head(tail(nats)), y0)))) -> c13(MARK(s(cons(head(tail(nats)), y0)))) MARK(incr(s(cons(tail(tail(nats)), y0)))) -> c13(MARK(s(cons(tail(tail(nats)), y0)))) MARK(incr(s(incr(cons(nats, y0))))) -> c13(MARK(s(incr(cons(nats, y0))))) MARK(incr(s(incr(cons(pairs, y0))))) -> c13(MARK(s(incr(cons(pairs, y0))))) MARK(incr(s(incr(cons(odds, y0))))) -> c13(MARK(s(incr(cons(odds, y0))))) MARK(incr(s(head(cons(nats, y0))))) -> c13(MARK(s(head(cons(nats, y0))))) MARK(incr(s(head(cons(pairs, y0))))) -> c13(MARK(s(head(cons(pairs, y0))))) MARK(incr(s(head(cons(odds, y0))))) -> c13(MARK(s(head(cons(odds, y0))))) S tuples: MARK(head(s(z0))) -> c16(MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c17(MARK(s(z0))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(tail(incr(incr(z0)))) -> c17(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(odds))) -> c17(A__TAIL(a__incr(a__odds)), MARK(incr(odds))) MARK(tail(incr(head(z0)))) -> c17(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c17(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c17(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c17(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c17(MARK(incr(x0))) MARK(tail(head(incr(z0)))) -> c17(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(odds))) -> c17(A__TAIL(a__head(a__odds)), MARK(head(odds))) MARK(tail(head(head(z0)))) -> c17(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c17(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c17(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c17(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) 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MARK(cons(head(tail(y0)), z1)) -> c18(MARK(head(tail(y0)))) MARK(cons(tail(incr(incr(y0))), z1)) -> c18(MARK(tail(incr(incr(y0))))) MARK(cons(tail(incr(odds)), z1)) -> c18(MARK(tail(incr(odds)))) MARK(cons(tail(incr(head(y0))), z1)) -> c18(MARK(tail(incr(head(y0))))) MARK(cons(tail(incr(tail(y0))), z1)) -> c18(MARK(tail(incr(tail(y0))))) MARK(cons(tail(incr(cons(y0, y1))), z1)) -> c18(MARK(tail(incr(cons(y0, y1))))) MARK(cons(tail(incr(s(y0))), z1)) -> c18(MARK(tail(incr(s(y0))))) MARK(cons(tail(incr(y0)), z1)) -> c18(MARK(tail(incr(y0)))) MARK(cons(tail(head(incr(y0))), z1)) -> c18(MARK(tail(head(incr(y0))))) MARK(cons(tail(head(odds)), z1)) -> c18(MARK(tail(head(odds)))) MARK(cons(tail(head(head(y0))), z1)) -> c18(MARK(tail(head(head(y0))))) MARK(cons(tail(head(tail(y0))), z1)) -> c18(MARK(tail(head(tail(y0))))) MARK(cons(tail(head(cons(y0, y1))), z1)) -> c18(MARK(tail(head(cons(y0, y1))))) MARK(cons(tail(head(s(y0))), z1)) -> c18(MARK(tail(head(s(y0))))) MARK(cons(tail(head(y0)), z1)) -> c18(MARK(tail(head(y0)))) MARK(cons(tail(tail(incr(y0))), z1)) -> c18(MARK(tail(tail(incr(y0))))) MARK(cons(tail(tail(pairs)), z1)) -> c18(MARK(tail(tail(pairs)))) MARK(cons(tail(tail(odds)), z1)) -> c18(MARK(tail(tail(odds)))) MARK(cons(tail(tail(head(y0))), z1)) -> c18(MARK(tail(tail(head(y0))))) MARK(cons(tail(tail(tail(y0))), z1)) -> c18(MARK(tail(tail(tail(y0))))) MARK(cons(tail(tail(cons(y0, y1))), z1)) -> c18(MARK(tail(tail(cons(y0, y1))))) MARK(cons(tail(tail(s(y0))), z1)) -> c18(MARK(tail(tail(s(y0))))) MARK(cons(tail(tail(y0)), z1)) -> c18(MARK(tail(tail(y0)))) MARK(cons(tail(pairs), z1)) -> c18(MARK(tail(pairs))) MARK(cons(incr(incr(nats)), z1)) -> c18(MARK(incr(incr(nats)))) MARK(cons(incr(incr(pairs)), z1)) -> c18(MARK(incr(incr(pairs)))) MARK(cons(incr(odds), z1)) -> c18(MARK(incr(odds))) MARK(cons(head(incr(nats)), z1)) -> c18(MARK(head(incr(nats)))) MARK(cons(head(incr(pairs)), z1)) -> c18(MARK(head(incr(pairs)))) MARK(cons(head(odds), z1)) -> c18(MARK(head(odds))) MARK(cons(tail(incr(nats)), z1)) -> c18(MARK(tail(incr(nats)))) MARK(cons(tail(incr(pairs)), z1)) -> c18(MARK(tail(incr(pairs)))) MARK(cons(tail(odds), z1)) -> c18(MARK(tail(odds))) MARK(cons(incr(cons(nats, y0)), z1)) -> c18(MARK(incr(cons(nats, y0)))) MARK(cons(incr(cons(incr(y0), y1)), z1)) -> c18(MARK(incr(cons(incr(y0), y1)))) MARK(cons(incr(cons(pairs, y0)), z1)) -> c18(MARK(incr(cons(pairs, y0)))) MARK(cons(incr(cons(odds, y0)), z1)) -> c18(MARK(incr(cons(odds, y0)))) MARK(cons(incr(cons(head(y0), y1)), z1)) -> c18(MARK(incr(cons(head(y0), y1)))) MARK(cons(incr(cons(tail(y0), y1)), z1)) -> c18(MARK(incr(cons(tail(y0), y1)))) MARK(cons(incr(cons(cons(y0, y1), y2)), z1)) -> c18(MARK(incr(cons(cons(y0, y1), y2)))) MARK(cons(incr(cons(s(y0), y1)), z1)) -> c18(MARK(incr(cons(s(y0), y1)))) MARK(cons(incr(cons(y0, y1)), z1)) -> c18(MARK(incr(cons(y0, y1)))) MARK(cons(incr(cons(0, y0)), z1)) -> c18(MARK(incr(cons(0, y0)))) MARK(cons(incr(head(nats)), z1)) -> c18(MARK(incr(head(nats)))) MARK(cons(incr(head(pairs)), z1)) -> c18(MARK(incr(head(pairs)))) MARK(cons(head(head(nats)), z1)) -> c18(MARK(head(head(nats)))) MARK(cons(head(head(pairs)), z1)) -> c18(MARK(head(head(pairs)))) MARK(cons(tail(head(nats)), z1)) -> c18(MARK(tail(head(nats)))) MARK(cons(tail(head(pairs)), z1)) -> c18(MARK(tail(head(pairs)))) MARK(cons(head(cons(nats, y0)), z1)) -> c18(MARK(head(cons(nats, y0)))) MARK(cons(head(cons(incr(y0), y1)), z1)) -> c18(MARK(head(cons(incr(y0), y1)))) MARK(cons(head(cons(pairs, y0)), z1)) -> c18(MARK(head(cons(pairs, y0)))) MARK(cons(head(cons(odds, y0)), z1)) -> c18(MARK(head(cons(odds, y0)))) MARK(cons(head(cons(head(y0), y1)), z1)) -> c18(MARK(head(cons(head(y0), y1)))) MARK(cons(head(cons(tail(y0), y1)), z1)) -> c18(MARK(head(cons(tail(y0), y1)))) MARK(cons(head(cons(cons(y0, y1), y2)), z1)) -> c18(MARK(head(cons(cons(y0, y1), y2)))) MARK(cons(head(cons(s(y0), y1)), z1)) -> c18(MARK(head(cons(s(y0), y1)))) MARK(cons(head(cons(y0, 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MARK(s(incr(incr(odds)))) -> c20(MARK(incr(incr(odds)))) MARK(s(incr(incr(head(y0))))) -> c20(MARK(incr(incr(head(y0))))) MARK(s(incr(incr(tail(y0))))) -> c20(MARK(incr(incr(tail(y0))))) MARK(s(incr(incr(cons(y0, y1))))) -> c20(MARK(incr(incr(cons(y0, y1))))) MARK(s(incr(incr(s(y0))))) -> c20(MARK(incr(incr(s(y0))))) MARK(s(incr(incr(y0)))) -> c20(MARK(incr(incr(y0)))) MARK(s(incr(head(incr(y0))))) -> c20(MARK(incr(head(incr(y0))))) MARK(s(incr(head(odds)))) -> c20(MARK(incr(head(odds)))) MARK(s(incr(head(head(y0))))) -> c20(MARK(incr(head(head(y0))))) MARK(s(incr(head(tail(y0))))) -> c20(MARK(incr(head(tail(y0))))) MARK(s(incr(head(cons(y0, y1))))) -> c20(MARK(incr(head(cons(y0, y1))))) MARK(s(incr(head(s(y0))))) -> c20(MARK(incr(head(s(y0))))) MARK(s(incr(head(y0)))) -> c20(MARK(incr(head(y0)))) MARK(s(incr(tail(incr(y0))))) -> c20(MARK(incr(tail(incr(y0))))) MARK(s(incr(tail(pairs)))) -> c20(MARK(incr(tail(pairs)))) MARK(s(incr(tail(odds)))) -> c20(MARK(incr(tail(odds)))) MARK(s(incr(tail(head(y0))))) -> c20(MARK(incr(tail(head(y0))))) MARK(s(incr(tail(tail(y0))))) -> c20(MARK(incr(tail(tail(y0))))) MARK(s(incr(tail(cons(y0, y1))))) -> c20(MARK(incr(tail(cons(y0, y1))))) MARK(s(incr(tail(s(y0))))) -> c20(MARK(incr(tail(s(y0))))) MARK(s(incr(tail(y0)))) -> c20(MARK(incr(tail(y0)))) MARK(s(head(incr(incr(y0))))) -> c20(MARK(head(incr(incr(y0))))) MARK(s(head(incr(odds)))) -> c20(MARK(head(incr(odds)))) MARK(s(head(incr(head(y0))))) -> c20(MARK(head(incr(head(y0))))) MARK(s(head(incr(tail(y0))))) -> c20(MARK(head(incr(tail(y0))))) MARK(s(head(incr(cons(y0, y1))))) -> c20(MARK(head(incr(cons(y0, y1))))) MARK(s(head(incr(s(y0))))) -> c20(MARK(head(incr(s(y0))))) MARK(s(head(incr(y0)))) -> c20(MARK(head(incr(y0)))) MARK(s(head(head(incr(y0))))) -> c20(MARK(head(head(incr(y0))))) MARK(s(head(head(odds)))) -> c20(MARK(head(head(odds)))) MARK(s(head(head(head(y0))))) -> c20(MARK(head(head(head(y0))))) MARK(s(head(head(tail(y0))))) -> c20(MARK(head(head(tail(y0))))) MARK(s(head(head(cons(y0, y1))))) -> c20(MARK(head(head(cons(y0, y1))))) MARK(s(head(head(s(y0))))) -> c20(MARK(head(head(s(y0))))) MARK(s(head(head(y0)))) -> c20(MARK(head(head(y0)))) MARK(s(head(tail(incr(y0))))) -> c20(MARK(head(tail(incr(y0))))) MARK(s(head(tail(pairs)))) -> c20(MARK(head(tail(pairs)))) MARK(s(head(tail(odds)))) -> c20(MARK(head(tail(odds)))) MARK(s(head(tail(head(y0))))) -> c20(MARK(head(tail(head(y0))))) MARK(s(head(tail(tail(y0))))) -> c20(MARK(head(tail(tail(y0))))) MARK(s(head(tail(cons(y0, y1))))) -> c20(MARK(head(tail(cons(y0, y1))))) MARK(s(head(tail(s(y0))))) -> c20(MARK(head(tail(s(y0))))) MARK(s(head(tail(y0)))) -> c20(MARK(head(tail(y0)))) MARK(s(tail(incr(incr(y0))))) -> c20(MARK(tail(incr(incr(y0))))) MARK(s(tail(incr(odds)))) -> c20(MARK(tail(incr(odds)))) MARK(s(tail(incr(head(y0))))) -> c20(MARK(tail(incr(head(y0))))) MARK(s(tail(incr(tail(y0))))) -> c20(MARK(tail(incr(tail(y0))))) MARK(s(tail(incr(cons(y0, y1))))) -> c20(MARK(tail(incr(cons(y0, y1))))) MARK(s(tail(incr(s(y0))))) -> c20(MARK(tail(incr(s(y0))))) MARK(s(tail(incr(y0)))) -> c20(MARK(tail(incr(y0)))) MARK(s(tail(head(incr(y0))))) -> c20(MARK(tail(head(incr(y0))))) MARK(s(tail(head(odds)))) -> c20(MARK(tail(head(odds)))) MARK(s(tail(head(head(y0))))) -> c20(MARK(tail(head(head(y0))))) MARK(s(tail(head(tail(y0))))) -> c20(MARK(tail(head(tail(y0))))) MARK(s(tail(head(cons(y0, y1))))) -> c20(MARK(tail(head(cons(y0, y1))))) MARK(s(tail(head(s(y0))))) -> c20(MARK(tail(head(s(y0))))) MARK(s(tail(head(y0)))) -> c20(MARK(tail(head(y0)))) MARK(s(tail(tail(incr(y0))))) -> c20(MARK(tail(tail(incr(y0))))) MARK(s(tail(tail(pairs)))) -> c20(MARK(tail(tail(pairs)))) MARK(s(tail(tail(odds)))) -> c20(MARK(tail(tail(odds)))) MARK(s(tail(tail(head(y0))))) -> c20(MARK(tail(tail(head(y0))))) MARK(s(tail(tail(tail(y0))))) -> c20(MARK(tail(tail(tail(y0))))) MARK(s(tail(tail(cons(y0, y1))))) -> c20(MARK(tail(tail(cons(y0, y1))))) MARK(s(tail(tail(s(y0))))) -> c20(MARK(tail(tail(s(y0))))) MARK(s(tail(tail(y0)))) -> c20(MARK(tail(tail(y0)))) MARK(s(tail(pairs))) -> c20(MARK(tail(pairs))) MARK(s(incr(incr(nats)))) -> c20(MARK(incr(incr(nats)))) MARK(s(incr(incr(pairs)))) -> c20(MARK(incr(incr(pairs)))) MARK(s(incr(odds))) -> c20(MARK(incr(odds))) MARK(s(head(incr(nats)))) -> c20(MARK(head(incr(nats)))) MARK(s(head(incr(pairs)))) -> c20(MARK(head(incr(pairs)))) MARK(s(head(odds))) -> c20(MARK(head(odds))) MARK(s(tail(incr(nats)))) -> c20(MARK(tail(incr(nats)))) MARK(s(tail(incr(pairs)))) -> c20(MARK(tail(incr(pairs)))) MARK(s(tail(odds))) -> c20(MARK(tail(odds))) MARK(s(incr(cons(incr(y0), y1)))) -> c20(MARK(incr(cons(incr(y0), y1)))) MARK(s(incr(cons(head(y0), y1)))) -> c20(MARK(incr(cons(head(y0), y1)))) MARK(s(incr(cons(tail(y0), y1)))) -> c20(MARK(incr(cons(tail(y0), y1)))) MARK(s(incr(cons(cons(y0, y1), y2)))) -> c20(MARK(incr(cons(cons(y0, y1), y2)))) MARK(s(incr(cons(s(y0), y1)))) -> c20(MARK(incr(cons(s(y0), y1)))) MARK(s(incr(cons(y0, y1)))) -> c20(MARK(incr(cons(y0, y1)))) MARK(s(incr(head(nats)))) -> c20(MARK(incr(head(nats)))) MARK(s(incr(head(pairs)))) -> c20(MARK(incr(head(pairs)))) MARK(s(head(head(nats)))) -> c20(MARK(head(head(nats)))) MARK(s(head(head(pairs)))) -> c20(MARK(head(head(pairs)))) MARK(s(tail(head(nats)))) -> c20(MARK(tail(head(nats)))) MARK(s(tail(head(pairs)))) -> c20(MARK(tail(head(pairs)))) MARK(s(head(cons(incr(y0), y1)))) -> c20(MARK(head(cons(incr(y0), y1)))) MARK(s(head(cons(head(y0), y1)))) -> c20(MARK(head(cons(head(y0), y1)))) MARK(s(head(cons(tail(y0), y1)))) -> c20(MARK(head(cons(tail(y0), y1)))) MARK(s(head(cons(cons(y0, y1), y2)))) -> c20(MARK(head(cons(cons(y0, y1), y2)))) MARK(s(head(cons(s(y0), y1)))) -> c20(MARK(head(cons(s(y0), y1)))) MARK(s(head(cons(y0, y1)))) -> c20(MARK(head(cons(y0, y1)))) MARK(s(incr(tail(nats)))) -> c20(MARK(incr(tail(nats)))) MARK(s(head(tail(nats)))) -> c20(MARK(head(tail(nats)))) MARK(s(tail(tail(nats)))) -> c20(MARK(tail(tail(nats)))) MARK(s(cons(cons(y0, y1), y2))) -> c20(MARK(cons(cons(y0, y1), y2))) MARK(s(cons(s(y0), y1))) -> c20(MARK(cons(s(y0), y1))) MARK(s(cons(incr(s(y0)), y1))) -> c20(MARK(cons(incr(s(y0)), y1))) MARK(s(cons(head(s(y0)), y1))) -> c20(MARK(cons(head(s(y0)), y1))) MARK(s(cons(tail(cons(y0, y1)), y2))) -> c20(MARK(cons(tail(cons(y0, y1)), y2))) MARK(s(cons(tail(s(y0)), y1))) -> c20(MARK(cons(tail(s(y0)), y1))) MARK(s(cons(incr(incr(incr(y0))), y1))) -> c20(MARK(cons(incr(incr(incr(y0))), y1))) MARK(s(cons(incr(incr(odds)), y0))) -> c20(MARK(cons(incr(incr(odds)), y0))) MARK(s(cons(incr(incr(head(y0))), y1))) -> c20(MARK(cons(incr(incr(head(y0))), y1))) MARK(s(cons(incr(incr(tail(y0))), y1))) -> c20(MARK(cons(incr(incr(tail(y0))), y1))) MARK(s(cons(incr(incr(cons(y0, y1))), y2))) -> c20(MARK(cons(incr(incr(cons(y0, y1))), y2))) MARK(s(cons(incr(incr(s(y0))), y1))) -> c20(MARK(cons(incr(incr(s(y0))), y1))) MARK(s(cons(incr(incr(y0)), y1))) -> c20(MARK(cons(incr(incr(y0)), y1))) MARK(s(cons(incr(head(incr(y0))), y1))) -> c20(MARK(cons(incr(head(incr(y0))), y1))) MARK(s(cons(incr(head(odds)), y0))) -> c20(MARK(cons(incr(head(odds)), y0))) MARK(s(cons(incr(head(head(y0))), y1))) -> c20(MARK(cons(incr(head(head(y0))), y1))) MARK(s(cons(incr(head(tail(y0))), y1))) -> c20(MARK(cons(incr(head(tail(y0))), y1))) MARK(s(cons(incr(head(cons(y0, y1))), y2))) -> c20(MARK(cons(incr(head(cons(y0, y1))), y2))) MARK(s(cons(incr(head(s(y0))), y1))) -> c20(MARK(cons(incr(head(s(y0))), y1))) MARK(s(cons(incr(head(y0)), y1))) -> c20(MARK(cons(incr(head(y0)), y1))) MARK(s(cons(incr(tail(incr(y0))), y1))) -> c20(MARK(cons(incr(tail(incr(y0))), y1))) MARK(s(cons(incr(tail(pairs)), y0))) -> c20(MARK(cons(incr(tail(pairs)), y0))) MARK(s(cons(incr(tail(odds)), y0))) -> c20(MARK(cons(incr(tail(odds)), y0))) MARK(s(cons(incr(tail(head(y0))), y1))) -> c20(MARK(cons(incr(tail(head(y0))), y1))) MARK(s(cons(incr(tail(tail(y0))), y1))) -> c20(MARK(cons(incr(tail(tail(y0))), y1))) MARK(s(cons(incr(tail(cons(y0, y1))), y2))) -> c20(MARK(cons(incr(tail(cons(y0, y1))), y2))) MARK(s(cons(incr(tail(s(y0))), y1))) -> c20(MARK(cons(incr(tail(s(y0))), y1))) MARK(s(cons(incr(tail(y0)), y1))) -> c20(MARK(cons(incr(tail(y0)), y1))) MARK(s(cons(head(incr(incr(y0))), y1))) -> c20(MARK(cons(head(incr(incr(y0))), y1))) MARK(s(cons(head(incr(odds)), y0))) -> c20(MARK(cons(head(incr(odds)), y0))) MARK(s(cons(head(incr(head(y0))), y1))) -> c20(MARK(cons(head(incr(head(y0))), y1))) MARK(s(cons(head(incr(tail(y0))), y1))) -> c20(MARK(cons(head(incr(tail(y0))), y1))) MARK(s(cons(head(incr(cons(y0, y1))), y2))) -> c20(MARK(cons(head(incr(cons(y0, y1))), y2))) MARK(s(cons(head(incr(s(y0))), y1))) -> c20(MARK(cons(head(incr(s(y0))), y1))) MARK(s(cons(head(incr(y0)), y1))) -> c20(MARK(cons(head(incr(y0)), y1))) MARK(s(cons(head(head(incr(y0))), y1))) -> c20(MARK(cons(head(head(incr(y0))), y1))) MARK(s(cons(head(head(odds)), y0))) -> c20(MARK(cons(head(head(odds)), y0))) MARK(s(cons(head(head(head(y0))), y1))) -> c20(MARK(cons(head(head(head(y0))), y1))) MARK(s(cons(head(head(tail(y0))), y1))) -> c20(MARK(cons(head(head(tail(y0))), y1))) MARK(s(cons(head(head(cons(y0, y1))), y2))) -> c20(MARK(cons(head(head(cons(y0, y1))), y2))) MARK(s(cons(head(head(s(y0))), y1))) -> c20(MARK(cons(head(head(s(y0))), y1))) MARK(s(cons(head(head(y0)), y1))) -> c20(MARK(cons(head(head(y0)), y1))) MARK(s(cons(head(tail(incr(y0))), y1))) -> c20(MARK(cons(head(tail(incr(y0))), y1))) MARK(s(cons(head(tail(pairs)), y0))) -> c20(MARK(cons(head(tail(pairs)), y0))) MARK(s(cons(head(tail(odds)), y0))) -> c20(MARK(cons(head(tail(odds)), y0))) MARK(s(cons(head(tail(head(y0))), y1))) -> c20(MARK(cons(head(tail(head(y0))), y1))) MARK(s(cons(head(tail(tail(y0))), y1))) -> c20(MARK(cons(head(tail(tail(y0))), y1))) MARK(s(cons(head(tail(cons(y0, y1))), y2))) -> c20(MARK(cons(head(tail(cons(y0, y1))), y2))) MARK(s(cons(head(tail(s(y0))), y1))) -> c20(MARK(cons(head(tail(s(y0))), y1))) MARK(s(cons(head(tail(y0)), y1))) -> c20(MARK(cons(head(tail(y0)), y1))) MARK(s(cons(tail(incr(incr(y0))), y1))) -> c20(MARK(cons(tail(incr(incr(y0))), y1))) MARK(s(cons(tail(incr(odds)), y0))) -> c20(MARK(cons(tail(incr(odds)), y0))) MARK(s(cons(tail(incr(head(y0))), y1))) -> c20(MARK(cons(tail(incr(head(y0))), y1))) MARK(s(cons(tail(incr(tail(y0))), y1))) -> c20(MARK(cons(tail(incr(tail(y0))), y1))) MARK(s(cons(tail(incr(cons(y0, y1))), y2))) -> c20(MARK(cons(tail(incr(cons(y0, y1))), y2))) MARK(s(cons(tail(incr(s(y0))), y1))) -> c20(MARK(cons(tail(incr(s(y0))), y1))) MARK(s(cons(tail(incr(y0)), y1))) -> c20(MARK(cons(tail(incr(y0)), y1))) MARK(s(cons(tail(head(incr(y0))), y1))) -> c20(MARK(cons(tail(head(incr(y0))), y1))) MARK(s(cons(tail(head(odds)), y0))) -> c20(MARK(cons(tail(head(odds)), y0))) MARK(s(cons(tail(head(head(y0))), y1))) -> c20(MARK(cons(tail(head(head(y0))), y1))) MARK(s(cons(tail(head(tail(y0))), y1))) -> c20(MARK(cons(tail(head(tail(y0))), y1))) MARK(s(cons(tail(head(cons(y0, y1))), y2))) -> c20(MARK(cons(tail(head(cons(y0, y1))), y2))) MARK(s(cons(tail(head(s(y0))), y1))) -> c20(MARK(cons(tail(head(s(y0))), y1))) MARK(s(cons(tail(head(y0)), y1))) -> c20(MARK(cons(tail(head(y0)), y1))) MARK(s(cons(tail(tail(incr(y0))), y1))) -> c20(MARK(cons(tail(tail(incr(y0))), y1))) MARK(s(cons(tail(tail(pairs)), y0))) -> c20(MARK(cons(tail(tail(pairs)), y0))) MARK(s(cons(tail(tail(odds)), y0))) -> c20(MARK(cons(tail(tail(odds)), y0))) MARK(s(cons(tail(tail(head(y0))), y1))) -> c20(MARK(cons(tail(tail(head(y0))), y1))) MARK(s(cons(tail(tail(tail(y0))), y1))) -> c20(MARK(cons(tail(tail(tail(y0))), y1))) MARK(s(cons(tail(tail(cons(y0, y1))), y2))) -> c20(MARK(cons(tail(tail(cons(y0, y1))), y2))) MARK(s(cons(tail(tail(s(y0))), y1))) -> c20(MARK(cons(tail(tail(s(y0))), y1))) MARK(s(cons(tail(tail(y0)), y1))) -> c20(MARK(cons(tail(tail(y0)), y1))) MARK(s(cons(tail(pairs), y0))) -> c20(MARK(cons(tail(pairs), y0))) MARK(s(cons(incr(incr(nats)), y0))) -> c20(MARK(cons(incr(incr(nats)), y0))) MARK(s(cons(incr(incr(pairs)), y0))) -> c20(MARK(cons(incr(incr(pairs)), y0))) MARK(s(cons(incr(odds), y0))) -> c20(MARK(cons(incr(odds), y0))) MARK(s(cons(head(incr(nats)), y0))) -> c20(MARK(cons(head(incr(nats)), y0))) MARK(s(cons(head(incr(pairs)), y0))) -> c20(MARK(cons(head(incr(pairs)), y0))) MARK(s(cons(head(odds), y0))) -> c20(MARK(cons(head(odds), y0))) MARK(s(cons(tail(incr(nats)), y0))) -> c20(MARK(cons(tail(incr(nats)), y0))) MARK(s(cons(tail(incr(pairs)), y0))) -> c20(MARK(cons(tail(incr(pairs)), y0))) MARK(s(cons(tail(odds), y0))) -> c20(MARK(cons(tail(odds), y0))) MARK(s(cons(incr(cons(nats, y0)), y1))) -> c20(MARK(cons(incr(cons(nats, y0)), y1))) MARK(s(cons(incr(cons(incr(y0), y1)), y2))) -> c20(MARK(cons(incr(cons(incr(y0), y1)), y2))) MARK(s(cons(incr(cons(pairs, y0)), y1))) -> c20(MARK(cons(incr(cons(pairs, y0)), y1))) MARK(s(cons(incr(cons(odds, y0)), y1))) -> c20(MARK(cons(incr(cons(odds, y0)), y1))) MARK(s(cons(incr(cons(head(y0), y1)), y2))) -> c20(MARK(cons(incr(cons(head(y0), y1)), y2))) MARK(s(cons(incr(cons(tail(y0), y1)), y2))) -> c20(MARK(cons(incr(cons(tail(y0), y1)), y2))) MARK(s(cons(incr(cons(cons(y0, y1), y2)), y3))) -> c20(MARK(cons(incr(cons(cons(y0, y1), y2)), y3))) MARK(s(cons(incr(cons(s(y0), y1)), y2))) -> c20(MARK(cons(incr(cons(s(y0), y1)), y2))) MARK(s(cons(incr(cons(y0, y1)), y2))) -> c20(MARK(cons(incr(cons(y0, y1)), y2))) MARK(s(cons(incr(cons(0, y0)), y1))) -> c20(MARK(cons(incr(cons(0, y0)), y1))) MARK(s(cons(incr(head(nats)), y0))) -> c20(MARK(cons(incr(head(nats)), y0))) MARK(s(cons(incr(head(pairs)), y0))) -> c20(MARK(cons(incr(head(pairs)), y0))) MARK(s(cons(head(head(nats)), y0))) -> c20(MARK(cons(head(head(nats)), y0))) MARK(s(cons(head(head(pairs)), y0))) -> c20(MARK(cons(head(head(pairs)), y0))) MARK(s(cons(tail(head(nats)), y0))) -> c20(MARK(cons(tail(head(nats)), y0))) MARK(s(cons(tail(head(pairs)), y0))) -> c20(MARK(cons(tail(head(pairs)), y0))) MARK(s(cons(head(cons(nats, y0)), y1))) -> c20(MARK(cons(head(cons(nats, y0)), y1))) MARK(s(cons(head(cons(incr(y0), y1)), y2))) -> c20(MARK(cons(head(cons(incr(y0), y1)), y2))) MARK(s(cons(head(cons(pairs, y0)), y1))) -> c20(MARK(cons(head(cons(pairs, y0)), y1))) MARK(s(cons(head(cons(odds, y0)), y1))) -> c20(MARK(cons(head(cons(odds, y0)), y1))) MARK(s(cons(head(cons(head(y0), y1)), y2))) -> c20(MARK(cons(head(cons(head(y0), y1)), y2))) MARK(s(cons(head(cons(tail(y0), y1)), y2))) -> c20(MARK(cons(head(cons(tail(y0), y1)), y2))) MARK(s(cons(head(cons(cons(y0, y1), y2)), y3))) -> c20(MARK(cons(head(cons(cons(y0, y1), y2)), y3))) MARK(s(cons(head(cons(s(y0), y1)), y2))) -> c20(MARK(cons(head(cons(s(y0), y1)), y2))) MARK(s(cons(head(cons(y0, y1)), y2))) -> c20(MARK(cons(head(cons(y0, y1)), y2))) MARK(s(cons(head(cons(0, y0)), y1))) -> c20(MARK(cons(head(cons(0, y0)), y1))) MARK(s(cons(incr(tail(nats)), y0))) -> c20(MARK(cons(incr(tail(nats)), y0))) MARK(s(cons(head(tail(nats)), y0))) -> c20(MARK(cons(head(tail(nats)), y0))) MARK(s(cons(tail(tail(nats)), y0))) -> c20(MARK(cons(tail(tail(nats)), y0))) MARK(s(incr(cons(nats, y0)))) -> c20(MARK(incr(cons(nats, y0)))) MARK(s(incr(cons(pairs, y0)))) -> c20(MARK(incr(cons(pairs, y0)))) MARK(s(incr(cons(odds, y0)))) -> c20(MARK(incr(cons(odds, y0)))) MARK(s(head(cons(nats, y0)))) -> c20(MARK(head(cons(nats, y0)))) MARK(s(head(cons(pairs, y0)))) -> c20(MARK(head(cons(pairs, y0)))) MARK(s(head(cons(odds, y0)))) -> c20(MARK(head(cons(odds, y0)))) MARK(incr(s(s(y0)))) -> c13(MARK(s(s(y0)))) MARK(incr(s(incr(s(y0))))) -> c13(MARK(s(incr(s(y0))))) MARK(incr(s(head(s(y0))))) -> c13(MARK(s(head(s(y0))))) MARK(incr(s(tail(cons(y0, y1))))) -> c13(MARK(s(tail(cons(y0, y1))))) MARK(incr(s(tail(s(y0))))) -> c13(MARK(s(tail(s(y0))))) MARK(incr(s(incr(incr(incr(y0)))))) -> c13(MARK(s(incr(incr(incr(y0)))))) MARK(incr(s(incr(incr(odds))))) -> c13(MARK(s(incr(incr(odds))))) MARK(incr(s(incr(incr(head(y0)))))) -> c13(MARK(s(incr(incr(head(y0)))))) MARK(incr(s(incr(incr(tail(y0)))))) -> c13(MARK(s(incr(incr(tail(y0)))))) MARK(incr(s(incr(incr(cons(y0, y1)))))) -> c13(MARK(s(incr(incr(cons(y0, y1)))))) MARK(incr(s(incr(incr(s(y0)))))) -> c13(MARK(s(incr(incr(s(y0)))))) MARK(incr(s(incr(incr(y0))))) -> c13(MARK(s(incr(incr(y0))))) MARK(incr(s(incr(head(incr(y0)))))) -> c13(MARK(s(incr(head(incr(y0)))))) MARK(incr(s(incr(head(odds))))) -> c13(MARK(s(incr(head(odds))))) MARK(incr(s(incr(head(head(y0)))))) -> c13(MARK(s(incr(head(head(y0)))))) MARK(incr(s(incr(head(tail(y0)))))) -> c13(MARK(s(incr(head(tail(y0)))))) MARK(incr(s(incr(head(cons(y0, y1)))))) -> c13(MARK(s(incr(head(cons(y0, y1)))))) MARK(incr(s(incr(head(s(y0)))))) -> c13(MARK(s(incr(head(s(y0)))))) MARK(incr(s(incr(head(y0))))) -> c13(MARK(s(incr(head(y0))))) MARK(incr(s(incr(tail(incr(y0)))))) -> c13(MARK(s(incr(tail(incr(y0)))))) MARK(incr(s(incr(tail(pairs))))) -> c13(MARK(s(incr(tail(pairs))))) MARK(incr(s(incr(tail(odds))))) -> c13(MARK(s(incr(tail(odds))))) MARK(incr(s(incr(tail(head(y0)))))) -> c13(MARK(s(incr(tail(head(y0)))))) MARK(incr(s(incr(tail(tail(y0)))))) -> c13(MARK(s(incr(tail(tail(y0)))))) MARK(incr(s(incr(tail(cons(y0, y1)))))) -> c13(MARK(s(incr(tail(cons(y0, y1)))))) MARK(incr(s(incr(tail(s(y0)))))) -> c13(MARK(s(incr(tail(s(y0)))))) MARK(incr(s(incr(tail(y0))))) -> c13(MARK(s(incr(tail(y0))))) MARK(incr(s(head(incr(incr(y0)))))) -> c13(MARK(s(head(incr(incr(y0)))))) MARK(incr(s(head(incr(odds))))) -> c13(MARK(s(head(incr(odds))))) MARK(incr(s(head(incr(head(y0)))))) -> c13(MARK(s(head(incr(head(y0)))))) MARK(incr(s(head(incr(tail(y0)))))) -> c13(MARK(s(head(incr(tail(y0)))))) MARK(incr(s(head(incr(cons(y0, y1)))))) -> c13(MARK(s(head(incr(cons(y0, y1)))))) MARK(incr(s(head(incr(s(y0)))))) -> c13(MARK(s(head(incr(s(y0)))))) MARK(incr(s(head(incr(y0))))) -> c13(MARK(s(head(incr(y0))))) MARK(incr(s(head(head(incr(y0)))))) -> c13(MARK(s(head(head(incr(y0)))))) MARK(incr(s(head(head(odds))))) -> c13(MARK(s(head(head(odds))))) MARK(incr(s(head(head(head(y0)))))) -> c13(MARK(s(head(head(head(y0)))))) MARK(incr(s(head(head(tail(y0)))))) -> c13(MARK(s(head(head(tail(y0)))))) MARK(incr(s(head(head(cons(y0, y1)))))) -> c13(MARK(s(head(head(cons(y0, y1)))))) MARK(incr(s(head(head(s(y0)))))) -> c13(MARK(s(head(head(s(y0)))))) MARK(incr(s(head(head(y0))))) -> c13(MARK(s(head(head(y0))))) MARK(incr(s(head(tail(incr(y0)))))) -> c13(MARK(s(head(tail(incr(y0)))))) MARK(incr(s(head(tail(pairs))))) -> c13(MARK(s(head(tail(pairs))))) MARK(incr(s(head(tail(odds))))) -> c13(MARK(s(head(tail(odds))))) MARK(incr(s(head(tail(head(y0)))))) -> c13(MARK(s(head(tail(head(y0)))))) MARK(incr(s(head(tail(tail(y0)))))) -> c13(MARK(s(head(tail(tail(y0)))))) MARK(incr(s(head(tail(cons(y0, y1)))))) -> c13(MARK(s(head(tail(cons(y0, y1)))))) MARK(incr(s(head(tail(s(y0)))))) -> c13(MARK(s(head(tail(s(y0)))))) MARK(incr(s(head(tail(y0))))) -> c13(MARK(s(head(tail(y0))))) MARK(incr(s(tail(incr(incr(y0)))))) -> c13(MARK(s(tail(incr(incr(y0)))))) MARK(incr(s(tail(incr(odds))))) -> c13(MARK(s(tail(incr(odds))))) MARK(incr(s(tail(incr(head(y0)))))) -> c13(MARK(s(tail(incr(head(y0)))))) MARK(incr(s(tail(incr(tail(y0)))))) -> c13(MARK(s(tail(incr(tail(y0)))))) MARK(incr(s(tail(incr(cons(y0, y1)))))) -> c13(MARK(s(tail(incr(cons(y0, y1)))))) MARK(incr(s(tail(incr(s(y0)))))) -> c13(MARK(s(tail(incr(s(y0)))))) MARK(incr(s(tail(incr(y0))))) -> c13(MARK(s(tail(incr(y0))))) MARK(incr(s(tail(head(incr(y0)))))) -> c13(MARK(s(tail(head(incr(y0)))))) MARK(incr(s(tail(head(odds))))) -> c13(MARK(s(tail(head(odds))))) MARK(incr(s(tail(head(head(y0)))))) -> c13(MARK(s(tail(head(head(y0)))))) MARK(incr(s(tail(head(tail(y0)))))) -> c13(MARK(s(tail(head(tail(y0)))))) MARK(incr(s(tail(head(cons(y0, y1)))))) -> c13(MARK(s(tail(head(cons(y0, y1)))))) MARK(incr(s(tail(head(s(y0)))))) -> c13(MARK(s(tail(head(s(y0)))))) MARK(incr(s(tail(head(y0))))) -> c13(MARK(s(tail(head(y0))))) MARK(incr(s(tail(tail(incr(y0)))))) -> c13(MARK(s(tail(tail(incr(y0)))))) MARK(incr(s(tail(tail(pairs))))) -> c13(MARK(s(tail(tail(pairs))))) MARK(incr(s(tail(tail(odds))))) -> c13(MARK(s(tail(tail(odds))))) MARK(incr(s(tail(tail(head(y0)))))) -> c13(MARK(s(tail(tail(head(y0)))))) MARK(incr(s(tail(tail(tail(y0)))))) -> c13(MARK(s(tail(tail(tail(y0)))))) MARK(incr(s(tail(tail(cons(y0, y1)))))) -> c13(MARK(s(tail(tail(cons(y0, y1)))))) MARK(incr(s(tail(tail(s(y0)))))) -> c13(MARK(s(tail(tail(s(y0)))))) MARK(incr(s(tail(tail(y0))))) -> c13(MARK(s(tail(tail(y0))))) MARK(incr(s(tail(pairs)))) -> c13(MARK(s(tail(pairs)))) MARK(incr(s(incr(incr(nats))))) -> c13(MARK(s(incr(incr(nats))))) MARK(incr(s(incr(incr(pairs))))) -> c13(MARK(s(incr(incr(pairs))))) MARK(incr(s(incr(odds)))) -> c13(MARK(s(incr(odds)))) MARK(incr(s(head(incr(nats))))) -> c13(MARK(s(head(incr(nats))))) MARK(incr(s(head(incr(pairs))))) -> c13(MARK(s(head(incr(pairs))))) MARK(incr(s(head(odds)))) -> c13(MARK(s(head(odds)))) MARK(incr(s(tail(incr(nats))))) -> c13(MARK(s(tail(incr(nats))))) MARK(incr(s(tail(incr(pairs))))) -> c13(MARK(s(tail(incr(pairs))))) MARK(incr(s(tail(odds)))) -> c13(MARK(s(tail(odds)))) MARK(incr(s(incr(cons(incr(y0), y1))))) -> c13(MARK(s(incr(cons(incr(y0), y1))))) MARK(incr(s(incr(cons(head(y0), y1))))) -> c13(MARK(s(incr(cons(head(y0), y1))))) MARK(incr(s(incr(cons(tail(y0), y1))))) -> c13(MARK(s(incr(cons(tail(y0), y1))))) MARK(incr(s(incr(cons(cons(y0, y1), y2))))) -> c13(MARK(s(incr(cons(cons(y0, y1), y2))))) MARK(incr(s(incr(cons(s(y0), y1))))) -> c13(MARK(s(incr(cons(s(y0), y1))))) MARK(incr(s(incr(cons(y0, y1))))) -> c13(MARK(s(incr(cons(y0, y1))))) MARK(incr(s(incr(head(nats))))) -> c13(MARK(s(incr(head(nats))))) MARK(incr(s(incr(head(pairs))))) -> c13(MARK(s(incr(head(pairs))))) MARK(incr(s(head(head(nats))))) -> c13(MARK(s(head(head(nats))))) MARK(incr(s(head(head(pairs))))) -> c13(MARK(s(head(head(pairs))))) MARK(incr(s(tail(head(nats))))) -> c13(MARK(s(tail(head(nats))))) MARK(incr(s(tail(head(pairs))))) -> c13(MARK(s(tail(head(pairs))))) MARK(incr(s(head(cons(incr(y0), y1))))) -> c13(MARK(s(head(cons(incr(y0), y1))))) MARK(incr(s(head(cons(head(y0), y1))))) -> c13(MARK(s(head(cons(head(y0), y1))))) MARK(incr(s(head(cons(tail(y0), y1))))) -> c13(MARK(s(head(cons(tail(y0), y1))))) MARK(incr(s(head(cons(cons(y0, y1), y2))))) -> c13(MARK(s(head(cons(cons(y0, y1), y2))))) MARK(incr(s(head(cons(s(y0), y1))))) -> c13(MARK(s(head(cons(s(y0), y1))))) MARK(incr(s(head(cons(y0, y1))))) -> c13(MARK(s(head(cons(y0, y1))))) MARK(incr(s(incr(tail(nats))))) -> c13(MARK(s(incr(tail(nats))))) MARK(incr(s(head(tail(nats))))) -> c13(MARK(s(head(tail(nats))))) MARK(incr(s(tail(tail(nats))))) -> c13(MARK(s(tail(tail(nats))))) MARK(incr(s(cons(cons(y0, y1), y2)))) -> c13(MARK(s(cons(cons(y0, y1), y2)))) MARK(incr(s(cons(s(y0), y1)))) -> c13(MARK(s(cons(s(y0), y1)))) MARK(incr(s(cons(incr(s(y0)), y1)))) -> c13(MARK(s(cons(incr(s(y0)), y1)))) MARK(incr(s(cons(head(s(y0)), y1)))) -> c13(MARK(s(cons(head(s(y0)), y1)))) MARK(incr(s(cons(tail(cons(y0, y1)), y2)))) -> c13(MARK(s(cons(tail(cons(y0, y1)), y2)))) MARK(incr(s(cons(tail(s(y0)), y1)))) -> c13(MARK(s(cons(tail(s(y0)), y1)))) MARK(incr(s(cons(incr(incr(incr(y0))), y1)))) -> c13(MARK(s(cons(incr(incr(incr(y0))), y1)))) MARK(incr(s(cons(incr(incr(odds)), y0)))) -> c13(MARK(s(cons(incr(incr(odds)), y0)))) MARK(incr(s(cons(incr(incr(head(y0))), y1)))) -> c13(MARK(s(cons(incr(incr(head(y0))), y1)))) MARK(incr(s(cons(incr(incr(tail(y0))), y1)))) -> c13(MARK(s(cons(incr(incr(tail(y0))), y1)))) MARK(incr(s(cons(incr(incr(cons(y0, y1))), y2)))) -> c13(MARK(s(cons(incr(incr(cons(y0, y1))), y2)))) MARK(incr(s(cons(incr(incr(s(y0))), y1)))) -> c13(MARK(s(cons(incr(incr(s(y0))), y1)))) MARK(incr(s(cons(incr(incr(y0)), y1)))) -> c13(MARK(s(cons(incr(incr(y0)), y1)))) MARK(incr(s(cons(incr(head(incr(y0))), y1)))) -> c13(MARK(s(cons(incr(head(incr(y0))), y1)))) MARK(incr(s(cons(incr(head(odds)), y0)))) -> c13(MARK(s(cons(incr(head(odds)), y0)))) MARK(incr(s(cons(incr(head(head(y0))), y1)))) -> c13(MARK(s(cons(incr(head(head(y0))), y1)))) MARK(incr(s(cons(incr(head(tail(y0))), y1)))) -> c13(MARK(s(cons(incr(head(tail(y0))), y1)))) MARK(incr(s(cons(incr(head(cons(y0, y1))), y2)))) -> c13(MARK(s(cons(incr(head(cons(y0, y1))), y2)))) MARK(incr(s(cons(incr(head(s(y0))), y1)))) -> c13(MARK(s(cons(incr(head(s(y0))), y1)))) MARK(incr(s(cons(incr(head(y0)), y1)))) -> c13(MARK(s(cons(incr(head(y0)), y1)))) MARK(incr(s(cons(incr(tail(incr(y0))), y1)))) -> c13(MARK(s(cons(incr(tail(incr(y0))), y1)))) MARK(incr(s(cons(incr(tail(pairs)), y0)))) -> c13(MARK(s(cons(incr(tail(pairs)), y0)))) MARK(incr(s(cons(incr(tail(odds)), y0)))) -> c13(MARK(s(cons(incr(tail(odds)), y0)))) MARK(incr(s(cons(incr(tail(head(y0))), y1)))) -> c13(MARK(s(cons(incr(tail(head(y0))), y1)))) MARK(incr(s(cons(incr(tail(tail(y0))), y1)))) -> c13(MARK(s(cons(incr(tail(tail(y0))), y1)))) MARK(incr(s(cons(incr(tail(cons(y0, y1))), y2)))) -> c13(MARK(s(cons(incr(tail(cons(y0, y1))), y2)))) MARK(incr(s(cons(incr(tail(s(y0))), y1)))) -> c13(MARK(s(cons(incr(tail(s(y0))), y1)))) MARK(incr(s(cons(incr(tail(y0)), y1)))) -> c13(MARK(s(cons(incr(tail(y0)), y1)))) MARK(incr(s(cons(head(incr(incr(y0))), y1)))) -> c13(MARK(s(cons(head(incr(incr(y0))), y1)))) MARK(incr(s(cons(head(incr(odds)), y0)))) -> c13(MARK(s(cons(head(incr(odds)), y0)))) MARK(incr(s(cons(head(incr(head(y0))), y1)))) -> c13(MARK(s(cons(head(incr(head(y0))), y1)))) MARK(incr(s(cons(head(incr(tail(y0))), y1)))) -> c13(MARK(s(cons(head(incr(tail(y0))), y1)))) MARK(incr(s(cons(head(incr(cons(y0, y1))), y2)))) -> c13(MARK(s(cons(head(incr(cons(y0, y1))), y2)))) MARK(incr(s(cons(head(incr(s(y0))), y1)))) -> c13(MARK(s(cons(head(incr(s(y0))), y1)))) MARK(incr(s(cons(head(incr(y0)), y1)))) -> c13(MARK(s(cons(head(incr(y0)), y1)))) MARK(incr(s(cons(head(head(incr(y0))), y1)))) -> c13(MARK(s(cons(head(head(incr(y0))), y1)))) MARK(incr(s(cons(head(head(odds)), y0)))) -> c13(MARK(s(cons(head(head(odds)), y0)))) MARK(incr(s(cons(head(head(head(y0))), y1)))) -> c13(MARK(s(cons(head(head(head(y0))), y1)))) MARK(incr(s(cons(head(head(tail(y0))), y1)))) -> c13(MARK(s(cons(head(head(tail(y0))), y1)))) MARK(incr(s(cons(head(head(cons(y0, y1))), y2)))) -> c13(MARK(s(cons(head(head(cons(y0, y1))), y2)))) MARK(incr(s(cons(head(head(s(y0))), y1)))) -> c13(MARK(s(cons(head(head(s(y0))), y1)))) MARK(incr(s(cons(head(head(y0)), y1)))) -> c13(MARK(s(cons(head(head(y0)), y1)))) MARK(incr(s(cons(head(tail(incr(y0))), y1)))) -> c13(MARK(s(cons(head(tail(incr(y0))), y1)))) MARK(incr(s(cons(head(tail(pairs)), y0)))) -> c13(MARK(s(cons(head(tail(pairs)), y0)))) MARK(incr(s(cons(head(tail(odds)), y0)))) -> c13(MARK(s(cons(head(tail(odds)), y0)))) MARK(incr(s(cons(head(tail(head(y0))), y1)))) -> c13(MARK(s(cons(head(tail(head(y0))), y1)))) MARK(incr(s(cons(head(tail(tail(y0))), y1)))) -> c13(MARK(s(cons(head(tail(tail(y0))), y1)))) MARK(incr(s(cons(head(tail(cons(y0, y1))), y2)))) -> c13(MARK(s(cons(head(tail(cons(y0, y1))), y2)))) MARK(incr(s(cons(head(tail(s(y0))), y1)))) -> c13(MARK(s(cons(head(tail(s(y0))), y1)))) MARK(incr(s(cons(head(tail(y0)), y1)))) -> c13(MARK(s(cons(head(tail(y0)), y1)))) MARK(incr(s(cons(tail(incr(incr(y0))), y1)))) -> c13(MARK(s(cons(tail(incr(incr(y0))), y1)))) MARK(incr(s(cons(tail(incr(odds)), y0)))) -> c13(MARK(s(cons(tail(incr(odds)), y0)))) MARK(incr(s(cons(tail(incr(head(y0))), y1)))) -> c13(MARK(s(cons(tail(incr(head(y0))), y1)))) MARK(incr(s(cons(tail(incr(tail(y0))), y1)))) -> c13(MARK(s(cons(tail(incr(tail(y0))), y1)))) MARK(incr(s(cons(tail(incr(cons(y0, y1))), y2)))) -> c13(MARK(s(cons(tail(incr(cons(y0, y1))), y2)))) MARK(incr(s(cons(tail(incr(s(y0))), y1)))) -> c13(MARK(s(cons(tail(incr(s(y0))), y1)))) MARK(incr(s(cons(tail(incr(y0)), y1)))) -> c13(MARK(s(cons(tail(incr(y0)), y1)))) MARK(incr(s(cons(tail(head(incr(y0))), y1)))) -> c13(MARK(s(cons(tail(head(incr(y0))), y1)))) MARK(incr(s(cons(tail(head(odds)), y0)))) -> c13(MARK(s(cons(tail(head(odds)), y0)))) MARK(incr(s(cons(tail(head(head(y0))), y1)))) -> c13(MARK(s(cons(tail(head(head(y0))), y1)))) MARK(incr(s(cons(tail(head(tail(y0))), y1)))) -> c13(MARK(s(cons(tail(head(tail(y0))), y1)))) MARK(incr(s(cons(tail(head(cons(y0, y1))), y2)))) -> c13(MARK(s(cons(tail(head(cons(y0, y1))), y2)))) MARK(incr(s(cons(tail(head(s(y0))), y1)))) -> c13(MARK(s(cons(tail(head(s(y0))), y1)))) MARK(incr(s(cons(tail(head(y0)), y1)))) -> c13(MARK(s(cons(tail(head(y0)), y1)))) MARK(incr(s(cons(tail(tail(incr(y0))), y1)))) -> c13(MARK(s(cons(tail(tail(incr(y0))), y1)))) MARK(incr(s(cons(tail(tail(pairs)), y0)))) -> c13(MARK(s(cons(tail(tail(pairs)), y0)))) MARK(incr(s(cons(tail(tail(odds)), y0)))) -> c13(MARK(s(cons(tail(tail(odds)), y0)))) MARK(incr(s(cons(tail(tail(head(y0))), y1)))) -> c13(MARK(s(cons(tail(tail(head(y0))), y1)))) MARK(incr(s(cons(tail(tail(tail(y0))), y1)))) -> c13(MARK(s(cons(tail(tail(tail(y0))), y1)))) MARK(incr(s(cons(tail(tail(cons(y0, y1))), y2)))) -> c13(MARK(s(cons(tail(tail(cons(y0, y1))), y2)))) MARK(incr(s(cons(tail(tail(s(y0))), y1)))) -> c13(MARK(s(cons(tail(tail(s(y0))), y1)))) MARK(incr(s(cons(tail(tail(y0)), y1)))) -> c13(MARK(s(cons(tail(tail(y0)), y1)))) MARK(incr(s(cons(tail(pairs), y0)))) -> c13(MARK(s(cons(tail(pairs), y0)))) MARK(incr(s(cons(incr(incr(nats)), y0)))) -> c13(MARK(s(cons(incr(incr(nats)), y0)))) MARK(incr(s(cons(incr(incr(pairs)), y0)))) -> c13(MARK(s(cons(incr(incr(pairs)), y0)))) MARK(incr(s(cons(incr(odds), y0)))) -> c13(MARK(s(cons(incr(odds), y0)))) MARK(incr(s(cons(head(incr(nats)), y0)))) -> c13(MARK(s(cons(head(incr(nats)), y0)))) MARK(incr(s(cons(head(incr(pairs)), y0)))) -> c13(MARK(s(cons(head(incr(pairs)), y0)))) MARK(incr(s(cons(head(odds), y0)))) -> c13(MARK(s(cons(head(odds), y0)))) MARK(incr(s(cons(tail(incr(nats)), y0)))) -> c13(MARK(s(cons(tail(incr(nats)), y0)))) MARK(incr(s(cons(tail(incr(pairs)), y0)))) -> c13(MARK(s(cons(tail(incr(pairs)), y0)))) MARK(incr(s(cons(tail(odds), y0)))) -> c13(MARK(s(cons(tail(odds), y0)))) MARK(incr(s(cons(incr(cons(nats, y0)), y1)))) -> c13(MARK(s(cons(incr(cons(nats, y0)), y1)))) MARK(incr(s(cons(incr(cons(incr(y0), y1)), y2)))) -> c13(MARK(s(cons(incr(cons(incr(y0), y1)), y2)))) MARK(incr(s(cons(incr(cons(pairs, y0)), y1)))) -> c13(MARK(s(cons(incr(cons(pairs, y0)), y1)))) MARK(incr(s(cons(incr(cons(odds, y0)), y1)))) -> c13(MARK(s(cons(incr(cons(odds, y0)), y1)))) MARK(incr(s(cons(incr(cons(head(y0), y1)), y2)))) -> c13(MARK(s(cons(incr(cons(head(y0), y1)), y2)))) MARK(incr(s(cons(incr(cons(tail(y0), y1)), y2)))) -> c13(MARK(s(cons(incr(cons(tail(y0), y1)), y2)))) MARK(incr(s(cons(incr(cons(cons(y0, y1), y2)), y3)))) -> c13(MARK(s(cons(incr(cons(cons(y0, y1), y2)), y3)))) MARK(incr(s(cons(incr(cons(s(y0), y1)), y2)))) -> c13(MARK(s(cons(incr(cons(s(y0), y1)), y2)))) MARK(incr(s(cons(incr(cons(y0, y1)), y2)))) -> c13(MARK(s(cons(incr(cons(y0, y1)), y2)))) MARK(incr(s(cons(incr(cons(0, y0)), y1)))) -> c13(MARK(s(cons(incr(cons(0, y0)), y1)))) MARK(incr(s(cons(incr(head(nats)), y0)))) -> c13(MARK(s(cons(incr(head(nats)), y0)))) MARK(incr(s(cons(incr(head(pairs)), y0)))) -> c13(MARK(s(cons(incr(head(pairs)), y0)))) MARK(incr(s(cons(head(head(nats)), y0)))) -> c13(MARK(s(cons(head(head(nats)), y0)))) MARK(incr(s(cons(head(head(pairs)), y0)))) -> c13(MARK(s(cons(head(head(pairs)), y0)))) MARK(incr(s(cons(tail(head(nats)), y0)))) -> c13(MARK(s(cons(tail(head(nats)), y0)))) MARK(incr(s(cons(tail(head(pairs)), y0)))) -> c13(MARK(s(cons(tail(head(pairs)), y0)))) MARK(incr(s(cons(head(cons(nats, y0)), y1)))) -> c13(MARK(s(cons(head(cons(nats, y0)), y1)))) MARK(incr(s(cons(head(cons(incr(y0), y1)), y2)))) -> c13(MARK(s(cons(head(cons(incr(y0), y1)), y2)))) MARK(incr(s(cons(head(cons(pairs, y0)), y1)))) -> c13(MARK(s(cons(head(cons(pairs, y0)), y1)))) MARK(incr(s(cons(head(cons(odds, y0)), y1)))) -> c13(MARK(s(cons(head(cons(odds, y0)), y1)))) MARK(incr(s(cons(head(cons(head(y0), y1)), y2)))) -> c13(MARK(s(cons(head(cons(head(y0), y1)), y2)))) MARK(incr(s(cons(head(cons(tail(y0), y1)), y2)))) -> c13(MARK(s(cons(head(cons(tail(y0), y1)), y2)))) MARK(incr(s(cons(head(cons(cons(y0, y1), y2)), y3)))) -> c13(MARK(s(cons(head(cons(cons(y0, y1), y2)), y3)))) MARK(incr(s(cons(head(cons(s(y0), y1)), y2)))) -> c13(MARK(s(cons(head(cons(s(y0), y1)), y2)))) MARK(incr(s(cons(head(cons(y0, y1)), y2)))) -> c13(MARK(s(cons(head(cons(y0, y1)), y2)))) MARK(incr(s(cons(head(cons(0, y0)), y1)))) -> c13(MARK(s(cons(head(cons(0, y0)), y1)))) MARK(incr(s(cons(incr(tail(nats)), y0)))) -> c13(MARK(s(cons(incr(tail(nats)), y0)))) MARK(incr(s(cons(head(tail(nats)), y0)))) -> c13(MARK(s(cons(head(tail(nats)), y0)))) MARK(incr(s(cons(tail(tail(nats)), y0)))) -> c13(MARK(s(cons(tail(tail(nats)), y0)))) MARK(incr(s(incr(cons(nats, y0))))) -> c13(MARK(s(incr(cons(nats, y0))))) MARK(incr(s(incr(cons(pairs, y0))))) -> c13(MARK(s(incr(cons(pairs, y0))))) MARK(incr(s(incr(cons(odds, y0))))) -> c13(MARK(s(incr(cons(odds, y0))))) MARK(incr(s(head(cons(nats, y0))))) -> c13(MARK(s(head(cons(nats, y0))))) MARK(incr(s(head(cons(pairs, y0))))) -> c13(MARK(s(head(cons(pairs, y0))))) MARK(incr(s(head(cons(odds, y0))))) -> c13(MARK(s(head(cons(odds, y0))))) K tuples:none Defined Rule Symbols: a__nats, a__pairs, a__odds, a__incr_1, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: MARK_1, A__INCR_1, A__HEAD_1, A__TAIL_1 Compound Symbols: c16_1, c17_2, c17_1, c13_2, c13_1, c16_2, c6_1, c8_1, c10_1, c18_1, c20_1 ---------------------------------------- (195) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace MARK(head(s(z0))) -> c16(MARK(s(z0))) by MARK(head(s(s(y0)))) -> c16(MARK(s(s(y0)))) MARK(head(s(incr(s(y0))))) -> c16(MARK(s(incr(s(y0))))) MARK(head(s(head(s(y0))))) -> c16(MARK(s(head(s(y0))))) MARK(head(s(tail(cons(y0, y1))))) -> c16(MARK(s(tail(cons(y0, y1))))) MARK(head(s(tail(s(y0))))) -> c16(MARK(s(tail(s(y0))))) MARK(head(s(incr(incr(incr(y0)))))) -> c16(MARK(s(incr(incr(incr(y0)))))) MARK(head(s(incr(incr(odds))))) -> c16(MARK(s(incr(incr(odds))))) MARK(head(s(incr(incr(head(y0)))))) -> c16(MARK(s(incr(incr(head(y0)))))) MARK(head(s(incr(incr(tail(y0)))))) -> c16(MARK(s(incr(incr(tail(y0)))))) MARK(head(s(incr(incr(cons(y0, y1)))))) -> c16(MARK(s(incr(incr(cons(y0, y1)))))) MARK(head(s(incr(incr(s(y0)))))) -> c16(MARK(s(incr(incr(s(y0)))))) MARK(head(s(incr(incr(y0))))) -> c16(MARK(s(incr(incr(y0))))) MARK(head(s(incr(head(incr(y0)))))) -> c16(MARK(s(incr(head(incr(y0)))))) MARK(head(s(incr(head(odds))))) -> c16(MARK(s(incr(head(odds))))) MARK(head(s(incr(head(head(y0)))))) -> c16(MARK(s(incr(head(head(y0)))))) MARK(head(s(incr(head(tail(y0)))))) -> c16(MARK(s(incr(head(tail(y0)))))) MARK(head(s(incr(head(cons(y0, y1)))))) -> c16(MARK(s(incr(head(cons(y0, y1)))))) MARK(head(s(incr(head(s(y0)))))) -> c16(MARK(s(incr(head(s(y0)))))) MARK(head(s(incr(head(y0))))) -> c16(MARK(s(incr(head(y0))))) MARK(head(s(incr(tail(incr(y0)))))) -> c16(MARK(s(incr(tail(incr(y0)))))) MARK(head(s(incr(tail(pairs))))) -> c16(MARK(s(incr(tail(pairs))))) MARK(head(s(incr(tail(odds))))) -> c16(MARK(s(incr(tail(odds))))) MARK(head(s(incr(tail(head(y0)))))) -> c16(MARK(s(incr(tail(head(y0)))))) MARK(head(s(incr(tail(tail(y0)))))) -> c16(MARK(s(incr(tail(tail(y0)))))) MARK(head(s(incr(tail(cons(y0, y1)))))) -> c16(MARK(s(incr(tail(cons(y0, y1)))))) MARK(head(s(incr(tail(s(y0)))))) -> c16(MARK(s(incr(tail(s(y0)))))) MARK(head(s(incr(tail(y0))))) -> c16(MARK(s(incr(tail(y0))))) MARK(head(s(head(incr(incr(y0)))))) -> c16(MARK(s(head(incr(incr(y0)))))) MARK(head(s(head(incr(odds))))) -> c16(MARK(s(head(incr(odds))))) MARK(head(s(head(incr(head(y0)))))) -> c16(MARK(s(head(incr(head(y0)))))) MARK(head(s(head(incr(tail(y0)))))) -> c16(MARK(s(head(incr(tail(y0)))))) MARK(head(s(head(incr(cons(y0, y1)))))) -> c16(MARK(s(head(incr(cons(y0, y1)))))) MARK(head(s(head(incr(s(y0)))))) -> c16(MARK(s(head(incr(s(y0)))))) MARK(head(s(head(incr(y0))))) -> c16(MARK(s(head(incr(y0))))) MARK(head(s(head(head(incr(y0)))))) -> c16(MARK(s(head(head(incr(y0)))))) MARK(head(s(head(head(odds))))) -> c16(MARK(s(head(head(odds))))) MARK(head(s(head(head(head(y0)))))) -> c16(MARK(s(head(head(head(y0)))))) MARK(head(s(head(head(tail(y0)))))) -> c16(MARK(s(head(head(tail(y0)))))) MARK(head(s(head(head(cons(y0, y1)))))) -> c16(MARK(s(head(head(cons(y0, y1)))))) MARK(head(s(head(head(s(y0)))))) -> c16(MARK(s(head(head(s(y0)))))) MARK(head(s(head(head(y0))))) -> c16(MARK(s(head(head(y0))))) MARK(head(s(head(tail(incr(y0)))))) -> c16(MARK(s(head(tail(incr(y0)))))) MARK(head(s(head(tail(pairs))))) -> c16(MARK(s(head(tail(pairs))))) MARK(head(s(head(tail(odds))))) -> c16(MARK(s(head(tail(odds))))) MARK(head(s(head(tail(head(y0)))))) -> c16(MARK(s(head(tail(head(y0)))))) MARK(head(s(head(tail(tail(y0)))))) -> c16(MARK(s(head(tail(tail(y0)))))) MARK(head(s(head(tail(cons(y0, y1)))))) -> c16(MARK(s(head(tail(cons(y0, y1)))))) MARK(head(s(head(tail(s(y0)))))) -> c16(MARK(s(head(tail(s(y0)))))) MARK(head(s(head(tail(y0))))) -> c16(MARK(s(head(tail(y0))))) MARK(head(s(tail(incr(incr(y0)))))) -> c16(MARK(s(tail(incr(incr(y0)))))) MARK(head(s(tail(incr(odds))))) -> c16(MARK(s(tail(incr(odds))))) MARK(head(s(tail(incr(head(y0)))))) -> c16(MARK(s(tail(incr(head(y0)))))) MARK(head(s(tail(incr(tail(y0)))))) -> c16(MARK(s(tail(incr(tail(y0)))))) MARK(head(s(tail(incr(cons(y0, y1)))))) -> c16(MARK(s(tail(incr(cons(y0, y1)))))) MARK(head(s(tail(incr(s(y0)))))) -> c16(MARK(s(tail(incr(s(y0)))))) MARK(head(s(tail(incr(y0))))) -> c16(MARK(s(tail(incr(y0))))) MARK(head(s(tail(head(incr(y0)))))) -> c16(MARK(s(tail(head(incr(y0)))))) MARK(head(s(tail(head(odds))))) -> c16(MARK(s(tail(head(odds))))) MARK(head(s(tail(head(head(y0)))))) -> c16(MARK(s(tail(head(head(y0)))))) MARK(head(s(tail(head(tail(y0)))))) -> c16(MARK(s(tail(head(tail(y0)))))) MARK(head(s(tail(head(cons(y0, y1)))))) -> c16(MARK(s(tail(head(cons(y0, y1)))))) MARK(head(s(tail(head(s(y0)))))) -> c16(MARK(s(tail(head(s(y0)))))) MARK(head(s(tail(head(y0))))) -> c16(MARK(s(tail(head(y0))))) MARK(head(s(tail(tail(incr(y0)))))) -> c16(MARK(s(tail(tail(incr(y0)))))) MARK(head(s(tail(tail(pairs))))) -> c16(MARK(s(tail(tail(pairs))))) MARK(head(s(tail(tail(odds))))) -> c16(MARK(s(tail(tail(odds))))) MARK(head(s(tail(tail(head(y0)))))) -> c16(MARK(s(tail(tail(head(y0)))))) MARK(head(s(tail(tail(tail(y0)))))) -> c16(MARK(s(tail(tail(tail(y0)))))) MARK(head(s(tail(tail(cons(y0, y1)))))) -> c16(MARK(s(tail(tail(cons(y0, y1)))))) MARK(head(s(tail(tail(s(y0)))))) -> c16(MARK(s(tail(tail(s(y0)))))) MARK(head(s(tail(tail(y0))))) -> c16(MARK(s(tail(tail(y0))))) MARK(head(s(tail(pairs)))) -> c16(MARK(s(tail(pairs)))) MARK(head(s(incr(incr(nats))))) -> c16(MARK(s(incr(incr(nats))))) MARK(head(s(incr(incr(pairs))))) -> c16(MARK(s(incr(incr(pairs))))) MARK(head(s(incr(odds)))) -> c16(MARK(s(incr(odds)))) MARK(head(s(head(incr(nats))))) -> c16(MARK(s(head(incr(nats))))) MARK(head(s(head(incr(pairs))))) -> c16(MARK(s(head(incr(pairs))))) MARK(head(s(head(odds)))) -> c16(MARK(s(head(odds)))) MARK(head(s(tail(incr(nats))))) -> c16(MARK(s(tail(incr(nats))))) MARK(head(s(tail(incr(pairs))))) -> c16(MARK(s(tail(incr(pairs))))) MARK(head(s(tail(odds)))) -> c16(MARK(s(tail(odds)))) MARK(head(s(incr(cons(incr(y0), y1))))) -> c16(MARK(s(incr(cons(incr(y0), y1))))) MARK(head(s(incr(cons(head(y0), y1))))) -> c16(MARK(s(incr(cons(head(y0), y1))))) MARK(head(s(incr(cons(tail(y0), y1))))) -> c16(MARK(s(incr(cons(tail(y0), y1))))) MARK(head(s(incr(cons(cons(y0, y1), y2))))) -> c16(MARK(s(incr(cons(cons(y0, y1), y2))))) MARK(head(s(incr(cons(s(y0), y1))))) -> c16(MARK(s(incr(cons(s(y0), y1))))) MARK(head(s(incr(cons(y0, y1))))) -> c16(MARK(s(incr(cons(y0, y1))))) MARK(head(s(incr(head(nats))))) -> c16(MARK(s(incr(head(nats))))) MARK(head(s(incr(head(pairs))))) -> c16(MARK(s(incr(head(pairs))))) MARK(head(s(head(head(nats))))) -> c16(MARK(s(head(head(nats))))) MARK(head(s(head(head(pairs))))) -> c16(MARK(s(head(head(pairs))))) MARK(head(s(tail(head(nats))))) -> c16(MARK(s(tail(head(nats))))) MARK(head(s(tail(head(pairs))))) -> c16(MARK(s(tail(head(pairs))))) MARK(head(s(head(cons(incr(y0), y1))))) -> c16(MARK(s(head(cons(incr(y0), y1))))) MARK(head(s(head(cons(head(y0), y1))))) -> c16(MARK(s(head(cons(head(y0), y1))))) MARK(head(s(head(cons(tail(y0), y1))))) -> c16(MARK(s(head(cons(tail(y0), y1))))) MARK(head(s(head(cons(cons(y0, y1), y2))))) -> c16(MARK(s(head(cons(cons(y0, y1), y2))))) MARK(head(s(head(cons(s(y0), y1))))) -> c16(MARK(s(head(cons(s(y0), y1))))) MARK(head(s(head(cons(y0, y1))))) -> c16(MARK(s(head(cons(y0, y1))))) MARK(head(s(incr(tail(nats))))) -> c16(MARK(s(incr(tail(nats))))) MARK(head(s(head(tail(nats))))) -> c16(MARK(s(head(tail(nats))))) MARK(head(s(tail(tail(nats))))) -> c16(MARK(s(tail(tail(nats))))) MARK(head(s(cons(cons(y0, y1), y2)))) -> c16(MARK(s(cons(cons(y0, y1), y2)))) MARK(head(s(cons(s(y0), y1)))) -> c16(MARK(s(cons(s(y0), y1)))) MARK(head(s(cons(incr(s(y0)), y1)))) -> c16(MARK(s(cons(incr(s(y0)), y1)))) MARK(head(s(cons(head(s(y0)), y1)))) -> c16(MARK(s(cons(head(s(y0)), y1)))) MARK(head(s(cons(tail(cons(y0, y1)), y2)))) -> c16(MARK(s(cons(tail(cons(y0, y1)), y2)))) MARK(head(s(cons(tail(s(y0)), y1)))) -> c16(MARK(s(cons(tail(s(y0)), y1)))) MARK(head(s(cons(incr(incr(incr(y0))), y1)))) -> c16(MARK(s(cons(incr(incr(incr(y0))), y1)))) MARK(head(s(cons(incr(incr(odds)), y0)))) -> c16(MARK(s(cons(incr(incr(odds)), y0)))) MARK(head(s(cons(incr(incr(head(y0))), y1)))) -> c16(MARK(s(cons(incr(incr(head(y0))), y1)))) MARK(head(s(cons(incr(incr(tail(y0))), y1)))) -> c16(MARK(s(cons(incr(incr(tail(y0))), y1)))) MARK(head(s(cons(incr(incr(cons(y0, y1))), y2)))) -> c16(MARK(s(cons(incr(incr(cons(y0, y1))), y2)))) MARK(head(s(cons(incr(incr(s(y0))), y1)))) -> c16(MARK(s(cons(incr(incr(s(y0))), y1)))) MARK(head(s(cons(incr(incr(y0)), y1)))) -> c16(MARK(s(cons(incr(incr(y0)), y1)))) MARK(head(s(cons(incr(head(incr(y0))), y1)))) -> c16(MARK(s(cons(incr(head(incr(y0))), y1)))) MARK(head(s(cons(incr(head(odds)), y0)))) -> c16(MARK(s(cons(incr(head(odds)), y0)))) MARK(head(s(cons(incr(head(head(y0))), y1)))) -> c16(MARK(s(cons(incr(head(head(y0))), y1)))) MARK(head(s(cons(incr(head(tail(y0))), y1)))) -> c16(MARK(s(cons(incr(head(tail(y0))), y1)))) MARK(head(s(cons(incr(head(cons(y0, y1))), y2)))) -> c16(MARK(s(cons(incr(head(cons(y0, y1))), y2)))) MARK(head(s(cons(incr(head(s(y0))), y1)))) -> c16(MARK(s(cons(incr(head(s(y0))), y1)))) MARK(head(s(cons(incr(head(y0)), y1)))) -> c16(MARK(s(cons(incr(head(y0)), y1)))) MARK(head(s(cons(incr(tail(incr(y0))), y1)))) -> c16(MARK(s(cons(incr(tail(incr(y0))), y1)))) MARK(head(s(cons(incr(tail(pairs)), y0)))) -> c16(MARK(s(cons(incr(tail(pairs)), y0)))) MARK(head(s(cons(incr(tail(odds)), y0)))) -> c16(MARK(s(cons(incr(tail(odds)), y0)))) MARK(head(s(cons(incr(tail(head(y0))), y1)))) -> c16(MARK(s(cons(incr(tail(head(y0))), y1)))) MARK(head(s(cons(incr(tail(tail(y0))), y1)))) -> c16(MARK(s(cons(incr(tail(tail(y0))), y1)))) MARK(head(s(cons(incr(tail(cons(y0, y1))), y2)))) -> c16(MARK(s(cons(incr(tail(cons(y0, y1))), y2)))) MARK(head(s(cons(incr(tail(s(y0))), y1)))) -> c16(MARK(s(cons(incr(tail(s(y0))), y1)))) MARK(head(s(cons(incr(tail(y0)), y1)))) -> c16(MARK(s(cons(incr(tail(y0)), y1)))) MARK(head(s(cons(head(incr(incr(y0))), y1)))) -> c16(MARK(s(cons(head(incr(incr(y0))), y1)))) MARK(head(s(cons(head(incr(odds)), y0)))) -> c16(MARK(s(cons(head(incr(odds)), y0)))) MARK(head(s(cons(head(incr(head(y0))), y1)))) -> c16(MARK(s(cons(head(incr(head(y0))), y1)))) MARK(head(s(cons(head(incr(tail(y0))), y1)))) -> c16(MARK(s(cons(head(incr(tail(y0))), y1)))) MARK(head(s(cons(head(incr(cons(y0, y1))), y2)))) -> c16(MARK(s(cons(head(incr(cons(y0, y1))), y2)))) MARK(head(s(cons(head(incr(s(y0))), y1)))) -> c16(MARK(s(cons(head(incr(s(y0))), y1)))) MARK(head(s(cons(head(incr(y0)), y1)))) -> c16(MARK(s(cons(head(incr(y0)), y1)))) MARK(head(s(cons(head(head(incr(y0))), y1)))) -> c16(MARK(s(cons(head(head(incr(y0))), y1)))) MARK(head(s(cons(head(head(odds)), y0)))) -> c16(MARK(s(cons(head(head(odds)), y0)))) MARK(head(s(cons(head(head(head(y0))), y1)))) -> c16(MARK(s(cons(head(head(head(y0))), y1)))) MARK(head(s(cons(head(head(tail(y0))), y1)))) -> c16(MARK(s(cons(head(head(tail(y0))), y1)))) MARK(head(s(cons(head(head(cons(y0, y1))), y2)))) -> c16(MARK(s(cons(head(head(cons(y0, y1))), y2)))) MARK(head(s(cons(head(head(s(y0))), y1)))) -> c16(MARK(s(cons(head(head(s(y0))), y1)))) MARK(head(s(cons(head(head(y0)), y1)))) -> c16(MARK(s(cons(head(head(y0)), y1)))) MARK(head(s(cons(head(tail(incr(y0))), y1)))) -> c16(MARK(s(cons(head(tail(incr(y0))), y1)))) MARK(head(s(cons(head(tail(pairs)), y0)))) -> c16(MARK(s(cons(head(tail(pairs)), y0)))) MARK(head(s(cons(head(tail(odds)), y0)))) -> c16(MARK(s(cons(head(tail(odds)), y0)))) MARK(head(s(cons(head(tail(head(y0))), y1)))) -> c16(MARK(s(cons(head(tail(head(y0))), y1)))) MARK(head(s(cons(head(tail(tail(y0))), y1)))) -> c16(MARK(s(cons(head(tail(tail(y0))), y1)))) MARK(head(s(cons(head(tail(cons(y0, y1))), y2)))) -> c16(MARK(s(cons(head(tail(cons(y0, y1))), y2)))) MARK(head(s(cons(head(tail(s(y0))), y1)))) -> c16(MARK(s(cons(head(tail(s(y0))), y1)))) MARK(head(s(cons(head(tail(y0)), y1)))) -> c16(MARK(s(cons(head(tail(y0)), y1)))) MARK(head(s(cons(tail(incr(incr(y0))), y1)))) -> c16(MARK(s(cons(tail(incr(incr(y0))), y1)))) MARK(head(s(cons(tail(incr(odds)), y0)))) -> c16(MARK(s(cons(tail(incr(odds)), y0)))) MARK(head(s(cons(tail(incr(head(y0))), y1)))) -> c16(MARK(s(cons(tail(incr(head(y0))), y1)))) MARK(head(s(cons(tail(incr(tail(y0))), y1)))) -> c16(MARK(s(cons(tail(incr(tail(y0))), y1)))) MARK(head(s(cons(tail(incr(cons(y0, y1))), y2)))) -> c16(MARK(s(cons(tail(incr(cons(y0, y1))), y2)))) MARK(head(s(cons(tail(incr(s(y0))), y1)))) -> c16(MARK(s(cons(tail(incr(s(y0))), y1)))) MARK(head(s(cons(tail(incr(y0)), y1)))) -> c16(MARK(s(cons(tail(incr(y0)), y1)))) MARK(head(s(cons(tail(head(incr(y0))), y1)))) -> c16(MARK(s(cons(tail(head(incr(y0))), y1)))) MARK(head(s(cons(tail(head(odds)), y0)))) -> c16(MARK(s(cons(tail(head(odds)), y0)))) MARK(head(s(cons(tail(head(head(y0))), y1)))) -> c16(MARK(s(cons(tail(head(head(y0))), y1)))) MARK(head(s(cons(tail(head(tail(y0))), y1)))) -> c16(MARK(s(cons(tail(head(tail(y0))), y1)))) MARK(head(s(cons(tail(head(cons(y0, y1))), y2)))) -> c16(MARK(s(cons(tail(head(cons(y0, y1))), y2)))) MARK(head(s(cons(tail(head(s(y0))), y1)))) -> c16(MARK(s(cons(tail(head(s(y0))), y1)))) MARK(head(s(cons(tail(head(y0)), y1)))) -> c16(MARK(s(cons(tail(head(y0)), y1)))) MARK(head(s(cons(tail(tail(incr(y0))), y1)))) -> c16(MARK(s(cons(tail(tail(incr(y0))), y1)))) MARK(head(s(cons(tail(tail(pairs)), y0)))) -> c16(MARK(s(cons(tail(tail(pairs)), y0)))) MARK(head(s(cons(tail(tail(odds)), y0)))) -> c16(MARK(s(cons(tail(tail(odds)), y0)))) MARK(head(s(cons(tail(tail(head(y0))), y1)))) -> c16(MARK(s(cons(tail(tail(head(y0))), y1)))) MARK(head(s(cons(tail(tail(tail(y0))), y1)))) -> c16(MARK(s(cons(tail(tail(tail(y0))), y1)))) MARK(head(s(cons(tail(tail(cons(y0, y1))), y2)))) -> c16(MARK(s(cons(tail(tail(cons(y0, y1))), y2)))) MARK(head(s(cons(tail(tail(s(y0))), y1)))) -> c16(MARK(s(cons(tail(tail(s(y0))), y1)))) MARK(head(s(cons(tail(tail(y0)), y1)))) -> c16(MARK(s(cons(tail(tail(y0)), y1)))) MARK(head(s(cons(tail(pairs), y0)))) -> c16(MARK(s(cons(tail(pairs), y0)))) MARK(head(s(cons(incr(incr(nats)), y0)))) -> c16(MARK(s(cons(incr(incr(nats)), y0)))) MARK(head(s(cons(incr(incr(pairs)), y0)))) -> c16(MARK(s(cons(incr(incr(pairs)), y0)))) MARK(head(s(cons(incr(odds), y0)))) -> c16(MARK(s(cons(incr(odds), y0)))) MARK(head(s(cons(head(incr(nats)), y0)))) -> c16(MARK(s(cons(head(incr(nats)), y0)))) MARK(head(s(cons(head(incr(pairs)), y0)))) -> c16(MARK(s(cons(head(incr(pairs)), y0)))) MARK(head(s(cons(head(odds), y0)))) -> c16(MARK(s(cons(head(odds), y0)))) MARK(head(s(cons(tail(incr(nats)), y0)))) -> c16(MARK(s(cons(tail(incr(nats)), y0)))) MARK(head(s(cons(tail(incr(pairs)), y0)))) -> c16(MARK(s(cons(tail(incr(pairs)), y0)))) MARK(head(s(cons(tail(odds), y0)))) -> c16(MARK(s(cons(tail(odds), y0)))) MARK(head(s(cons(incr(cons(nats, y0)), y1)))) -> c16(MARK(s(cons(incr(cons(nats, y0)), y1)))) MARK(head(s(cons(incr(cons(incr(y0), y1)), y2)))) -> c16(MARK(s(cons(incr(cons(incr(y0), y1)), y2)))) MARK(head(s(cons(incr(cons(pairs, y0)), y1)))) -> c16(MARK(s(cons(incr(cons(pairs, y0)), y1)))) MARK(head(s(cons(incr(cons(odds, y0)), y1)))) -> c16(MARK(s(cons(incr(cons(odds, y0)), y1)))) MARK(head(s(cons(incr(cons(head(y0), y1)), y2)))) -> c16(MARK(s(cons(incr(cons(head(y0), y1)), y2)))) MARK(head(s(cons(incr(cons(tail(y0), y1)), y2)))) -> c16(MARK(s(cons(incr(cons(tail(y0), y1)), y2)))) MARK(head(s(cons(incr(cons(cons(y0, y1), y2)), y3)))) -> c16(MARK(s(cons(incr(cons(cons(y0, y1), y2)), y3)))) MARK(head(s(cons(incr(cons(s(y0), y1)), y2)))) -> c16(MARK(s(cons(incr(cons(s(y0), y1)), y2)))) MARK(head(s(cons(incr(cons(y0, y1)), y2)))) -> c16(MARK(s(cons(incr(cons(y0, y1)), y2)))) MARK(head(s(cons(incr(cons(0, y0)), y1)))) -> c16(MARK(s(cons(incr(cons(0, y0)), y1)))) MARK(head(s(cons(incr(head(nats)), y0)))) -> c16(MARK(s(cons(incr(head(nats)), y0)))) MARK(head(s(cons(incr(head(pairs)), y0)))) -> c16(MARK(s(cons(incr(head(pairs)), y0)))) MARK(head(s(cons(head(head(nats)), y0)))) -> c16(MARK(s(cons(head(head(nats)), y0)))) MARK(head(s(cons(head(head(pairs)), y0)))) -> c16(MARK(s(cons(head(head(pairs)), y0)))) MARK(head(s(cons(tail(head(nats)), y0)))) -> c16(MARK(s(cons(tail(head(nats)), y0)))) MARK(head(s(cons(tail(head(pairs)), y0)))) -> c16(MARK(s(cons(tail(head(pairs)), y0)))) MARK(head(s(cons(head(cons(nats, y0)), y1)))) -> c16(MARK(s(cons(head(cons(nats, y0)), y1)))) MARK(head(s(cons(head(cons(incr(y0), y1)), y2)))) -> c16(MARK(s(cons(head(cons(incr(y0), y1)), y2)))) MARK(head(s(cons(head(cons(pairs, y0)), y1)))) -> c16(MARK(s(cons(head(cons(pairs, y0)), y1)))) MARK(head(s(cons(head(cons(odds, y0)), y1)))) -> c16(MARK(s(cons(head(cons(odds, y0)), y1)))) MARK(head(s(cons(head(cons(head(y0), y1)), y2)))) -> c16(MARK(s(cons(head(cons(head(y0), y1)), y2)))) MARK(head(s(cons(head(cons(tail(y0), y1)), y2)))) -> c16(MARK(s(cons(head(cons(tail(y0), y1)), y2)))) MARK(head(s(cons(head(cons(cons(y0, y1), y2)), y3)))) -> c16(MARK(s(cons(head(cons(cons(y0, y1), y2)), y3)))) MARK(head(s(cons(head(cons(s(y0), y1)), y2)))) -> c16(MARK(s(cons(head(cons(s(y0), y1)), y2)))) MARK(head(s(cons(head(cons(y0, y1)), y2)))) -> c16(MARK(s(cons(head(cons(y0, y1)), y2)))) MARK(head(s(cons(head(cons(0, y0)), y1)))) -> c16(MARK(s(cons(head(cons(0, y0)), y1)))) MARK(head(s(cons(incr(tail(nats)), y0)))) -> c16(MARK(s(cons(incr(tail(nats)), y0)))) MARK(head(s(cons(head(tail(nats)), y0)))) -> c16(MARK(s(cons(head(tail(nats)), y0)))) MARK(head(s(cons(tail(tail(nats)), y0)))) -> c16(MARK(s(cons(tail(tail(nats)), y0)))) MARK(head(s(incr(cons(nats, y0))))) -> c16(MARK(s(incr(cons(nats, y0))))) MARK(head(s(incr(cons(pairs, y0))))) -> c16(MARK(s(incr(cons(pairs, y0))))) MARK(head(s(incr(cons(odds, y0))))) -> c16(MARK(s(incr(cons(odds, y0))))) MARK(head(s(head(cons(nats, y0))))) -> c16(MARK(s(head(cons(nats, y0))))) MARK(head(s(head(cons(pairs, y0))))) -> c16(MARK(s(head(cons(pairs, y0))))) MARK(head(s(head(cons(odds, y0))))) -> c16(MARK(s(head(cons(odds, y0))))) ---------------------------------------- (196) Obligation: Complexity Dependency Tuples Problem Rules: a__nats -> cons(0, incr(nats)) a__nats -> nats a__pairs -> cons(0, incr(odds)) a__pairs -> pairs a__odds -> a__incr(a__pairs) a__odds -> odds a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) 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(nats) -> a__nats mark(incr(z0)) -> a__incr(mark(z0)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c17(MARK(s(z0))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(tail(incr(incr(z0)))) -> c17(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(odds))) -> c17(A__TAIL(a__incr(a__odds)), MARK(incr(odds))) MARK(tail(incr(head(z0)))) -> c17(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c17(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c17(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c17(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c17(MARK(incr(x0))) MARK(tail(head(incr(z0)))) -> c17(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(odds))) -> c17(A__TAIL(a__head(a__odds)), MARK(head(odds))) MARK(tail(head(head(z0)))) -> c17(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c17(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c17(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c17(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c17(MARK(head(x0))) MARK(tail(tail(incr(z0)))) -> c17(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(pairs))) -> c17(A__TAIL(a__tail(a__pairs)), MARK(tail(pairs))) MARK(tail(tail(odds))) -> c17(A__TAIL(a__tail(a__odds)), MARK(tail(odds))) MARK(tail(tail(head(z0)))) -> c17(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c17(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c17(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c17(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c17(MARK(tail(x0))) MARK(tail(pairs)) -> c17(A__TAIL(cons(0, incr(odds)))) A__INCR(cons(cons(y0, y1), z1)) -> c6(MARK(cons(y0, y1))) A__INCR(cons(s(y0), z1)) -> c6(MARK(s(y0))) A__INCR(cons(incr(cons(y0, y1)), z1)) -> c6(MARK(incr(cons(y0, y1)))) A__INCR(cons(incr(s(y0)), z1)) -> c6(MARK(incr(s(y0)))) A__INCR(cons(head(cons(y0, y1)), z1)) -> c6(MARK(head(cons(y0, y1)))) A__INCR(cons(head(s(y0)), z1)) -> c6(MARK(head(s(y0)))) A__INCR(cons(tail(cons(y0, y1)), z1)) -> c6(MARK(tail(cons(y0, y1)))) A__INCR(cons(tail(s(y0)), z1)) -> c6(MARK(tail(s(y0)))) A__INCR(cons(incr(incr(nats)), z1)) -> c6(MARK(incr(incr(nats)))) A__INCR(cons(incr(incr(incr(y0))), z1)) -> c6(MARK(incr(incr(incr(y0))))) A__INCR(cons(incr(incr(pairs)), z1)) -> c6(MARK(incr(incr(pairs)))) A__INCR(cons(incr(incr(odds)), z1)) -> c6(MARK(incr(incr(odds)))) A__INCR(cons(incr(incr(head(y0))), z1)) -> c6(MARK(incr(incr(head(y0))))) A__INCR(cons(incr(incr(tail(y0))), z1)) -> c6(MARK(incr(incr(tail(y0))))) A__INCR(cons(incr(incr(cons(y0, y1))), z1)) -> c6(MARK(incr(incr(cons(y0, y1))))) A__INCR(cons(incr(incr(s(y0))), z1)) -> c6(MARK(incr(incr(s(y0))))) A__INCR(cons(incr(incr(y0)), z1)) -> c6(MARK(incr(incr(y0)))) A__INCR(cons(incr(odds), z1)) -> c6(MARK(incr(odds))) A__INCR(cons(incr(head(nats)), z1)) -> c6(MARK(incr(head(nats)))) A__INCR(cons(incr(head(incr(y0))), z1)) -> c6(MARK(incr(head(incr(y0))))) A__INCR(cons(incr(head(pairs)), z1)) -> c6(MARK(incr(head(pairs)))) A__INCR(cons(incr(head(odds)), z1)) -> c6(MARK(incr(head(odds)))) A__INCR(cons(incr(head(head(y0))), z1)) -> c6(MARK(incr(head(head(y0))))) A__INCR(cons(incr(head(tail(y0))), z1)) -> c6(MARK(incr(head(tail(y0))))) A__INCR(cons(incr(head(cons(y0, y1))), z1)) -> c6(MARK(incr(head(cons(y0, y1))))) A__INCR(cons(incr(head(s(y0))), z1)) -> c6(MARK(incr(head(s(y0))))) A__INCR(cons(incr(head(y0)), z1)) -> c6(MARK(incr(head(y0)))) A__INCR(cons(incr(tail(nats)), z1)) -> c6(MARK(incr(tail(nats)))) A__INCR(cons(incr(tail(incr(y0))), z1)) -> c6(MARK(incr(tail(incr(y0))))) A__INCR(cons(incr(tail(pairs)), z1)) -> c6(MARK(incr(tail(pairs)))) A__INCR(cons(incr(tail(odds)), z1)) -> c6(MARK(incr(tail(odds)))) A__INCR(cons(incr(tail(head(y0))), z1)) -> c6(MARK(incr(tail(head(y0))))) A__INCR(cons(incr(tail(tail(y0))), z1)) -> c6(MARK(incr(tail(tail(y0))))) A__INCR(cons(incr(tail(cons(y0, y1))), z1)) -> c6(MARK(incr(tail(cons(y0, y1))))) A__INCR(cons(incr(tail(s(y0))), z1)) -> c6(MARK(incr(tail(s(y0))))) A__INCR(cons(incr(tail(y0)), z1)) -> c6(MARK(incr(tail(y0)))) A__INCR(cons(head(incr(nats)), z1)) -> c6(MARK(head(incr(nats)))) A__INCR(cons(head(incr(incr(y0))), z1)) -> c6(MARK(head(incr(incr(y0))))) A__INCR(cons(head(incr(pairs)), z1)) -> c6(MARK(head(incr(pairs)))) A__INCR(cons(head(incr(odds)), z1)) -> c6(MARK(head(incr(odds)))) A__INCR(cons(head(incr(head(y0))), z1)) -> c6(MARK(head(incr(head(y0))))) A__INCR(cons(head(incr(tail(y0))), z1)) -> c6(MARK(head(incr(tail(y0))))) A__INCR(cons(head(incr(cons(y0, y1))), z1)) -> c6(MARK(head(incr(cons(y0, y1))))) A__INCR(cons(head(incr(s(y0))), z1)) -> c6(MARK(head(incr(s(y0))))) A__INCR(cons(head(incr(y0)), z1)) -> c6(MARK(head(incr(y0)))) A__INCR(cons(head(odds), z1)) -> c6(MARK(head(odds))) A__INCR(cons(head(head(nats)), z1)) -> c6(MARK(head(head(nats)))) A__INCR(cons(head(head(incr(y0))), z1)) -> c6(MARK(head(head(incr(y0))))) A__INCR(cons(head(head(pairs)), z1)) -> c6(MARK(head(head(pairs)))) A__INCR(cons(head(head(odds)), z1)) -> c6(MARK(head(head(odds)))) A__INCR(cons(head(head(head(y0))), z1)) -> c6(MARK(head(head(head(y0))))) A__INCR(cons(head(head(tail(y0))), z1)) -> c6(MARK(head(head(tail(y0))))) A__INCR(cons(head(head(cons(y0, y1))), z1)) -> c6(MARK(head(head(cons(y0, y1))))) A__INCR(cons(head(head(s(y0))), z1)) -> c6(MARK(head(head(s(y0))))) A__INCR(cons(head(head(y0)), z1)) -> c6(MARK(head(head(y0)))) A__INCR(cons(head(tail(nats)), z1)) -> c6(MARK(head(tail(nats)))) A__INCR(cons(head(tail(incr(y0))), z1)) -> c6(MARK(head(tail(incr(y0))))) A__INCR(cons(head(tail(pairs)), z1)) -> c6(MARK(head(tail(pairs)))) A__INCR(cons(head(tail(odds)), z1)) -> c6(MARK(head(tail(odds)))) A__INCR(cons(head(tail(head(y0))), z1)) -> c6(MARK(head(tail(head(y0))))) A__INCR(cons(head(tail(tail(y0))), z1)) -> c6(MARK(head(tail(tail(y0))))) A__INCR(cons(head(tail(cons(y0, y1))), z1)) -> c6(MARK(head(tail(cons(y0, y1))))) A__INCR(cons(head(tail(s(y0))), z1)) -> c6(MARK(head(tail(s(y0))))) A__INCR(cons(head(tail(y0)), z1)) -> c6(MARK(head(tail(y0)))) A__INCR(cons(tail(incr(nats)), z1)) -> c6(MARK(tail(incr(nats)))) A__INCR(cons(tail(incr(incr(y0))), z1)) -> c6(MARK(tail(incr(incr(y0))))) A__INCR(cons(tail(incr(pairs)), z1)) -> c6(MARK(tail(incr(pairs)))) A__INCR(cons(tail(incr(odds)), z1)) -> c6(MARK(tail(incr(odds)))) A__INCR(cons(tail(incr(head(y0))), z1)) -> c6(MARK(tail(incr(head(y0))))) A__INCR(cons(tail(incr(tail(y0))), z1)) -> c6(MARK(tail(incr(tail(y0))))) A__INCR(cons(tail(incr(cons(y0, y1))), z1)) -> c6(MARK(tail(incr(cons(y0, y1))))) A__INCR(cons(tail(incr(s(y0))), z1)) -> c6(MARK(tail(incr(s(y0))))) A__INCR(cons(tail(incr(y0)), z1)) -> c6(MARK(tail(incr(y0)))) A__INCR(cons(tail(odds), z1)) -> c6(MARK(tail(odds))) A__INCR(cons(tail(head(nats)), z1)) -> c6(MARK(tail(head(nats)))) A__INCR(cons(tail(head(incr(y0))), z1)) -> c6(MARK(tail(head(incr(y0))))) A__INCR(cons(tail(head(pairs)), z1)) -> c6(MARK(tail(head(pairs)))) A__INCR(cons(tail(head(odds)), z1)) -> c6(MARK(tail(head(odds)))) A__INCR(cons(tail(head(head(y0))), z1)) -> c6(MARK(tail(head(head(y0))))) A__INCR(cons(tail(head(tail(y0))), z1)) -> c6(MARK(tail(head(tail(y0))))) A__INCR(cons(tail(head(cons(y0, y1))), z1)) -> c6(MARK(tail(head(cons(y0, y1))))) A__INCR(cons(tail(head(s(y0))), z1)) -> c6(MARK(tail(head(s(y0))))) A__INCR(cons(tail(head(y0)), z1)) -> c6(MARK(tail(head(y0)))) A__INCR(cons(tail(tail(nats)), z1)) -> c6(MARK(tail(tail(nats)))) A__INCR(cons(tail(tail(incr(y0))), z1)) -> c6(MARK(tail(tail(incr(y0))))) A__INCR(cons(tail(tail(pairs)), z1)) -> c6(MARK(tail(tail(pairs)))) A__INCR(cons(tail(tail(odds)), z1)) -> c6(MARK(tail(tail(odds)))) A__INCR(cons(tail(tail(head(y0))), z1)) -> c6(MARK(tail(tail(head(y0))))) A__INCR(cons(tail(tail(tail(y0))), z1)) -> c6(MARK(tail(tail(tail(y0))))) A__INCR(cons(tail(tail(cons(y0, y1))), z1)) -> c6(MARK(tail(tail(cons(y0, y1))))) A__INCR(cons(tail(tail(s(y0))), z1)) -> c6(MARK(tail(tail(s(y0))))) A__INCR(cons(tail(tail(y0)), z1)) -> c6(MARK(tail(tail(y0)))) A__INCR(cons(tail(pairs), z1)) -> c6(MARK(tail(pairs))) MARK(incr(incr(nats))) -> c13(A__INCR(a__incr(a__nats))) MARK(incr(incr(pairs))) -> c13(A__INCR(a__incr(a__pairs))) MARK(incr(odds)) -> c13(A__INCR(a__incr(a__pairs))) MARK(head(incr(nats))) -> c16(A__HEAD(a__incr(a__nats))) MARK(head(incr(pairs))) -> c16(A__HEAD(a__incr(a__pairs))) MARK(head(odds)) -> c16(A__HEAD(a__incr(a__pairs))) MARK(tail(incr(nats))) -> c17(A__TAIL(a__incr(a__nats))) MARK(tail(incr(pairs))) -> c17(A__TAIL(a__incr(a__pairs))) MARK(tail(odds)) -> c17(A__TAIL(a__incr(a__pairs))) 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-> c18(MARK(incr(odds))) MARK(cons(head(incr(nats)), z1)) -> c18(MARK(head(incr(nats)))) MARK(cons(head(incr(pairs)), z1)) -> c18(MARK(head(incr(pairs)))) MARK(cons(head(odds), z1)) -> c18(MARK(head(odds))) MARK(cons(tail(incr(nats)), z1)) -> c18(MARK(tail(incr(nats)))) MARK(cons(tail(incr(pairs)), z1)) -> c18(MARK(tail(incr(pairs)))) MARK(cons(tail(odds), z1)) -> c18(MARK(tail(odds))) MARK(cons(incr(cons(nats, y0)), z1)) -> c18(MARK(incr(cons(nats, y0)))) MARK(cons(incr(cons(incr(y0), y1)), z1)) -> c18(MARK(incr(cons(incr(y0), y1)))) MARK(cons(incr(cons(pairs, y0)), z1)) -> c18(MARK(incr(cons(pairs, y0)))) MARK(cons(incr(cons(odds, y0)), z1)) -> c18(MARK(incr(cons(odds, y0)))) MARK(cons(incr(cons(head(y0), y1)), z1)) -> c18(MARK(incr(cons(head(y0), y1)))) MARK(cons(incr(cons(tail(y0), y1)), z1)) -> c18(MARK(incr(cons(tail(y0), y1)))) MARK(cons(incr(cons(cons(y0, y1), y2)), z1)) -> c18(MARK(incr(cons(cons(y0, y1), y2)))) MARK(cons(incr(cons(s(y0), y1)), z1)) -> c18(MARK(incr(cons(s(y0), y1)))) MARK(cons(incr(cons(y0, y1)), z1)) -> c18(MARK(incr(cons(y0, y1)))) MARK(cons(incr(cons(0, y0)), z1)) -> c18(MARK(incr(cons(0, y0)))) MARK(cons(incr(head(nats)), z1)) -> c18(MARK(incr(head(nats)))) MARK(cons(incr(head(pairs)), z1)) -> c18(MARK(incr(head(pairs)))) MARK(cons(head(head(nats)), z1)) -> c18(MARK(head(head(nats)))) MARK(cons(head(head(pairs)), z1)) -> c18(MARK(head(head(pairs)))) MARK(cons(tail(head(nats)), z1)) -> c18(MARK(tail(head(nats)))) MARK(cons(tail(head(pairs)), z1)) -> c18(MARK(tail(head(pairs)))) MARK(cons(head(cons(nats, y0)), z1)) -> c18(MARK(head(cons(nats, y0)))) MARK(cons(head(cons(incr(y0), y1)), z1)) -> c18(MARK(head(cons(incr(y0), y1)))) MARK(cons(head(cons(pairs, y0)), z1)) -> c18(MARK(head(cons(pairs, y0)))) MARK(cons(head(cons(odds, y0)), z1)) -> c18(MARK(head(cons(odds, y0)))) MARK(cons(head(cons(head(y0), y1)), z1)) -> c18(MARK(head(cons(head(y0), y1)))) MARK(cons(head(cons(tail(y0), y1)), z1)) -> c18(MARK(head(cons(tail(y0), y1)))) MARK(cons(head(cons(cons(y0, y1), y2)), z1)) -> c18(MARK(head(cons(cons(y0, y1), y2)))) MARK(cons(head(cons(s(y0), y1)), z1)) -> c18(MARK(head(cons(s(y0), y1)))) MARK(cons(head(cons(y0, y1)), z1)) -> c18(MARK(head(cons(y0, y1)))) MARK(cons(head(cons(0, y0)), z1)) -> c18(MARK(head(cons(0, y0)))) MARK(cons(incr(tail(nats)), z1)) -> c18(MARK(incr(tail(nats)))) MARK(cons(head(tail(nats)), z1)) -> c18(MARK(head(tail(nats)))) MARK(cons(tail(tail(nats)), z1)) -> c18(MARK(tail(tail(nats)))) MARK(incr(cons(nats, x1))) -> c13(A__INCR(cons(a__nats, x1))) MARK(incr(cons(pairs, x1))) -> c13(A__INCR(cons(a__pairs, x1))) MARK(incr(cons(odds, x1))) -> c13(A__INCR(cons(a__odds, x1))) MARK(head(cons(nats, x1))) -> c16(A__HEAD(cons(a__nats, x1))) MARK(head(cons(pairs, x1))) -> c16(A__HEAD(cons(a__pairs, x1))) MARK(head(cons(odds, x1))) -> c16(A__HEAD(cons(a__odds, x1))) MARK(s(s(y0))) -> c20(MARK(s(y0))) MARK(s(incr(s(y0)))) -> c20(MARK(incr(s(y0)))) MARK(s(head(s(y0)))) -> c20(MARK(head(s(y0)))) MARK(s(tail(cons(y0, y1)))) -> c20(MARK(tail(cons(y0, y1)))) MARK(s(tail(s(y0)))) -> c20(MARK(tail(s(y0)))) MARK(s(incr(incr(incr(y0))))) -> c20(MARK(incr(incr(incr(y0))))) MARK(s(incr(incr(odds)))) -> c20(MARK(incr(incr(odds)))) MARK(s(incr(incr(head(y0))))) -> c20(MARK(incr(incr(head(y0))))) MARK(s(incr(incr(tail(y0))))) -> c20(MARK(incr(incr(tail(y0))))) MARK(s(incr(incr(cons(y0, y1))))) -> c20(MARK(incr(incr(cons(y0, y1))))) MARK(s(incr(incr(s(y0))))) -> c20(MARK(incr(incr(s(y0))))) MARK(s(incr(incr(y0)))) -> c20(MARK(incr(incr(y0)))) MARK(s(incr(head(incr(y0))))) -> c20(MARK(incr(head(incr(y0))))) MARK(s(incr(head(odds)))) -> c20(MARK(incr(head(odds)))) MARK(s(incr(head(head(y0))))) -> c20(MARK(incr(head(head(y0))))) MARK(s(incr(head(tail(y0))))) -> c20(MARK(incr(head(tail(y0))))) MARK(s(incr(head(cons(y0, y1))))) -> c20(MARK(incr(head(cons(y0, y1))))) MARK(s(incr(head(s(y0))))) -> c20(MARK(incr(head(s(y0))))) MARK(s(incr(head(y0)))) -> c20(MARK(incr(head(y0)))) MARK(s(incr(tail(incr(y0))))) -> c20(MARK(incr(tail(incr(y0))))) MARK(s(incr(tail(pairs)))) -> c20(MARK(incr(tail(pairs)))) MARK(s(incr(tail(odds)))) -> c20(MARK(incr(tail(odds)))) MARK(s(incr(tail(head(y0))))) -> c20(MARK(incr(tail(head(y0))))) MARK(s(incr(tail(tail(y0))))) -> c20(MARK(incr(tail(tail(y0))))) MARK(s(incr(tail(cons(y0, y1))))) -> c20(MARK(incr(tail(cons(y0, y1))))) MARK(s(incr(tail(s(y0))))) -> c20(MARK(incr(tail(s(y0))))) MARK(s(incr(tail(y0)))) -> c20(MARK(incr(tail(y0)))) MARK(s(head(incr(incr(y0))))) -> c20(MARK(head(incr(incr(y0))))) MARK(s(head(incr(odds)))) -> c20(MARK(head(incr(odds)))) MARK(s(head(incr(head(y0))))) -> c20(MARK(head(incr(head(y0))))) MARK(s(head(incr(tail(y0))))) -> c20(MARK(head(incr(tail(y0))))) MARK(s(head(incr(cons(y0, y1))))) -> c20(MARK(head(incr(cons(y0, y1))))) MARK(s(head(incr(s(y0))))) -> c20(MARK(head(incr(s(y0))))) MARK(s(head(incr(y0)))) -> c20(MARK(head(incr(y0)))) MARK(s(head(head(incr(y0))))) -> c20(MARK(head(head(incr(y0))))) MARK(s(head(head(odds)))) -> c20(MARK(head(head(odds)))) MARK(s(head(head(head(y0))))) -> c20(MARK(head(head(head(y0))))) MARK(s(head(head(tail(y0))))) -> c20(MARK(head(head(tail(y0))))) MARK(s(head(head(cons(y0, y1))))) -> c20(MARK(head(head(cons(y0, y1))))) MARK(s(head(head(s(y0))))) -> c20(MARK(head(head(s(y0))))) MARK(s(head(head(y0)))) -> c20(MARK(head(head(y0)))) MARK(s(head(tail(incr(y0))))) -> c20(MARK(head(tail(incr(y0))))) MARK(s(head(tail(pairs)))) -> c20(MARK(head(tail(pairs)))) MARK(s(head(tail(odds)))) -> c20(MARK(head(tail(odds)))) MARK(s(head(tail(head(y0))))) -> c20(MARK(head(tail(head(y0))))) MARK(s(head(tail(tail(y0))))) -> c20(MARK(head(tail(tail(y0))))) MARK(s(head(tail(cons(y0, y1))))) -> c20(MARK(head(tail(cons(y0, y1))))) MARK(s(head(tail(s(y0))))) -> c20(MARK(head(tail(s(y0))))) MARK(s(head(tail(y0)))) -> c20(MARK(head(tail(y0)))) MARK(s(tail(incr(incr(y0))))) -> c20(MARK(tail(incr(incr(y0))))) MARK(s(tail(incr(odds)))) -> c20(MARK(tail(incr(odds)))) MARK(s(tail(incr(head(y0))))) -> c20(MARK(tail(incr(head(y0))))) MARK(s(tail(incr(tail(y0))))) -> c20(MARK(tail(incr(tail(y0))))) MARK(s(tail(incr(cons(y0, y1))))) -> c20(MARK(tail(incr(cons(y0, y1))))) MARK(s(tail(incr(s(y0))))) -> c20(MARK(tail(incr(s(y0))))) MARK(s(tail(incr(y0)))) -> c20(MARK(tail(incr(y0)))) MARK(s(tail(head(incr(y0))))) -> c20(MARK(tail(head(incr(y0))))) MARK(s(tail(head(odds)))) -> c20(MARK(tail(head(odds)))) MARK(s(tail(head(head(y0))))) -> c20(MARK(tail(head(head(y0))))) MARK(s(tail(head(tail(y0))))) -> c20(MARK(tail(head(tail(y0))))) MARK(s(tail(head(cons(y0, y1))))) -> c20(MARK(tail(head(cons(y0, y1))))) MARK(s(tail(head(s(y0))))) -> c20(MARK(tail(head(s(y0))))) MARK(s(tail(head(y0)))) -> c20(MARK(tail(head(y0)))) MARK(s(tail(tail(incr(y0))))) -> c20(MARK(tail(tail(incr(y0))))) MARK(s(tail(tail(pairs)))) -> c20(MARK(tail(tail(pairs)))) MARK(s(tail(tail(odds)))) -> c20(MARK(tail(tail(odds)))) MARK(s(tail(tail(head(y0))))) -> c20(MARK(tail(tail(head(y0))))) MARK(s(tail(tail(tail(y0))))) -> c20(MARK(tail(tail(tail(y0))))) MARK(s(tail(tail(cons(y0, y1))))) -> c20(MARK(tail(tail(cons(y0, y1))))) MARK(s(tail(tail(s(y0))))) -> c20(MARK(tail(tail(s(y0))))) MARK(s(tail(tail(y0)))) -> c20(MARK(tail(tail(y0)))) MARK(s(tail(pairs))) -> c20(MARK(tail(pairs))) MARK(s(incr(incr(nats)))) -> c20(MARK(incr(incr(nats)))) MARK(s(incr(incr(pairs)))) -> c20(MARK(incr(incr(pairs)))) MARK(s(incr(odds))) -> c20(MARK(incr(odds))) MARK(s(head(incr(nats)))) -> c20(MARK(head(incr(nats)))) MARK(s(head(incr(pairs)))) -> c20(MARK(head(incr(pairs)))) MARK(s(head(odds))) -> c20(MARK(head(odds))) MARK(s(tail(incr(nats)))) -> c20(MARK(tail(incr(nats)))) MARK(s(tail(incr(pairs)))) -> c20(MARK(tail(incr(pairs)))) MARK(s(tail(odds))) -> c20(MARK(tail(odds))) MARK(s(incr(cons(incr(y0), y1)))) -> c20(MARK(incr(cons(incr(y0), y1)))) MARK(s(incr(cons(head(y0), y1)))) -> c20(MARK(incr(cons(head(y0), y1)))) MARK(s(incr(cons(tail(y0), y1)))) -> c20(MARK(incr(cons(tail(y0), y1)))) MARK(s(incr(cons(cons(y0, y1), y2)))) -> c20(MARK(incr(cons(cons(y0, y1), y2)))) MARK(s(incr(cons(s(y0), y1)))) -> c20(MARK(incr(cons(s(y0), y1)))) MARK(s(incr(cons(y0, y1)))) -> c20(MARK(incr(cons(y0, y1)))) MARK(s(incr(head(nats)))) -> c20(MARK(incr(head(nats)))) MARK(s(incr(head(pairs)))) -> c20(MARK(incr(head(pairs)))) MARK(s(head(head(nats)))) -> c20(MARK(head(head(nats)))) MARK(s(head(head(pairs)))) -> c20(MARK(head(head(pairs)))) MARK(s(tail(head(nats)))) -> c20(MARK(tail(head(nats)))) MARK(s(tail(head(pairs)))) -> c20(MARK(tail(head(pairs)))) MARK(s(head(cons(incr(y0), y1)))) -> c20(MARK(head(cons(incr(y0), y1)))) MARK(s(head(cons(head(y0), y1)))) -> c20(MARK(head(cons(head(y0), y1)))) MARK(s(head(cons(tail(y0), y1)))) -> c20(MARK(head(cons(tail(y0), y1)))) MARK(s(head(cons(cons(y0, y1), y2)))) -> c20(MARK(head(cons(cons(y0, y1), y2)))) MARK(s(head(cons(s(y0), y1)))) -> c20(MARK(head(cons(s(y0), y1)))) MARK(s(head(cons(y0, y1)))) -> c20(MARK(head(cons(y0, y1)))) MARK(s(incr(tail(nats)))) -> c20(MARK(incr(tail(nats)))) MARK(s(head(tail(nats)))) -> c20(MARK(head(tail(nats)))) MARK(s(tail(tail(nats)))) -> c20(MARK(tail(tail(nats)))) MARK(s(cons(cons(y0, y1), y2))) -> c20(MARK(cons(cons(y0, y1), y2))) MARK(s(cons(s(y0), y1))) -> c20(MARK(cons(s(y0), y1))) MARK(s(cons(incr(s(y0)), y1))) -> c20(MARK(cons(incr(s(y0)), y1))) MARK(s(cons(head(s(y0)), y1))) -> c20(MARK(cons(head(s(y0)), y1))) MARK(s(cons(tail(cons(y0, y1)), y2))) -> c20(MARK(cons(tail(cons(y0, y1)), y2))) MARK(s(cons(tail(s(y0)), y1))) -> c20(MARK(cons(tail(s(y0)), y1))) MARK(s(cons(incr(incr(incr(y0))), y1))) -> c20(MARK(cons(incr(incr(incr(y0))), y1))) MARK(s(cons(incr(incr(odds)), y0))) -> c20(MARK(cons(incr(incr(odds)), y0))) MARK(s(cons(incr(incr(head(y0))), y1))) -> c20(MARK(cons(incr(incr(head(y0))), y1))) MARK(s(cons(incr(incr(tail(y0))), y1))) -> c20(MARK(cons(incr(incr(tail(y0))), y1))) MARK(s(cons(incr(incr(cons(y0, y1))), y2))) -> c20(MARK(cons(incr(incr(cons(y0, y1))), y2))) MARK(s(cons(incr(incr(s(y0))), y1))) -> c20(MARK(cons(incr(incr(s(y0))), y1))) MARK(s(cons(incr(incr(y0)), y1))) -> c20(MARK(cons(incr(incr(y0)), y1))) MARK(s(cons(incr(head(incr(y0))), y1))) -> c20(MARK(cons(incr(head(incr(y0))), y1))) MARK(s(cons(incr(head(odds)), y0))) -> c20(MARK(cons(incr(head(odds)), y0))) MARK(s(cons(incr(head(head(y0))), y1))) -> c20(MARK(cons(incr(head(head(y0))), y1))) MARK(s(cons(incr(head(tail(y0))), y1))) -> c20(MARK(cons(incr(head(tail(y0))), y1))) MARK(s(cons(incr(head(cons(y0, y1))), y2))) -> c20(MARK(cons(incr(head(cons(y0, y1))), y2))) MARK(s(cons(incr(head(s(y0))), y1))) -> c20(MARK(cons(incr(head(s(y0))), y1))) MARK(s(cons(incr(head(y0)), y1))) -> c20(MARK(cons(incr(head(y0)), y1))) MARK(s(cons(incr(tail(incr(y0))), y1))) -> c20(MARK(cons(incr(tail(incr(y0))), y1))) MARK(s(cons(incr(tail(pairs)), y0))) -> c20(MARK(cons(incr(tail(pairs)), y0))) MARK(s(cons(incr(tail(odds)), y0))) -> c20(MARK(cons(incr(tail(odds)), y0))) MARK(s(cons(incr(tail(head(y0))), y1))) -> c20(MARK(cons(incr(tail(head(y0))), y1))) MARK(s(cons(incr(tail(tail(y0))), y1))) -> c20(MARK(cons(incr(tail(tail(y0))), y1))) MARK(s(cons(incr(tail(cons(y0, y1))), y2))) -> c20(MARK(cons(incr(tail(cons(y0, y1))), y2))) MARK(s(cons(incr(tail(s(y0))), y1))) -> c20(MARK(cons(incr(tail(s(y0))), y1))) MARK(s(cons(incr(tail(y0)), y1))) -> c20(MARK(cons(incr(tail(y0)), y1))) MARK(s(cons(head(incr(incr(y0))), y1))) -> c20(MARK(cons(head(incr(incr(y0))), y1))) MARK(s(cons(head(incr(odds)), y0))) -> c20(MARK(cons(head(incr(odds)), y0))) MARK(s(cons(head(incr(head(y0))), y1))) -> c20(MARK(cons(head(incr(head(y0))), y1))) MARK(s(cons(head(incr(tail(y0))), y1))) -> c20(MARK(cons(head(incr(tail(y0))), y1))) MARK(s(cons(head(incr(cons(y0, y1))), y2))) -> c20(MARK(cons(head(incr(cons(y0, y1))), y2))) MARK(s(cons(head(incr(s(y0))), y1))) -> c20(MARK(cons(head(incr(s(y0))), y1))) MARK(s(cons(head(incr(y0)), y1))) -> c20(MARK(cons(head(incr(y0)), y1))) MARK(s(cons(head(head(incr(y0))), y1))) -> c20(MARK(cons(head(head(incr(y0))), y1))) MARK(s(cons(head(head(odds)), y0))) -> c20(MARK(cons(head(head(odds)), y0))) MARK(s(cons(head(head(head(y0))), y1))) -> c20(MARK(cons(head(head(head(y0))), y1))) MARK(s(cons(head(head(tail(y0))), y1))) -> c20(MARK(cons(head(head(tail(y0))), y1))) MARK(s(cons(head(head(cons(y0, y1))), y2))) -> c20(MARK(cons(head(head(cons(y0, y1))), y2))) MARK(s(cons(head(head(s(y0))), y1))) -> c20(MARK(cons(head(head(s(y0))), y1))) MARK(s(cons(head(head(y0)), y1))) -> c20(MARK(cons(head(head(y0)), y1))) MARK(s(cons(head(tail(incr(y0))), y1))) -> c20(MARK(cons(head(tail(incr(y0))), y1))) MARK(s(cons(head(tail(pairs)), y0))) -> c20(MARK(cons(head(tail(pairs)), y0))) MARK(s(cons(head(tail(odds)), y0))) -> c20(MARK(cons(head(tail(odds)), y0))) MARK(s(cons(head(tail(head(y0))), y1))) -> c20(MARK(cons(head(tail(head(y0))), y1))) MARK(s(cons(head(tail(tail(y0))), y1))) -> c20(MARK(cons(head(tail(tail(y0))), y1))) MARK(s(cons(head(tail(cons(y0, y1))), y2))) -> c20(MARK(cons(head(tail(cons(y0, y1))), y2))) MARK(s(cons(head(tail(s(y0))), y1))) -> c20(MARK(cons(head(tail(s(y0))), y1))) MARK(s(cons(head(tail(y0)), y1))) -> c20(MARK(cons(head(tail(y0)), y1))) MARK(s(cons(tail(incr(incr(y0))), y1))) -> c20(MARK(cons(tail(incr(incr(y0))), y1))) MARK(s(cons(tail(incr(odds)), y0))) -> c20(MARK(cons(tail(incr(odds)), y0))) MARK(s(cons(tail(incr(head(y0))), y1))) -> c20(MARK(cons(tail(incr(head(y0))), y1))) MARK(s(cons(tail(incr(tail(y0))), y1))) -> c20(MARK(cons(tail(incr(tail(y0))), y1))) MARK(s(cons(tail(incr(cons(y0, y1))), y2))) -> c20(MARK(cons(tail(incr(cons(y0, y1))), y2))) MARK(s(cons(tail(incr(s(y0))), y1))) -> c20(MARK(cons(tail(incr(s(y0))), y1))) MARK(s(cons(tail(incr(y0)), y1))) -> c20(MARK(cons(tail(incr(y0)), y1))) MARK(s(cons(tail(head(incr(y0))), y1))) -> c20(MARK(cons(tail(head(incr(y0))), y1))) MARK(s(cons(tail(head(odds)), y0))) -> c20(MARK(cons(tail(head(odds)), y0))) MARK(s(cons(tail(head(head(y0))), y1))) -> c20(MARK(cons(tail(head(head(y0))), y1))) MARK(s(cons(tail(head(tail(y0))), y1))) -> c20(MARK(cons(tail(head(tail(y0))), y1))) MARK(s(cons(tail(head(cons(y0, y1))), y2))) -> c20(MARK(cons(tail(head(cons(y0, y1))), y2))) MARK(s(cons(tail(head(s(y0))), y1))) -> c20(MARK(cons(tail(head(s(y0))), y1))) MARK(s(cons(tail(head(y0)), y1))) -> c20(MARK(cons(tail(head(y0)), y1))) MARK(s(cons(tail(tail(incr(y0))), y1))) -> c20(MARK(cons(tail(tail(incr(y0))), y1))) MARK(s(cons(tail(tail(pairs)), y0))) -> c20(MARK(cons(tail(tail(pairs)), y0))) MARK(s(cons(tail(tail(odds)), y0))) -> c20(MARK(cons(tail(tail(odds)), y0))) MARK(s(cons(tail(tail(head(y0))), y1))) -> c20(MARK(cons(tail(tail(head(y0))), y1))) MARK(s(cons(tail(tail(tail(y0))), y1))) -> c20(MARK(cons(tail(tail(tail(y0))), y1))) MARK(s(cons(tail(tail(cons(y0, y1))), y2))) -> c20(MARK(cons(tail(tail(cons(y0, y1))), y2))) MARK(s(cons(tail(tail(s(y0))), y1))) -> c20(MARK(cons(tail(tail(s(y0))), y1))) MARK(s(cons(tail(tail(y0)), y1))) -> c20(MARK(cons(tail(tail(y0)), y1))) MARK(s(cons(tail(pairs), y0))) -> c20(MARK(cons(tail(pairs), y0))) MARK(s(cons(incr(incr(nats)), y0))) -> c20(MARK(cons(incr(incr(nats)), y0))) MARK(s(cons(incr(incr(pairs)), y0))) -> c20(MARK(cons(incr(incr(pairs)), y0))) MARK(s(cons(incr(odds), y0))) -> c20(MARK(cons(incr(odds), y0))) MARK(s(cons(head(incr(nats)), y0))) -> c20(MARK(cons(head(incr(nats)), y0))) MARK(s(cons(head(incr(pairs)), y0))) -> c20(MARK(cons(head(incr(pairs)), y0))) MARK(s(cons(head(odds), y0))) -> c20(MARK(cons(head(odds), y0))) MARK(s(cons(tail(incr(nats)), y0))) -> c20(MARK(cons(tail(incr(nats)), y0))) MARK(s(cons(tail(incr(pairs)), y0))) -> c20(MARK(cons(tail(incr(pairs)), y0))) MARK(s(cons(tail(odds), y0))) -> c20(MARK(cons(tail(odds), y0))) MARK(s(cons(incr(cons(nats, y0)), y1))) -> c20(MARK(cons(incr(cons(nats, y0)), y1))) MARK(s(cons(incr(cons(incr(y0), y1)), y2))) -> c20(MARK(cons(incr(cons(incr(y0), y1)), y2))) MARK(s(cons(incr(cons(pairs, y0)), y1))) -> c20(MARK(cons(incr(cons(pairs, y0)), y1))) MARK(s(cons(incr(cons(odds, y0)), y1))) -> c20(MARK(cons(incr(cons(odds, y0)), y1))) MARK(s(cons(incr(cons(head(y0), y1)), y2))) -> c20(MARK(cons(incr(cons(head(y0), y1)), y2))) MARK(s(cons(incr(cons(tail(y0), y1)), y2))) -> c20(MARK(cons(incr(cons(tail(y0), y1)), y2))) MARK(s(cons(incr(cons(cons(y0, y1), y2)), y3))) -> c20(MARK(cons(incr(cons(cons(y0, y1), y2)), y3))) MARK(s(cons(incr(cons(s(y0), y1)), y2))) -> c20(MARK(cons(incr(cons(s(y0), y1)), y2))) MARK(s(cons(incr(cons(y0, y1)), y2))) -> c20(MARK(cons(incr(cons(y0, y1)), y2))) MARK(s(cons(incr(cons(0, y0)), y1))) -> c20(MARK(cons(incr(cons(0, y0)), y1))) MARK(s(cons(incr(head(nats)), y0))) -> c20(MARK(cons(incr(head(nats)), y0))) MARK(s(cons(incr(head(pairs)), y0))) -> c20(MARK(cons(incr(head(pairs)), y0))) MARK(s(cons(head(head(nats)), y0))) -> c20(MARK(cons(head(head(nats)), y0))) MARK(s(cons(head(head(pairs)), y0))) -> c20(MARK(cons(head(head(pairs)), y0))) MARK(s(cons(tail(head(nats)), y0))) -> c20(MARK(cons(tail(head(nats)), y0))) MARK(s(cons(tail(head(pairs)), y0))) -> c20(MARK(cons(tail(head(pairs)), y0))) MARK(s(cons(head(cons(nats, y0)), y1))) -> c20(MARK(cons(head(cons(nats, y0)), y1))) MARK(s(cons(head(cons(incr(y0), y1)), y2))) -> c20(MARK(cons(head(cons(incr(y0), y1)), y2))) MARK(s(cons(head(cons(pairs, y0)), y1))) -> c20(MARK(cons(head(cons(pairs, y0)), y1))) MARK(s(cons(head(cons(odds, y0)), y1))) -> c20(MARK(cons(head(cons(odds, y0)), y1))) MARK(s(cons(head(cons(head(y0), y1)), y2))) -> c20(MARK(cons(head(cons(head(y0), y1)), y2))) MARK(s(cons(head(cons(tail(y0), y1)), y2))) -> c20(MARK(cons(head(cons(tail(y0), y1)), y2))) MARK(s(cons(head(cons(cons(y0, y1), y2)), y3))) -> c20(MARK(cons(head(cons(cons(y0, y1), y2)), y3))) MARK(s(cons(head(cons(s(y0), y1)), y2))) -> c20(MARK(cons(head(cons(s(y0), y1)), y2))) MARK(s(cons(head(cons(y0, y1)), y2))) -> c20(MARK(cons(head(cons(y0, y1)), y2))) MARK(s(cons(head(cons(0, y0)), y1))) -> c20(MARK(cons(head(cons(0, y0)), y1))) MARK(s(cons(incr(tail(nats)), y0))) -> c20(MARK(cons(incr(tail(nats)), y0))) MARK(s(cons(head(tail(nats)), y0))) -> c20(MARK(cons(head(tail(nats)), y0))) MARK(s(cons(tail(tail(nats)), y0))) -> c20(MARK(cons(tail(tail(nats)), y0))) MARK(s(incr(cons(nats, y0)))) -> c20(MARK(incr(cons(nats, y0)))) MARK(s(incr(cons(pairs, y0)))) -> c20(MARK(incr(cons(pairs, y0)))) MARK(s(incr(cons(odds, y0)))) -> c20(MARK(incr(cons(odds, y0)))) MARK(s(head(cons(nats, y0)))) -> c20(MARK(head(cons(nats, y0)))) MARK(s(head(cons(pairs, y0)))) -> c20(MARK(head(cons(pairs, y0)))) MARK(s(head(cons(odds, y0)))) -> c20(MARK(head(cons(odds, y0)))) MARK(incr(s(s(y0)))) -> c13(MARK(s(s(y0)))) MARK(incr(s(incr(s(y0))))) -> c13(MARK(s(incr(s(y0))))) MARK(incr(s(head(s(y0))))) -> c13(MARK(s(head(s(y0))))) MARK(incr(s(tail(cons(y0, y1))))) -> c13(MARK(s(tail(cons(y0, y1))))) MARK(incr(s(tail(s(y0))))) -> c13(MARK(s(tail(s(y0))))) MARK(incr(s(incr(incr(incr(y0)))))) -> c13(MARK(s(incr(incr(incr(y0)))))) MARK(incr(s(incr(incr(odds))))) -> c13(MARK(s(incr(incr(odds))))) MARK(incr(s(incr(incr(head(y0)))))) -> c13(MARK(s(incr(incr(head(y0)))))) MARK(incr(s(incr(incr(tail(y0)))))) -> c13(MARK(s(incr(incr(tail(y0)))))) MARK(incr(s(incr(incr(cons(y0, y1)))))) -> c13(MARK(s(incr(incr(cons(y0, y1)))))) MARK(incr(s(incr(incr(s(y0)))))) -> c13(MARK(s(incr(incr(s(y0)))))) MARK(incr(s(incr(incr(y0))))) -> c13(MARK(s(incr(incr(y0))))) MARK(incr(s(incr(head(incr(y0)))))) -> c13(MARK(s(incr(head(incr(y0)))))) MARK(incr(s(incr(head(odds))))) -> c13(MARK(s(incr(head(odds))))) MARK(incr(s(incr(head(head(y0)))))) -> c13(MARK(s(incr(head(head(y0)))))) MARK(incr(s(incr(head(tail(y0)))))) -> c13(MARK(s(incr(head(tail(y0)))))) MARK(incr(s(incr(head(cons(y0, y1)))))) -> c13(MARK(s(incr(head(cons(y0, y1)))))) MARK(incr(s(incr(head(s(y0)))))) -> c13(MARK(s(incr(head(s(y0)))))) MARK(incr(s(incr(head(y0))))) -> c13(MARK(s(incr(head(y0))))) MARK(incr(s(incr(tail(incr(y0)))))) -> c13(MARK(s(incr(tail(incr(y0)))))) MARK(incr(s(incr(tail(pairs))))) -> c13(MARK(s(incr(tail(pairs))))) MARK(incr(s(incr(tail(odds))))) -> c13(MARK(s(incr(tail(odds))))) MARK(incr(s(incr(tail(head(y0)))))) -> c13(MARK(s(incr(tail(head(y0)))))) MARK(incr(s(incr(tail(tail(y0)))))) -> c13(MARK(s(incr(tail(tail(y0)))))) MARK(incr(s(incr(tail(cons(y0, y1)))))) -> c13(MARK(s(incr(tail(cons(y0, y1)))))) MARK(incr(s(incr(tail(s(y0)))))) -> c13(MARK(s(incr(tail(s(y0)))))) MARK(incr(s(incr(tail(y0))))) -> c13(MARK(s(incr(tail(y0))))) MARK(incr(s(head(incr(incr(y0)))))) -> c13(MARK(s(head(incr(incr(y0)))))) MARK(incr(s(head(incr(odds))))) -> c13(MARK(s(head(incr(odds))))) MARK(incr(s(head(incr(head(y0)))))) -> c13(MARK(s(head(incr(head(y0)))))) MARK(incr(s(head(incr(tail(y0)))))) -> c13(MARK(s(head(incr(tail(y0)))))) MARK(incr(s(head(incr(cons(y0, y1)))))) -> c13(MARK(s(head(incr(cons(y0, y1)))))) MARK(incr(s(head(incr(s(y0)))))) -> c13(MARK(s(head(incr(s(y0)))))) MARK(incr(s(head(incr(y0))))) -> c13(MARK(s(head(incr(y0))))) MARK(incr(s(head(head(incr(y0)))))) -> c13(MARK(s(head(head(incr(y0)))))) MARK(incr(s(head(head(odds))))) -> c13(MARK(s(head(head(odds))))) MARK(incr(s(head(head(head(y0)))))) -> c13(MARK(s(head(head(head(y0)))))) MARK(incr(s(head(head(tail(y0)))))) -> c13(MARK(s(head(head(tail(y0)))))) MARK(incr(s(head(head(cons(y0, y1)))))) -> c13(MARK(s(head(head(cons(y0, y1)))))) MARK(incr(s(head(head(s(y0)))))) -> c13(MARK(s(head(head(s(y0)))))) MARK(incr(s(head(head(y0))))) -> c13(MARK(s(head(head(y0))))) MARK(incr(s(head(tail(incr(y0)))))) -> c13(MARK(s(head(tail(incr(y0)))))) MARK(incr(s(head(tail(pairs))))) -> c13(MARK(s(head(tail(pairs))))) MARK(incr(s(head(tail(odds))))) -> c13(MARK(s(head(tail(odds))))) MARK(incr(s(head(tail(head(y0)))))) -> c13(MARK(s(head(tail(head(y0)))))) MARK(incr(s(head(tail(tail(y0)))))) -> c13(MARK(s(head(tail(tail(y0)))))) MARK(incr(s(head(tail(cons(y0, y1)))))) -> c13(MARK(s(head(tail(cons(y0, y1)))))) MARK(incr(s(head(tail(s(y0)))))) -> c13(MARK(s(head(tail(s(y0)))))) MARK(incr(s(head(tail(y0))))) -> c13(MARK(s(head(tail(y0))))) MARK(incr(s(tail(incr(incr(y0)))))) -> c13(MARK(s(tail(incr(incr(y0)))))) MARK(incr(s(tail(incr(odds))))) -> c13(MARK(s(tail(incr(odds))))) MARK(incr(s(tail(incr(head(y0)))))) -> c13(MARK(s(tail(incr(head(y0)))))) MARK(incr(s(tail(incr(tail(y0)))))) -> c13(MARK(s(tail(incr(tail(y0)))))) MARK(incr(s(tail(incr(cons(y0, y1)))))) -> c13(MARK(s(tail(incr(cons(y0, y1)))))) MARK(incr(s(tail(incr(s(y0)))))) -> c13(MARK(s(tail(incr(s(y0)))))) MARK(incr(s(tail(incr(y0))))) -> c13(MARK(s(tail(incr(y0))))) MARK(incr(s(tail(head(incr(y0)))))) -> c13(MARK(s(tail(head(incr(y0)))))) MARK(incr(s(tail(head(odds))))) -> c13(MARK(s(tail(head(odds))))) MARK(incr(s(tail(head(head(y0)))))) -> c13(MARK(s(tail(head(head(y0)))))) MARK(incr(s(tail(head(tail(y0)))))) -> c13(MARK(s(tail(head(tail(y0)))))) MARK(incr(s(tail(head(cons(y0, y1)))))) -> c13(MARK(s(tail(head(cons(y0, y1)))))) MARK(incr(s(tail(head(s(y0)))))) -> c13(MARK(s(tail(head(s(y0)))))) MARK(incr(s(tail(head(y0))))) -> c13(MARK(s(tail(head(y0))))) MARK(incr(s(tail(tail(incr(y0)))))) -> c13(MARK(s(tail(tail(incr(y0)))))) MARK(incr(s(tail(tail(pairs))))) -> c13(MARK(s(tail(tail(pairs))))) MARK(incr(s(tail(tail(odds))))) -> c13(MARK(s(tail(tail(odds))))) MARK(incr(s(tail(tail(head(y0)))))) -> c13(MARK(s(tail(tail(head(y0)))))) MARK(incr(s(tail(tail(tail(y0)))))) -> c13(MARK(s(tail(tail(tail(y0)))))) MARK(incr(s(tail(tail(cons(y0, y1)))))) -> c13(MARK(s(tail(tail(cons(y0, y1)))))) MARK(incr(s(tail(tail(s(y0)))))) -> c13(MARK(s(tail(tail(s(y0)))))) MARK(incr(s(tail(tail(y0))))) -> c13(MARK(s(tail(tail(y0))))) MARK(incr(s(tail(pairs)))) -> c13(MARK(s(tail(pairs)))) MARK(incr(s(incr(incr(nats))))) -> c13(MARK(s(incr(incr(nats))))) MARK(incr(s(incr(incr(pairs))))) -> c13(MARK(s(incr(incr(pairs))))) MARK(incr(s(incr(odds)))) -> c13(MARK(s(incr(odds)))) MARK(incr(s(head(incr(nats))))) -> c13(MARK(s(head(incr(nats))))) MARK(incr(s(head(incr(pairs))))) -> c13(MARK(s(head(incr(pairs))))) MARK(incr(s(head(odds)))) -> c13(MARK(s(head(odds)))) MARK(incr(s(tail(incr(nats))))) -> c13(MARK(s(tail(incr(nats))))) MARK(incr(s(tail(incr(pairs))))) -> c13(MARK(s(tail(incr(pairs))))) MARK(incr(s(tail(odds)))) -> c13(MARK(s(tail(odds)))) MARK(incr(s(incr(cons(incr(y0), y1))))) -> c13(MARK(s(incr(cons(incr(y0), y1))))) MARK(incr(s(incr(cons(head(y0), y1))))) -> c13(MARK(s(incr(cons(head(y0), y1))))) MARK(incr(s(incr(cons(tail(y0), y1))))) -> c13(MARK(s(incr(cons(tail(y0), y1))))) MARK(incr(s(incr(cons(cons(y0, y1), y2))))) -> c13(MARK(s(incr(cons(cons(y0, y1), y2))))) MARK(incr(s(incr(cons(s(y0), y1))))) -> c13(MARK(s(incr(cons(s(y0), y1))))) MARK(incr(s(incr(cons(y0, y1))))) -> c13(MARK(s(incr(cons(y0, y1))))) MARK(incr(s(incr(head(nats))))) -> c13(MARK(s(incr(head(nats))))) MARK(incr(s(incr(head(pairs))))) -> c13(MARK(s(incr(head(pairs))))) MARK(incr(s(head(head(nats))))) -> c13(MARK(s(head(head(nats))))) MARK(incr(s(head(head(pairs))))) -> c13(MARK(s(head(head(pairs))))) MARK(incr(s(tail(head(nats))))) -> c13(MARK(s(tail(head(nats))))) MARK(incr(s(tail(head(pairs))))) -> c13(MARK(s(tail(head(pairs))))) MARK(incr(s(head(cons(incr(y0), y1))))) -> c13(MARK(s(head(cons(incr(y0), y1))))) MARK(incr(s(head(cons(head(y0), y1))))) -> c13(MARK(s(head(cons(head(y0), y1))))) MARK(incr(s(head(cons(tail(y0), y1))))) -> c13(MARK(s(head(cons(tail(y0), y1))))) MARK(incr(s(head(cons(cons(y0, y1), y2))))) -> c13(MARK(s(head(cons(cons(y0, y1), y2))))) MARK(incr(s(head(cons(s(y0), y1))))) -> c13(MARK(s(head(cons(s(y0), y1))))) MARK(incr(s(head(cons(y0, y1))))) -> c13(MARK(s(head(cons(y0, y1))))) MARK(incr(s(incr(tail(nats))))) -> c13(MARK(s(incr(tail(nats))))) MARK(incr(s(head(tail(nats))))) -> c13(MARK(s(head(tail(nats))))) MARK(incr(s(tail(tail(nats))))) -> c13(MARK(s(tail(tail(nats))))) MARK(incr(s(cons(cons(y0, y1), y2)))) -> c13(MARK(s(cons(cons(y0, y1), y2)))) MARK(incr(s(cons(s(y0), y1)))) -> c13(MARK(s(cons(s(y0), y1)))) MARK(incr(s(cons(incr(s(y0)), y1)))) -> c13(MARK(s(cons(incr(s(y0)), y1)))) MARK(incr(s(cons(head(s(y0)), y1)))) -> c13(MARK(s(cons(head(s(y0)), y1)))) MARK(incr(s(cons(tail(cons(y0, y1)), y2)))) -> c13(MARK(s(cons(tail(cons(y0, y1)), y2)))) MARK(incr(s(cons(tail(s(y0)), y1)))) -> c13(MARK(s(cons(tail(s(y0)), y1)))) MARK(incr(s(cons(incr(incr(incr(y0))), y1)))) -> c13(MARK(s(cons(incr(incr(incr(y0))), y1)))) MARK(incr(s(cons(incr(incr(odds)), y0)))) -> c13(MARK(s(cons(incr(incr(odds)), y0)))) MARK(incr(s(cons(incr(incr(head(y0))), y1)))) -> c13(MARK(s(cons(incr(incr(head(y0))), y1)))) MARK(incr(s(cons(incr(incr(tail(y0))), y1)))) -> c13(MARK(s(cons(incr(incr(tail(y0))), y1)))) MARK(incr(s(cons(incr(incr(cons(y0, y1))), y2)))) -> c13(MARK(s(cons(incr(incr(cons(y0, y1))), y2)))) MARK(incr(s(cons(incr(incr(s(y0))), y1)))) -> c13(MARK(s(cons(incr(incr(s(y0))), y1)))) MARK(incr(s(cons(incr(incr(y0)), y1)))) -> c13(MARK(s(cons(incr(incr(y0)), y1)))) MARK(incr(s(cons(incr(head(incr(y0))), y1)))) -> c13(MARK(s(cons(incr(head(incr(y0))), y1)))) MARK(incr(s(cons(incr(head(odds)), y0)))) -> c13(MARK(s(cons(incr(head(odds)), y0)))) MARK(incr(s(cons(incr(head(head(y0))), y1)))) -> c13(MARK(s(cons(incr(head(head(y0))), y1)))) MARK(incr(s(cons(incr(head(tail(y0))), y1)))) -> c13(MARK(s(cons(incr(head(tail(y0))), y1)))) MARK(incr(s(cons(incr(head(cons(y0, y1))), y2)))) -> c13(MARK(s(cons(incr(head(cons(y0, y1))), y2)))) MARK(incr(s(cons(incr(head(s(y0))), y1)))) -> c13(MARK(s(cons(incr(head(s(y0))), y1)))) MARK(incr(s(cons(incr(head(y0)), y1)))) -> c13(MARK(s(cons(incr(head(y0)), y1)))) MARK(incr(s(cons(incr(tail(incr(y0))), y1)))) -> c13(MARK(s(cons(incr(tail(incr(y0))), y1)))) MARK(incr(s(cons(incr(tail(pairs)), y0)))) -> c13(MARK(s(cons(incr(tail(pairs)), y0)))) MARK(incr(s(cons(incr(tail(odds)), y0)))) -> c13(MARK(s(cons(incr(tail(odds)), y0)))) MARK(incr(s(cons(incr(tail(head(y0))), y1)))) -> c13(MARK(s(cons(incr(tail(head(y0))), y1)))) MARK(incr(s(cons(incr(tail(tail(y0))), y1)))) -> c13(MARK(s(cons(incr(tail(tail(y0))), y1)))) MARK(incr(s(cons(incr(tail(cons(y0, y1))), y2)))) -> c13(MARK(s(cons(incr(tail(cons(y0, y1))), y2)))) MARK(incr(s(cons(incr(tail(s(y0))), y1)))) -> c13(MARK(s(cons(incr(tail(s(y0))), y1)))) MARK(incr(s(cons(incr(tail(y0)), y1)))) -> c13(MARK(s(cons(incr(tail(y0)), y1)))) MARK(incr(s(cons(head(incr(incr(y0))), y1)))) -> c13(MARK(s(cons(head(incr(incr(y0))), y1)))) MARK(incr(s(cons(head(incr(odds)), y0)))) -> c13(MARK(s(cons(head(incr(odds)), y0)))) MARK(incr(s(cons(head(incr(head(y0))), y1)))) -> c13(MARK(s(cons(head(incr(head(y0))), y1)))) MARK(incr(s(cons(head(incr(tail(y0))), y1)))) -> c13(MARK(s(cons(head(incr(tail(y0))), y1)))) MARK(incr(s(cons(head(incr(cons(y0, y1))), y2)))) -> c13(MARK(s(cons(head(incr(cons(y0, y1))), y2)))) MARK(incr(s(cons(head(incr(s(y0))), y1)))) -> c13(MARK(s(cons(head(incr(s(y0))), y1)))) MARK(incr(s(cons(head(incr(y0)), y1)))) -> c13(MARK(s(cons(head(incr(y0)), y1)))) MARK(incr(s(cons(head(head(incr(y0))), y1)))) -> c13(MARK(s(cons(head(head(incr(y0))), y1)))) MARK(incr(s(cons(head(head(odds)), y0)))) -> c13(MARK(s(cons(head(head(odds)), y0)))) MARK(incr(s(cons(head(head(head(y0))), y1)))) -> c13(MARK(s(cons(head(head(head(y0))), y1)))) MARK(incr(s(cons(head(head(tail(y0))), y1)))) -> c13(MARK(s(cons(head(head(tail(y0))), y1)))) MARK(incr(s(cons(head(head(cons(y0, y1))), y2)))) -> c13(MARK(s(cons(head(head(cons(y0, y1))), y2)))) MARK(incr(s(cons(head(head(s(y0))), y1)))) -> c13(MARK(s(cons(head(head(s(y0))), y1)))) MARK(incr(s(cons(head(head(y0)), y1)))) -> c13(MARK(s(cons(head(head(y0)), y1)))) MARK(incr(s(cons(head(tail(incr(y0))), y1)))) -> c13(MARK(s(cons(head(tail(incr(y0))), y1)))) MARK(incr(s(cons(head(tail(pairs)), y0)))) -> c13(MARK(s(cons(head(tail(pairs)), y0)))) MARK(incr(s(cons(head(tail(odds)), y0)))) -> c13(MARK(s(cons(head(tail(odds)), y0)))) MARK(incr(s(cons(head(tail(head(y0))), y1)))) -> c13(MARK(s(cons(head(tail(head(y0))), y1)))) MARK(incr(s(cons(head(tail(tail(y0))), y1)))) -> c13(MARK(s(cons(head(tail(tail(y0))), y1)))) MARK(incr(s(cons(head(tail(cons(y0, y1))), y2)))) -> c13(MARK(s(cons(head(tail(cons(y0, y1))), y2)))) MARK(incr(s(cons(head(tail(s(y0))), y1)))) -> c13(MARK(s(cons(head(tail(s(y0))), y1)))) MARK(incr(s(cons(head(tail(y0)), y1)))) -> c13(MARK(s(cons(head(tail(y0)), y1)))) MARK(incr(s(cons(tail(incr(incr(y0))), y1)))) -> c13(MARK(s(cons(tail(incr(incr(y0))), y1)))) MARK(incr(s(cons(tail(incr(odds)), y0)))) -> c13(MARK(s(cons(tail(incr(odds)), y0)))) MARK(incr(s(cons(tail(incr(head(y0))), y1)))) -> c13(MARK(s(cons(tail(incr(head(y0))), y1)))) MARK(incr(s(cons(tail(incr(tail(y0))), y1)))) -> c13(MARK(s(cons(tail(incr(tail(y0))), y1)))) MARK(incr(s(cons(tail(incr(cons(y0, y1))), y2)))) -> c13(MARK(s(cons(tail(incr(cons(y0, y1))), y2)))) MARK(incr(s(cons(tail(incr(s(y0))), y1)))) -> c13(MARK(s(cons(tail(incr(s(y0))), y1)))) MARK(incr(s(cons(tail(incr(y0)), y1)))) -> c13(MARK(s(cons(tail(incr(y0)), y1)))) MARK(incr(s(cons(tail(head(incr(y0))), y1)))) -> c13(MARK(s(cons(tail(head(incr(y0))), y1)))) MARK(incr(s(cons(tail(head(odds)), y0)))) -> c13(MARK(s(cons(tail(head(odds)), y0)))) MARK(incr(s(cons(tail(head(head(y0))), y1)))) -> c13(MARK(s(cons(tail(head(head(y0))), y1)))) MARK(incr(s(cons(tail(head(tail(y0))), y1)))) -> c13(MARK(s(cons(tail(head(tail(y0))), y1)))) MARK(incr(s(cons(tail(head(cons(y0, y1))), y2)))) -> c13(MARK(s(cons(tail(head(cons(y0, y1))), y2)))) MARK(incr(s(cons(tail(head(s(y0))), y1)))) -> c13(MARK(s(cons(tail(head(s(y0))), y1)))) MARK(incr(s(cons(tail(head(y0)), y1)))) -> c13(MARK(s(cons(tail(head(y0)), y1)))) MARK(incr(s(cons(tail(tail(incr(y0))), y1)))) -> c13(MARK(s(cons(tail(tail(incr(y0))), y1)))) MARK(incr(s(cons(tail(tail(pairs)), y0)))) -> c13(MARK(s(cons(tail(tail(pairs)), y0)))) MARK(incr(s(cons(tail(tail(odds)), y0)))) -> c13(MARK(s(cons(tail(tail(odds)), y0)))) MARK(incr(s(cons(tail(tail(head(y0))), y1)))) -> c13(MARK(s(cons(tail(tail(head(y0))), y1)))) MARK(incr(s(cons(tail(tail(tail(y0))), y1)))) -> c13(MARK(s(cons(tail(tail(tail(y0))), y1)))) MARK(incr(s(cons(tail(tail(cons(y0, y1))), y2)))) -> c13(MARK(s(cons(tail(tail(cons(y0, y1))), y2)))) MARK(incr(s(cons(tail(tail(s(y0))), y1)))) -> c13(MARK(s(cons(tail(tail(s(y0))), y1)))) MARK(incr(s(cons(tail(tail(y0)), y1)))) -> c13(MARK(s(cons(tail(tail(y0)), y1)))) MARK(incr(s(cons(tail(pairs), y0)))) -> c13(MARK(s(cons(tail(pairs), y0)))) MARK(incr(s(cons(incr(incr(nats)), y0)))) -> c13(MARK(s(cons(incr(incr(nats)), y0)))) MARK(incr(s(cons(incr(incr(pairs)), y0)))) -> c13(MARK(s(cons(incr(incr(pairs)), y0)))) MARK(incr(s(cons(incr(odds), y0)))) -> c13(MARK(s(cons(incr(odds), y0)))) MARK(incr(s(cons(head(incr(nats)), y0)))) -> c13(MARK(s(cons(head(incr(nats)), y0)))) MARK(incr(s(cons(head(incr(pairs)), y0)))) -> c13(MARK(s(cons(head(incr(pairs)), y0)))) MARK(incr(s(cons(head(odds), y0)))) -> c13(MARK(s(cons(head(odds), y0)))) MARK(incr(s(cons(tail(incr(nats)), y0)))) -> c13(MARK(s(cons(tail(incr(nats)), y0)))) MARK(incr(s(cons(tail(incr(pairs)), y0)))) -> c13(MARK(s(cons(tail(incr(pairs)), y0)))) MARK(incr(s(cons(tail(odds), y0)))) -> c13(MARK(s(cons(tail(odds), y0)))) MARK(incr(s(cons(incr(cons(nats, y0)), y1)))) -> c13(MARK(s(cons(incr(cons(nats, y0)), y1)))) MARK(incr(s(cons(incr(cons(incr(y0), y1)), y2)))) -> c13(MARK(s(cons(incr(cons(incr(y0), y1)), y2)))) MARK(incr(s(cons(incr(cons(pairs, y0)), y1)))) -> c13(MARK(s(cons(incr(cons(pairs, y0)), y1)))) MARK(incr(s(cons(incr(cons(odds, y0)), y1)))) -> c13(MARK(s(cons(incr(cons(odds, y0)), y1)))) MARK(incr(s(cons(incr(cons(head(y0), y1)), y2)))) -> c13(MARK(s(cons(incr(cons(head(y0), y1)), y2)))) MARK(incr(s(cons(incr(cons(tail(y0), y1)), y2)))) -> c13(MARK(s(cons(incr(cons(tail(y0), y1)), y2)))) MARK(incr(s(cons(incr(cons(cons(y0, y1), y2)), y3)))) -> c13(MARK(s(cons(incr(cons(cons(y0, y1), y2)), y3)))) MARK(incr(s(cons(incr(cons(s(y0), y1)), y2)))) -> c13(MARK(s(cons(incr(cons(s(y0), y1)), y2)))) MARK(incr(s(cons(incr(cons(y0, y1)), y2)))) -> c13(MARK(s(cons(incr(cons(y0, y1)), y2)))) MARK(incr(s(cons(incr(cons(0, y0)), y1)))) -> c13(MARK(s(cons(incr(cons(0, y0)), y1)))) MARK(incr(s(cons(incr(head(nats)), y0)))) -> c13(MARK(s(cons(incr(head(nats)), y0)))) MARK(incr(s(cons(incr(head(pairs)), y0)))) -> c13(MARK(s(cons(incr(head(pairs)), y0)))) MARK(incr(s(cons(head(head(nats)), y0)))) -> c13(MARK(s(cons(head(head(nats)), y0)))) MARK(incr(s(cons(head(head(pairs)), y0)))) -> c13(MARK(s(cons(head(head(pairs)), y0)))) MARK(incr(s(cons(tail(head(nats)), y0)))) -> c13(MARK(s(cons(tail(head(nats)), y0)))) MARK(incr(s(cons(tail(head(pairs)), y0)))) -> c13(MARK(s(cons(tail(head(pairs)), y0)))) MARK(incr(s(cons(head(cons(nats, y0)), y1)))) -> c13(MARK(s(cons(head(cons(nats, y0)), y1)))) MARK(incr(s(cons(head(cons(incr(y0), y1)), y2)))) -> c13(MARK(s(cons(head(cons(incr(y0), y1)), y2)))) MARK(incr(s(cons(head(cons(pairs, y0)), y1)))) -> c13(MARK(s(cons(head(cons(pairs, y0)), y1)))) MARK(incr(s(cons(head(cons(odds, y0)), y1)))) -> c13(MARK(s(cons(head(cons(odds, y0)), y1)))) MARK(incr(s(cons(head(cons(head(y0), y1)), y2)))) -> c13(MARK(s(cons(head(cons(head(y0), y1)), y2)))) MARK(incr(s(cons(head(cons(tail(y0), y1)), y2)))) -> c13(MARK(s(cons(head(cons(tail(y0), y1)), y2)))) MARK(incr(s(cons(head(cons(cons(y0, y1), y2)), y3)))) -> c13(MARK(s(cons(head(cons(cons(y0, y1), y2)), y3)))) MARK(incr(s(cons(head(cons(s(y0), y1)), y2)))) -> c13(MARK(s(cons(head(cons(s(y0), y1)), y2)))) MARK(incr(s(cons(head(cons(y0, y1)), y2)))) -> c13(MARK(s(cons(head(cons(y0, y1)), y2)))) MARK(incr(s(cons(head(cons(0, y0)), y1)))) -> c13(MARK(s(cons(head(cons(0, y0)), y1)))) MARK(incr(s(cons(incr(tail(nats)), y0)))) -> c13(MARK(s(cons(incr(tail(nats)), y0)))) MARK(incr(s(cons(head(tail(nats)), y0)))) -> c13(MARK(s(cons(head(tail(nats)), y0)))) MARK(incr(s(cons(tail(tail(nats)), y0)))) -> c13(MARK(s(cons(tail(tail(nats)), y0)))) MARK(incr(s(incr(cons(nats, y0))))) -> c13(MARK(s(incr(cons(nats, y0))))) MARK(incr(s(incr(cons(pairs, y0))))) -> c13(MARK(s(incr(cons(pairs, y0))))) MARK(incr(s(incr(cons(odds, y0))))) -> c13(MARK(s(incr(cons(odds, y0))))) MARK(incr(s(head(cons(nats, y0))))) -> c13(MARK(s(head(cons(nats, y0))))) MARK(incr(s(head(cons(pairs, y0))))) -> c13(MARK(s(head(cons(pairs, y0))))) MARK(incr(s(head(cons(odds, y0))))) -> c13(MARK(s(head(cons(odds, y0))))) MARK(head(s(s(y0)))) -> c16(MARK(s(s(y0)))) MARK(head(s(incr(s(y0))))) -> c16(MARK(s(incr(s(y0))))) MARK(head(s(head(s(y0))))) -> c16(MARK(s(head(s(y0))))) MARK(head(s(tail(cons(y0, y1))))) -> c16(MARK(s(tail(cons(y0, y1))))) MARK(head(s(tail(s(y0))))) -> c16(MARK(s(tail(s(y0))))) MARK(head(s(incr(incr(incr(y0)))))) -> c16(MARK(s(incr(incr(incr(y0)))))) MARK(head(s(incr(incr(odds))))) -> c16(MARK(s(incr(incr(odds))))) MARK(head(s(incr(incr(head(y0)))))) -> c16(MARK(s(incr(incr(head(y0)))))) MARK(head(s(incr(incr(tail(y0)))))) -> c16(MARK(s(incr(incr(tail(y0)))))) MARK(head(s(incr(incr(cons(y0, y1)))))) -> c16(MARK(s(incr(incr(cons(y0, y1)))))) MARK(head(s(incr(incr(s(y0)))))) -> c16(MARK(s(incr(incr(s(y0)))))) MARK(head(s(incr(incr(y0))))) -> c16(MARK(s(incr(incr(y0))))) MARK(head(s(incr(head(incr(y0)))))) -> c16(MARK(s(incr(head(incr(y0)))))) MARK(head(s(incr(head(odds))))) -> c16(MARK(s(incr(head(odds))))) MARK(head(s(incr(head(head(y0)))))) -> c16(MARK(s(incr(head(head(y0)))))) MARK(head(s(incr(head(tail(y0)))))) -> c16(MARK(s(incr(head(tail(y0)))))) MARK(head(s(incr(head(cons(y0, y1)))))) -> c16(MARK(s(incr(head(cons(y0, y1)))))) MARK(head(s(incr(head(s(y0)))))) -> c16(MARK(s(incr(head(s(y0)))))) MARK(head(s(incr(head(y0))))) -> c16(MARK(s(incr(head(y0))))) MARK(head(s(incr(tail(incr(y0)))))) -> c16(MARK(s(incr(tail(incr(y0)))))) MARK(head(s(incr(tail(pairs))))) -> c16(MARK(s(incr(tail(pairs))))) MARK(head(s(incr(tail(odds))))) -> c16(MARK(s(incr(tail(odds))))) MARK(head(s(incr(tail(head(y0)))))) -> c16(MARK(s(incr(tail(head(y0)))))) MARK(head(s(incr(tail(tail(y0)))))) -> c16(MARK(s(incr(tail(tail(y0)))))) MARK(head(s(incr(tail(cons(y0, y1)))))) -> c16(MARK(s(incr(tail(cons(y0, y1)))))) MARK(head(s(incr(tail(s(y0)))))) -> c16(MARK(s(incr(tail(s(y0)))))) MARK(head(s(incr(tail(y0))))) -> c16(MARK(s(incr(tail(y0))))) MARK(head(s(head(incr(incr(y0)))))) -> c16(MARK(s(head(incr(incr(y0)))))) MARK(head(s(head(incr(odds))))) -> c16(MARK(s(head(incr(odds))))) MARK(head(s(head(incr(head(y0)))))) -> c16(MARK(s(head(incr(head(y0)))))) MARK(head(s(head(incr(tail(y0)))))) -> c16(MARK(s(head(incr(tail(y0)))))) MARK(head(s(head(incr(cons(y0, y1)))))) -> c16(MARK(s(head(incr(cons(y0, y1)))))) MARK(head(s(head(incr(s(y0)))))) -> c16(MARK(s(head(incr(s(y0)))))) MARK(head(s(head(incr(y0))))) -> c16(MARK(s(head(incr(y0))))) MARK(head(s(head(head(incr(y0)))))) -> c16(MARK(s(head(head(incr(y0)))))) MARK(head(s(head(head(odds))))) -> c16(MARK(s(head(head(odds))))) MARK(head(s(head(head(head(y0)))))) -> c16(MARK(s(head(head(head(y0)))))) MARK(head(s(head(head(tail(y0)))))) -> c16(MARK(s(head(head(tail(y0)))))) MARK(head(s(head(head(cons(y0, y1)))))) -> c16(MARK(s(head(head(cons(y0, y1)))))) MARK(head(s(head(head(s(y0)))))) -> c16(MARK(s(head(head(s(y0)))))) MARK(head(s(head(head(y0))))) -> c16(MARK(s(head(head(y0))))) MARK(head(s(head(tail(incr(y0)))))) -> c16(MARK(s(head(tail(incr(y0)))))) MARK(head(s(head(tail(pairs))))) -> c16(MARK(s(head(tail(pairs))))) MARK(head(s(head(tail(odds))))) -> c16(MARK(s(head(tail(odds))))) MARK(head(s(head(tail(head(y0)))))) -> c16(MARK(s(head(tail(head(y0)))))) MARK(head(s(head(tail(tail(y0)))))) -> c16(MARK(s(head(tail(tail(y0)))))) MARK(head(s(head(tail(cons(y0, y1)))))) -> c16(MARK(s(head(tail(cons(y0, y1)))))) MARK(head(s(head(tail(s(y0)))))) -> c16(MARK(s(head(tail(s(y0)))))) MARK(head(s(head(tail(y0))))) -> c16(MARK(s(head(tail(y0))))) MARK(head(s(tail(incr(incr(y0)))))) -> c16(MARK(s(tail(incr(incr(y0)))))) MARK(head(s(tail(incr(odds))))) -> c16(MARK(s(tail(incr(odds))))) MARK(head(s(tail(incr(head(y0)))))) -> c16(MARK(s(tail(incr(head(y0)))))) MARK(head(s(tail(incr(tail(y0)))))) -> c16(MARK(s(tail(incr(tail(y0)))))) MARK(head(s(tail(incr(cons(y0, y1)))))) -> c16(MARK(s(tail(incr(cons(y0, y1)))))) MARK(head(s(tail(incr(s(y0)))))) -> c16(MARK(s(tail(incr(s(y0)))))) MARK(head(s(tail(incr(y0))))) -> c16(MARK(s(tail(incr(y0))))) MARK(head(s(tail(head(incr(y0)))))) -> c16(MARK(s(tail(head(incr(y0)))))) MARK(head(s(tail(head(odds))))) -> c16(MARK(s(tail(head(odds))))) MARK(head(s(tail(head(head(y0)))))) -> c16(MARK(s(tail(head(head(y0)))))) MARK(head(s(tail(head(tail(y0)))))) -> c16(MARK(s(tail(head(tail(y0)))))) MARK(head(s(tail(head(cons(y0, y1)))))) -> c16(MARK(s(tail(head(cons(y0, y1)))))) MARK(head(s(tail(head(s(y0)))))) -> c16(MARK(s(tail(head(s(y0)))))) MARK(head(s(tail(head(y0))))) -> c16(MARK(s(tail(head(y0))))) MARK(head(s(tail(tail(incr(y0)))))) -> c16(MARK(s(tail(tail(incr(y0)))))) MARK(head(s(tail(tail(pairs))))) -> c16(MARK(s(tail(tail(pairs))))) MARK(head(s(tail(tail(odds))))) -> c16(MARK(s(tail(tail(odds))))) MARK(head(s(tail(tail(head(y0)))))) -> c16(MARK(s(tail(tail(head(y0)))))) MARK(head(s(tail(tail(tail(y0)))))) -> c16(MARK(s(tail(tail(tail(y0)))))) MARK(head(s(tail(tail(cons(y0, y1)))))) -> c16(MARK(s(tail(tail(cons(y0, y1)))))) MARK(head(s(tail(tail(s(y0)))))) -> c16(MARK(s(tail(tail(s(y0)))))) MARK(head(s(tail(tail(y0))))) -> c16(MARK(s(tail(tail(y0))))) MARK(head(s(tail(pairs)))) -> c16(MARK(s(tail(pairs)))) MARK(head(s(incr(incr(nats))))) -> c16(MARK(s(incr(incr(nats))))) MARK(head(s(incr(incr(pairs))))) -> c16(MARK(s(incr(incr(pairs))))) MARK(head(s(incr(odds)))) -> c16(MARK(s(incr(odds)))) MARK(head(s(head(incr(nats))))) -> c16(MARK(s(head(incr(nats))))) MARK(head(s(head(incr(pairs))))) -> c16(MARK(s(head(incr(pairs))))) MARK(head(s(head(odds)))) -> c16(MARK(s(head(odds)))) MARK(head(s(tail(incr(nats))))) -> c16(MARK(s(tail(incr(nats))))) MARK(head(s(tail(incr(pairs))))) -> c16(MARK(s(tail(incr(pairs))))) MARK(head(s(tail(odds)))) -> c16(MARK(s(tail(odds)))) MARK(head(s(incr(cons(incr(y0), y1))))) -> c16(MARK(s(incr(cons(incr(y0), y1))))) MARK(head(s(incr(cons(head(y0), y1))))) -> c16(MARK(s(incr(cons(head(y0), y1))))) MARK(head(s(incr(cons(tail(y0), y1))))) -> c16(MARK(s(incr(cons(tail(y0), y1))))) MARK(head(s(incr(cons(cons(y0, y1), y2))))) -> c16(MARK(s(incr(cons(cons(y0, y1), y2))))) MARK(head(s(incr(cons(s(y0), y1))))) -> c16(MARK(s(incr(cons(s(y0), y1))))) MARK(head(s(incr(cons(y0, y1))))) -> c16(MARK(s(incr(cons(y0, y1))))) MARK(head(s(incr(head(nats))))) -> c16(MARK(s(incr(head(nats))))) MARK(head(s(incr(head(pairs))))) -> c16(MARK(s(incr(head(pairs))))) MARK(head(s(head(head(nats))))) -> c16(MARK(s(head(head(nats))))) MARK(head(s(head(head(pairs))))) -> c16(MARK(s(head(head(pairs))))) MARK(head(s(tail(head(nats))))) -> c16(MARK(s(tail(head(nats))))) MARK(head(s(tail(head(pairs))))) -> c16(MARK(s(tail(head(pairs))))) MARK(head(s(head(cons(incr(y0), y1))))) -> c16(MARK(s(head(cons(incr(y0), y1))))) MARK(head(s(head(cons(head(y0), y1))))) -> c16(MARK(s(head(cons(head(y0), y1))))) MARK(head(s(head(cons(tail(y0), y1))))) -> c16(MARK(s(head(cons(tail(y0), y1))))) MARK(head(s(head(cons(cons(y0, y1), y2))))) -> c16(MARK(s(head(cons(cons(y0, y1), y2))))) MARK(head(s(head(cons(s(y0), y1))))) -> c16(MARK(s(head(cons(s(y0), y1))))) MARK(head(s(head(cons(y0, y1))))) -> c16(MARK(s(head(cons(y0, y1))))) MARK(head(s(incr(tail(nats))))) -> c16(MARK(s(incr(tail(nats))))) MARK(head(s(head(tail(nats))))) -> c16(MARK(s(head(tail(nats))))) MARK(head(s(tail(tail(nats))))) -> c16(MARK(s(tail(tail(nats))))) MARK(head(s(cons(cons(y0, y1), y2)))) -> c16(MARK(s(cons(cons(y0, y1), y2)))) MARK(head(s(cons(s(y0), y1)))) -> c16(MARK(s(cons(s(y0), y1)))) MARK(head(s(cons(incr(s(y0)), y1)))) -> c16(MARK(s(cons(incr(s(y0)), y1)))) MARK(head(s(cons(head(s(y0)), y1)))) -> c16(MARK(s(cons(head(s(y0)), y1)))) MARK(head(s(cons(tail(cons(y0, y1)), y2)))) -> c16(MARK(s(cons(tail(cons(y0, y1)), y2)))) MARK(head(s(cons(tail(s(y0)), y1)))) -> c16(MARK(s(cons(tail(s(y0)), y1)))) MARK(head(s(cons(incr(incr(incr(y0))), y1)))) -> c16(MARK(s(cons(incr(incr(incr(y0))), y1)))) MARK(head(s(cons(incr(incr(odds)), y0)))) -> c16(MARK(s(cons(incr(incr(odds)), y0)))) MARK(head(s(cons(incr(incr(head(y0))), y1)))) -> c16(MARK(s(cons(incr(incr(head(y0))), y1)))) MARK(head(s(cons(incr(incr(tail(y0))), y1)))) -> c16(MARK(s(cons(incr(incr(tail(y0))), y1)))) MARK(head(s(cons(incr(incr(cons(y0, y1))), y2)))) -> c16(MARK(s(cons(incr(incr(cons(y0, y1))), y2)))) MARK(head(s(cons(incr(incr(s(y0))), y1)))) -> c16(MARK(s(cons(incr(incr(s(y0))), y1)))) MARK(head(s(cons(incr(incr(y0)), y1)))) -> c16(MARK(s(cons(incr(incr(y0)), y1)))) MARK(head(s(cons(incr(head(incr(y0))), y1)))) -> c16(MARK(s(cons(incr(head(incr(y0))), y1)))) MARK(head(s(cons(incr(head(odds)), y0)))) -> c16(MARK(s(cons(incr(head(odds)), y0)))) MARK(head(s(cons(incr(head(head(y0))), y1)))) -> c16(MARK(s(cons(incr(head(head(y0))), y1)))) MARK(head(s(cons(incr(head(tail(y0))), y1)))) -> c16(MARK(s(cons(incr(head(tail(y0))), y1)))) MARK(head(s(cons(incr(head(cons(y0, y1))), y2)))) -> c16(MARK(s(cons(incr(head(cons(y0, y1))), y2)))) MARK(head(s(cons(incr(head(s(y0))), y1)))) -> c16(MARK(s(cons(incr(head(s(y0))), y1)))) MARK(head(s(cons(incr(head(y0)), y1)))) -> c16(MARK(s(cons(incr(head(y0)), y1)))) MARK(head(s(cons(incr(tail(incr(y0))), y1)))) -> c16(MARK(s(cons(incr(tail(incr(y0))), y1)))) MARK(head(s(cons(incr(tail(pairs)), y0)))) -> c16(MARK(s(cons(incr(tail(pairs)), y0)))) MARK(head(s(cons(incr(tail(odds)), y0)))) -> c16(MARK(s(cons(incr(tail(odds)), y0)))) MARK(head(s(cons(incr(tail(head(y0))), y1)))) -> c16(MARK(s(cons(incr(tail(head(y0))), y1)))) MARK(head(s(cons(incr(tail(tail(y0))), y1)))) -> c16(MARK(s(cons(incr(tail(tail(y0))), y1)))) MARK(head(s(cons(incr(tail(cons(y0, y1))), y2)))) -> c16(MARK(s(cons(incr(tail(cons(y0, y1))), y2)))) MARK(head(s(cons(incr(tail(s(y0))), y1)))) -> c16(MARK(s(cons(incr(tail(s(y0))), y1)))) MARK(head(s(cons(incr(tail(y0)), y1)))) -> c16(MARK(s(cons(incr(tail(y0)), y1)))) MARK(head(s(cons(head(incr(incr(y0))), y1)))) -> c16(MARK(s(cons(head(incr(incr(y0))), y1)))) MARK(head(s(cons(head(incr(odds)), y0)))) -> c16(MARK(s(cons(head(incr(odds)), y0)))) MARK(head(s(cons(head(incr(head(y0))), y1)))) -> c16(MARK(s(cons(head(incr(head(y0))), y1)))) MARK(head(s(cons(head(incr(tail(y0))), y1)))) -> c16(MARK(s(cons(head(incr(tail(y0))), y1)))) MARK(head(s(cons(head(incr(cons(y0, y1))), y2)))) -> c16(MARK(s(cons(head(incr(cons(y0, y1))), y2)))) MARK(head(s(cons(head(incr(s(y0))), y1)))) -> c16(MARK(s(cons(head(incr(s(y0))), y1)))) MARK(head(s(cons(head(incr(y0)), y1)))) -> c16(MARK(s(cons(head(incr(y0)), y1)))) MARK(head(s(cons(head(head(incr(y0))), y1)))) -> c16(MARK(s(cons(head(head(incr(y0))), y1)))) MARK(head(s(cons(head(head(odds)), y0)))) -> c16(MARK(s(cons(head(head(odds)), y0)))) MARK(head(s(cons(head(head(head(y0))), y1)))) -> c16(MARK(s(cons(head(head(head(y0))), y1)))) MARK(head(s(cons(head(head(tail(y0))), y1)))) -> c16(MARK(s(cons(head(head(tail(y0))), y1)))) MARK(head(s(cons(head(head(cons(y0, y1))), y2)))) -> c16(MARK(s(cons(head(head(cons(y0, y1))), y2)))) MARK(head(s(cons(head(head(s(y0))), y1)))) -> c16(MARK(s(cons(head(head(s(y0))), y1)))) MARK(head(s(cons(head(head(y0)), y1)))) -> c16(MARK(s(cons(head(head(y0)), y1)))) MARK(head(s(cons(head(tail(incr(y0))), y1)))) -> c16(MARK(s(cons(head(tail(incr(y0))), y1)))) MARK(head(s(cons(head(tail(pairs)), y0)))) -> c16(MARK(s(cons(head(tail(pairs)), y0)))) MARK(head(s(cons(head(tail(odds)), y0)))) -> c16(MARK(s(cons(head(tail(odds)), y0)))) MARK(head(s(cons(head(tail(head(y0))), y1)))) -> c16(MARK(s(cons(head(tail(head(y0))), y1)))) MARK(head(s(cons(head(tail(tail(y0))), y1)))) -> c16(MARK(s(cons(head(tail(tail(y0))), y1)))) MARK(head(s(cons(head(tail(cons(y0, y1))), y2)))) -> c16(MARK(s(cons(head(tail(cons(y0, y1))), y2)))) MARK(head(s(cons(head(tail(s(y0))), y1)))) -> c16(MARK(s(cons(head(tail(s(y0))), y1)))) MARK(head(s(cons(head(tail(y0)), y1)))) -> c16(MARK(s(cons(head(tail(y0)), y1)))) MARK(head(s(cons(tail(incr(incr(y0))), y1)))) -> c16(MARK(s(cons(tail(incr(incr(y0))), y1)))) MARK(head(s(cons(tail(incr(odds)), y0)))) -> c16(MARK(s(cons(tail(incr(odds)), y0)))) MARK(head(s(cons(tail(incr(head(y0))), y1)))) -> c16(MARK(s(cons(tail(incr(head(y0))), y1)))) MARK(head(s(cons(tail(incr(tail(y0))), y1)))) -> c16(MARK(s(cons(tail(incr(tail(y0))), y1)))) MARK(head(s(cons(tail(incr(cons(y0, y1))), y2)))) -> c16(MARK(s(cons(tail(incr(cons(y0, y1))), y2)))) MARK(head(s(cons(tail(incr(s(y0))), y1)))) -> c16(MARK(s(cons(tail(incr(s(y0))), y1)))) MARK(head(s(cons(tail(incr(y0)), y1)))) -> c16(MARK(s(cons(tail(incr(y0)), y1)))) MARK(head(s(cons(tail(head(incr(y0))), y1)))) -> c16(MARK(s(cons(tail(head(incr(y0))), y1)))) MARK(head(s(cons(tail(head(odds)), y0)))) -> c16(MARK(s(cons(tail(head(odds)), y0)))) MARK(head(s(cons(tail(head(head(y0))), y1)))) -> c16(MARK(s(cons(tail(head(head(y0))), y1)))) MARK(head(s(cons(tail(head(tail(y0))), y1)))) -> c16(MARK(s(cons(tail(head(tail(y0))), y1)))) MARK(head(s(cons(tail(head(cons(y0, y1))), y2)))) -> c16(MARK(s(cons(tail(head(cons(y0, y1))), y2)))) MARK(head(s(cons(tail(head(s(y0))), y1)))) -> c16(MARK(s(cons(tail(head(s(y0))), y1)))) MARK(head(s(cons(tail(head(y0)), y1)))) -> c16(MARK(s(cons(tail(head(y0)), y1)))) MARK(head(s(cons(tail(tail(incr(y0))), y1)))) -> c16(MARK(s(cons(tail(tail(incr(y0))), y1)))) MARK(head(s(cons(tail(tail(pairs)), y0)))) -> c16(MARK(s(cons(tail(tail(pairs)), y0)))) MARK(head(s(cons(tail(tail(odds)), y0)))) -> c16(MARK(s(cons(tail(tail(odds)), y0)))) MARK(head(s(cons(tail(tail(head(y0))), y1)))) -> c16(MARK(s(cons(tail(tail(head(y0))), y1)))) MARK(head(s(cons(tail(tail(tail(y0))), y1)))) -> c16(MARK(s(cons(tail(tail(tail(y0))), y1)))) MARK(head(s(cons(tail(tail(cons(y0, y1))), y2)))) -> c16(MARK(s(cons(tail(tail(cons(y0, y1))), y2)))) MARK(head(s(cons(tail(tail(s(y0))), y1)))) -> c16(MARK(s(cons(tail(tail(s(y0))), y1)))) MARK(head(s(cons(tail(tail(y0)), y1)))) -> c16(MARK(s(cons(tail(tail(y0)), y1)))) MARK(head(s(cons(tail(pairs), y0)))) -> c16(MARK(s(cons(tail(pairs), y0)))) MARK(head(s(cons(incr(incr(nats)), y0)))) -> c16(MARK(s(cons(incr(incr(nats)), y0)))) MARK(head(s(cons(incr(incr(pairs)), y0)))) -> c16(MARK(s(cons(incr(incr(pairs)), y0)))) MARK(head(s(cons(incr(odds), y0)))) -> c16(MARK(s(cons(incr(odds), y0)))) MARK(head(s(cons(head(incr(nats)), y0)))) -> c16(MARK(s(cons(head(incr(nats)), y0)))) MARK(head(s(cons(head(incr(pairs)), y0)))) -> c16(MARK(s(cons(head(incr(pairs)), y0)))) MARK(head(s(cons(head(odds), y0)))) -> c16(MARK(s(cons(head(odds), y0)))) MARK(head(s(cons(tail(incr(nats)), y0)))) -> c16(MARK(s(cons(tail(incr(nats)), y0)))) MARK(head(s(cons(tail(incr(pairs)), y0)))) -> c16(MARK(s(cons(tail(incr(pairs)), y0)))) MARK(head(s(cons(tail(odds), y0)))) -> c16(MARK(s(cons(tail(odds), y0)))) MARK(head(s(cons(incr(cons(nats, y0)), y1)))) -> c16(MARK(s(cons(incr(cons(nats, y0)), y1)))) MARK(head(s(cons(incr(cons(incr(y0), y1)), y2)))) -> c16(MARK(s(cons(incr(cons(incr(y0), y1)), y2)))) MARK(head(s(cons(incr(cons(pairs, y0)), y1)))) -> c16(MARK(s(cons(incr(cons(pairs, y0)), y1)))) MARK(head(s(cons(incr(cons(odds, y0)), y1)))) -> c16(MARK(s(cons(incr(cons(odds, y0)), y1)))) MARK(head(s(cons(incr(cons(head(y0), y1)), y2)))) -> c16(MARK(s(cons(incr(cons(head(y0), y1)), y2)))) MARK(head(s(cons(incr(cons(tail(y0), y1)), y2)))) -> c16(MARK(s(cons(incr(cons(tail(y0), y1)), y2)))) MARK(head(s(cons(incr(cons(cons(y0, y1), y2)), y3)))) -> c16(MARK(s(cons(incr(cons(cons(y0, y1), y2)), y3)))) MARK(head(s(cons(incr(cons(s(y0), y1)), y2)))) -> c16(MARK(s(cons(incr(cons(s(y0), y1)), y2)))) MARK(head(s(cons(incr(cons(y0, y1)), y2)))) -> c16(MARK(s(cons(incr(cons(y0, y1)), y2)))) MARK(head(s(cons(incr(cons(0, y0)), y1)))) -> c16(MARK(s(cons(incr(cons(0, y0)), y1)))) MARK(head(s(cons(incr(head(nats)), y0)))) -> c16(MARK(s(cons(incr(head(nats)), y0)))) MARK(head(s(cons(incr(head(pairs)), y0)))) -> c16(MARK(s(cons(incr(head(pairs)), y0)))) MARK(head(s(cons(head(head(nats)), y0)))) -> c16(MARK(s(cons(head(head(nats)), y0)))) MARK(head(s(cons(head(head(pairs)), y0)))) -> c16(MARK(s(cons(head(head(pairs)), y0)))) MARK(head(s(cons(tail(head(nats)), y0)))) -> c16(MARK(s(cons(tail(head(nats)), y0)))) MARK(head(s(cons(tail(head(pairs)), y0)))) -> c16(MARK(s(cons(tail(head(pairs)), y0)))) MARK(head(s(cons(head(cons(nats, y0)), y1)))) -> c16(MARK(s(cons(head(cons(nats, y0)), y1)))) MARK(head(s(cons(head(cons(incr(y0), y1)), y2)))) -> c16(MARK(s(cons(head(cons(incr(y0), y1)), y2)))) MARK(head(s(cons(head(cons(pairs, y0)), y1)))) -> c16(MARK(s(cons(head(cons(pairs, y0)), y1)))) MARK(head(s(cons(head(cons(odds, y0)), y1)))) -> c16(MARK(s(cons(head(cons(odds, y0)), y1)))) MARK(head(s(cons(head(cons(head(y0), y1)), y2)))) -> c16(MARK(s(cons(head(cons(head(y0), y1)), y2)))) MARK(head(s(cons(head(cons(tail(y0), y1)), y2)))) -> c16(MARK(s(cons(head(cons(tail(y0), y1)), y2)))) MARK(head(s(cons(head(cons(cons(y0, y1), y2)), y3)))) -> c16(MARK(s(cons(head(cons(cons(y0, y1), y2)), y3)))) MARK(head(s(cons(head(cons(s(y0), y1)), y2)))) -> c16(MARK(s(cons(head(cons(s(y0), y1)), y2)))) MARK(head(s(cons(head(cons(y0, y1)), y2)))) -> c16(MARK(s(cons(head(cons(y0, y1)), y2)))) MARK(head(s(cons(head(cons(0, y0)), y1)))) -> c16(MARK(s(cons(head(cons(0, y0)), y1)))) MARK(head(s(cons(incr(tail(nats)), y0)))) -> c16(MARK(s(cons(incr(tail(nats)), y0)))) MARK(head(s(cons(head(tail(nats)), y0)))) -> c16(MARK(s(cons(head(tail(nats)), y0)))) MARK(head(s(cons(tail(tail(nats)), y0)))) -> c16(MARK(s(cons(tail(tail(nats)), y0)))) MARK(head(s(incr(cons(nats, y0))))) -> c16(MARK(s(incr(cons(nats, y0))))) MARK(head(s(incr(cons(pairs, y0))))) -> c16(MARK(s(incr(cons(pairs, y0))))) MARK(head(s(incr(cons(odds, y0))))) -> c16(MARK(s(incr(cons(odds, y0))))) MARK(head(s(head(cons(nats, y0))))) -> c16(MARK(s(head(cons(nats, y0))))) MARK(head(s(head(cons(pairs, y0))))) -> c16(MARK(s(head(cons(pairs, y0))))) MARK(head(s(head(cons(odds, y0))))) -> c16(MARK(s(head(cons(odds, y0))))) S tuples: MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c17(MARK(s(z0))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(tail(incr(incr(z0)))) -> c17(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(odds))) -> c17(A__TAIL(a__incr(a__odds)), MARK(incr(odds))) MARK(tail(incr(head(z0)))) -> c17(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c17(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c17(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c17(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c17(MARK(incr(x0))) MARK(tail(head(incr(z0)))) -> c17(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(odds))) -> c17(A__TAIL(a__head(a__odds)), MARK(head(odds))) MARK(tail(head(head(z0)))) -> c17(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c17(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c17(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c17(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c17(MARK(head(x0))) MARK(tail(tail(incr(z0)))) -> c17(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(pairs))) -> c17(A__TAIL(a__tail(a__pairs)), MARK(tail(pairs))) MARK(tail(tail(odds))) -> c17(A__TAIL(a__tail(a__odds)), MARK(tail(odds))) MARK(tail(tail(head(z0)))) -> c17(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c17(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c17(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c17(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c17(MARK(tail(x0))) MARK(tail(pairs)) -> c17(A__TAIL(cons(0, incr(odds)))) A__INCR(cons(cons(y0, y1), z1)) -> c6(MARK(cons(y0, y1))) A__INCR(cons(s(y0), z1)) -> c6(MARK(s(y0))) A__INCR(cons(incr(cons(y0, y1)), z1)) -> c6(MARK(incr(cons(y0, y1)))) A__INCR(cons(incr(s(y0)), z1)) -> c6(MARK(incr(s(y0)))) A__INCR(cons(head(cons(y0, y1)), z1)) -> c6(MARK(head(cons(y0, y1)))) A__INCR(cons(head(s(y0)), z1)) -> c6(MARK(head(s(y0)))) A__INCR(cons(tail(cons(y0, y1)), z1)) -> c6(MARK(tail(cons(y0, y1)))) A__INCR(cons(tail(s(y0)), z1)) -> c6(MARK(tail(s(y0)))) A__INCR(cons(incr(incr(nats)), z1)) -> c6(MARK(incr(incr(nats)))) A__INCR(cons(incr(incr(incr(y0))), z1)) -> c6(MARK(incr(incr(incr(y0))))) A__INCR(cons(incr(incr(pairs)), z1)) -> c6(MARK(incr(incr(pairs)))) A__INCR(cons(incr(incr(odds)), z1)) -> c6(MARK(incr(incr(odds)))) A__INCR(cons(incr(incr(head(y0))), z1)) -> c6(MARK(incr(incr(head(y0))))) A__INCR(cons(incr(incr(tail(y0))), z1)) -> c6(MARK(incr(incr(tail(y0))))) A__INCR(cons(incr(incr(cons(y0, y1))), z1)) -> c6(MARK(incr(incr(cons(y0, y1))))) A__INCR(cons(incr(incr(s(y0))), z1)) -> c6(MARK(incr(incr(s(y0))))) A__INCR(cons(incr(incr(y0)), z1)) -> c6(MARK(incr(incr(y0)))) A__INCR(cons(incr(odds), z1)) -> c6(MARK(incr(odds))) A__INCR(cons(incr(head(nats)), z1)) -> c6(MARK(incr(head(nats)))) A__INCR(cons(incr(head(incr(y0))), z1)) -> c6(MARK(incr(head(incr(y0))))) A__INCR(cons(incr(head(pairs)), z1)) -> c6(MARK(incr(head(pairs)))) A__INCR(cons(incr(head(odds)), z1)) -> c6(MARK(incr(head(odds)))) A__INCR(cons(incr(head(head(y0))), z1)) -> c6(MARK(incr(head(head(y0))))) A__INCR(cons(incr(head(tail(y0))), z1)) -> c6(MARK(incr(head(tail(y0))))) A__INCR(cons(incr(head(cons(y0, y1))), z1)) -> c6(MARK(incr(head(cons(y0, y1))))) A__INCR(cons(incr(head(s(y0))), z1)) -> c6(MARK(incr(head(s(y0))))) A__INCR(cons(incr(head(y0)), z1)) -> c6(MARK(incr(head(y0)))) A__INCR(cons(incr(tail(nats)), z1)) -> c6(MARK(incr(tail(nats)))) A__INCR(cons(incr(tail(incr(y0))), z1)) -> c6(MARK(incr(tail(incr(y0))))) A__INCR(cons(incr(tail(pairs)), z1)) -> c6(MARK(incr(tail(pairs)))) A__INCR(cons(incr(tail(odds)), z1)) -> c6(MARK(incr(tail(odds)))) A__INCR(cons(incr(tail(head(y0))), z1)) -> c6(MARK(incr(tail(head(y0))))) A__INCR(cons(incr(tail(tail(y0))), z1)) -> c6(MARK(incr(tail(tail(y0))))) A__INCR(cons(incr(tail(cons(y0, y1))), z1)) -> c6(MARK(incr(tail(cons(y0, y1))))) A__INCR(cons(incr(tail(s(y0))), z1)) -> c6(MARK(incr(tail(s(y0))))) A__INCR(cons(incr(tail(y0)), z1)) -> c6(MARK(incr(tail(y0)))) A__INCR(cons(head(incr(nats)), z1)) -> c6(MARK(head(incr(nats)))) A__INCR(cons(head(incr(incr(y0))), z1)) -> c6(MARK(head(incr(incr(y0))))) A__INCR(cons(head(incr(pairs)), z1)) -> c6(MARK(head(incr(pairs)))) A__INCR(cons(head(incr(odds)), z1)) -> c6(MARK(head(incr(odds)))) A__INCR(cons(head(incr(head(y0))), z1)) -> c6(MARK(head(incr(head(y0))))) A__INCR(cons(head(incr(tail(y0))), z1)) -> c6(MARK(head(incr(tail(y0))))) A__INCR(cons(head(incr(cons(y0, y1))), z1)) -> c6(MARK(head(incr(cons(y0, y1))))) A__INCR(cons(head(incr(s(y0))), z1)) -> c6(MARK(head(incr(s(y0))))) A__INCR(cons(head(incr(y0)), z1)) -> c6(MARK(head(incr(y0)))) A__INCR(cons(head(odds), z1)) -> c6(MARK(head(odds))) A__INCR(cons(head(head(nats)), z1)) -> c6(MARK(head(head(nats)))) A__INCR(cons(head(head(incr(y0))), z1)) -> c6(MARK(head(head(incr(y0))))) A__INCR(cons(head(head(pairs)), z1)) -> c6(MARK(head(head(pairs)))) A__INCR(cons(head(head(odds)), z1)) -> c6(MARK(head(head(odds)))) A__INCR(cons(head(head(head(y0))), z1)) -> c6(MARK(head(head(head(y0))))) A__INCR(cons(head(head(tail(y0))), z1)) -> c6(MARK(head(head(tail(y0))))) A__INCR(cons(head(head(cons(y0, y1))), z1)) -> c6(MARK(head(head(cons(y0, y1))))) A__INCR(cons(head(head(s(y0))), z1)) -> c6(MARK(head(head(s(y0))))) A__INCR(cons(head(head(y0)), z1)) -> c6(MARK(head(head(y0)))) 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-> c10(MARK(head(incr(odds)))) A__TAIL(cons(z0, head(incr(head(y0))))) -> c10(MARK(head(incr(head(y0))))) A__TAIL(cons(z0, head(incr(tail(y0))))) -> c10(MARK(head(incr(tail(y0))))) A__TAIL(cons(z0, head(incr(cons(y0, y1))))) -> c10(MARK(head(incr(cons(y0, y1))))) A__TAIL(cons(z0, head(incr(s(y0))))) -> c10(MARK(head(incr(s(y0))))) A__TAIL(cons(z0, head(incr(y0)))) -> c10(MARK(head(incr(y0)))) A__TAIL(cons(z0, head(head(incr(y0))))) -> c10(MARK(head(head(incr(y0))))) A__TAIL(cons(z0, head(head(odds)))) -> c10(MARK(head(head(odds)))) A__TAIL(cons(z0, head(head(head(y0))))) -> c10(MARK(head(head(head(y0))))) A__TAIL(cons(z0, head(head(tail(y0))))) -> c10(MARK(head(head(tail(y0))))) A__TAIL(cons(z0, head(head(cons(y0, y1))))) -> c10(MARK(head(head(cons(y0, y1))))) A__TAIL(cons(z0, head(head(s(y0))))) -> c10(MARK(head(head(s(y0))))) A__TAIL(cons(z0, head(head(y0)))) -> c10(MARK(head(head(y0)))) A__TAIL(cons(z0, head(tail(nats)))) -> c10(MARK(head(tail(nats)))) A__TAIL(cons(z0, 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c10(MARK(tail(incr(s(y0))))) A__TAIL(cons(z0, tail(incr(y0)))) -> c10(MARK(tail(incr(y0)))) A__TAIL(cons(z0, tail(head(incr(y0))))) -> c10(MARK(tail(head(incr(y0))))) A__TAIL(cons(z0, tail(head(odds)))) -> c10(MARK(tail(head(odds)))) A__TAIL(cons(z0, tail(head(head(y0))))) -> c10(MARK(tail(head(head(y0))))) A__TAIL(cons(z0, tail(head(tail(y0))))) -> c10(MARK(tail(head(tail(y0))))) A__TAIL(cons(z0, tail(head(cons(y0, y1))))) -> c10(MARK(tail(head(cons(y0, y1))))) A__TAIL(cons(z0, tail(head(s(y0))))) -> c10(MARK(tail(head(s(y0))))) A__TAIL(cons(z0, tail(head(y0)))) -> c10(MARK(tail(head(y0)))) A__TAIL(cons(z0, tail(tail(nats)))) -> c10(MARK(tail(tail(nats)))) A__TAIL(cons(z0, tail(tail(incr(y0))))) -> c10(MARK(tail(tail(incr(y0))))) A__TAIL(cons(z0, tail(tail(pairs)))) -> c10(MARK(tail(tail(pairs)))) A__TAIL(cons(z0, tail(tail(odds)))) -> c10(MARK(tail(tail(odds)))) A__TAIL(cons(z0, tail(tail(head(y0))))) -> c10(MARK(tail(tail(head(y0))))) A__TAIL(cons(z0, tail(tail(tail(y0))))) -> c10(MARK(tail(tail(tail(y0))))) A__TAIL(cons(z0, tail(tail(cons(y0, y1))))) -> c10(MARK(tail(tail(cons(y0, y1))))) A__TAIL(cons(z0, tail(tail(s(y0))))) -> c10(MARK(tail(tail(s(y0))))) A__TAIL(cons(z0, tail(tail(y0)))) -> c10(MARK(tail(tail(y0)))) A__TAIL(cons(z0, tail(pairs))) -> c10(MARK(tail(pairs))) A__TAIL(cons(z0, incr(incr(nats)))) -> c10(MARK(incr(incr(nats)))) A__TAIL(cons(z0, incr(incr(pairs)))) -> c10(MARK(incr(incr(pairs)))) A__TAIL(cons(z0, incr(odds))) -> c10(MARK(incr(odds))) A__TAIL(cons(z0, head(incr(nats)))) -> c10(MARK(head(incr(nats)))) A__TAIL(cons(z0, head(incr(pairs)))) -> c10(MARK(head(incr(pairs)))) A__TAIL(cons(z0, head(odds))) -> c10(MARK(head(odds))) A__TAIL(cons(z0, tail(incr(nats)))) -> c10(MARK(tail(incr(nats)))) A__TAIL(cons(z0, tail(incr(pairs)))) -> c10(MARK(tail(incr(pairs)))) A__TAIL(cons(z0, tail(odds))) -> c10(MARK(tail(odds))) A__TAIL(cons(z0, incr(cons(nats, y0)))) -> c10(MARK(incr(cons(nats, y0)))) A__TAIL(cons(z0, incr(cons(incr(y0), y1)))) -> c10(MARK(incr(cons(incr(y0), y1)))) A__TAIL(cons(z0, incr(cons(pairs, y0)))) -> c10(MARK(incr(cons(pairs, y0)))) A__TAIL(cons(z0, incr(cons(odds, y0)))) -> c10(MARK(incr(cons(odds, y0)))) A__TAIL(cons(z0, incr(cons(head(y0), y1)))) -> c10(MARK(incr(cons(head(y0), y1)))) A__TAIL(cons(z0, incr(cons(tail(y0), y1)))) -> c10(MARK(incr(cons(tail(y0), y1)))) A__TAIL(cons(z0, incr(cons(cons(y0, y1), y2)))) -> c10(MARK(incr(cons(cons(y0, y1), y2)))) A__TAIL(cons(z0, incr(cons(s(y0), y1)))) -> c10(MARK(incr(cons(s(y0), y1)))) A__TAIL(cons(z0, incr(cons(y0, y1)))) -> c10(MARK(incr(cons(y0, y1)))) A__TAIL(cons(z0, incr(cons(0, y0)))) -> c10(MARK(incr(cons(0, y0)))) A__TAIL(cons(z0, incr(head(nats)))) -> c10(MARK(incr(head(nats)))) A__TAIL(cons(z0, incr(head(pairs)))) -> c10(MARK(incr(head(pairs)))) A__TAIL(cons(z0, head(head(nats)))) -> c10(MARK(head(head(nats)))) A__TAIL(cons(z0, head(head(pairs)))) -> c10(MARK(head(head(pairs)))) A__TAIL(cons(z0, tail(head(nats)))) -> c10(MARK(tail(head(nats)))) A__TAIL(cons(z0, tail(head(pairs)))) -> c10(MARK(tail(head(pairs)))) A__TAIL(cons(z0, head(cons(nats, y0)))) -> c10(MARK(head(cons(nats, y0)))) A__TAIL(cons(z0, head(cons(incr(y0), y1)))) -> c10(MARK(head(cons(incr(y0), y1)))) A__TAIL(cons(z0, head(cons(pairs, y0)))) -> c10(MARK(head(cons(pairs, y0)))) A__TAIL(cons(z0, head(cons(odds, y0)))) -> c10(MARK(head(cons(odds, y0)))) A__TAIL(cons(z0, head(cons(head(y0), y1)))) -> c10(MARK(head(cons(head(y0), y1)))) A__TAIL(cons(z0, head(cons(tail(y0), y1)))) -> c10(MARK(head(cons(tail(y0), y1)))) A__TAIL(cons(z0, head(cons(cons(y0, y1), y2)))) -> c10(MARK(head(cons(cons(y0, y1), y2)))) A__TAIL(cons(z0, head(cons(s(y0), y1)))) -> c10(MARK(head(cons(s(y0), y1)))) A__TAIL(cons(z0, head(cons(y0, y1)))) -> c10(MARK(head(cons(y0, y1)))) A__TAIL(cons(z0, head(cons(0, y0)))) -> c10(MARK(head(cons(0, y0)))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats))) MARK(tail(tail(nats))) -> c17(A__TAIL(a__tail(a__nats))) MARK(cons(cons(y0, y1), z1)) -> c18(MARK(cons(y0, y1))) MARK(cons(s(y0), z1)) -> c18(MARK(s(y0))) MARK(cons(incr(s(y0)), z1)) -> c18(MARK(incr(s(y0)))) MARK(cons(head(s(y0)), z1)) -> c18(MARK(head(s(y0)))) MARK(cons(tail(cons(y0, y1)), z1)) -> c18(MARK(tail(cons(y0, y1)))) MARK(cons(tail(s(y0)), z1)) -> c18(MARK(tail(s(y0)))) MARK(cons(incr(incr(incr(y0))), z1)) -> c18(MARK(incr(incr(incr(y0))))) MARK(cons(incr(incr(odds)), z1)) -> c18(MARK(incr(incr(odds)))) MARK(cons(incr(incr(head(y0))), z1)) -> c18(MARK(incr(incr(head(y0))))) MARK(cons(incr(incr(tail(y0))), z1)) -> c18(MARK(incr(incr(tail(y0))))) MARK(cons(incr(incr(cons(y0, y1))), z1)) -> c18(MARK(incr(incr(cons(y0, y1))))) MARK(cons(incr(incr(s(y0))), z1)) -> c18(MARK(incr(incr(s(y0))))) MARK(cons(incr(incr(y0)), z1)) -> c18(MARK(incr(incr(y0)))) MARK(cons(incr(head(incr(y0))), z1)) -> c18(MARK(incr(head(incr(y0))))) MARK(cons(incr(head(odds)), z1)) -> c18(MARK(incr(head(odds)))) MARK(cons(incr(head(head(y0))), z1)) -> c18(MARK(incr(head(head(y0))))) MARK(cons(incr(head(tail(y0))), z1)) -> c18(MARK(incr(head(tail(y0))))) MARK(cons(incr(head(cons(y0, y1))), z1)) -> c18(MARK(incr(head(cons(y0, y1))))) MARK(cons(incr(head(s(y0))), z1)) -> c18(MARK(incr(head(s(y0))))) MARK(cons(incr(head(y0)), z1)) -> c18(MARK(incr(head(y0)))) MARK(cons(incr(tail(incr(y0))), z1)) -> c18(MARK(incr(tail(incr(y0))))) MARK(cons(incr(tail(pairs)), z1)) -> c18(MARK(incr(tail(pairs)))) MARK(cons(incr(tail(odds)), z1)) -> c18(MARK(incr(tail(odds)))) MARK(cons(incr(tail(head(y0))), z1)) -> c18(MARK(incr(tail(head(y0))))) MARK(cons(incr(tail(tail(y0))), z1)) -> c18(MARK(incr(tail(tail(y0))))) MARK(cons(incr(tail(cons(y0, y1))), z1)) -> c18(MARK(incr(tail(cons(y0, y1))))) MARK(cons(incr(tail(s(y0))), z1)) -> c18(MARK(incr(tail(s(y0))))) MARK(cons(incr(tail(y0)), z1)) -> c18(MARK(incr(tail(y0)))) MARK(cons(head(incr(incr(y0))), z1)) -> c18(MARK(head(incr(incr(y0))))) MARK(cons(head(incr(odds)), z1)) -> c18(MARK(head(incr(odds)))) MARK(cons(head(incr(head(y0))), z1)) -> c18(MARK(head(incr(head(y0))))) MARK(cons(head(incr(tail(y0))), z1)) -> c18(MARK(head(incr(tail(y0))))) MARK(cons(head(incr(cons(y0, y1))), z1)) -> c18(MARK(head(incr(cons(y0, y1))))) MARK(cons(head(incr(s(y0))), z1)) -> c18(MARK(head(incr(s(y0))))) MARK(cons(head(incr(y0)), z1)) -> c18(MARK(head(incr(y0)))) MARK(cons(head(head(incr(y0))), z1)) -> c18(MARK(head(head(incr(y0))))) MARK(cons(head(head(odds)), z1)) -> c18(MARK(head(head(odds)))) MARK(cons(head(head(head(y0))), z1)) -> c18(MARK(head(head(head(y0))))) MARK(cons(head(head(tail(y0))), z1)) -> c18(MARK(head(head(tail(y0))))) MARK(cons(head(head(cons(y0, y1))), z1)) -> c18(MARK(head(head(cons(y0, y1))))) MARK(cons(head(head(s(y0))), z1)) -> c18(MARK(head(head(s(y0))))) MARK(cons(head(head(y0)), z1)) -> c18(MARK(head(head(y0)))) MARK(cons(head(tail(incr(y0))), z1)) -> c18(MARK(head(tail(incr(y0))))) MARK(cons(head(tail(pairs)), z1)) -> c18(MARK(head(tail(pairs)))) MARK(cons(head(tail(odds)), z1)) -> c18(MARK(head(tail(odds)))) MARK(cons(head(tail(head(y0))), z1)) -> c18(MARK(head(tail(head(y0))))) MARK(cons(head(tail(tail(y0))), z1)) -> c18(MARK(head(tail(tail(y0))))) MARK(cons(head(tail(cons(y0, y1))), z1)) -> c18(MARK(head(tail(cons(y0, y1))))) MARK(cons(head(tail(s(y0))), z1)) -> c18(MARK(head(tail(s(y0))))) MARK(cons(head(tail(y0)), z1)) -> c18(MARK(head(tail(y0)))) MARK(cons(tail(incr(incr(y0))), z1)) -> c18(MARK(tail(incr(incr(y0))))) MARK(cons(tail(incr(odds)), z1)) -> c18(MARK(tail(incr(odds)))) MARK(cons(tail(incr(head(y0))), z1)) -> c18(MARK(tail(incr(head(y0))))) MARK(cons(tail(incr(tail(y0))), z1)) -> c18(MARK(tail(incr(tail(y0))))) MARK(cons(tail(incr(cons(y0, y1))), z1)) -> c18(MARK(tail(incr(cons(y0, y1))))) MARK(cons(tail(incr(s(y0))), z1)) -> c18(MARK(tail(incr(s(y0))))) MARK(cons(tail(incr(y0)), z1)) -> c18(MARK(tail(incr(y0)))) MARK(cons(tail(head(incr(y0))), z1)) -> c18(MARK(tail(head(incr(y0))))) MARK(cons(tail(head(odds)), z1)) -> c18(MARK(tail(head(odds)))) MARK(cons(tail(head(head(y0))), z1)) -> c18(MARK(tail(head(head(y0))))) MARK(cons(tail(head(tail(y0))), z1)) -> c18(MARK(tail(head(tail(y0))))) MARK(cons(tail(head(cons(y0, y1))), z1)) -> c18(MARK(tail(head(cons(y0, y1))))) MARK(cons(tail(head(s(y0))), z1)) -> c18(MARK(tail(head(s(y0))))) MARK(cons(tail(head(y0)), z1)) -> c18(MARK(tail(head(y0)))) MARK(cons(tail(tail(incr(y0))), z1)) -> c18(MARK(tail(tail(incr(y0))))) MARK(cons(tail(tail(pairs)), z1)) -> c18(MARK(tail(tail(pairs)))) MARK(cons(tail(tail(odds)), z1)) -> c18(MARK(tail(tail(odds)))) MARK(cons(tail(tail(head(y0))), z1)) -> c18(MARK(tail(tail(head(y0))))) MARK(cons(tail(tail(tail(y0))), z1)) -> c18(MARK(tail(tail(tail(y0))))) MARK(cons(tail(tail(cons(y0, y1))), z1)) -> c18(MARK(tail(tail(cons(y0, y1))))) MARK(cons(tail(tail(s(y0))), z1)) -> c18(MARK(tail(tail(s(y0))))) MARK(cons(tail(tail(y0)), z1)) -> c18(MARK(tail(tail(y0)))) MARK(cons(tail(pairs), z1)) -> c18(MARK(tail(pairs))) MARK(cons(incr(incr(nats)), z1)) -> c18(MARK(incr(incr(nats)))) MARK(cons(incr(incr(pairs)), z1)) -> c18(MARK(incr(incr(pairs)))) MARK(cons(incr(odds), z1)) -> c18(MARK(incr(odds))) MARK(cons(head(incr(nats)), z1)) -> c18(MARK(head(incr(nats)))) MARK(cons(head(incr(pairs)), z1)) -> c18(MARK(head(incr(pairs)))) MARK(cons(head(odds), z1)) -> c18(MARK(head(odds))) MARK(cons(tail(incr(nats)), z1)) -> c18(MARK(tail(incr(nats)))) MARK(cons(tail(incr(pairs)), z1)) -> c18(MARK(tail(incr(pairs)))) MARK(cons(tail(odds), z1)) -> c18(MARK(tail(odds))) MARK(cons(incr(cons(nats, y0)), z1)) -> c18(MARK(incr(cons(nats, y0)))) MARK(cons(incr(cons(incr(y0), y1)), z1)) -> c18(MARK(incr(cons(incr(y0), y1)))) MARK(cons(incr(cons(pairs, y0)), z1)) -> c18(MARK(incr(cons(pairs, y0)))) MARK(cons(incr(cons(odds, y0)), z1)) -> c18(MARK(incr(cons(odds, y0)))) MARK(cons(incr(cons(head(y0), y1)), z1)) -> c18(MARK(incr(cons(head(y0), y1)))) MARK(cons(incr(cons(tail(y0), y1)), z1)) -> c18(MARK(incr(cons(tail(y0), y1)))) MARK(cons(incr(cons(cons(y0, y1), y2)), z1)) -> c18(MARK(incr(cons(cons(y0, y1), y2)))) MARK(cons(incr(cons(s(y0), y1)), z1)) -> c18(MARK(incr(cons(s(y0), y1)))) MARK(cons(incr(cons(y0, y1)), z1)) -> c18(MARK(incr(cons(y0, y1)))) MARK(cons(incr(cons(0, y0)), z1)) -> c18(MARK(incr(cons(0, y0)))) MARK(cons(incr(head(nats)), z1)) -> c18(MARK(incr(head(nats)))) MARK(cons(incr(head(pairs)), z1)) -> c18(MARK(incr(head(pairs)))) MARK(cons(head(head(nats)), z1)) -> c18(MARK(head(head(nats)))) MARK(cons(head(head(pairs)), z1)) -> c18(MARK(head(head(pairs)))) MARK(cons(tail(head(nats)), z1)) -> c18(MARK(tail(head(nats)))) MARK(cons(tail(head(pairs)), z1)) -> c18(MARK(tail(head(pairs)))) MARK(cons(head(cons(nats, y0)), z1)) -> c18(MARK(head(cons(nats, y0)))) MARK(cons(head(cons(incr(y0), y1)), z1)) -> c18(MARK(head(cons(incr(y0), y1)))) MARK(cons(head(cons(pairs, y0)), z1)) -> c18(MARK(head(cons(pairs, y0)))) MARK(cons(head(cons(odds, y0)), z1)) -> c18(MARK(head(cons(odds, y0)))) MARK(cons(head(cons(head(y0), y1)), z1)) -> c18(MARK(head(cons(head(y0), y1)))) MARK(cons(head(cons(tail(y0), y1)), z1)) -> c18(MARK(head(cons(tail(y0), y1)))) MARK(cons(head(cons(cons(y0, y1), y2)), z1)) -> c18(MARK(head(cons(cons(y0, y1), y2)))) MARK(cons(head(cons(s(y0), y1)), z1)) -> c18(MARK(head(cons(s(y0), y1)))) MARK(cons(head(cons(y0, y1)), z1)) -> c18(MARK(head(cons(y0, y1)))) MARK(cons(head(cons(0, y0)), z1)) -> c18(MARK(head(cons(0, y0)))) MARK(cons(incr(tail(nats)), z1)) -> c18(MARK(incr(tail(nats)))) MARK(cons(head(tail(nats)), z1)) -> c18(MARK(head(tail(nats)))) MARK(cons(tail(tail(nats)), z1)) -> c18(MARK(tail(tail(nats)))) MARK(incr(cons(nats, x1))) -> c13(A__INCR(cons(a__nats, x1))) MARK(incr(cons(pairs, x1))) -> c13(A__INCR(cons(a__pairs, x1))) MARK(incr(cons(odds, x1))) -> c13(A__INCR(cons(a__odds, x1))) MARK(head(cons(nats, x1))) -> c16(A__HEAD(cons(a__nats, x1))) MARK(head(cons(pairs, x1))) -> c16(A__HEAD(cons(a__pairs, x1))) MARK(head(cons(odds, x1))) -> c16(A__HEAD(cons(a__odds, x1))) MARK(s(s(y0))) -> c20(MARK(s(y0))) MARK(s(incr(s(y0)))) -> c20(MARK(incr(s(y0)))) MARK(s(head(s(y0)))) -> c20(MARK(head(s(y0)))) MARK(s(tail(cons(y0, y1)))) -> c20(MARK(tail(cons(y0, y1)))) MARK(s(tail(s(y0)))) -> c20(MARK(tail(s(y0)))) MARK(s(incr(incr(incr(y0))))) -> c20(MARK(incr(incr(incr(y0))))) MARK(s(incr(incr(odds)))) -> c20(MARK(incr(incr(odds)))) MARK(s(incr(incr(head(y0))))) -> c20(MARK(incr(incr(head(y0))))) MARK(s(incr(incr(tail(y0))))) -> c20(MARK(incr(incr(tail(y0))))) MARK(s(incr(incr(cons(y0, y1))))) -> c20(MARK(incr(incr(cons(y0, y1))))) MARK(s(incr(incr(s(y0))))) -> c20(MARK(incr(incr(s(y0))))) MARK(s(incr(incr(y0)))) -> c20(MARK(incr(incr(y0)))) MARK(s(incr(head(incr(y0))))) -> c20(MARK(incr(head(incr(y0))))) MARK(s(incr(head(odds)))) -> c20(MARK(incr(head(odds)))) MARK(s(incr(head(head(y0))))) -> c20(MARK(incr(head(head(y0))))) MARK(s(incr(head(tail(y0))))) -> c20(MARK(incr(head(tail(y0))))) MARK(s(incr(head(cons(y0, y1))))) -> c20(MARK(incr(head(cons(y0, y1))))) MARK(s(incr(head(s(y0))))) -> c20(MARK(incr(head(s(y0))))) MARK(s(incr(head(y0)))) -> c20(MARK(incr(head(y0)))) MARK(s(incr(tail(incr(y0))))) -> c20(MARK(incr(tail(incr(y0))))) MARK(s(incr(tail(pairs)))) -> c20(MARK(incr(tail(pairs)))) MARK(s(incr(tail(odds)))) -> c20(MARK(incr(tail(odds)))) MARK(s(incr(tail(head(y0))))) -> c20(MARK(incr(tail(head(y0))))) MARK(s(incr(tail(tail(y0))))) -> c20(MARK(incr(tail(tail(y0))))) MARK(s(incr(tail(cons(y0, y1))))) -> c20(MARK(incr(tail(cons(y0, y1))))) MARK(s(incr(tail(s(y0))))) -> c20(MARK(incr(tail(s(y0))))) MARK(s(incr(tail(y0)))) -> c20(MARK(incr(tail(y0)))) MARK(s(head(incr(incr(y0))))) -> c20(MARK(head(incr(incr(y0))))) MARK(s(head(incr(odds)))) -> c20(MARK(head(incr(odds)))) MARK(s(head(incr(head(y0))))) -> c20(MARK(head(incr(head(y0))))) MARK(s(head(incr(tail(y0))))) -> c20(MARK(head(incr(tail(y0))))) MARK(s(head(incr(cons(y0, y1))))) -> c20(MARK(head(incr(cons(y0, y1))))) MARK(s(head(incr(s(y0))))) -> c20(MARK(head(incr(s(y0))))) MARK(s(head(incr(y0)))) -> c20(MARK(head(incr(y0)))) MARK(s(head(head(incr(y0))))) -> c20(MARK(head(head(incr(y0))))) MARK(s(head(head(odds)))) -> c20(MARK(head(head(odds)))) MARK(s(head(head(head(y0))))) -> c20(MARK(head(head(head(y0))))) MARK(s(head(head(tail(y0))))) -> c20(MARK(head(head(tail(y0))))) MARK(s(head(head(cons(y0, y1))))) -> c20(MARK(head(head(cons(y0, y1))))) MARK(s(head(head(s(y0))))) -> c20(MARK(head(head(s(y0))))) MARK(s(head(head(y0)))) -> c20(MARK(head(head(y0)))) MARK(s(head(tail(incr(y0))))) -> c20(MARK(head(tail(incr(y0))))) MARK(s(head(tail(pairs)))) -> c20(MARK(head(tail(pairs)))) MARK(s(head(tail(odds)))) -> c20(MARK(head(tail(odds)))) MARK(s(head(tail(head(y0))))) -> c20(MARK(head(tail(head(y0))))) MARK(s(head(tail(tail(y0))))) -> c20(MARK(head(tail(tail(y0))))) MARK(s(head(tail(cons(y0, y1))))) -> c20(MARK(head(tail(cons(y0, y1))))) MARK(s(head(tail(s(y0))))) -> c20(MARK(head(tail(s(y0))))) MARK(s(head(tail(y0)))) -> c20(MARK(head(tail(y0)))) MARK(s(tail(incr(incr(y0))))) -> c20(MARK(tail(incr(incr(y0))))) MARK(s(tail(incr(odds)))) -> c20(MARK(tail(incr(odds)))) MARK(s(tail(incr(head(y0))))) -> c20(MARK(tail(incr(head(y0))))) MARK(s(tail(incr(tail(y0))))) -> c20(MARK(tail(incr(tail(y0))))) MARK(s(tail(incr(cons(y0, y1))))) -> c20(MARK(tail(incr(cons(y0, y1))))) MARK(s(tail(incr(s(y0))))) -> c20(MARK(tail(incr(s(y0))))) MARK(s(tail(incr(y0)))) -> c20(MARK(tail(incr(y0)))) MARK(s(tail(head(incr(y0))))) -> c20(MARK(tail(head(incr(y0))))) MARK(s(tail(head(odds)))) -> c20(MARK(tail(head(odds)))) MARK(s(tail(head(head(y0))))) -> c20(MARK(tail(head(head(y0))))) MARK(s(tail(head(tail(y0))))) -> c20(MARK(tail(head(tail(y0))))) MARK(s(tail(head(cons(y0, y1))))) -> c20(MARK(tail(head(cons(y0, y1))))) MARK(s(tail(head(s(y0))))) -> c20(MARK(tail(head(s(y0))))) MARK(s(tail(head(y0)))) -> c20(MARK(tail(head(y0)))) MARK(s(tail(tail(incr(y0))))) -> c20(MARK(tail(tail(incr(y0))))) MARK(s(tail(tail(pairs)))) -> c20(MARK(tail(tail(pairs)))) MARK(s(tail(tail(odds)))) -> c20(MARK(tail(tail(odds)))) MARK(s(tail(tail(head(y0))))) -> c20(MARK(tail(tail(head(y0))))) MARK(s(tail(tail(tail(y0))))) -> c20(MARK(tail(tail(tail(y0))))) MARK(s(tail(tail(cons(y0, y1))))) -> c20(MARK(tail(tail(cons(y0, y1))))) MARK(s(tail(tail(s(y0))))) -> c20(MARK(tail(tail(s(y0))))) MARK(s(tail(tail(y0)))) -> c20(MARK(tail(tail(y0)))) MARK(s(tail(pairs))) -> c20(MARK(tail(pairs))) MARK(s(incr(incr(nats)))) -> c20(MARK(incr(incr(nats)))) MARK(s(incr(incr(pairs)))) -> c20(MARK(incr(incr(pairs)))) MARK(s(incr(odds))) -> c20(MARK(incr(odds))) MARK(s(head(incr(nats)))) -> c20(MARK(head(incr(nats)))) MARK(s(head(incr(pairs)))) -> c20(MARK(head(incr(pairs)))) MARK(s(head(odds))) -> c20(MARK(head(odds))) MARK(s(tail(incr(nats)))) -> c20(MARK(tail(incr(nats)))) MARK(s(tail(incr(pairs)))) -> c20(MARK(tail(incr(pairs)))) MARK(s(tail(odds))) -> c20(MARK(tail(odds))) MARK(s(incr(cons(incr(y0), y1)))) -> c20(MARK(incr(cons(incr(y0), y1)))) MARK(s(incr(cons(head(y0), y1)))) -> c20(MARK(incr(cons(head(y0), y1)))) MARK(s(incr(cons(tail(y0), y1)))) -> c20(MARK(incr(cons(tail(y0), y1)))) MARK(s(incr(cons(cons(y0, y1), y2)))) -> c20(MARK(incr(cons(cons(y0, y1), y2)))) MARK(s(incr(cons(s(y0), y1)))) -> c20(MARK(incr(cons(s(y0), y1)))) MARK(s(incr(cons(y0, y1)))) -> c20(MARK(incr(cons(y0, y1)))) MARK(s(incr(head(nats)))) -> c20(MARK(incr(head(nats)))) MARK(s(incr(head(pairs)))) -> c20(MARK(incr(head(pairs)))) MARK(s(head(head(nats)))) -> c20(MARK(head(head(nats)))) MARK(s(head(head(pairs)))) -> c20(MARK(head(head(pairs)))) MARK(s(tail(head(nats)))) -> c20(MARK(tail(head(nats)))) MARK(s(tail(head(pairs)))) -> c20(MARK(tail(head(pairs)))) MARK(s(head(cons(incr(y0), y1)))) -> c20(MARK(head(cons(incr(y0), y1)))) MARK(s(head(cons(head(y0), y1)))) -> c20(MARK(head(cons(head(y0), y1)))) MARK(s(head(cons(tail(y0), y1)))) -> c20(MARK(head(cons(tail(y0), y1)))) MARK(s(head(cons(cons(y0, y1), y2)))) -> c20(MARK(head(cons(cons(y0, y1), y2)))) MARK(s(head(cons(s(y0), y1)))) -> c20(MARK(head(cons(s(y0), y1)))) MARK(s(head(cons(y0, y1)))) -> c20(MARK(head(cons(y0, y1)))) MARK(s(incr(tail(nats)))) -> c20(MARK(incr(tail(nats)))) MARK(s(head(tail(nats)))) -> c20(MARK(head(tail(nats)))) MARK(s(tail(tail(nats)))) -> c20(MARK(tail(tail(nats)))) MARK(s(cons(cons(y0, y1), y2))) -> c20(MARK(cons(cons(y0, y1), y2))) MARK(s(cons(s(y0), y1))) -> c20(MARK(cons(s(y0), y1))) MARK(s(cons(incr(s(y0)), y1))) -> c20(MARK(cons(incr(s(y0)), y1))) MARK(s(cons(head(s(y0)), y1))) -> c20(MARK(cons(head(s(y0)), y1))) MARK(s(cons(tail(cons(y0, y1)), y2))) -> c20(MARK(cons(tail(cons(y0, y1)), y2))) MARK(s(cons(tail(s(y0)), y1))) -> c20(MARK(cons(tail(s(y0)), y1))) MARK(s(cons(incr(incr(incr(y0))), y1))) -> c20(MARK(cons(incr(incr(incr(y0))), y1))) MARK(s(cons(incr(incr(odds)), y0))) -> c20(MARK(cons(incr(incr(odds)), y0))) MARK(s(cons(incr(incr(head(y0))), y1))) -> c20(MARK(cons(incr(incr(head(y0))), y1))) MARK(s(cons(incr(incr(tail(y0))), y1))) -> c20(MARK(cons(incr(incr(tail(y0))), y1))) MARK(s(cons(incr(incr(cons(y0, y1))), y2))) -> c20(MARK(cons(incr(incr(cons(y0, y1))), y2))) MARK(s(cons(incr(incr(s(y0))), y1))) -> c20(MARK(cons(incr(incr(s(y0))), y1))) MARK(s(cons(incr(incr(y0)), y1))) -> c20(MARK(cons(incr(incr(y0)), y1))) MARK(s(cons(incr(head(incr(y0))), y1))) -> c20(MARK(cons(incr(head(incr(y0))), y1))) MARK(s(cons(incr(head(odds)), y0))) -> c20(MARK(cons(incr(head(odds)), y0))) MARK(s(cons(incr(head(head(y0))), y1))) -> c20(MARK(cons(incr(head(head(y0))), y1))) MARK(s(cons(incr(head(tail(y0))), y1))) -> c20(MARK(cons(incr(head(tail(y0))), y1))) MARK(s(cons(incr(head(cons(y0, y1))), y2))) -> c20(MARK(cons(incr(head(cons(y0, y1))), y2))) MARK(s(cons(incr(head(s(y0))), y1))) -> c20(MARK(cons(incr(head(s(y0))), y1))) MARK(s(cons(incr(head(y0)), y1))) -> c20(MARK(cons(incr(head(y0)), y1))) MARK(s(cons(incr(tail(incr(y0))), y1))) -> c20(MARK(cons(incr(tail(incr(y0))), y1))) MARK(s(cons(incr(tail(pairs)), y0))) -> c20(MARK(cons(incr(tail(pairs)), y0))) MARK(s(cons(incr(tail(odds)), y0))) -> c20(MARK(cons(incr(tail(odds)), y0))) MARK(s(cons(incr(tail(head(y0))), y1))) -> c20(MARK(cons(incr(tail(head(y0))), y1))) MARK(s(cons(incr(tail(tail(y0))), y1))) -> c20(MARK(cons(incr(tail(tail(y0))), y1))) MARK(s(cons(incr(tail(cons(y0, y1))), y2))) -> c20(MARK(cons(incr(tail(cons(y0, y1))), y2))) MARK(s(cons(incr(tail(s(y0))), y1))) -> c20(MARK(cons(incr(tail(s(y0))), y1))) MARK(s(cons(incr(tail(y0)), y1))) -> c20(MARK(cons(incr(tail(y0)), y1))) MARK(s(cons(head(incr(incr(y0))), y1))) -> c20(MARK(cons(head(incr(incr(y0))), y1))) MARK(s(cons(head(incr(odds)), y0))) -> c20(MARK(cons(head(incr(odds)), y0))) MARK(s(cons(head(incr(head(y0))), y1))) -> c20(MARK(cons(head(incr(head(y0))), y1))) MARK(s(cons(head(incr(tail(y0))), y1))) -> c20(MARK(cons(head(incr(tail(y0))), y1))) MARK(s(cons(head(incr(cons(y0, y1))), y2))) -> c20(MARK(cons(head(incr(cons(y0, y1))), y2))) MARK(s(cons(head(incr(s(y0))), y1))) -> c20(MARK(cons(head(incr(s(y0))), y1))) MARK(s(cons(head(incr(y0)), y1))) -> c20(MARK(cons(head(incr(y0)), y1))) MARK(s(cons(head(head(incr(y0))), y1))) -> c20(MARK(cons(head(head(incr(y0))), y1))) MARK(s(cons(head(head(odds)), y0))) -> c20(MARK(cons(head(head(odds)), y0))) MARK(s(cons(head(head(head(y0))), y1))) -> c20(MARK(cons(head(head(head(y0))), y1))) MARK(s(cons(head(head(tail(y0))), y1))) -> c20(MARK(cons(head(head(tail(y0))), y1))) MARK(s(cons(head(head(cons(y0, y1))), y2))) -> c20(MARK(cons(head(head(cons(y0, y1))), y2))) MARK(s(cons(head(head(s(y0))), y1))) -> c20(MARK(cons(head(head(s(y0))), y1))) MARK(s(cons(head(head(y0)), y1))) -> c20(MARK(cons(head(head(y0)), y1))) MARK(s(cons(head(tail(incr(y0))), y1))) -> c20(MARK(cons(head(tail(incr(y0))), y1))) MARK(s(cons(head(tail(pairs)), y0))) -> c20(MARK(cons(head(tail(pairs)), y0))) MARK(s(cons(head(tail(odds)), y0))) -> c20(MARK(cons(head(tail(odds)), y0))) MARK(s(cons(head(tail(head(y0))), y1))) -> c20(MARK(cons(head(tail(head(y0))), y1))) MARK(s(cons(head(tail(tail(y0))), y1))) -> c20(MARK(cons(head(tail(tail(y0))), y1))) MARK(s(cons(head(tail(cons(y0, y1))), y2))) -> c20(MARK(cons(head(tail(cons(y0, y1))), y2))) MARK(s(cons(head(tail(s(y0))), y1))) -> c20(MARK(cons(head(tail(s(y0))), y1))) MARK(s(cons(head(tail(y0)), y1))) -> c20(MARK(cons(head(tail(y0)), y1))) MARK(s(cons(tail(incr(incr(y0))), y1))) -> c20(MARK(cons(tail(incr(incr(y0))), y1))) MARK(s(cons(tail(incr(odds)), y0))) -> c20(MARK(cons(tail(incr(odds)), y0))) MARK(s(cons(tail(incr(head(y0))), y1))) -> c20(MARK(cons(tail(incr(head(y0))), y1))) MARK(s(cons(tail(incr(tail(y0))), y1))) -> c20(MARK(cons(tail(incr(tail(y0))), y1))) MARK(s(cons(tail(incr(cons(y0, y1))), y2))) -> c20(MARK(cons(tail(incr(cons(y0, y1))), y2))) MARK(s(cons(tail(incr(s(y0))), y1))) -> c20(MARK(cons(tail(incr(s(y0))), y1))) MARK(s(cons(tail(incr(y0)), y1))) -> c20(MARK(cons(tail(incr(y0)), y1))) MARK(s(cons(tail(head(incr(y0))), y1))) -> c20(MARK(cons(tail(head(incr(y0))), y1))) MARK(s(cons(tail(head(odds)), y0))) -> c20(MARK(cons(tail(head(odds)), y0))) MARK(s(cons(tail(head(head(y0))), y1))) -> c20(MARK(cons(tail(head(head(y0))), y1))) MARK(s(cons(tail(head(tail(y0))), y1))) -> c20(MARK(cons(tail(head(tail(y0))), y1))) MARK(s(cons(tail(head(cons(y0, y1))), y2))) -> c20(MARK(cons(tail(head(cons(y0, y1))), y2))) MARK(s(cons(tail(head(s(y0))), y1))) -> c20(MARK(cons(tail(head(s(y0))), y1))) MARK(s(cons(tail(head(y0)), y1))) -> c20(MARK(cons(tail(head(y0)), y1))) MARK(s(cons(tail(tail(incr(y0))), y1))) -> c20(MARK(cons(tail(tail(incr(y0))), y1))) MARK(s(cons(tail(tail(pairs)), y0))) -> c20(MARK(cons(tail(tail(pairs)), y0))) MARK(s(cons(tail(tail(odds)), y0))) -> c20(MARK(cons(tail(tail(odds)), y0))) MARK(s(cons(tail(tail(head(y0))), y1))) -> c20(MARK(cons(tail(tail(head(y0))), y1))) MARK(s(cons(tail(tail(tail(y0))), y1))) -> c20(MARK(cons(tail(tail(tail(y0))), y1))) MARK(s(cons(tail(tail(cons(y0, y1))), y2))) -> c20(MARK(cons(tail(tail(cons(y0, y1))), y2))) MARK(s(cons(tail(tail(s(y0))), y1))) -> c20(MARK(cons(tail(tail(s(y0))), y1))) MARK(s(cons(tail(tail(y0)), y1))) -> c20(MARK(cons(tail(tail(y0)), y1))) MARK(s(cons(tail(pairs), y0))) -> c20(MARK(cons(tail(pairs), y0))) MARK(s(cons(incr(incr(nats)), y0))) -> c20(MARK(cons(incr(incr(nats)), y0))) MARK(s(cons(incr(incr(pairs)), y0))) -> c20(MARK(cons(incr(incr(pairs)), y0))) MARK(s(cons(incr(odds), y0))) -> c20(MARK(cons(incr(odds), y0))) MARK(s(cons(head(incr(nats)), y0))) -> c20(MARK(cons(head(incr(nats)), y0))) MARK(s(cons(head(incr(pairs)), y0))) -> c20(MARK(cons(head(incr(pairs)), y0))) MARK(s(cons(head(odds), y0))) -> c20(MARK(cons(head(odds), y0))) MARK(s(cons(tail(incr(nats)), y0))) -> c20(MARK(cons(tail(incr(nats)), y0))) MARK(s(cons(tail(incr(pairs)), y0))) -> c20(MARK(cons(tail(incr(pairs)), y0))) MARK(s(cons(tail(odds), y0))) -> c20(MARK(cons(tail(odds), y0))) MARK(s(cons(incr(cons(nats, y0)), y1))) -> c20(MARK(cons(incr(cons(nats, y0)), y1))) MARK(s(cons(incr(cons(incr(y0), y1)), y2))) -> c20(MARK(cons(incr(cons(incr(y0), y1)), y2))) MARK(s(cons(incr(cons(pairs, y0)), y1))) -> c20(MARK(cons(incr(cons(pairs, y0)), y1))) MARK(s(cons(incr(cons(odds, y0)), y1))) -> c20(MARK(cons(incr(cons(odds, y0)), y1))) MARK(s(cons(incr(cons(head(y0), y1)), y2))) -> c20(MARK(cons(incr(cons(head(y0), y1)), y2))) MARK(s(cons(incr(cons(tail(y0), y1)), y2))) -> c20(MARK(cons(incr(cons(tail(y0), y1)), y2))) MARK(s(cons(incr(cons(cons(y0, y1), y2)), y3))) -> c20(MARK(cons(incr(cons(cons(y0, y1), y2)), y3))) MARK(s(cons(incr(cons(s(y0), y1)), y2))) -> c20(MARK(cons(incr(cons(s(y0), y1)), y2))) MARK(s(cons(incr(cons(y0, y1)), y2))) -> c20(MARK(cons(incr(cons(y0, y1)), y2))) MARK(s(cons(incr(cons(0, y0)), y1))) -> c20(MARK(cons(incr(cons(0, y0)), y1))) MARK(s(cons(incr(head(nats)), y0))) -> c20(MARK(cons(incr(head(nats)), y0))) MARK(s(cons(incr(head(pairs)), y0))) -> c20(MARK(cons(incr(head(pairs)), y0))) MARK(s(cons(head(head(nats)), y0))) -> c20(MARK(cons(head(head(nats)), y0))) MARK(s(cons(head(head(pairs)), y0))) -> c20(MARK(cons(head(head(pairs)), y0))) MARK(s(cons(tail(head(nats)), y0))) -> c20(MARK(cons(tail(head(nats)), y0))) MARK(s(cons(tail(head(pairs)), y0))) -> c20(MARK(cons(tail(head(pairs)), y0))) MARK(s(cons(head(cons(nats, y0)), y1))) -> c20(MARK(cons(head(cons(nats, y0)), y1))) MARK(s(cons(head(cons(incr(y0), y1)), y2))) -> c20(MARK(cons(head(cons(incr(y0), y1)), y2))) MARK(s(cons(head(cons(pairs, y0)), y1))) -> c20(MARK(cons(head(cons(pairs, y0)), y1))) MARK(s(cons(head(cons(odds, y0)), y1))) -> c20(MARK(cons(head(cons(odds, y0)), y1))) MARK(s(cons(head(cons(head(y0), y1)), y2))) -> c20(MARK(cons(head(cons(head(y0), y1)), y2))) MARK(s(cons(head(cons(tail(y0), y1)), y2))) -> c20(MARK(cons(head(cons(tail(y0), y1)), y2))) MARK(s(cons(head(cons(cons(y0, y1), y2)), y3))) -> c20(MARK(cons(head(cons(cons(y0, y1), y2)), y3))) MARK(s(cons(head(cons(s(y0), y1)), y2))) -> c20(MARK(cons(head(cons(s(y0), y1)), y2))) MARK(s(cons(head(cons(y0, y1)), y2))) -> c20(MARK(cons(head(cons(y0, y1)), y2))) MARK(s(cons(head(cons(0, y0)), y1))) -> c20(MARK(cons(head(cons(0, y0)), y1))) MARK(s(cons(incr(tail(nats)), y0))) -> c20(MARK(cons(incr(tail(nats)), y0))) MARK(s(cons(head(tail(nats)), y0))) -> c20(MARK(cons(head(tail(nats)), y0))) MARK(s(cons(tail(tail(nats)), y0))) -> c20(MARK(cons(tail(tail(nats)), y0))) MARK(s(incr(cons(nats, y0)))) -> c20(MARK(incr(cons(nats, y0)))) MARK(s(incr(cons(pairs, y0)))) -> c20(MARK(incr(cons(pairs, y0)))) MARK(s(incr(cons(odds, y0)))) -> c20(MARK(incr(cons(odds, y0)))) MARK(s(head(cons(nats, y0)))) -> c20(MARK(head(cons(nats, y0)))) MARK(s(head(cons(pairs, y0)))) -> c20(MARK(head(cons(pairs, y0)))) MARK(s(head(cons(odds, y0)))) -> c20(MARK(head(cons(odds, y0)))) MARK(incr(s(s(y0)))) -> c13(MARK(s(s(y0)))) MARK(incr(s(incr(s(y0))))) -> c13(MARK(s(incr(s(y0))))) MARK(incr(s(head(s(y0))))) -> c13(MARK(s(head(s(y0))))) MARK(incr(s(tail(cons(y0, y1))))) -> c13(MARK(s(tail(cons(y0, y1))))) MARK(incr(s(tail(s(y0))))) -> c13(MARK(s(tail(s(y0))))) MARK(incr(s(incr(incr(incr(y0)))))) -> c13(MARK(s(incr(incr(incr(y0)))))) MARK(incr(s(incr(incr(odds))))) -> c13(MARK(s(incr(incr(odds))))) MARK(incr(s(incr(incr(head(y0)))))) -> c13(MARK(s(incr(incr(head(y0)))))) MARK(incr(s(incr(incr(tail(y0)))))) -> c13(MARK(s(incr(incr(tail(y0)))))) MARK(incr(s(incr(incr(cons(y0, y1)))))) -> c13(MARK(s(incr(incr(cons(y0, y1)))))) MARK(incr(s(incr(incr(s(y0)))))) -> c13(MARK(s(incr(incr(s(y0)))))) MARK(incr(s(incr(incr(y0))))) -> c13(MARK(s(incr(incr(y0))))) MARK(incr(s(incr(head(incr(y0)))))) -> c13(MARK(s(incr(head(incr(y0)))))) MARK(incr(s(incr(head(odds))))) -> c13(MARK(s(incr(head(odds))))) MARK(incr(s(incr(head(head(y0)))))) -> c13(MARK(s(incr(head(head(y0)))))) MARK(incr(s(incr(head(tail(y0)))))) -> c13(MARK(s(incr(head(tail(y0)))))) MARK(incr(s(incr(head(cons(y0, y1)))))) -> c13(MARK(s(incr(head(cons(y0, y1)))))) MARK(incr(s(incr(head(s(y0)))))) -> c13(MARK(s(incr(head(s(y0)))))) MARK(incr(s(incr(head(y0))))) -> c13(MARK(s(incr(head(y0))))) MARK(incr(s(incr(tail(incr(y0)))))) -> c13(MARK(s(incr(tail(incr(y0)))))) MARK(incr(s(incr(tail(pairs))))) -> c13(MARK(s(incr(tail(pairs))))) MARK(incr(s(incr(tail(odds))))) -> c13(MARK(s(incr(tail(odds))))) MARK(incr(s(incr(tail(head(y0)))))) -> c13(MARK(s(incr(tail(head(y0)))))) MARK(incr(s(incr(tail(tail(y0)))))) -> c13(MARK(s(incr(tail(tail(y0)))))) MARK(incr(s(incr(tail(cons(y0, y1)))))) -> c13(MARK(s(incr(tail(cons(y0, y1)))))) MARK(incr(s(incr(tail(s(y0)))))) -> c13(MARK(s(incr(tail(s(y0)))))) MARK(incr(s(incr(tail(y0))))) -> c13(MARK(s(incr(tail(y0))))) MARK(incr(s(head(incr(incr(y0)))))) -> c13(MARK(s(head(incr(incr(y0)))))) MARK(incr(s(head(incr(odds))))) -> c13(MARK(s(head(incr(odds))))) MARK(incr(s(head(incr(head(y0)))))) -> c13(MARK(s(head(incr(head(y0)))))) MARK(incr(s(head(incr(tail(y0)))))) -> c13(MARK(s(head(incr(tail(y0)))))) MARK(incr(s(head(incr(cons(y0, y1)))))) -> c13(MARK(s(head(incr(cons(y0, y1)))))) MARK(incr(s(head(incr(s(y0)))))) -> c13(MARK(s(head(incr(s(y0)))))) MARK(incr(s(head(incr(y0))))) -> c13(MARK(s(head(incr(y0))))) MARK(incr(s(head(head(incr(y0)))))) -> c13(MARK(s(head(head(incr(y0)))))) MARK(incr(s(head(head(odds))))) -> c13(MARK(s(head(head(odds))))) MARK(incr(s(head(head(head(y0)))))) -> c13(MARK(s(head(head(head(y0)))))) MARK(incr(s(head(head(tail(y0)))))) -> c13(MARK(s(head(head(tail(y0)))))) MARK(incr(s(head(head(cons(y0, y1)))))) -> c13(MARK(s(head(head(cons(y0, y1)))))) MARK(incr(s(head(head(s(y0)))))) -> c13(MARK(s(head(head(s(y0)))))) MARK(incr(s(head(head(y0))))) -> c13(MARK(s(head(head(y0))))) MARK(incr(s(head(tail(incr(y0)))))) -> c13(MARK(s(head(tail(incr(y0)))))) MARK(incr(s(head(tail(pairs))))) -> c13(MARK(s(head(tail(pairs))))) MARK(incr(s(head(tail(odds))))) -> c13(MARK(s(head(tail(odds))))) MARK(incr(s(head(tail(head(y0)))))) -> c13(MARK(s(head(tail(head(y0)))))) MARK(incr(s(head(tail(tail(y0)))))) -> c13(MARK(s(head(tail(tail(y0)))))) MARK(incr(s(head(tail(cons(y0, y1)))))) -> c13(MARK(s(head(tail(cons(y0, y1)))))) MARK(incr(s(head(tail(s(y0)))))) -> c13(MARK(s(head(tail(s(y0)))))) MARK(incr(s(head(tail(y0))))) -> c13(MARK(s(head(tail(y0))))) MARK(incr(s(tail(incr(incr(y0)))))) -> c13(MARK(s(tail(incr(incr(y0)))))) MARK(incr(s(tail(incr(odds))))) -> c13(MARK(s(tail(incr(odds))))) MARK(incr(s(tail(incr(head(y0)))))) -> c13(MARK(s(tail(incr(head(y0)))))) MARK(incr(s(tail(incr(tail(y0)))))) -> c13(MARK(s(tail(incr(tail(y0)))))) MARK(incr(s(tail(incr(cons(y0, y1)))))) -> c13(MARK(s(tail(incr(cons(y0, y1)))))) MARK(incr(s(tail(incr(s(y0)))))) -> c13(MARK(s(tail(incr(s(y0)))))) MARK(incr(s(tail(incr(y0))))) -> c13(MARK(s(tail(incr(y0))))) MARK(incr(s(tail(head(incr(y0)))))) -> c13(MARK(s(tail(head(incr(y0)))))) MARK(incr(s(tail(head(odds))))) -> c13(MARK(s(tail(head(odds))))) MARK(incr(s(tail(head(head(y0)))))) -> c13(MARK(s(tail(head(head(y0)))))) MARK(incr(s(tail(head(tail(y0)))))) -> c13(MARK(s(tail(head(tail(y0)))))) MARK(incr(s(tail(head(cons(y0, y1)))))) -> c13(MARK(s(tail(head(cons(y0, y1)))))) MARK(incr(s(tail(head(s(y0)))))) -> c13(MARK(s(tail(head(s(y0)))))) MARK(incr(s(tail(head(y0))))) -> c13(MARK(s(tail(head(y0))))) MARK(incr(s(tail(tail(incr(y0)))))) -> c13(MARK(s(tail(tail(incr(y0)))))) MARK(incr(s(tail(tail(pairs))))) -> c13(MARK(s(tail(tail(pairs))))) MARK(incr(s(tail(tail(odds))))) -> c13(MARK(s(tail(tail(odds))))) MARK(incr(s(tail(tail(head(y0)))))) -> c13(MARK(s(tail(tail(head(y0)))))) MARK(incr(s(tail(tail(tail(y0)))))) -> c13(MARK(s(tail(tail(tail(y0)))))) MARK(incr(s(tail(tail(cons(y0, y1)))))) -> c13(MARK(s(tail(tail(cons(y0, y1)))))) MARK(incr(s(tail(tail(s(y0)))))) -> c13(MARK(s(tail(tail(s(y0)))))) MARK(incr(s(tail(tail(y0))))) -> c13(MARK(s(tail(tail(y0))))) MARK(incr(s(tail(pairs)))) -> c13(MARK(s(tail(pairs)))) MARK(incr(s(incr(incr(nats))))) -> c13(MARK(s(incr(incr(nats))))) MARK(incr(s(incr(incr(pairs))))) -> c13(MARK(s(incr(incr(pairs))))) MARK(incr(s(incr(odds)))) -> c13(MARK(s(incr(odds)))) MARK(incr(s(head(incr(nats))))) -> c13(MARK(s(head(incr(nats))))) MARK(incr(s(head(incr(pairs))))) -> c13(MARK(s(head(incr(pairs))))) MARK(incr(s(head(odds)))) -> c13(MARK(s(head(odds)))) MARK(incr(s(tail(incr(nats))))) -> c13(MARK(s(tail(incr(nats))))) MARK(incr(s(tail(incr(pairs))))) -> c13(MARK(s(tail(incr(pairs))))) MARK(incr(s(tail(odds)))) -> c13(MARK(s(tail(odds)))) MARK(incr(s(incr(cons(incr(y0), y1))))) -> c13(MARK(s(incr(cons(incr(y0), y1))))) MARK(incr(s(incr(cons(head(y0), y1))))) -> c13(MARK(s(incr(cons(head(y0), y1))))) MARK(incr(s(incr(cons(tail(y0), y1))))) -> c13(MARK(s(incr(cons(tail(y0), y1))))) MARK(incr(s(incr(cons(cons(y0, y1), y2))))) -> c13(MARK(s(incr(cons(cons(y0, y1), y2))))) MARK(incr(s(incr(cons(s(y0), y1))))) -> c13(MARK(s(incr(cons(s(y0), y1))))) MARK(incr(s(incr(cons(y0, y1))))) -> c13(MARK(s(incr(cons(y0, y1))))) MARK(incr(s(incr(head(nats))))) -> c13(MARK(s(incr(head(nats))))) MARK(incr(s(incr(head(pairs))))) -> c13(MARK(s(incr(head(pairs))))) MARK(incr(s(head(head(nats))))) -> c13(MARK(s(head(head(nats))))) MARK(incr(s(head(head(pairs))))) -> c13(MARK(s(head(head(pairs))))) MARK(incr(s(tail(head(nats))))) -> c13(MARK(s(tail(head(nats))))) MARK(incr(s(tail(head(pairs))))) -> c13(MARK(s(tail(head(pairs))))) MARK(incr(s(head(cons(incr(y0), y1))))) -> c13(MARK(s(head(cons(incr(y0), y1))))) MARK(incr(s(head(cons(head(y0), y1))))) -> c13(MARK(s(head(cons(head(y0), y1))))) MARK(incr(s(head(cons(tail(y0), y1))))) -> c13(MARK(s(head(cons(tail(y0), y1))))) MARK(incr(s(head(cons(cons(y0, y1), y2))))) -> c13(MARK(s(head(cons(cons(y0, y1), y2))))) MARK(incr(s(head(cons(s(y0), y1))))) -> c13(MARK(s(head(cons(s(y0), y1))))) MARK(incr(s(head(cons(y0, y1))))) -> c13(MARK(s(head(cons(y0, y1))))) MARK(incr(s(incr(tail(nats))))) -> c13(MARK(s(incr(tail(nats))))) MARK(incr(s(head(tail(nats))))) -> c13(MARK(s(head(tail(nats))))) MARK(incr(s(tail(tail(nats))))) -> c13(MARK(s(tail(tail(nats))))) MARK(incr(s(cons(cons(y0, y1), y2)))) -> c13(MARK(s(cons(cons(y0, y1), y2)))) MARK(incr(s(cons(s(y0), y1)))) -> c13(MARK(s(cons(s(y0), y1)))) MARK(incr(s(cons(incr(s(y0)), y1)))) -> c13(MARK(s(cons(incr(s(y0)), y1)))) MARK(incr(s(cons(head(s(y0)), y1)))) -> c13(MARK(s(cons(head(s(y0)), y1)))) MARK(incr(s(cons(tail(cons(y0, y1)), y2)))) -> c13(MARK(s(cons(tail(cons(y0, y1)), y2)))) MARK(incr(s(cons(tail(s(y0)), y1)))) -> c13(MARK(s(cons(tail(s(y0)), y1)))) MARK(incr(s(cons(incr(incr(incr(y0))), y1)))) -> c13(MARK(s(cons(incr(incr(incr(y0))), y1)))) MARK(incr(s(cons(incr(incr(odds)), y0)))) -> c13(MARK(s(cons(incr(incr(odds)), y0)))) MARK(incr(s(cons(incr(incr(head(y0))), y1)))) -> c13(MARK(s(cons(incr(incr(head(y0))), y1)))) MARK(incr(s(cons(incr(incr(tail(y0))), y1)))) -> c13(MARK(s(cons(incr(incr(tail(y0))), y1)))) MARK(incr(s(cons(incr(incr(cons(y0, y1))), y2)))) -> c13(MARK(s(cons(incr(incr(cons(y0, y1))), y2)))) MARK(incr(s(cons(incr(incr(s(y0))), y1)))) -> c13(MARK(s(cons(incr(incr(s(y0))), y1)))) MARK(incr(s(cons(incr(incr(y0)), y1)))) -> c13(MARK(s(cons(incr(incr(y0)), y1)))) MARK(incr(s(cons(incr(head(incr(y0))), y1)))) -> c13(MARK(s(cons(incr(head(incr(y0))), y1)))) MARK(incr(s(cons(incr(head(odds)), y0)))) -> c13(MARK(s(cons(incr(head(odds)), y0)))) MARK(incr(s(cons(incr(head(head(y0))), y1)))) -> c13(MARK(s(cons(incr(head(head(y0))), y1)))) MARK(incr(s(cons(incr(head(tail(y0))), y1)))) -> c13(MARK(s(cons(incr(head(tail(y0))), y1)))) MARK(incr(s(cons(incr(head(cons(y0, y1))), y2)))) -> c13(MARK(s(cons(incr(head(cons(y0, y1))), y2)))) MARK(incr(s(cons(incr(head(s(y0))), y1)))) -> c13(MARK(s(cons(incr(head(s(y0))), y1)))) MARK(incr(s(cons(incr(head(y0)), y1)))) -> c13(MARK(s(cons(incr(head(y0)), y1)))) MARK(incr(s(cons(incr(tail(incr(y0))), y1)))) -> c13(MARK(s(cons(incr(tail(incr(y0))), y1)))) MARK(incr(s(cons(incr(tail(pairs)), y0)))) -> c13(MARK(s(cons(incr(tail(pairs)), y0)))) MARK(incr(s(cons(incr(tail(odds)), y0)))) -> c13(MARK(s(cons(incr(tail(odds)), y0)))) MARK(incr(s(cons(incr(tail(head(y0))), y1)))) -> c13(MARK(s(cons(incr(tail(head(y0))), y1)))) MARK(incr(s(cons(incr(tail(tail(y0))), y1)))) -> c13(MARK(s(cons(incr(tail(tail(y0))), y1)))) MARK(incr(s(cons(incr(tail(cons(y0, y1))), y2)))) -> c13(MARK(s(cons(incr(tail(cons(y0, y1))), y2)))) MARK(incr(s(cons(incr(tail(s(y0))), y1)))) -> c13(MARK(s(cons(incr(tail(s(y0))), y1)))) MARK(incr(s(cons(incr(tail(y0)), y1)))) -> c13(MARK(s(cons(incr(tail(y0)), y1)))) MARK(incr(s(cons(head(incr(incr(y0))), y1)))) -> c13(MARK(s(cons(head(incr(incr(y0))), y1)))) MARK(incr(s(cons(head(incr(odds)), y0)))) -> c13(MARK(s(cons(head(incr(odds)), y0)))) MARK(incr(s(cons(head(incr(head(y0))), y1)))) -> c13(MARK(s(cons(head(incr(head(y0))), y1)))) MARK(incr(s(cons(head(incr(tail(y0))), y1)))) -> c13(MARK(s(cons(head(incr(tail(y0))), y1)))) MARK(incr(s(cons(head(incr(cons(y0, y1))), y2)))) -> c13(MARK(s(cons(head(incr(cons(y0, y1))), y2)))) MARK(incr(s(cons(head(incr(s(y0))), y1)))) -> c13(MARK(s(cons(head(incr(s(y0))), y1)))) MARK(incr(s(cons(head(incr(y0)), y1)))) -> c13(MARK(s(cons(head(incr(y0)), y1)))) MARK(incr(s(cons(head(head(incr(y0))), y1)))) -> c13(MARK(s(cons(head(head(incr(y0))), y1)))) MARK(incr(s(cons(head(head(odds)), y0)))) -> c13(MARK(s(cons(head(head(odds)), y0)))) MARK(incr(s(cons(head(head(head(y0))), y1)))) -> c13(MARK(s(cons(head(head(head(y0))), y1)))) MARK(incr(s(cons(head(head(tail(y0))), y1)))) -> c13(MARK(s(cons(head(head(tail(y0))), y1)))) MARK(incr(s(cons(head(head(cons(y0, y1))), y2)))) -> c13(MARK(s(cons(head(head(cons(y0, y1))), y2)))) MARK(incr(s(cons(head(head(s(y0))), y1)))) -> c13(MARK(s(cons(head(head(s(y0))), y1)))) MARK(incr(s(cons(head(head(y0)), y1)))) -> c13(MARK(s(cons(head(head(y0)), y1)))) MARK(incr(s(cons(head(tail(incr(y0))), y1)))) -> c13(MARK(s(cons(head(tail(incr(y0))), y1)))) MARK(incr(s(cons(head(tail(pairs)), y0)))) -> c13(MARK(s(cons(head(tail(pairs)), y0)))) MARK(incr(s(cons(head(tail(odds)), y0)))) -> c13(MARK(s(cons(head(tail(odds)), y0)))) MARK(incr(s(cons(head(tail(head(y0))), y1)))) -> c13(MARK(s(cons(head(tail(head(y0))), y1)))) MARK(incr(s(cons(head(tail(tail(y0))), y1)))) -> c13(MARK(s(cons(head(tail(tail(y0))), y1)))) MARK(incr(s(cons(head(tail(cons(y0, y1))), y2)))) -> c13(MARK(s(cons(head(tail(cons(y0, y1))), y2)))) MARK(incr(s(cons(head(tail(s(y0))), y1)))) -> c13(MARK(s(cons(head(tail(s(y0))), y1)))) MARK(incr(s(cons(head(tail(y0)), y1)))) -> c13(MARK(s(cons(head(tail(y0)), y1)))) MARK(incr(s(cons(tail(incr(incr(y0))), y1)))) -> c13(MARK(s(cons(tail(incr(incr(y0))), y1)))) MARK(incr(s(cons(tail(incr(odds)), y0)))) -> c13(MARK(s(cons(tail(incr(odds)), y0)))) MARK(incr(s(cons(tail(incr(head(y0))), y1)))) -> c13(MARK(s(cons(tail(incr(head(y0))), y1)))) MARK(incr(s(cons(tail(incr(tail(y0))), y1)))) -> c13(MARK(s(cons(tail(incr(tail(y0))), y1)))) MARK(incr(s(cons(tail(incr(cons(y0, y1))), y2)))) -> c13(MARK(s(cons(tail(incr(cons(y0, y1))), y2)))) MARK(incr(s(cons(tail(incr(s(y0))), y1)))) -> c13(MARK(s(cons(tail(incr(s(y0))), y1)))) MARK(incr(s(cons(tail(incr(y0)), y1)))) -> c13(MARK(s(cons(tail(incr(y0)), y1)))) MARK(incr(s(cons(tail(head(incr(y0))), y1)))) -> c13(MARK(s(cons(tail(head(incr(y0))), y1)))) MARK(incr(s(cons(tail(head(odds)), y0)))) -> c13(MARK(s(cons(tail(head(odds)), y0)))) MARK(incr(s(cons(tail(head(head(y0))), y1)))) -> c13(MARK(s(cons(tail(head(head(y0))), y1)))) MARK(incr(s(cons(tail(head(tail(y0))), y1)))) -> c13(MARK(s(cons(tail(head(tail(y0))), y1)))) MARK(incr(s(cons(tail(head(cons(y0, y1))), y2)))) -> c13(MARK(s(cons(tail(head(cons(y0, y1))), y2)))) MARK(incr(s(cons(tail(head(s(y0))), y1)))) -> c13(MARK(s(cons(tail(head(s(y0))), y1)))) MARK(incr(s(cons(tail(head(y0)), y1)))) -> c13(MARK(s(cons(tail(head(y0)), y1)))) MARK(incr(s(cons(tail(tail(incr(y0))), y1)))) -> c13(MARK(s(cons(tail(tail(incr(y0))), y1)))) MARK(incr(s(cons(tail(tail(pairs)), y0)))) -> c13(MARK(s(cons(tail(tail(pairs)), y0)))) MARK(incr(s(cons(tail(tail(odds)), y0)))) -> c13(MARK(s(cons(tail(tail(odds)), y0)))) MARK(incr(s(cons(tail(tail(head(y0))), y1)))) -> c13(MARK(s(cons(tail(tail(head(y0))), y1)))) MARK(incr(s(cons(tail(tail(tail(y0))), y1)))) -> c13(MARK(s(cons(tail(tail(tail(y0))), y1)))) MARK(incr(s(cons(tail(tail(cons(y0, y1))), y2)))) -> c13(MARK(s(cons(tail(tail(cons(y0, y1))), y2)))) MARK(incr(s(cons(tail(tail(s(y0))), y1)))) -> c13(MARK(s(cons(tail(tail(s(y0))), y1)))) MARK(incr(s(cons(tail(tail(y0)), y1)))) -> c13(MARK(s(cons(tail(tail(y0)), y1)))) MARK(incr(s(cons(tail(pairs), y0)))) -> c13(MARK(s(cons(tail(pairs), y0)))) MARK(incr(s(cons(incr(incr(nats)), y0)))) -> c13(MARK(s(cons(incr(incr(nats)), y0)))) MARK(incr(s(cons(incr(incr(pairs)), y0)))) -> c13(MARK(s(cons(incr(incr(pairs)), y0)))) MARK(incr(s(cons(incr(odds), y0)))) -> c13(MARK(s(cons(incr(odds), y0)))) MARK(incr(s(cons(head(incr(nats)), y0)))) -> c13(MARK(s(cons(head(incr(nats)), y0)))) MARK(incr(s(cons(head(incr(pairs)), y0)))) -> c13(MARK(s(cons(head(incr(pairs)), y0)))) MARK(incr(s(cons(head(odds), y0)))) -> c13(MARK(s(cons(head(odds), y0)))) MARK(incr(s(cons(tail(incr(nats)), y0)))) -> c13(MARK(s(cons(tail(incr(nats)), y0)))) MARK(incr(s(cons(tail(incr(pairs)), y0)))) -> c13(MARK(s(cons(tail(incr(pairs)), y0)))) MARK(incr(s(cons(tail(odds), y0)))) -> c13(MARK(s(cons(tail(odds), y0)))) MARK(incr(s(cons(incr(cons(nats, y0)), y1)))) -> c13(MARK(s(cons(incr(cons(nats, y0)), y1)))) MARK(incr(s(cons(incr(cons(incr(y0), y1)), y2)))) -> c13(MARK(s(cons(incr(cons(incr(y0), y1)), y2)))) MARK(incr(s(cons(incr(cons(pairs, y0)), y1)))) -> c13(MARK(s(cons(incr(cons(pairs, y0)), y1)))) MARK(incr(s(cons(incr(cons(odds, y0)), y1)))) -> c13(MARK(s(cons(incr(cons(odds, y0)), y1)))) MARK(incr(s(cons(incr(cons(head(y0), y1)), y2)))) -> c13(MARK(s(cons(incr(cons(head(y0), y1)), y2)))) MARK(incr(s(cons(incr(cons(tail(y0), y1)), y2)))) -> c13(MARK(s(cons(incr(cons(tail(y0), y1)), y2)))) MARK(incr(s(cons(incr(cons(cons(y0, y1), y2)), y3)))) -> c13(MARK(s(cons(incr(cons(cons(y0, y1), y2)), y3)))) MARK(incr(s(cons(incr(cons(s(y0), y1)), y2)))) -> c13(MARK(s(cons(incr(cons(s(y0), y1)), y2)))) MARK(incr(s(cons(incr(cons(y0, y1)), y2)))) -> c13(MARK(s(cons(incr(cons(y0, y1)), y2)))) MARK(incr(s(cons(incr(cons(0, y0)), y1)))) -> c13(MARK(s(cons(incr(cons(0, y0)), y1)))) MARK(incr(s(cons(incr(head(nats)), y0)))) -> c13(MARK(s(cons(incr(head(nats)), y0)))) MARK(incr(s(cons(incr(head(pairs)), y0)))) -> c13(MARK(s(cons(incr(head(pairs)), y0)))) MARK(incr(s(cons(head(head(nats)), y0)))) -> c13(MARK(s(cons(head(head(nats)), y0)))) MARK(incr(s(cons(head(head(pairs)), y0)))) -> c13(MARK(s(cons(head(head(pairs)), y0)))) MARK(incr(s(cons(tail(head(nats)), y0)))) -> c13(MARK(s(cons(tail(head(nats)), y0)))) MARK(incr(s(cons(tail(head(pairs)), y0)))) -> c13(MARK(s(cons(tail(head(pairs)), y0)))) MARK(incr(s(cons(head(cons(nats, y0)), y1)))) -> c13(MARK(s(cons(head(cons(nats, y0)), y1)))) MARK(incr(s(cons(head(cons(incr(y0), y1)), y2)))) -> c13(MARK(s(cons(head(cons(incr(y0), y1)), y2)))) MARK(incr(s(cons(head(cons(pairs, y0)), y1)))) -> c13(MARK(s(cons(head(cons(pairs, y0)), y1)))) MARK(incr(s(cons(head(cons(odds, y0)), y1)))) -> c13(MARK(s(cons(head(cons(odds, y0)), y1)))) MARK(incr(s(cons(head(cons(head(y0), y1)), y2)))) -> c13(MARK(s(cons(head(cons(head(y0), y1)), y2)))) MARK(incr(s(cons(head(cons(tail(y0), y1)), y2)))) -> c13(MARK(s(cons(head(cons(tail(y0), y1)), y2)))) MARK(incr(s(cons(head(cons(cons(y0, y1), y2)), y3)))) -> c13(MARK(s(cons(head(cons(cons(y0, y1), y2)), y3)))) MARK(incr(s(cons(head(cons(s(y0), y1)), y2)))) -> c13(MARK(s(cons(head(cons(s(y0), y1)), y2)))) MARK(incr(s(cons(head(cons(y0, y1)), y2)))) -> c13(MARK(s(cons(head(cons(y0, y1)), y2)))) MARK(incr(s(cons(head(cons(0, y0)), y1)))) -> c13(MARK(s(cons(head(cons(0, y0)), y1)))) MARK(incr(s(cons(incr(tail(nats)), y0)))) -> c13(MARK(s(cons(incr(tail(nats)), y0)))) MARK(incr(s(cons(head(tail(nats)), y0)))) -> c13(MARK(s(cons(head(tail(nats)), y0)))) MARK(incr(s(cons(tail(tail(nats)), y0)))) -> c13(MARK(s(cons(tail(tail(nats)), y0)))) MARK(incr(s(incr(cons(nats, y0))))) -> c13(MARK(s(incr(cons(nats, y0))))) MARK(incr(s(incr(cons(pairs, y0))))) -> c13(MARK(s(incr(cons(pairs, y0))))) MARK(incr(s(incr(cons(odds, y0))))) -> c13(MARK(s(incr(cons(odds, y0))))) MARK(incr(s(head(cons(nats, y0))))) -> c13(MARK(s(head(cons(nats, y0))))) MARK(incr(s(head(cons(pairs, y0))))) -> c13(MARK(s(head(cons(pairs, y0))))) MARK(incr(s(head(cons(odds, y0))))) -> c13(MARK(s(head(cons(odds, y0))))) MARK(head(s(s(y0)))) -> c16(MARK(s(s(y0)))) MARK(head(s(incr(s(y0))))) -> c16(MARK(s(incr(s(y0))))) MARK(head(s(head(s(y0))))) -> c16(MARK(s(head(s(y0))))) MARK(head(s(tail(cons(y0, y1))))) -> c16(MARK(s(tail(cons(y0, y1))))) MARK(head(s(tail(s(y0))))) -> c16(MARK(s(tail(s(y0))))) MARK(head(s(incr(incr(incr(y0)))))) -> c16(MARK(s(incr(incr(incr(y0)))))) MARK(head(s(incr(incr(odds))))) -> c16(MARK(s(incr(incr(odds))))) MARK(head(s(incr(incr(head(y0)))))) -> c16(MARK(s(incr(incr(head(y0)))))) MARK(head(s(incr(incr(tail(y0)))))) -> c16(MARK(s(incr(incr(tail(y0)))))) MARK(head(s(incr(incr(cons(y0, y1)))))) -> c16(MARK(s(incr(incr(cons(y0, y1)))))) MARK(head(s(incr(incr(s(y0)))))) -> c16(MARK(s(incr(incr(s(y0)))))) MARK(head(s(incr(incr(y0))))) -> c16(MARK(s(incr(incr(y0))))) MARK(head(s(incr(head(incr(y0)))))) -> c16(MARK(s(incr(head(incr(y0)))))) MARK(head(s(incr(head(odds))))) -> c16(MARK(s(incr(head(odds))))) MARK(head(s(incr(head(head(y0)))))) -> c16(MARK(s(incr(head(head(y0)))))) MARK(head(s(incr(head(tail(y0)))))) -> c16(MARK(s(incr(head(tail(y0)))))) MARK(head(s(incr(head(cons(y0, y1)))))) -> c16(MARK(s(incr(head(cons(y0, y1)))))) MARK(head(s(incr(head(s(y0)))))) -> c16(MARK(s(incr(head(s(y0)))))) MARK(head(s(incr(head(y0))))) -> c16(MARK(s(incr(head(y0))))) MARK(head(s(incr(tail(incr(y0)))))) -> c16(MARK(s(incr(tail(incr(y0)))))) MARK(head(s(incr(tail(pairs))))) -> c16(MARK(s(incr(tail(pairs))))) MARK(head(s(incr(tail(odds))))) -> c16(MARK(s(incr(tail(odds))))) MARK(head(s(incr(tail(head(y0)))))) -> c16(MARK(s(incr(tail(head(y0)))))) MARK(head(s(incr(tail(tail(y0)))))) -> c16(MARK(s(incr(tail(tail(y0)))))) MARK(head(s(incr(tail(cons(y0, y1)))))) -> c16(MARK(s(incr(tail(cons(y0, y1)))))) MARK(head(s(incr(tail(s(y0)))))) -> c16(MARK(s(incr(tail(s(y0)))))) MARK(head(s(incr(tail(y0))))) -> c16(MARK(s(incr(tail(y0))))) MARK(head(s(head(incr(incr(y0)))))) -> c16(MARK(s(head(incr(incr(y0)))))) MARK(head(s(head(incr(odds))))) -> c16(MARK(s(head(incr(odds))))) MARK(head(s(head(incr(head(y0)))))) -> c16(MARK(s(head(incr(head(y0)))))) MARK(head(s(head(incr(tail(y0)))))) -> c16(MARK(s(head(incr(tail(y0)))))) MARK(head(s(head(incr(cons(y0, y1)))))) -> c16(MARK(s(head(incr(cons(y0, y1)))))) MARK(head(s(head(incr(s(y0)))))) -> c16(MARK(s(head(incr(s(y0)))))) MARK(head(s(head(incr(y0))))) -> c16(MARK(s(head(incr(y0))))) MARK(head(s(head(head(incr(y0)))))) -> c16(MARK(s(head(head(incr(y0)))))) MARK(head(s(head(head(odds))))) -> c16(MARK(s(head(head(odds))))) MARK(head(s(head(head(head(y0)))))) -> c16(MARK(s(head(head(head(y0)))))) MARK(head(s(head(head(tail(y0)))))) -> c16(MARK(s(head(head(tail(y0)))))) MARK(head(s(head(head(cons(y0, y1)))))) -> c16(MARK(s(head(head(cons(y0, y1)))))) MARK(head(s(head(head(s(y0)))))) -> c16(MARK(s(head(head(s(y0)))))) MARK(head(s(head(head(y0))))) -> c16(MARK(s(head(head(y0))))) MARK(head(s(head(tail(incr(y0)))))) -> c16(MARK(s(head(tail(incr(y0)))))) MARK(head(s(head(tail(pairs))))) -> c16(MARK(s(head(tail(pairs))))) MARK(head(s(head(tail(odds))))) -> c16(MARK(s(head(tail(odds))))) MARK(head(s(head(tail(head(y0)))))) -> c16(MARK(s(head(tail(head(y0)))))) MARK(head(s(head(tail(tail(y0)))))) -> c16(MARK(s(head(tail(tail(y0)))))) MARK(head(s(head(tail(cons(y0, y1)))))) -> c16(MARK(s(head(tail(cons(y0, y1)))))) MARK(head(s(head(tail(s(y0)))))) -> c16(MARK(s(head(tail(s(y0)))))) MARK(head(s(head(tail(y0))))) -> c16(MARK(s(head(tail(y0))))) MARK(head(s(tail(incr(incr(y0)))))) -> c16(MARK(s(tail(incr(incr(y0)))))) MARK(head(s(tail(incr(odds))))) -> c16(MARK(s(tail(incr(odds))))) MARK(head(s(tail(incr(head(y0)))))) -> c16(MARK(s(tail(incr(head(y0)))))) MARK(head(s(tail(incr(tail(y0)))))) -> c16(MARK(s(tail(incr(tail(y0)))))) MARK(head(s(tail(incr(cons(y0, y1)))))) -> c16(MARK(s(tail(incr(cons(y0, y1)))))) MARK(head(s(tail(incr(s(y0)))))) -> c16(MARK(s(tail(incr(s(y0)))))) MARK(head(s(tail(incr(y0))))) -> c16(MARK(s(tail(incr(y0))))) MARK(head(s(tail(head(incr(y0)))))) -> c16(MARK(s(tail(head(incr(y0)))))) MARK(head(s(tail(head(odds))))) -> c16(MARK(s(tail(head(odds))))) MARK(head(s(tail(head(head(y0)))))) -> c16(MARK(s(tail(head(head(y0)))))) MARK(head(s(tail(head(tail(y0)))))) -> c16(MARK(s(tail(head(tail(y0)))))) MARK(head(s(tail(head(cons(y0, y1)))))) -> c16(MARK(s(tail(head(cons(y0, y1)))))) MARK(head(s(tail(head(s(y0)))))) -> c16(MARK(s(tail(head(s(y0)))))) MARK(head(s(tail(head(y0))))) -> c16(MARK(s(tail(head(y0))))) MARK(head(s(tail(tail(incr(y0)))))) -> c16(MARK(s(tail(tail(incr(y0)))))) MARK(head(s(tail(tail(pairs))))) -> c16(MARK(s(tail(tail(pairs))))) MARK(head(s(tail(tail(odds))))) -> c16(MARK(s(tail(tail(odds))))) MARK(head(s(tail(tail(head(y0)))))) -> c16(MARK(s(tail(tail(head(y0)))))) MARK(head(s(tail(tail(tail(y0)))))) -> c16(MARK(s(tail(tail(tail(y0)))))) MARK(head(s(tail(tail(cons(y0, y1)))))) -> c16(MARK(s(tail(tail(cons(y0, y1)))))) MARK(head(s(tail(tail(s(y0)))))) -> c16(MARK(s(tail(tail(s(y0)))))) MARK(head(s(tail(tail(y0))))) -> c16(MARK(s(tail(tail(y0))))) MARK(head(s(tail(pairs)))) -> c16(MARK(s(tail(pairs)))) MARK(head(s(incr(incr(nats))))) -> c16(MARK(s(incr(incr(nats))))) MARK(head(s(incr(incr(pairs))))) -> c16(MARK(s(incr(incr(pairs))))) MARK(head(s(incr(odds)))) -> c16(MARK(s(incr(odds)))) MARK(head(s(head(incr(nats))))) -> c16(MARK(s(head(incr(nats))))) MARK(head(s(head(incr(pairs))))) -> c16(MARK(s(head(incr(pairs))))) MARK(head(s(head(odds)))) -> c16(MARK(s(head(odds)))) MARK(head(s(tail(incr(nats))))) -> c16(MARK(s(tail(incr(nats))))) MARK(head(s(tail(incr(pairs))))) -> c16(MARK(s(tail(incr(pairs))))) MARK(head(s(tail(odds)))) -> c16(MARK(s(tail(odds)))) MARK(head(s(incr(cons(incr(y0), y1))))) -> c16(MARK(s(incr(cons(incr(y0), y1))))) MARK(head(s(incr(cons(head(y0), y1))))) -> c16(MARK(s(incr(cons(head(y0), y1))))) MARK(head(s(incr(cons(tail(y0), y1))))) -> c16(MARK(s(incr(cons(tail(y0), y1))))) MARK(head(s(incr(cons(cons(y0, y1), y2))))) -> c16(MARK(s(incr(cons(cons(y0, y1), y2))))) MARK(head(s(incr(cons(s(y0), y1))))) -> c16(MARK(s(incr(cons(s(y0), y1))))) MARK(head(s(incr(cons(y0, y1))))) -> c16(MARK(s(incr(cons(y0, y1))))) MARK(head(s(incr(head(nats))))) -> c16(MARK(s(incr(head(nats))))) MARK(head(s(incr(head(pairs))))) -> c16(MARK(s(incr(head(pairs))))) MARK(head(s(head(head(nats))))) -> c16(MARK(s(head(head(nats))))) MARK(head(s(head(head(pairs))))) -> c16(MARK(s(head(head(pairs))))) MARK(head(s(tail(head(nats))))) -> c16(MARK(s(tail(head(nats))))) MARK(head(s(tail(head(pairs))))) -> c16(MARK(s(tail(head(pairs))))) MARK(head(s(head(cons(incr(y0), y1))))) -> c16(MARK(s(head(cons(incr(y0), y1))))) MARK(head(s(head(cons(head(y0), y1))))) -> c16(MARK(s(head(cons(head(y0), y1))))) MARK(head(s(head(cons(tail(y0), y1))))) -> c16(MARK(s(head(cons(tail(y0), y1))))) MARK(head(s(head(cons(cons(y0, y1), y2))))) -> c16(MARK(s(head(cons(cons(y0, y1), y2))))) MARK(head(s(head(cons(s(y0), y1))))) -> c16(MARK(s(head(cons(s(y0), y1))))) MARK(head(s(head(cons(y0, y1))))) -> c16(MARK(s(head(cons(y0, y1))))) MARK(head(s(incr(tail(nats))))) -> c16(MARK(s(incr(tail(nats))))) MARK(head(s(head(tail(nats))))) -> c16(MARK(s(head(tail(nats))))) MARK(head(s(tail(tail(nats))))) -> c16(MARK(s(tail(tail(nats))))) MARK(head(s(cons(cons(y0, y1), y2)))) -> c16(MARK(s(cons(cons(y0, y1), y2)))) MARK(head(s(cons(s(y0), y1)))) -> c16(MARK(s(cons(s(y0), y1)))) MARK(head(s(cons(incr(s(y0)), y1)))) -> c16(MARK(s(cons(incr(s(y0)), y1)))) MARK(head(s(cons(head(s(y0)), y1)))) -> c16(MARK(s(cons(head(s(y0)), y1)))) MARK(head(s(cons(tail(cons(y0, y1)), y2)))) -> c16(MARK(s(cons(tail(cons(y0, y1)), y2)))) MARK(head(s(cons(tail(s(y0)), y1)))) -> c16(MARK(s(cons(tail(s(y0)), y1)))) MARK(head(s(cons(incr(incr(incr(y0))), y1)))) -> c16(MARK(s(cons(incr(incr(incr(y0))), y1)))) MARK(head(s(cons(incr(incr(odds)), y0)))) -> c16(MARK(s(cons(incr(incr(odds)), y0)))) MARK(head(s(cons(incr(incr(head(y0))), y1)))) -> c16(MARK(s(cons(incr(incr(head(y0))), y1)))) MARK(head(s(cons(incr(incr(tail(y0))), y1)))) -> c16(MARK(s(cons(incr(incr(tail(y0))), y1)))) MARK(head(s(cons(incr(incr(cons(y0, y1))), y2)))) -> c16(MARK(s(cons(incr(incr(cons(y0, y1))), y2)))) MARK(head(s(cons(incr(incr(s(y0))), y1)))) -> c16(MARK(s(cons(incr(incr(s(y0))), y1)))) MARK(head(s(cons(incr(incr(y0)), y1)))) -> c16(MARK(s(cons(incr(incr(y0)), y1)))) MARK(head(s(cons(incr(head(incr(y0))), y1)))) -> c16(MARK(s(cons(incr(head(incr(y0))), y1)))) MARK(head(s(cons(incr(head(odds)), y0)))) -> c16(MARK(s(cons(incr(head(odds)), y0)))) MARK(head(s(cons(incr(head(head(y0))), y1)))) -> c16(MARK(s(cons(incr(head(head(y0))), y1)))) MARK(head(s(cons(incr(head(tail(y0))), y1)))) -> c16(MARK(s(cons(incr(head(tail(y0))), y1)))) MARK(head(s(cons(incr(head(cons(y0, y1))), y2)))) -> c16(MARK(s(cons(incr(head(cons(y0, y1))), y2)))) MARK(head(s(cons(incr(head(s(y0))), y1)))) -> c16(MARK(s(cons(incr(head(s(y0))), y1)))) MARK(head(s(cons(incr(head(y0)), y1)))) -> c16(MARK(s(cons(incr(head(y0)), y1)))) MARK(head(s(cons(incr(tail(incr(y0))), y1)))) -> c16(MARK(s(cons(incr(tail(incr(y0))), y1)))) MARK(head(s(cons(incr(tail(pairs)), y0)))) -> c16(MARK(s(cons(incr(tail(pairs)), y0)))) MARK(head(s(cons(incr(tail(odds)), y0)))) -> c16(MARK(s(cons(incr(tail(odds)), y0)))) MARK(head(s(cons(incr(tail(head(y0))), y1)))) -> c16(MARK(s(cons(incr(tail(head(y0))), y1)))) MARK(head(s(cons(incr(tail(tail(y0))), y1)))) -> c16(MARK(s(cons(incr(tail(tail(y0))), y1)))) MARK(head(s(cons(incr(tail(cons(y0, y1))), y2)))) -> c16(MARK(s(cons(incr(tail(cons(y0, y1))), y2)))) MARK(head(s(cons(incr(tail(s(y0))), y1)))) -> c16(MARK(s(cons(incr(tail(s(y0))), y1)))) MARK(head(s(cons(incr(tail(y0)), y1)))) -> c16(MARK(s(cons(incr(tail(y0)), y1)))) MARK(head(s(cons(head(incr(incr(y0))), y1)))) -> c16(MARK(s(cons(head(incr(incr(y0))), y1)))) MARK(head(s(cons(head(incr(odds)), y0)))) -> c16(MARK(s(cons(head(incr(odds)), y0)))) MARK(head(s(cons(head(incr(head(y0))), y1)))) -> c16(MARK(s(cons(head(incr(head(y0))), y1)))) MARK(head(s(cons(head(incr(tail(y0))), y1)))) -> c16(MARK(s(cons(head(incr(tail(y0))), y1)))) MARK(head(s(cons(head(incr(cons(y0, y1))), y2)))) -> c16(MARK(s(cons(head(incr(cons(y0, y1))), y2)))) MARK(head(s(cons(head(incr(s(y0))), y1)))) -> c16(MARK(s(cons(head(incr(s(y0))), y1)))) MARK(head(s(cons(head(incr(y0)), y1)))) -> c16(MARK(s(cons(head(incr(y0)), y1)))) MARK(head(s(cons(head(head(incr(y0))), y1)))) -> c16(MARK(s(cons(head(head(incr(y0))), y1)))) MARK(head(s(cons(head(head(odds)), y0)))) -> c16(MARK(s(cons(head(head(odds)), y0)))) MARK(head(s(cons(head(head(head(y0))), y1)))) -> c16(MARK(s(cons(head(head(head(y0))), y1)))) MARK(head(s(cons(head(head(tail(y0))), y1)))) -> c16(MARK(s(cons(head(head(tail(y0))), y1)))) MARK(head(s(cons(head(head(cons(y0, y1))), y2)))) -> c16(MARK(s(cons(head(head(cons(y0, y1))), y2)))) MARK(head(s(cons(head(head(s(y0))), y1)))) -> c16(MARK(s(cons(head(head(s(y0))), y1)))) MARK(head(s(cons(head(head(y0)), y1)))) -> c16(MARK(s(cons(head(head(y0)), y1)))) MARK(head(s(cons(head(tail(incr(y0))), y1)))) -> c16(MARK(s(cons(head(tail(incr(y0))), y1)))) MARK(head(s(cons(head(tail(pairs)), y0)))) -> c16(MARK(s(cons(head(tail(pairs)), y0)))) MARK(head(s(cons(head(tail(odds)), y0)))) -> c16(MARK(s(cons(head(tail(odds)), y0)))) MARK(head(s(cons(head(tail(head(y0))), y1)))) -> c16(MARK(s(cons(head(tail(head(y0))), y1)))) MARK(head(s(cons(head(tail(tail(y0))), y1)))) -> c16(MARK(s(cons(head(tail(tail(y0))), y1)))) MARK(head(s(cons(head(tail(cons(y0, y1))), y2)))) -> c16(MARK(s(cons(head(tail(cons(y0, y1))), y2)))) MARK(head(s(cons(head(tail(s(y0))), y1)))) -> c16(MARK(s(cons(head(tail(s(y0))), y1)))) MARK(head(s(cons(head(tail(y0)), y1)))) -> c16(MARK(s(cons(head(tail(y0)), y1)))) MARK(head(s(cons(tail(incr(incr(y0))), y1)))) -> c16(MARK(s(cons(tail(incr(incr(y0))), y1)))) MARK(head(s(cons(tail(incr(odds)), y0)))) -> c16(MARK(s(cons(tail(incr(odds)), y0)))) MARK(head(s(cons(tail(incr(head(y0))), y1)))) -> c16(MARK(s(cons(tail(incr(head(y0))), y1)))) MARK(head(s(cons(tail(incr(tail(y0))), y1)))) -> c16(MARK(s(cons(tail(incr(tail(y0))), y1)))) MARK(head(s(cons(tail(incr(cons(y0, y1))), y2)))) -> c16(MARK(s(cons(tail(incr(cons(y0, y1))), y2)))) MARK(head(s(cons(tail(incr(s(y0))), y1)))) -> c16(MARK(s(cons(tail(incr(s(y0))), y1)))) MARK(head(s(cons(tail(incr(y0)), y1)))) -> c16(MARK(s(cons(tail(incr(y0)), y1)))) MARK(head(s(cons(tail(head(incr(y0))), y1)))) -> c16(MARK(s(cons(tail(head(incr(y0))), y1)))) MARK(head(s(cons(tail(head(odds)), y0)))) -> c16(MARK(s(cons(tail(head(odds)), y0)))) MARK(head(s(cons(tail(head(head(y0))), y1)))) -> c16(MARK(s(cons(tail(head(head(y0))), y1)))) MARK(head(s(cons(tail(head(tail(y0))), y1)))) -> c16(MARK(s(cons(tail(head(tail(y0))), y1)))) MARK(head(s(cons(tail(head(cons(y0, y1))), y2)))) -> c16(MARK(s(cons(tail(head(cons(y0, y1))), y2)))) MARK(head(s(cons(tail(head(s(y0))), y1)))) -> c16(MARK(s(cons(tail(head(s(y0))), y1)))) MARK(head(s(cons(tail(head(y0)), y1)))) -> c16(MARK(s(cons(tail(head(y0)), y1)))) MARK(head(s(cons(tail(tail(incr(y0))), y1)))) -> c16(MARK(s(cons(tail(tail(incr(y0))), y1)))) MARK(head(s(cons(tail(tail(pairs)), y0)))) -> c16(MARK(s(cons(tail(tail(pairs)), y0)))) MARK(head(s(cons(tail(tail(odds)), y0)))) -> c16(MARK(s(cons(tail(tail(odds)), y0)))) MARK(head(s(cons(tail(tail(head(y0))), y1)))) -> c16(MARK(s(cons(tail(tail(head(y0))), y1)))) MARK(head(s(cons(tail(tail(tail(y0))), y1)))) -> c16(MARK(s(cons(tail(tail(tail(y0))), y1)))) MARK(head(s(cons(tail(tail(cons(y0, y1))), y2)))) -> c16(MARK(s(cons(tail(tail(cons(y0, y1))), y2)))) MARK(head(s(cons(tail(tail(s(y0))), y1)))) -> c16(MARK(s(cons(tail(tail(s(y0))), y1)))) MARK(head(s(cons(tail(tail(y0)), y1)))) -> c16(MARK(s(cons(tail(tail(y0)), y1)))) MARK(head(s(cons(tail(pairs), y0)))) -> c16(MARK(s(cons(tail(pairs), y0)))) MARK(head(s(cons(incr(incr(nats)), y0)))) -> c16(MARK(s(cons(incr(incr(nats)), y0)))) MARK(head(s(cons(incr(incr(pairs)), y0)))) -> c16(MARK(s(cons(incr(incr(pairs)), y0)))) MARK(head(s(cons(incr(odds), y0)))) -> c16(MARK(s(cons(incr(odds), y0)))) MARK(head(s(cons(head(incr(nats)), y0)))) -> c16(MARK(s(cons(head(incr(nats)), y0)))) MARK(head(s(cons(head(incr(pairs)), y0)))) -> c16(MARK(s(cons(head(incr(pairs)), y0)))) MARK(head(s(cons(head(odds), y0)))) -> c16(MARK(s(cons(head(odds), y0)))) MARK(head(s(cons(tail(incr(nats)), y0)))) -> c16(MARK(s(cons(tail(incr(nats)), y0)))) MARK(head(s(cons(tail(incr(pairs)), y0)))) -> c16(MARK(s(cons(tail(incr(pairs)), y0)))) MARK(head(s(cons(tail(odds), y0)))) -> c16(MARK(s(cons(tail(odds), y0)))) MARK(head(s(cons(incr(cons(nats, y0)), y1)))) -> c16(MARK(s(cons(incr(cons(nats, y0)), y1)))) MARK(head(s(cons(incr(cons(incr(y0), y1)), y2)))) -> c16(MARK(s(cons(incr(cons(incr(y0), y1)), y2)))) MARK(head(s(cons(incr(cons(pairs, y0)), y1)))) -> c16(MARK(s(cons(incr(cons(pairs, y0)), y1)))) MARK(head(s(cons(incr(cons(odds, y0)), y1)))) -> c16(MARK(s(cons(incr(cons(odds, y0)), y1)))) MARK(head(s(cons(incr(cons(head(y0), y1)), y2)))) -> c16(MARK(s(cons(incr(cons(head(y0), y1)), y2)))) MARK(head(s(cons(incr(cons(tail(y0), y1)), y2)))) -> c16(MARK(s(cons(incr(cons(tail(y0), y1)), y2)))) MARK(head(s(cons(incr(cons(cons(y0, y1), y2)), y3)))) -> c16(MARK(s(cons(incr(cons(cons(y0, y1), y2)), y3)))) MARK(head(s(cons(incr(cons(s(y0), y1)), y2)))) -> c16(MARK(s(cons(incr(cons(s(y0), y1)), y2)))) MARK(head(s(cons(incr(cons(y0, y1)), y2)))) -> c16(MARK(s(cons(incr(cons(y0, y1)), y2)))) MARK(head(s(cons(incr(cons(0, y0)), y1)))) -> c16(MARK(s(cons(incr(cons(0, y0)), y1)))) MARK(head(s(cons(incr(head(nats)), y0)))) -> c16(MARK(s(cons(incr(head(nats)), y0)))) MARK(head(s(cons(incr(head(pairs)), y0)))) -> c16(MARK(s(cons(incr(head(pairs)), y0)))) MARK(head(s(cons(head(head(nats)), y0)))) -> c16(MARK(s(cons(head(head(nats)), y0)))) MARK(head(s(cons(head(head(pairs)), y0)))) -> c16(MARK(s(cons(head(head(pairs)), y0)))) MARK(head(s(cons(tail(head(nats)), y0)))) -> c16(MARK(s(cons(tail(head(nats)), y0)))) MARK(head(s(cons(tail(head(pairs)), y0)))) -> c16(MARK(s(cons(tail(head(pairs)), y0)))) MARK(head(s(cons(head(cons(nats, y0)), y1)))) -> c16(MARK(s(cons(head(cons(nats, y0)), y1)))) MARK(head(s(cons(head(cons(incr(y0), y1)), y2)))) -> c16(MARK(s(cons(head(cons(incr(y0), y1)), y2)))) MARK(head(s(cons(head(cons(pairs, y0)), y1)))) -> c16(MARK(s(cons(head(cons(pairs, y0)), y1)))) MARK(head(s(cons(head(cons(odds, y0)), y1)))) -> c16(MARK(s(cons(head(cons(odds, y0)), y1)))) MARK(head(s(cons(head(cons(head(y0), y1)), y2)))) -> c16(MARK(s(cons(head(cons(head(y0), y1)), y2)))) MARK(head(s(cons(head(cons(tail(y0), y1)), y2)))) -> c16(MARK(s(cons(head(cons(tail(y0), y1)), y2)))) MARK(head(s(cons(head(cons(cons(y0, y1), y2)), y3)))) -> c16(MARK(s(cons(head(cons(cons(y0, y1), y2)), y3)))) MARK(head(s(cons(head(cons(s(y0), y1)), y2)))) -> c16(MARK(s(cons(head(cons(s(y0), y1)), y2)))) MARK(head(s(cons(head(cons(y0, y1)), y2)))) -> c16(MARK(s(cons(head(cons(y0, y1)), y2)))) MARK(head(s(cons(head(cons(0, y0)), y1)))) -> c16(MARK(s(cons(head(cons(0, y0)), y1)))) MARK(head(s(cons(incr(tail(nats)), y0)))) -> c16(MARK(s(cons(incr(tail(nats)), y0)))) MARK(head(s(cons(head(tail(nats)), y0)))) -> c16(MARK(s(cons(head(tail(nats)), y0)))) MARK(head(s(cons(tail(tail(nats)), y0)))) -> c16(MARK(s(cons(tail(tail(nats)), y0)))) MARK(head(s(incr(cons(nats, y0))))) -> c16(MARK(s(incr(cons(nats, y0))))) MARK(head(s(incr(cons(pairs, y0))))) -> c16(MARK(s(incr(cons(pairs, y0))))) MARK(head(s(incr(cons(odds, y0))))) -> c16(MARK(s(incr(cons(odds, y0))))) MARK(head(s(head(cons(nats, y0))))) -> c16(MARK(s(head(cons(nats, y0))))) MARK(head(s(head(cons(pairs, y0))))) -> c16(MARK(s(head(cons(pairs, y0))))) MARK(head(s(head(cons(odds, y0))))) -> c16(MARK(s(head(cons(odds, y0))))) K tuples:none Defined Rule Symbols: a__nats, a__pairs, a__odds, a__incr_1, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: MARK_1, A__INCR_1, A__HEAD_1, A__TAIL_1 Compound Symbols: c17_2, c17_1, c13_2, c13_1, c16_2, c16_1, c6_1, c8_1, c10_1, c18_1, c20_1 ---------------------------------------- (197) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace MARK(tail(s(z0))) -> c17(MARK(s(z0))) by MARK(tail(s(s(y0)))) -> c17(MARK(s(s(y0)))) MARK(tail(s(incr(s(y0))))) -> c17(MARK(s(incr(s(y0))))) MARK(tail(s(head(s(y0))))) -> c17(MARK(s(head(s(y0))))) MARK(tail(s(tail(cons(y0, y1))))) -> c17(MARK(s(tail(cons(y0, y1))))) MARK(tail(s(tail(s(y0))))) -> c17(MARK(s(tail(s(y0))))) MARK(tail(s(incr(incr(incr(y0)))))) -> c17(MARK(s(incr(incr(incr(y0)))))) MARK(tail(s(incr(incr(odds))))) -> c17(MARK(s(incr(incr(odds))))) MARK(tail(s(incr(incr(head(y0)))))) -> c17(MARK(s(incr(incr(head(y0)))))) MARK(tail(s(incr(incr(tail(y0)))))) -> c17(MARK(s(incr(incr(tail(y0)))))) MARK(tail(s(incr(incr(cons(y0, y1)))))) -> c17(MARK(s(incr(incr(cons(y0, y1)))))) MARK(tail(s(incr(incr(s(y0)))))) -> c17(MARK(s(incr(incr(s(y0)))))) MARK(tail(s(incr(incr(y0))))) -> c17(MARK(s(incr(incr(y0))))) MARK(tail(s(incr(head(incr(y0)))))) -> c17(MARK(s(incr(head(incr(y0)))))) MARK(tail(s(incr(head(odds))))) -> c17(MARK(s(incr(head(odds))))) MARK(tail(s(incr(head(head(y0)))))) -> c17(MARK(s(incr(head(head(y0)))))) MARK(tail(s(incr(head(tail(y0)))))) -> c17(MARK(s(incr(head(tail(y0)))))) MARK(tail(s(incr(head(cons(y0, y1)))))) -> c17(MARK(s(incr(head(cons(y0, y1)))))) MARK(tail(s(incr(head(s(y0)))))) -> c17(MARK(s(incr(head(s(y0)))))) MARK(tail(s(incr(head(y0))))) -> c17(MARK(s(incr(head(y0))))) MARK(tail(s(incr(tail(incr(y0)))))) -> c17(MARK(s(incr(tail(incr(y0)))))) MARK(tail(s(incr(tail(pairs))))) -> c17(MARK(s(incr(tail(pairs))))) MARK(tail(s(incr(tail(odds))))) -> c17(MARK(s(incr(tail(odds))))) MARK(tail(s(incr(tail(head(y0)))))) -> c17(MARK(s(incr(tail(head(y0)))))) MARK(tail(s(incr(tail(tail(y0)))))) -> c17(MARK(s(incr(tail(tail(y0)))))) MARK(tail(s(incr(tail(cons(y0, y1)))))) -> c17(MARK(s(incr(tail(cons(y0, y1)))))) MARK(tail(s(incr(tail(s(y0)))))) -> c17(MARK(s(incr(tail(s(y0)))))) MARK(tail(s(incr(tail(y0))))) -> c17(MARK(s(incr(tail(y0))))) MARK(tail(s(head(incr(incr(y0)))))) -> c17(MARK(s(head(incr(incr(y0)))))) MARK(tail(s(head(incr(odds))))) -> c17(MARK(s(head(incr(odds))))) MARK(tail(s(head(incr(head(y0)))))) -> c17(MARK(s(head(incr(head(y0)))))) MARK(tail(s(head(incr(tail(y0)))))) -> c17(MARK(s(head(incr(tail(y0)))))) MARK(tail(s(head(incr(cons(y0, y1)))))) -> c17(MARK(s(head(incr(cons(y0, y1)))))) MARK(tail(s(head(incr(s(y0)))))) -> c17(MARK(s(head(incr(s(y0)))))) MARK(tail(s(head(incr(y0))))) -> c17(MARK(s(head(incr(y0))))) MARK(tail(s(head(head(incr(y0)))))) -> c17(MARK(s(head(head(incr(y0)))))) MARK(tail(s(head(head(odds))))) -> c17(MARK(s(head(head(odds))))) MARK(tail(s(head(head(head(y0)))))) -> c17(MARK(s(head(head(head(y0)))))) MARK(tail(s(head(head(tail(y0)))))) -> c17(MARK(s(head(head(tail(y0)))))) MARK(tail(s(head(head(cons(y0, y1)))))) -> c17(MARK(s(head(head(cons(y0, y1)))))) MARK(tail(s(head(head(s(y0)))))) -> c17(MARK(s(head(head(s(y0)))))) MARK(tail(s(head(head(y0))))) -> c17(MARK(s(head(head(y0))))) MARK(tail(s(head(tail(incr(y0)))))) -> c17(MARK(s(head(tail(incr(y0)))))) MARK(tail(s(head(tail(pairs))))) -> c17(MARK(s(head(tail(pairs))))) MARK(tail(s(head(tail(odds))))) -> c17(MARK(s(head(tail(odds))))) MARK(tail(s(head(tail(head(y0)))))) -> c17(MARK(s(head(tail(head(y0)))))) MARK(tail(s(head(tail(tail(y0)))))) -> c17(MARK(s(head(tail(tail(y0)))))) MARK(tail(s(head(tail(cons(y0, y1)))))) -> c17(MARK(s(head(tail(cons(y0, y1)))))) MARK(tail(s(head(tail(s(y0)))))) -> c17(MARK(s(head(tail(s(y0)))))) MARK(tail(s(head(tail(y0))))) -> c17(MARK(s(head(tail(y0))))) MARK(tail(s(tail(incr(incr(y0)))))) -> c17(MARK(s(tail(incr(incr(y0)))))) MARK(tail(s(tail(incr(odds))))) -> c17(MARK(s(tail(incr(odds))))) MARK(tail(s(tail(incr(head(y0)))))) -> c17(MARK(s(tail(incr(head(y0)))))) MARK(tail(s(tail(incr(tail(y0)))))) -> c17(MARK(s(tail(incr(tail(y0)))))) MARK(tail(s(tail(incr(cons(y0, y1)))))) -> c17(MARK(s(tail(incr(cons(y0, y1)))))) MARK(tail(s(tail(incr(s(y0)))))) -> c17(MARK(s(tail(incr(s(y0)))))) MARK(tail(s(tail(incr(y0))))) -> c17(MARK(s(tail(incr(y0))))) MARK(tail(s(tail(head(incr(y0)))))) -> c17(MARK(s(tail(head(incr(y0)))))) MARK(tail(s(tail(head(odds))))) -> c17(MARK(s(tail(head(odds))))) MARK(tail(s(tail(head(head(y0)))))) -> c17(MARK(s(tail(head(head(y0)))))) MARK(tail(s(tail(head(tail(y0)))))) -> c17(MARK(s(tail(head(tail(y0)))))) MARK(tail(s(tail(head(cons(y0, y1)))))) -> c17(MARK(s(tail(head(cons(y0, y1)))))) MARK(tail(s(tail(head(s(y0)))))) -> c17(MARK(s(tail(head(s(y0)))))) MARK(tail(s(tail(head(y0))))) -> c17(MARK(s(tail(head(y0))))) MARK(tail(s(tail(tail(incr(y0)))))) -> c17(MARK(s(tail(tail(incr(y0)))))) MARK(tail(s(tail(tail(pairs))))) -> c17(MARK(s(tail(tail(pairs))))) MARK(tail(s(tail(tail(odds))))) -> c17(MARK(s(tail(tail(odds))))) MARK(tail(s(tail(tail(head(y0)))))) -> c17(MARK(s(tail(tail(head(y0)))))) MARK(tail(s(tail(tail(tail(y0)))))) -> c17(MARK(s(tail(tail(tail(y0)))))) MARK(tail(s(tail(tail(cons(y0, y1)))))) -> c17(MARK(s(tail(tail(cons(y0, y1)))))) MARK(tail(s(tail(tail(s(y0)))))) -> c17(MARK(s(tail(tail(s(y0)))))) MARK(tail(s(tail(tail(y0))))) -> c17(MARK(s(tail(tail(y0))))) MARK(tail(s(tail(pairs)))) -> c17(MARK(s(tail(pairs)))) MARK(tail(s(incr(incr(nats))))) -> c17(MARK(s(incr(incr(nats))))) MARK(tail(s(incr(incr(pairs))))) -> c17(MARK(s(incr(incr(pairs))))) MARK(tail(s(incr(odds)))) -> c17(MARK(s(incr(odds)))) MARK(tail(s(head(incr(nats))))) -> c17(MARK(s(head(incr(nats))))) MARK(tail(s(head(incr(pairs))))) -> c17(MARK(s(head(incr(pairs))))) MARK(tail(s(head(odds)))) -> c17(MARK(s(head(odds)))) MARK(tail(s(tail(incr(nats))))) -> c17(MARK(s(tail(incr(nats))))) MARK(tail(s(tail(incr(pairs))))) -> c17(MARK(s(tail(incr(pairs))))) MARK(tail(s(tail(odds)))) -> c17(MARK(s(tail(odds)))) MARK(tail(s(incr(cons(incr(y0), y1))))) -> c17(MARK(s(incr(cons(incr(y0), y1))))) MARK(tail(s(incr(cons(head(y0), y1))))) -> c17(MARK(s(incr(cons(head(y0), y1))))) MARK(tail(s(incr(cons(tail(y0), y1))))) -> c17(MARK(s(incr(cons(tail(y0), y1))))) MARK(tail(s(incr(cons(cons(y0, y1), y2))))) -> c17(MARK(s(incr(cons(cons(y0, y1), y2))))) MARK(tail(s(incr(cons(s(y0), y1))))) -> c17(MARK(s(incr(cons(s(y0), y1))))) MARK(tail(s(incr(cons(y0, y1))))) -> c17(MARK(s(incr(cons(y0, y1))))) MARK(tail(s(incr(head(nats))))) -> c17(MARK(s(incr(head(nats))))) MARK(tail(s(incr(head(pairs))))) -> c17(MARK(s(incr(head(pairs))))) MARK(tail(s(head(head(nats))))) -> c17(MARK(s(head(head(nats))))) MARK(tail(s(head(head(pairs))))) -> c17(MARK(s(head(head(pairs))))) MARK(tail(s(tail(head(nats))))) -> c17(MARK(s(tail(head(nats))))) MARK(tail(s(tail(head(pairs))))) -> c17(MARK(s(tail(head(pairs))))) MARK(tail(s(head(cons(incr(y0), y1))))) -> c17(MARK(s(head(cons(incr(y0), y1))))) MARK(tail(s(head(cons(head(y0), y1))))) -> c17(MARK(s(head(cons(head(y0), y1))))) MARK(tail(s(head(cons(tail(y0), y1))))) -> c17(MARK(s(head(cons(tail(y0), y1))))) MARK(tail(s(head(cons(cons(y0, y1), y2))))) -> c17(MARK(s(head(cons(cons(y0, y1), y2))))) MARK(tail(s(head(cons(s(y0), y1))))) -> c17(MARK(s(head(cons(s(y0), y1))))) MARK(tail(s(head(cons(y0, y1))))) -> c17(MARK(s(head(cons(y0, y1))))) MARK(tail(s(incr(tail(nats))))) -> c17(MARK(s(incr(tail(nats))))) MARK(tail(s(head(tail(nats))))) -> c17(MARK(s(head(tail(nats))))) MARK(tail(s(tail(tail(nats))))) -> c17(MARK(s(tail(tail(nats))))) MARK(tail(s(cons(cons(y0, y1), y2)))) -> c17(MARK(s(cons(cons(y0, y1), y2)))) MARK(tail(s(cons(s(y0), y1)))) -> c17(MARK(s(cons(s(y0), y1)))) MARK(tail(s(cons(incr(s(y0)), y1)))) -> c17(MARK(s(cons(incr(s(y0)), y1)))) MARK(tail(s(cons(head(s(y0)), y1)))) -> c17(MARK(s(cons(head(s(y0)), y1)))) MARK(tail(s(cons(tail(cons(y0, y1)), y2)))) -> c17(MARK(s(cons(tail(cons(y0, y1)), y2)))) MARK(tail(s(cons(tail(s(y0)), y1)))) -> c17(MARK(s(cons(tail(s(y0)), y1)))) MARK(tail(s(cons(incr(incr(incr(y0))), y1)))) -> c17(MARK(s(cons(incr(incr(incr(y0))), y1)))) MARK(tail(s(cons(incr(incr(odds)), y0)))) -> c17(MARK(s(cons(incr(incr(odds)), y0)))) MARK(tail(s(cons(incr(incr(head(y0))), y1)))) -> c17(MARK(s(cons(incr(incr(head(y0))), y1)))) MARK(tail(s(cons(incr(incr(tail(y0))), y1)))) -> c17(MARK(s(cons(incr(incr(tail(y0))), y1)))) MARK(tail(s(cons(incr(incr(cons(y0, y1))), y2)))) -> c17(MARK(s(cons(incr(incr(cons(y0, y1))), y2)))) MARK(tail(s(cons(incr(incr(s(y0))), y1)))) -> c17(MARK(s(cons(incr(incr(s(y0))), y1)))) MARK(tail(s(cons(incr(incr(y0)), y1)))) -> c17(MARK(s(cons(incr(incr(y0)), y1)))) MARK(tail(s(cons(incr(head(incr(y0))), y1)))) -> c17(MARK(s(cons(incr(head(incr(y0))), y1)))) MARK(tail(s(cons(incr(head(odds)), y0)))) -> c17(MARK(s(cons(incr(head(odds)), y0)))) MARK(tail(s(cons(incr(head(head(y0))), y1)))) -> c17(MARK(s(cons(incr(head(head(y0))), y1)))) MARK(tail(s(cons(incr(head(tail(y0))), y1)))) -> c17(MARK(s(cons(incr(head(tail(y0))), y1)))) MARK(tail(s(cons(incr(head(cons(y0, y1))), y2)))) -> c17(MARK(s(cons(incr(head(cons(y0, y1))), y2)))) MARK(tail(s(cons(incr(head(s(y0))), y1)))) -> c17(MARK(s(cons(incr(head(s(y0))), y1)))) MARK(tail(s(cons(incr(head(y0)), y1)))) -> c17(MARK(s(cons(incr(head(y0)), y1)))) MARK(tail(s(cons(incr(tail(incr(y0))), y1)))) -> c17(MARK(s(cons(incr(tail(incr(y0))), y1)))) MARK(tail(s(cons(incr(tail(pairs)), y0)))) -> c17(MARK(s(cons(incr(tail(pairs)), y0)))) MARK(tail(s(cons(incr(tail(odds)), y0)))) -> c17(MARK(s(cons(incr(tail(odds)), y0)))) MARK(tail(s(cons(incr(tail(head(y0))), y1)))) -> c17(MARK(s(cons(incr(tail(head(y0))), y1)))) MARK(tail(s(cons(incr(tail(tail(y0))), y1)))) -> c17(MARK(s(cons(incr(tail(tail(y0))), y1)))) MARK(tail(s(cons(incr(tail(cons(y0, y1))), y2)))) -> c17(MARK(s(cons(incr(tail(cons(y0, y1))), y2)))) MARK(tail(s(cons(incr(tail(s(y0))), y1)))) -> c17(MARK(s(cons(incr(tail(s(y0))), y1)))) MARK(tail(s(cons(incr(tail(y0)), y1)))) -> c17(MARK(s(cons(incr(tail(y0)), y1)))) MARK(tail(s(cons(head(incr(incr(y0))), y1)))) -> c17(MARK(s(cons(head(incr(incr(y0))), y1)))) MARK(tail(s(cons(head(incr(odds)), y0)))) -> c17(MARK(s(cons(head(incr(odds)), y0)))) MARK(tail(s(cons(head(incr(head(y0))), y1)))) -> c17(MARK(s(cons(head(incr(head(y0))), y1)))) MARK(tail(s(cons(head(incr(tail(y0))), y1)))) -> c17(MARK(s(cons(head(incr(tail(y0))), y1)))) MARK(tail(s(cons(head(incr(cons(y0, y1))), y2)))) -> c17(MARK(s(cons(head(incr(cons(y0, y1))), y2)))) MARK(tail(s(cons(head(incr(s(y0))), y1)))) -> c17(MARK(s(cons(head(incr(s(y0))), y1)))) MARK(tail(s(cons(head(incr(y0)), y1)))) -> c17(MARK(s(cons(head(incr(y0)), y1)))) MARK(tail(s(cons(head(head(incr(y0))), y1)))) -> c17(MARK(s(cons(head(head(incr(y0))), y1)))) MARK(tail(s(cons(head(head(odds)), y0)))) -> c17(MARK(s(cons(head(head(odds)), y0)))) MARK(tail(s(cons(head(head(head(y0))), y1)))) -> c17(MARK(s(cons(head(head(head(y0))), y1)))) MARK(tail(s(cons(head(head(tail(y0))), y1)))) -> c17(MARK(s(cons(head(head(tail(y0))), y1)))) MARK(tail(s(cons(head(head(cons(y0, y1))), y2)))) -> c17(MARK(s(cons(head(head(cons(y0, y1))), y2)))) MARK(tail(s(cons(head(head(s(y0))), y1)))) -> c17(MARK(s(cons(head(head(s(y0))), y1)))) MARK(tail(s(cons(head(head(y0)), y1)))) -> c17(MARK(s(cons(head(head(y0)), y1)))) MARK(tail(s(cons(head(tail(incr(y0))), y1)))) -> c17(MARK(s(cons(head(tail(incr(y0))), y1)))) MARK(tail(s(cons(head(tail(pairs)), y0)))) -> c17(MARK(s(cons(head(tail(pairs)), y0)))) MARK(tail(s(cons(head(tail(odds)), y0)))) -> c17(MARK(s(cons(head(tail(odds)), y0)))) MARK(tail(s(cons(head(tail(head(y0))), y1)))) -> c17(MARK(s(cons(head(tail(head(y0))), y1)))) MARK(tail(s(cons(head(tail(tail(y0))), y1)))) -> c17(MARK(s(cons(head(tail(tail(y0))), y1)))) MARK(tail(s(cons(head(tail(cons(y0, y1))), y2)))) -> c17(MARK(s(cons(head(tail(cons(y0, y1))), y2)))) MARK(tail(s(cons(head(tail(s(y0))), y1)))) -> c17(MARK(s(cons(head(tail(s(y0))), y1)))) MARK(tail(s(cons(head(tail(y0)), y1)))) -> c17(MARK(s(cons(head(tail(y0)), y1)))) MARK(tail(s(cons(tail(incr(incr(y0))), y1)))) -> c17(MARK(s(cons(tail(incr(incr(y0))), y1)))) MARK(tail(s(cons(tail(incr(odds)), y0)))) -> c17(MARK(s(cons(tail(incr(odds)), y0)))) MARK(tail(s(cons(tail(incr(head(y0))), y1)))) -> c17(MARK(s(cons(tail(incr(head(y0))), y1)))) MARK(tail(s(cons(tail(incr(tail(y0))), y1)))) -> c17(MARK(s(cons(tail(incr(tail(y0))), y1)))) MARK(tail(s(cons(tail(incr(cons(y0, y1))), y2)))) -> c17(MARK(s(cons(tail(incr(cons(y0, y1))), y2)))) MARK(tail(s(cons(tail(incr(s(y0))), y1)))) -> c17(MARK(s(cons(tail(incr(s(y0))), y1)))) MARK(tail(s(cons(tail(incr(y0)), y1)))) -> c17(MARK(s(cons(tail(incr(y0)), y1)))) MARK(tail(s(cons(tail(head(incr(y0))), y1)))) -> c17(MARK(s(cons(tail(head(incr(y0))), y1)))) MARK(tail(s(cons(tail(head(odds)), y0)))) -> c17(MARK(s(cons(tail(head(odds)), y0)))) MARK(tail(s(cons(tail(head(head(y0))), y1)))) -> c17(MARK(s(cons(tail(head(head(y0))), y1)))) MARK(tail(s(cons(tail(head(tail(y0))), y1)))) -> c17(MARK(s(cons(tail(head(tail(y0))), y1)))) MARK(tail(s(cons(tail(head(cons(y0, y1))), y2)))) -> c17(MARK(s(cons(tail(head(cons(y0, y1))), y2)))) MARK(tail(s(cons(tail(head(s(y0))), y1)))) -> c17(MARK(s(cons(tail(head(s(y0))), y1)))) MARK(tail(s(cons(tail(head(y0)), y1)))) -> c17(MARK(s(cons(tail(head(y0)), y1)))) MARK(tail(s(cons(tail(tail(incr(y0))), y1)))) -> c17(MARK(s(cons(tail(tail(incr(y0))), y1)))) MARK(tail(s(cons(tail(tail(pairs)), y0)))) -> c17(MARK(s(cons(tail(tail(pairs)), y0)))) MARK(tail(s(cons(tail(tail(odds)), y0)))) -> c17(MARK(s(cons(tail(tail(odds)), y0)))) MARK(tail(s(cons(tail(tail(head(y0))), y1)))) -> c17(MARK(s(cons(tail(tail(head(y0))), y1)))) MARK(tail(s(cons(tail(tail(tail(y0))), y1)))) -> c17(MARK(s(cons(tail(tail(tail(y0))), y1)))) MARK(tail(s(cons(tail(tail(cons(y0, y1))), y2)))) -> c17(MARK(s(cons(tail(tail(cons(y0, y1))), y2)))) MARK(tail(s(cons(tail(tail(s(y0))), y1)))) -> c17(MARK(s(cons(tail(tail(s(y0))), y1)))) MARK(tail(s(cons(tail(tail(y0)), y1)))) -> c17(MARK(s(cons(tail(tail(y0)), y1)))) MARK(tail(s(cons(tail(pairs), y0)))) -> c17(MARK(s(cons(tail(pairs), y0)))) MARK(tail(s(cons(incr(incr(nats)), y0)))) -> c17(MARK(s(cons(incr(incr(nats)), y0)))) MARK(tail(s(cons(incr(incr(pairs)), y0)))) -> c17(MARK(s(cons(incr(incr(pairs)), y0)))) MARK(tail(s(cons(incr(odds), y0)))) -> c17(MARK(s(cons(incr(odds), y0)))) MARK(tail(s(cons(head(incr(nats)), y0)))) -> c17(MARK(s(cons(head(incr(nats)), y0)))) MARK(tail(s(cons(head(incr(pairs)), y0)))) -> c17(MARK(s(cons(head(incr(pairs)), y0)))) MARK(tail(s(cons(head(odds), y0)))) -> c17(MARK(s(cons(head(odds), y0)))) MARK(tail(s(cons(tail(incr(nats)), y0)))) -> c17(MARK(s(cons(tail(incr(nats)), y0)))) MARK(tail(s(cons(tail(incr(pairs)), y0)))) -> c17(MARK(s(cons(tail(incr(pairs)), y0)))) MARK(tail(s(cons(tail(odds), y0)))) -> c17(MARK(s(cons(tail(odds), y0)))) MARK(tail(s(cons(incr(cons(nats, y0)), y1)))) -> c17(MARK(s(cons(incr(cons(nats, y0)), y1)))) MARK(tail(s(cons(incr(cons(incr(y0), y1)), y2)))) -> c17(MARK(s(cons(incr(cons(incr(y0), y1)), y2)))) MARK(tail(s(cons(incr(cons(pairs, y0)), y1)))) -> c17(MARK(s(cons(incr(cons(pairs, y0)), y1)))) MARK(tail(s(cons(incr(cons(odds, y0)), y1)))) -> c17(MARK(s(cons(incr(cons(odds, y0)), y1)))) MARK(tail(s(cons(incr(cons(head(y0), y1)), y2)))) -> c17(MARK(s(cons(incr(cons(head(y0), y1)), y2)))) MARK(tail(s(cons(incr(cons(tail(y0), y1)), y2)))) -> c17(MARK(s(cons(incr(cons(tail(y0), y1)), y2)))) MARK(tail(s(cons(incr(cons(cons(y0, y1), y2)), y3)))) -> c17(MARK(s(cons(incr(cons(cons(y0, y1), y2)), y3)))) MARK(tail(s(cons(incr(cons(s(y0), y1)), y2)))) -> c17(MARK(s(cons(incr(cons(s(y0), y1)), y2)))) MARK(tail(s(cons(incr(cons(y0, y1)), y2)))) -> c17(MARK(s(cons(incr(cons(y0, y1)), y2)))) MARK(tail(s(cons(incr(cons(0, y0)), y1)))) -> c17(MARK(s(cons(incr(cons(0, y0)), y1)))) MARK(tail(s(cons(incr(head(nats)), y0)))) -> c17(MARK(s(cons(incr(head(nats)), y0)))) MARK(tail(s(cons(incr(head(pairs)), y0)))) -> c17(MARK(s(cons(incr(head(pairs)), y0)))) MARK(tail(s(cons(head(head(nats)), y0)))) -> c17(MARK(s(cons(head(head(nats)), y0)))) MARK(tail(s(cons(head(head(pairs)), y0)))) -> c17(MARK(s(cons(head(head(pairs)), y0)))) MARK(tail(s(cons(tail(head(nats)), y0)))) -> c17(MARK(s(cons(tail(head(nats)), y0)))) MARK(tail(s(cons(tail(head(pairs)), y0)))) -> c17(MARK(s(cons(tail(head(pairs)), y0)))) MARK(tail(s(cons(head(cons(nats, y0)), y1)))) -> c17(MARK(s(cons(head(cons(nats, y0)), y1)))) MARK(tail(s(cons(head(cons(incr(y0), y1)), y2)))) -> c17(MARK(s(cons(head(cons(incr(y0), y1)), y2)))) MARK(tail(s(cons(head(cons(pairs, y0)), y1)))) -> c17(MARK(s(cons(head(cons(pairs, y0)), y1)))) MARK(tail(s(cons(head(cons(odds, y0)), y1)))) -> c17(MARK(s(cons(head(cons(odds, y0)), y1)))) MARK(tail(s(cons(head(cons(head(y0), y1)), y2)))) -> c17(MARK(s(cons(head(cons(head(y0), y1)), y2)))) MARK(tail(s(cons(head(cons(tail(y0), y1)), y2)))) -> c17(MARK(s(cons(head(cons(tail(y0), y1)), y2)))) MARK(tail(s(cons(head(cons(cons(y0, y1), y2)), y3)))) -> c17(MARK(s(cons(head(cons(cons(y0, y1), y2)), y3)))) MARK(tail(s(cons(head(cons(s(y0), y1)), y2)))) -> c17(MARK(s(cons(head(cons(s(y0), y1)), y2)))) MARK(tail(s(cons(head(cons(y0, y1)), y2)))) -> c17(MARK(s(cons(head(cons(y0, y1)), y2)))) MARK(tail(s(cons(head(cons(0, y0)), y1)))) -> c17(MARK(s(cons(head(cons(0, y0)), y1)))) MARK(tail(s(cons(incr(tail(nats)), y0)))) -> c17(MARK(s(cons(incr(tail(nats)), y0)))) MARK(tail(s(cons(head(tail(nats)), y0)))) -> c17(MARK(s(cons(head(tail(nats)), y0)))) MARK(tail(s(cons(tail(tail(nats)), y0)))) -> c17(MARK(s(cons(tail(tail(nats)), y0)))) MARK(tail(s(incr(cons(nats, y0))))) -> c17(MARK(s(incr(cons(nats, y0))))) MARK(tail(s(incr(cons(pairs, y0))))) -> c17(MARK(s(incr(cons(pairs, y0))))) MARK(tail(s(incr(cons(odds, y0))))) -> c17(MARK(s(incr(cons(odds, y0))))) MARK(tail(s(head(cons(nats, y0))))) -> c17(MARK(s(head(cons(nats, y0))))) MARK(tail(s(head(cons(pairs, y0))))) -> c17(MARK(s(head(cons(pairs, y0))))) MARK(tail(s(head(cons(odds, y0))))) -> c17(MARK(s(head(cons(odds, y0))))) ---------------------------------------- (198) Obligation: Complexity Dependency Tuples Problem Rules: a__nats -> cons(0, incr(nats)) a__nats -> nats a__pairs -> cons(0, incr(odds)) a__pairs -> pairs a__odds -> a__incr(a__pairs) a__odds -> odds a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) 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(nats) -> a__nats mark(incr(z0)) -> a__incr(mark(z0)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(tail(incr(incr(z0)))) -> c17(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(odds))) -> c17(A__TAIL(a__incr(a__odds)), MARK(incr(odds))) MARK(tail(incr(head(z0)))) -> c17(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c17(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c17(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c17(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c17(MARK(incr(x0))) 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MARK(cons(head(cons(pairs, y0)), z1)) -> c18(MARK(head(cons(pairs, y0)))) MARK(cons(head(cons(odds, y0)), z1)) -> c18(MARK(head(cons(odds, y0)))) MARK(cons(head(cons(head(y0), y1)), z1)) -> c18(MARK(head(cons(head(y0), y1)))) MARK(cons(head(cons(tail(y0), y1)), z1)) -> c18(MARK(head(cons(tail(y0), y1)))) MARK(cons(head(cons(cons(y0, y1), y2)), z1)) -> c18(MARK(head(cons(cons(y0, y1), y2)))) MARK(cons(head(cons(s(y0), y1)), z1)) -> c18(MARK(head(cons(s(y0), y1)))) MARK(cons(head(cons(y0, y1)), z1)) -> c18(MARK(head(cons(y0, y1)))) MARK(cons(head(cons(0, y0)), z1)) -> c18(MARK(head(cons(0, y0)))) MARK(cons(incr(tail(nats)), z1)) -> c18(MARK(incr(tail(nats)))) MARK(cons(head(tail(nats)), z1)) -> c18(MARK(head(tail(nats)))) MARK(cons(tail(tail(nats)), z1)) -> c18(MARK(tail(tail(nats)))) MARK(incr(cons(nats, x1))) -> c13(A__INCR(cons(a__nats, x1))) MARK(incr(cons(pairs, x1))) -> c13(A__INCR(cons(a__pairs, x1))) MARK(incr(cons(odds, x1))) -> c13(A__INCR(cons(a__odds, x1))) 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MARK(s(incr(head(head(y0))))) -> c20(MARK(incr(head(head(y0))))) MARK(s(incr(head(tail(y0))))) -> c20(MARK(incr(head(tail(y0))))) MARK(s(incr(head(cons(y0, y1))))) -> c20(MARK(incr(head(cons(y0, y1))))) MARK(s(incr(head(s(y0))))) -> c20(MARK(incr(head(s(y0))))) MARK(s(incr(head(y0)))) -> c20(MARK(incr(head(y0)))) MARK(s(incr(tail(incr(y0))))) -> c20(MARK(incr(tail(incr(y0))))) MARK(s(incr(tail(pairs)))) -> c20(MARK(incr(tail(pairs)))) MARK(s(incr(tail(odds)))) -> c20(MARK(incr(tail(odds)))) MARK(s(incr(tail(head(y0))))) -> c20(MARK(incr(tail(head(y0))))) MARK(s(incr(tail(tail(y0))))) -> c20(MARK(incr(tail(tail(y0))))) MARK(s(incr(tail(cons(y0, y1))))) -> c20(MARK(incr(tail(cons(y0, y1))))) MARK(s(incr(tail(s(y0))))) -> c20(MARK(incr(tail(s(y0))))) MARK(s(incr(tail(y0)))) -> c20(MARK(incr(tail(y0)))) MARK(s(head(incr(incr(y0))))) -> c20(MARK(head(incr(incr(y0))))) MARK(s(head(incr(odds)))) -> c20(MARK(head(incr(odds)))) MARK(s(head(incr(head(y0))))) -> c20(MARK(head(incr(head(y0))))) MARK(s(head(incr(tail(y0))))) -> c20(MARK(head(incr(tail(y0))))) MARK(s(head(incr(cons(y0, y1))))) -> c20(MARK(head(incr(cons(y0, y1))))) MARK(s(head(incr(s(y0))))) -> c20(MARK(head(incr(s(y0))))) MARK(s(head(incr(y0)))) -> c20(MARK(head(incr(y0)))) MARK(s(head(head(incr(y0))))) -> c20(MARK(head(head(incr(y0))))) MARK(s(head(head(odds)))) -> c20(MARK(head(head(odds)))) MARK(s(head(head(head(y0))))) -> c20(MARK(head(head(head(y0))))) MARK(s(head(head(tail(y0))))) -> c20(MARK(head(head(tail(y0))))) MARK(s(head(head(cons(y0, y1))))) -> c20(MARK(head(head(cons(y0, y1))))) MARK(s(head(head(s(y0))))) -> c20(MARK(head(head(s(y0))))) MARK(s(head(head(y0)))) -> c20(MARK(head(head(y0)))) MARK(s(head(tail(incr(y0))))) -> c20(MARK(head(tail(incr(y0))))) MARK(s(head(tail(pairs)))) -> c20(MARK(head(tail(pairs)))) MARK(s(head(tail(odds)))) -> c20(MARK(head(tail(odds)))) MARK(s(head(tail(head(y0))))) -> c20(MARK(head(tail(head(y0))))) MARK(s(head(tail(tail(y0))))) -> c20(MARK(head(tail(tail(y0))))) MARK(s(head(tail(cons(y0, y1))))) -> c20(MARK(head(tail(cons(y0, y1))))) MARK(s(head(tail(s(y0))))) -> c20(MARK(head(tail(s(y0))))) MARK(s(head(tail(y0)))) -> c20(MARK(head(tail(y0)))) MARK(s(tail(incr(incr(y0))))) -> c20(MARK(tail(incr(incr(y0))))) MARK(s(tail(incr(odds)))) -> c20(MARK(tail(incr(odds)))) MARK(s(tail(incr(head(y0))))) -> c20(MARK(tail(incr(head(y0))))) MARK(s(tail(incr(tail(y0))))) -> c20(MARK(tail(incr(tail(y0))))) MARK(s(tail(incr(cons(y0, y1))))) -> c20(MARK(tail(incr(cons(y0, y1))))) MARK(s(tail(incr(s(y0))))) -> c20(MARK(tail(incr(s(y0))))) MARK(s(tail(incr(y0)))) -> c20(MARK(tail(incr(y0)))) MARK(s(tail(head(incr(y0))))) -> c20(MARK(tail(head(incr(y0))))) MARK(s(tail(head(odds)))) -> c20(MARK(tail(head(odds)))) MARK(s(tail(head(head(y0))))) -> c20(MARK(tail(head(head(y0))))) MARK(s(tail(head(tail(y0))))) -> c20(MARK(tail(head(tail(y0))))) MARK(s(tail(head(cons(y0, y1))))) -> c20(MARK(tail(head(cons(y0, y1))))) MARK(s(tail(head(s(y0))))) -> c20(MARK(tail(head(s(y0))))) MARK(s(tail(head(y0)))) -> c20(MARK(tail(head(y0)))) MARK(s(tail(tail(incr(y0))))) -> c20(MARK(tail(tail(incr(y0))))) MARK(s(tail(tail(pairs)))) -> c20(MARK(tail(tail(pairs)))) MARK(s(tail(tail(odds)))) -> c20(MARK(tail(tail(odds)))) MARK(s(tail(tail(head(y0))))) -> c20(MARK(tail(tail(head(y0))))) MARK(s(tail(tail(tail(y0))))) -> c20(MARK(tail(tail(tail(y0))))) MARK(s(tail(tail(cons(y0, y1))))) -> c20(MARK(tail(tail(cons(y0, y1))))) MARK(s(tail(tail(s(y0))))) -> c20(MARK(tail(tail(s(y0))))) MARK(s(tail(tail(y0)))) -> c20(MARK(tail(tail(y0)))) MARK(s(tail(pairs))) -> c20(MARK(tail(pairs))) MARK(s(incr(incr(nats)))) -> c20(MARK(incr(incr(nats)))) MARK(s(incr(incr(pairs)))) -> c20(MARK(incr(incr(pairs)))) MARK(s(incr(odds))) -> c20(MARK(incr(odds))) MARK(s(head(incr(nats)))) -> c20(MARK(head(incr(nats)))) MARK(s(head(incr(pairs)))) -> c20(MARK(head(incr(pairs)))) MARK(s(head(odds))) -> c20(MARK(head(odds))) MARK(s(tail(incr(nats)))) -> c20(MARK(tail(incr(nats)))) MARK(s(tail(incr(pairs)))) -> c20(MARK(tail(incr(pairs)))) MARK(s(tail(odds))) -> c20(MARK(tail(odds))) MARK(s(incr(cons(incr(y0), y1)))) -> c20(MARK(incr(cons(incr(y0), y1)))) MARK(s(incr(cons(head(y0), y1)))) -> c20(MARK(incr(cons(head(y0), y1)))) MARK(s(incr(cons(tail(y0), y1)))) -> c20(MARK(incr(cons(tail(y0), y1)))) MARK(s(incr(cons(cons(y0, y1), y2)))) -> c20(MARK(incr(cons(cons(y0, y1), y2)))) MARK(s(incr(cons(s(y0), y1)))) -> c20(MARK(incr(cons(s(y0), y1)))) MARK(s(incr(cons(y0, y1)))) -> c20(MARK(incr(cons(y0, y1)))) MARK(s(incr(head(nats)))) -> c20(MARK(incr(head(nats)))) MARK(s(incr(head(pairs)))) -> c20(MARK(incr(head(pairs)))) MARK(s(head(head(nats)))) -> c20(MARK(head(head(nats)))) MARK(s(head(head(pairs)))) -> c20(MARK(head(head(pairs)))) MARK(s(tail(head(nats)))) -> c20(MARK(tail(head(nats)))) MARK(s(tail(head(pairs)))) -> c20(MARK(tail(head(pairs)))) MARK(s(head(cons(incr(y0), y1)))) -> c20(MARK(head(cons(incr(y0), y1)))) MARK(s(head(cons(head(y0), y1)))) -> c20(MARK(head(cons(head(y0), y1)))) MARK(s(head(cons(tail(y0), y1)))) -> c20(MARK(head(cons(tail(y0), y1)))) MARK(s(head(cons(cons(y0, y1), y2)))) -> c20(MARK(head(cons(cons(y0, y1), y2)))) MARK(s(head(cons(s(y0), y1)))) -> c20(MARK(head(cons(s(y0), y1)))) MARK(s(head(cons(y0, y1)))) -> c20(MARK(head(cons(y0, y1)))) MARK(s(incr(tail(nats)))) -> c20(MARK(incr(tail(nats)))) MARK(s(head(tail(nats)))) -> c20(MARK(head(tail(nats)))) MARK(s(tail(tail(nats)))) -> c20(MARK(tail(tail(nats)))) MARK(s(cons(cons(y0, y1), y2))) -> c20(MARK(cons(cons(y0, y1), y2))) MARK(s(cons(s(y0), y1))) -> c20(MARK(cons(s(y0), y1))) MARK(s(cons(incr(s(y0)), y1))) -> c20(MARK(cons(incr(s(y0)), y1))) MARK(s(cons(head(s(y0)), y1))) -> c20(MARK(cons(head(s(y0)), y1))) MARK(s(cons(tail(cons(y0, y1)), y2))) -> c20(MARK(cons(tail(cons(y0, y1)), y2))) MARK(s(cons(tail(s(y0)), y1))) -> c20(MARK(cons(tail(s(y0)), y1))) MARK(s(cons(incr(incr(incr(y0))), y1))) -> 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c20(MARK(cons(incr(head(s(y0))), y1))) MARK(s(cons(incr(head(y0)), y1))) -> c20(MARK(cons(incr(head(y0)), y1))) MARK(s(cons(incr(tail(incr(y0))), y1))) -> c20(MARK(cons(incr(tail(incr(y0))), y1))) MARK(s(cons(incr(tail(pairs)), y0))) -> c20(MARK(cons(incr(tail(pairs)), y0))) MARK(s(cons(incr(tail(odds)), y0))) -> c20(MARK(cons(incr(tail(odds)), y0))) MARK(s(cons(incr(tail(head(y0))), y1))) -> c20(MARK(cons(incr(tail(head(y0))), y1))) MARK(s(cons(incr(tail(tail(y0))), y1))) -> c20(MARK(cons(incr(tail(tail(y0))), y1))) MARK(s(cons(incr(tail(cons(y0, y1))), y2))) -> c20(MARK(cons(incr(tail(cons(y0, y1))), y2))) MARK(s(cons(incr(tail(s(y0))), y1))) -> c20(MARK(cons(incr(tail(s(y0))), y1))) MARK(s(cons(incr(tail(y0)), y1))) -> c20(MARK(cons(incr(tail(y0)), y1))) MARK(s(cons(head(incr(incr(y0))), y1))) -> c20(MARK(cons(head(incr(incr(y0))), y1))) MARK(s(cons(head(incr(odds)), y0))) -> c20(MARK(cons(head(incr(odds)), y0))) MARK(s(cons(head(incr(head(y0))), y1))) -> c20(MARK(cons(head(incr(head(y0))), y1))) MARK(s(cons(head(incr(tail(y0))), y1))) -> c20(MARK(cons(head(incr(tail(y0))), y1))) MARK(s(cons(head(incr(cons(y0, y1))), y2))) -> c20(MARK(cons(head(incr(cons(y0, y1))), y2))) MARK(s(cons(head(incr(s(y0))), y1))) -> c20(MARK(cons(head(incr(s(y0))), y1))) MARK(s(cons(head(incr(y0)), y1))) -> c20(MARK(cons(head(incr(y0)), y1))) MARK(s(cons(head(head(incr(y0))), y1))) -> c20(MARK(cons(head(head(incr(y0))), y1))) MARK(s(cons(head(head(odds)), y0))) -> c20(MARK(cons(head(head(odds)), y0))) MARK(s(cons(head(head(head(y0))), y1))) -> c20(MARK(cons(head(head(head(y0))), y1))) MARK(s(cons(head(head(tail(y0))), y1))) -> c20(MARK(cons(head(head(tail(y0))), y1))) MARK(s(cons(head(head(cons(y0, y1))), y2))) -> c20(MARK(cons(head(head(cons(y0, y1))), y2))) MARK(s(cons(head(head(s(y0))), y1))) -> c20(MARK(cons(head(head(s(y0))), y1))) MARK(s(cons(head(head(y0)), y1))) -> c20(MARK(cons(head(head(y0)), y1))) MARK(s(cons(head(tail(incr(y0))), y1))) -> c20(MARK(cons(head(tail(incr(y0))), y1))) MARK(s(cons(head(tail(pairs)), y0))) -> c20(MARK(cons(head(tail(pairs)), y0))) MARK(s(cons(head(tail(odds)), y0))) -> c20(MARK(cons(head(tail(odds)), y0))) MARK(s(cons(head(tail(head(y0))), y1))) -> c20(MARK(cons(head(tail(head(y0))), y1))) MARK(s(cons(head(tail(tail(y0))), y1))) -> c20(MARK(cons(head(tail(tail(y0))), y1))) MARK(s(cons(head(tail(cons(y0, y1))), y2))) -> c20(MARK(cons(head(tail(cons(y0, y1))), y2))) MARK(s(cons(head(tail(s(y0))), y1))) -> c20(MARK(cons(head(tail(s(y0))), y1))) MARK(s(cons(head(tail(y0)), y1))) -> c20(MARK(cons(head(tail(y0)), y1))) MARK(s(cons(tail(incr(incr(y0))), y1))) -> c20(MARK(cons(tail(incr(incr(y0))), y1))) MARK(s(cons(tail(incr(odds)), y0))) -> c20(MARK(cons(tail(incr(odds)), y0))) MARK(s(cons(tail(incr(head(y0))), y1))) -> c20(MARK(cons(tail(incr(head(y0))), y1))) MARK(s(cons(tail(incr(tail(y0))), y1))) -> c20(MARK(cons(tail(incr(tail(y0))), y1))) MARK(s(cons(tail(incr(cons(y0, y1))), y2))) -> c20(MARK(cons(tail(incr(cons(y0, y1))), y2))) MARK(s(cons(tail(incr(s(y0))), y1))) -> c20(MARK(cons(tail(incr(s(y0))), y1))) MARK(s(cons(tail(incr(y0)), y1))) -> c20(MARK(cons(tail(incr(y0)), y1))) MARK(s(cons(tail(head(incr(y0))), y1))) -> c20(MARK(cons(tail(head(incr(y0))), y1))) MARK(s(cons(tail(head(odds)), y0))) -> c20(MARK(cons(tail(head(odds)), y0))) MARK(s(cons(tail(head(head(y0))), y1))) -> c20(MARK(cons(tail(head(head(y0))), y1))) MARK(s(cons(tail(head(tail(y0))), y1))) -> c20(MARK(cons(tail(head(tail(y0))), y1))) MARK(s(cons(tail(head(cons(y0, y1))), y2))) -> c20(MARK(cons(tail(head(cons(y0, y1))), y2))) MARK(s(cons(tail(head(s(y0))), y1))) -> c20(MARK(cons(tail(head(s(y0))), y1))) MARK(s(cons(tail(head(y0)), y1))) -> c20(MARK(cons(tail(head(y0)), y1))) MARK(s(cons(tail(tail(incr(y0))), y1))) -> c20(MARK(cons(tail(tail(incr(y0))), y1))) MARK(s(cons(tail(tail(pairs)), y0))) -> c20(MARK(cons(tail(tail(pairs)), y0))) MARK(s(cons(tail(tail(odds)), y0))) -> c20(MARK(cons(tail(tail(odds)), y0))) MARK(s(cons(tail(tail(head(y0))), y1))) -> c20(MARK(cons(tail(tail(head(y0))), y1))) MARK(s(cons(tail(tail(tail(y0))), y1))) -> c20(MARK(cons(tail(tail(tail(y0))), y1))) MARK(s(cons(tail(tail(cons(y0, y1))), y2))) -> c20(MARK(cons(tail(tail(cons(y0, y1))), y2))) MARK(s(cons(tail(tail(s(y0))), y1))) -> c20(MARK(cons(tail(tail(s(y0))), y1))) MARK(s(cons(tail(tail(y0)), y1))) -> c20(MARK(cons(tail(tail(y0)), y1))) MARK(s(cons(tail(pairs), y0))) -> c20(MARK(cons(tail(pairs), y0))) MARK(s(cons(incr(incr(nats)), y0))) -> c20(MARK(cons(incr(incr(nats)), y0))) MARK(s(cons(incr(incr(pairs)), y0))) -> c20(MARK(cons(incr(incr(pairs)), y0))) MARK(s(cons(incr(odds), y0))) -> c20(MARK(cons(incr(odds), y0))) MARK(s(cons(head(incr(nats)), y0))) -> c20(MARK(cons(head(incr(nats)), y0))) MARK(s(cons(head(incr(pairs)), y0))) -> c20(MARK(cons(head(incr(pairs)), y0))) MARK(s(cons(head(odds), y0))) -> c20(MARK(cons(head(odds), y0))) MARK(s(cons(tail(incr(nats)), y0))) -> c20(MARK(cons(tail(incr(nats)), y0))) MARK(s(cons(tail(incr(pairs)), y0))) -> c20(MARK(cons(tail(incr(pairs)), y0))) MARK(s(cons(tail(odds), y0))) -> c20(MARK(cons(tail(odds), y0))) MARK(s(cons(incr(cons(nats, y0)), y1))) -> c20(MARK(cons(incr(cons(nats, y0)), y1))) MARK(s(cons(incr(cons(incr(y0), y1)), y2))) -> c20(MARK(cons(incr(cons(incr(y0), y1)), y2))) MARK(s(cons(incr(cons(pairs, y0)), y1))) -> c20(MARK(cons(incr(cons(pairs, y0)), y1))) MARK(s(cons(incr(cons(odds, y0)), y1))) -> c20(MARK(cons(incr(cons(odds, y0)), y1))) MARK(s(cons(incr(cons(head(y0), y1)), y2))) -> c20(MARK(cons(incr(cons(head(y0), y1)), y2))) MARK(s(cons(incr(cons(tail(y0), y1)), y2))) -> c20(MARK(cons(incr(cons(tail(y0), y1)), y2))) MARK(s(cons(incr(cons(cons(y0, y1), y2)), y3))) -> c20(MARK(cons(incr(cons(cons(y0, y1), y2)), y3))) MARK(s(cons(incr(cons(s(y0), y1)), y2))) -> c20(MARK(cons(incr(cons(s(y0), y1)), y2))) MARK(s(cons(incr(cons(y0, y1)), y2))) -> c20(MARK(cons(incr(cons(y0, y1)), y2))) MARK(s(cons(incr(cons(0, y0)), y1))) -> c20(MARK(cons(incr(cons(0, y0)), y1))) MARK(s(cons(incr(head(nats)), y0))) -> c20(MARK(cons(incr(head(nats)), y0))) MARK(s(cons(incr(head(pairs)), y0))) -> c20(MARK(cons(incr(head(pairs)), y0))) MARK(s(cons(head(head(nats)), y0))) -> c20(MARK(cons(head(head(nats)), y0))) MARK(s(cons(head(head(pairs)), y0))) -> c20(MARK(cons(head(head(pairs)), y0))) MARK(s(cons(tail(head(nats)), y0))) -> c20(MARK(cons(tail(head(nats)), y0))) MARK(s(cons(tail(head(pairs)), y0))) -> c20(MARK(cons(tail(head(pairs)), y0))) MARK(s(cons(head(cons(nats, y0)), y1))) -> c20(MARK(cons(head(cons(nats, y0)), y1))) MARK(s(cons(head(cons(incr(y0), y1)), y2))) -> c20(MARK(cons(head(cons(incr(y0), y1)), y2))) MARK(s(cons(head(cons(pairs, y0)), y1))) -> c20(MARK(cons(head(cons(pairs, y0)), y1))) MARK(s(cons(head(cons(odds, y0)), y1))) -> c20(MARK(cons(head(cons(odds, y0)), y1))) MARK(s(cons(head(cons(head(y0), y1)), y2))) -> c20(MARK(cons(head(cons(head(y0), y1)), y2))) MARK(s(cons(head(cons(tail(y0), y1)), y2))) -> c20(MARK(cons(head(cons(tail(y0), y1)), y2))) MARK(s(cons(head(cons(cons(y0, y1), y2)), y3))) -> c20(MARK(cons(head(cons(cons(y0, y1), y2)), y3))) MARK(s(cons(head(cons(s(y0), y1)), y2))) -> c20(MARK(cons(head(cons(s(y0), y1)), y2))) MARK(s(cons(head(cons(y0, y1)), y2))) -> c20(MARK(cons(head(cons(y0, y1)), y2))) MARK(s(cons(head(cons(0, y0)), y1))) -> c20(MARK(cons(head(cons(0, y0)), y1))) MARK(s(cons(incr(tail(nats)), y0))) -> c20(MARK(cons(incr(tail(nats)), y0))) MARK(s(cons(head(tail(nats)), y0))) -> c20(MARK(cons(head(tail(nats)), y0))) MARK(s(cons(tail(tail(nats)), y0))) -> c20(MARK(cons(tail(tail(nats)), y0))) MARK(s(incr(cons(nats, y0)))) -> c20(MARK(incr(cons(nats, y0)))) MARK(s(incr(cons(pairs, y0)))) -> c20(MARK(incr(cons(pairs, y0)))) MARK(s(incr(cons(odds, y0)))) -> c20(MARK(incr(cons(odds, y0)))) MARK(s(head(cons(nats, y0)))) -> c20(MARK(head(cons(nats, y0)))) MARK(s(head(cons(pairs, y0)))) -> c20(MARK(head(cons(pairs, y0)))) MARK(s(head(cons(odds, y0)))) -> c20(MARK(head(cons(odds, y0)))) MARK(incr(s(s(y0)))) -> c13(MARK(s(s(y0)))) MARK(incr(s(incr(s(y0))))) -> c13(MARK(s(incr(s(y0))))) MARK(incr(s(head(s(y0))))) -> c13(MARK(s(head(s(y0))))) MARK(incr(s(tail(cons(y0, y1))))) -> c13(MARK(s(tail(cons(y0, y1))))) MARK(incr(s(tail(s(y0))))) -> c13(MARK(s(tail(s(y0))))) MARK(incr(s(incr(incr(incr(y0)))))) -> c13(MARK(s(incr(incr(incr(y0)))))) MARK(incr(s(incr(incr(odds))))) -> c13(MARK(s(incr(incr(odds))))) MARK(incr(s(incr(incr(head(y0)))))) -> c13(MARK(s(incr(incr(head(y0)))))) MARK(incr(s(incr(incr(tail(y0)))))) -> c13(MARK(s(incr(incr(tail(y0)))))) MARK(incr(s(incr(incr(cons(y0, y1)))))) -> c13(MARK(s(incr(incr(cons(y0, y1)))))) MARK(incr(s(incr(incr(s(y0)))))) -> c13(MARK(s(incr(incr(s(y0)))))) MARK(incr(s(incr(incr(y0))))) -> c13(MARK(s(incr(incr(y0))))) MARK(incr(s(incr(head(incr(y0)))))) -> c13(MARK(s(incr(head(incr(y0)))))) MARK(incr(s(incr(head(odds))))) -> c13(MARK(s(incr(head(odds))))) MARK(incr(s(incr(head(head(y0)))))) -> c13(MARK(s(incr(head(head(y0)))))) MARK(incr(s(incr(head(tail(y0)))))) -> c13(MARK(s(incr(head(tail(y0)))))) MARK(incr(s(incr(head(cons(y0, y1)))))) -> c13(MARK(s(incr(head(cons(y0, y1)))))) MARK(incr(s(incr(head(s(y0)))))) -> c13(MARK(s(incr(head(s(y0)))))) MARK(incr(s(incr(head(y0))))) -> c13(MARK(s(incr(head(y0))))) MARK(incr(s(incr(tail(incr(y0)))))) -> c13(MARK(s(incr(tail(incr(y0)))))) MARK(incr(s(incr(tail(pairs))))) -> c13(MARK(s(incr(tail(pairs))))) MARK(incr(s(incr(tail(odds))))) -> c13(MARK(s(incr(tail(odds))))) MARK(incr(s(incr(tail(head(y0)))))) -> c13(MARK(s(incr(tail(head(y0)))))) MARK(incr(s(incr(tail(tail(y0)))))) -> c13(MARK(s(incr(tail(tail(y0)))))) MARK(incr(s(incr(tail(cons(y0, y1)))))) -> c13(MARK(s(incr(tail(cons(y0, y1)))))) MARK(incr(s(incr(tail(s(y0)))))) -> c13(MARK(s(incr(tail(s(y0)))))) MARK(incr(s(incr(tail(y0))))) -> c13(MARK(s(incr(tail(y0))))) MARK(incr(s(head(incr(incr(y0)))))) -> c13(MARK(s(head(incr(incr(y0)))))) MARK(incr(s(head(incr(odds))))) -> c13(MARK(s(head(incr(odds))))) MARK(incr(s(head(incr(head(y0)))))) -> c13(MARK(s(head(incr(head(y0)))))) MARK(incr(s(head(incr(tail(y0)))))) -> c13(MARK(s(head(incr(tail(y0)))))) MARK(incr(s(head(incr(cons(y0, y1)))))) -> c13(MARK(s(head(incr(cons(y0, y1)))))) MARK(incr(s(head(incr(s(y0)))))) -> c13(MARK(s(head(incr(s(y0)))))) MARK(incr(s(head(incr(y0))))) -> c13(MARK(s(head(incr(y0))))) MARK(incr(s(head(head(incr(y0)))))) -> c13(MARK(s(head(head(incr(y0)))))) MARK(incr(s(head(head(odds))))) -> c13(MARK(s(head(head(odds))))) MARK(incr(s(head(head(head(y0)))))) -> c13(MARK(s(head(head(head(y0)))))) MARK(incr(s(head(head(tail(y0)))))) -> c13(MARK(s(head(head(tail(y0)))))) MARK(incr(s(head(head(cons(y0, y1)))))) -> c13(MARK(s(head(head(cons(y0, y1)))))) MARK(incr(s(head(head(s(y0)))))) -> c13(MARK(s(head(head(s(y0)))))) MARK(incr(s(head(head(y0))))) -> c13(MARK(s(head(head(y0))))) MARK(incr(s(head(tail(incr(y0)))))) -> c13(MARK(s(head(tail(incr(y0)))))) MARK(incr(s(head(tail(pairs))))) -> c13(MARK(s(head(tail(pairs))))) MARK(incr(s(head(tail(odds))))) -> c13(MARK(s(head(tail(odds))))) MARK(incr(s(head(tail(head(y0)))))) -> c13(MARK(s(head(tail(head(y0)))))) MARK(incr(s(head(tail(tail(y0)))))) -> c13(MARK(s(head(tail(tail(y0)))))) MARK(incr(s(head(tail(cons(y0, y1)))))) -> c13(MARK(s(head(tail(cons(y0, y1)))))) MARK(incr(s(head(tail(s(y0)))))) -> c13(MARK(s(head(tail(s(y0)))))) MARK(incr(s(head(tail(y0))))) -> c13(MARK(s(head(tail(y0))))) MARK(incr(s(tail(incr(incr(y0)))))) -> c13(MARK(s(tail(incr(incr(y0)))))) MARK(incr(s(tail(incr(odds))))) -> c13(MARK(s(tail(incr(odds))))) MARK(incr(s(tail(incr(head(y0)))))) -> c13(MARK(s(tail(incr(head(y0)))))) MARK(incr(s(tail(incr(tail(y0)))))) -> c13(MARK(s(tail(incr(tail(y0)))))) MARK(incr(s(tail(incr(cons(y0, y1)))))) -> c13(MARK(s(tail(incr(cons(y0, y1)))))) MARK(incr(s(tail(incr(s(y0)))))) -> c13(MARK(s(tail(incr(s(y0)))))) MARK(incr(s(tail(incr(y0))))) -> c13(MARK(s(tail(incr(y0))))) MARK(incr(s(tail(head(incr(y0)))))) -> c13(MARK(s(tail(head(incr(y0)))))) MARK(incr(s(tail(head(odds))))) -> c13(MARK(s(tail(head(odds))))) MARK(incr(s(tail(head(head(y0)))))) -> c13(MARK(s(tail(head(head(y0)))))) MARK(incr(s(tail(head(tail(y0)))))) -> c13(MARK(s(tail(head(tail(y0)))))) MARK(incr(s(tail(head(cons(y0, y1)))))) -> c13(MARK(s(tail(head(cons(y0, y1)))))) MARK(incr(s(tail(head(s(y0)))))) -> c13(MARK(s(tail(head(s(y0)))))) MARK(incr(s(tail(head(y0))))) -> c13(MARK(s(tail(head(y0))))) MARK(incr(s(tail(tail(incr(y0)))))) -> c13(MARK(s(tail(tail(incr(y0)))))) MARK(incr(s(tail(tail(pairs))))) -> c13(MARK(s(tail(tail(pairs))))) MARK(incr(s(tail(tail(odds))))) -> c13(MARK(s(tail(tail(odds))))) MARK(incr(s(tail(tail(head(y0)))))) -> c13(MARK(s(tail(tail(head(y0)))))) MARK(incr(s(tail(tail(tail(y0)))))) -> c13(MARK(s(tail(tail(tail(y0)))))) MARK(incr(s(tail(tail(cons(y0, y1)))))) -> c13(MARK(s(tail(tail(cons(y0, y1)))))) MARK(incr(s(tail(tail(s(y0)))))) -> c13(MARK(s(tail(tail(s(y0)))))) MARK(incr(s(tail(tail(y0))))) -> c13(MARK(s(tail(tail(y0))))) MARK(incr(s(tail(pairs)))) -> c13(MARK(s(tail(pairs)))) MARK(incr(s(incr(incr(nats))))) -> c13(MARK(s(incr(incr(nats))))) MARK(incr(s(incr(incr(pairs))))) -> c13(MARK(s(incr(incr(pairs))))) MARK(incr(s(incr(odds)))) -> c13(MARK(s(incr(odds)))) MARK(incr(s(head(incr(nats))))) -> c13(MARK(s(head(incr(nats))))) MARK(incr(s(head(incr(pairs))))) -> c13(MARK(s(head(incr(pairs))))) MARK(incr(s(head(odds)))) -> c13(MARK(s(head(odds)))) MARK(incr(s(tail(incr(nats))))) -> c13(MARK(s(tail(incr(nats))))) MARK(incr(s(tail(incr(pairs))))) -> c13(MARK(s(tail(incr(pairs))))) MARK(incr(s(tail(odds)))) -> c13(MARK(s(tail(odds)))) MARK(incr(s(incr(cons(incr(y0), y1))))) -> c13(MARK(s(incr(cons(incr(y0), y1))))) MARK(incr(s(incr(cons(head(y0), y1))))) -> c13(MARK(s(incr(cons(head(y0), y1))))) MARK(incr(s(incr(cons(tail(y0), y1))))) -> c13(MARK(s(incr(cons(tail(y0), y1))))) MARK(incr(s(incr(cons(cons(y0, y1), y2))))) -> c13(MARK(s(incr(cons(cons(y0, y1), y2))))) MARK(incr(s(incr(cons(s(y0), y1))))) -> c13(MARK(s(incr(cons(s(y0), y1))))) MARK(incr(s(incr(cons(y0, y1))))) -> c13(MARK(s(incr(cons(y0, y1))))) MARK(incr(s(incr(head(nats))))) -> c13(MARK(s(incr(head(nats))))) MARK(incr(s(incr(head(pairs))))) -> c13(MARK(s(incr(head(pairs))))) MARK(incr(s(head(head(nats))))) -> c13(MARK(s(head(head(nats))))) MARK(incr(s(head(head(pairs))))) -> c13(MARK(s(head(head(pairs))))) MARK(incr(s(tail(head(nats))))) -> c13(MARK(s(tail(head(nats))))) MARK(incr(s(tail(head(pairs))))) -> c13(MARK(s(tail(head(pairs))))) MARK(incr(s(head(cons(incr(y0), y1))))) -> c13(MARK(s(head(cons(incr(y0), y1))))) MARK(incr(s(head(cons(head(y0), y1))))) -> c13(MARK(s(head(cons(head(y0), y1))))) MARK(incr(s(head(cons(tail(y0), y1))))) -> c13(MARK(s(head(cons(tail(y0), y1))))) MARK(incr(s(head(cons(cons(y0, y1), y2))))) -> c13(MARK(s(head(cons(cons(y0, y1), y2))))) MARK(incr(s(head(cons(s(y0), y1))))) -> c13(MARK(s(head(cons(s(y0), y1))))) MARK(incr(s(head(cons(y0, y1))))) -> c13(MARK(s(head(cons(y0, y1))))) MARK(incr(s(incr(tail(nats))))) -> c13(MARK(s(incr(tail(nats))))) MARK(incr(s(head(tail(nats))))) -> c13(MARK(s(head(tail(nats))))) MARK(incr(s(tail(tail(nats))))) -> c13(MARK(s(tail(tail(nats))))) MARK(incr(s(cons(cons(y0, y1), y2)))) -> c13(MARK(s(cons(cons(y0, y1), y2)))) MARK(incr(s(cons(s(y0), y1)))) -> c13(MARK(s(cons(s(y0), y1)))) MARK(incr(s(cons(incr(s(y0)), y1)))) -> c13(MARK(s(cons(incr(s(y0)), y1)))) MARK(incr(s(cons(head(s(y0)), y1)))) -> c13(MARK(s(cons(head(s(y0)), y1)))) MARK(incr(s(cons(tail(cons(y0, y1)), y2)))) -> c13(MARK(s(cons(tail(cons(y0, y1)), y2)))) MARK(incr(s(cons(tail(s(y0)), y1)))) -> c13(MARK(s(cons(tail(s(y0)), y1)))) MARK(incr(s(cons(incr(incr(incr(y0))), y1)))) -> c13(MARK(s(cons(incr(incr(incr(y0))), y1)))) MARK(incr(s(cons(incr(incr(odds)), y0)))) -> c13(MARK(s(cons(incr(incr(odds)), y0)))) MARK(incr(s(cons(incr(incr(head(y0))), y1)))) -> c13(MARK(s(cons(incr(incr(head(y0))), y1)))) MARK(incr(s(cons(incr(incr(tail(y0))), y1)))) -> c13(MARK(s(cons(incr(incr(tail(y0))), y1)))) MARK(incr(s(cons(incr(incr(cons(y0, y1))), y2)))) -> c13(MARK(s(cons(incr(incr(cons(y0, y1))), y2)))) MARK(incr(s(cons(incr(incr(s(y0))), y1)))) -> c13(MARK(s(cons(incr(incr(s(y0))), y1)))) MARK(incr(s(cons(incr(incr(y0)), y1)))) -> c13(MARK(s(cons(incr(incr(y0)), y1)))) MARK(incr(s(cons(incr(head(incr(y0))), y1)))) -> c13(MARK(s(cons(incr(head(incr(y0))), y1)))) MARK(incr(s(cons(incr(head(odds)), y0)))) -> c13(MARK(s(cons(incr(head(odds)), y0)))) MARK(incr(s(cons(incr(head(head(y0))), y1)))) -> c13(MARK(s(cons(incr(head(head(y0))), y1)))) MARK(incr(s(cons(incr(head(tail(y0))), y1)))) -> c13(MARK(s(cons(incr(head(tail(y0))), y1)))) MARK(incr(s(cons(incr(head(cons(y0, y1))), y2)))) -> c13(MARK(s(cons(incr(head(cons(y0, y1))), y2)))) MARK(incr(s(cons(incr(head(s(y0))), y1)))) -> c13(MARK(s(cons(incr(head(s(y0))), y1)))) MARK(incr(s(cons(incr(head(y0)), y1)))) -> c13(MARK(s(cons(incr(head(y0)), y1)))) MARK(incr(s(cons(incr(tail(incr(y0))), y1)))) -> c13(MARK(s(cons(incr(tail(incr(y0))), y1)))) MARK(incr(s(cons(incr(tail(pairs)), y0)))) -> c13(MARK(s(cons(incr(tail(pairs)), y0)))) MARK(incr(s(cons(incr(tail(odds)), y0)))) -> c13(MARK(s(cons(incr(tail(odds)), y0)))) MARK(incr(s(cons(incr(tail(head(y0))), y1)))) -> c13(MARK(s(cons(incr(tail(head(y0))), y1)))) MARK(incr(s(cons(incr(tail(tail(y0))), y1)))) -> c13(MARK(s(cons(incr(tail(tail(y0))), y1)))) MARK(incr(s(cons(incr(tail(cons(y0, y1))), y2)))) -> c13(MARK(s(cons(incr(tail(cons(y0, y1))), y2)))) MARK(incr(s(cons(incr(tail(s(y0))), y1)))) -> c13(MARK(s(cons(incr(tail(s(y0))), y1)))) MARK(incr(s(cons(incr(tail(y0)), y1)))) -> c13(MARK(s(cons(incr(tail(y0)), y1)))) MARK(incr(s(cons(head(incr(incr(y0))), y1)))) -> c13(MARK(s(cons(head(incr(incr(y0))), y1)))) MARK(incr(s(cons(head(incr(odds)), y0)))) -> c13(MARK(s(cons(head(incr(odds)), y0)))) MARK(incr(s(cons(head(incr(head(y0))), y1)))) -> c13(MARK(s(cons(head(incr(head(y0))), y1)))) MARK(incr(s(cons(head(incr(tail(y0))), y1)))) -> c13(MARK(s(cons(head(incr(tail(y0))), y1)))) MARK(incr(s(cons(head(incr(cons(y0, y1))), y2)))) -> c13(MARK(s(cons(head(incr(cons(y0, y1))), y2)))) MARK(incr(s(cons(head(incr(s(y0))), y1)))) -> c13(MARK(s(cons(head(incr(s(y0))), y1)))) MARK(incr(s(cons(head(incr(y0)), y1)))) -> c13(MARK(s(cons(head(incr(y0)), y1)))) MARK(incr(s(cons(head(head(incr(y0))), y1)))) -> c13(MARK(s(cons(head(head(incr(y0))), y1)))) MARK(incr(s(cons(head(head(odds)), y0)))) -> c13(MARK(s(cons(head(head(odds)), y0)))) MARK(incr(s(cons(head(head(head(y0))), y1)))) -> c13(MARK(s(cons(head(head(head(y0))), y1)))) MARK(incr(s(cons(head(head(tail(y0))), y1)))) -> c13(MARK(s(cons(head(head(tail(y0))), y1)))) MARK(incr(s(cons(head(head(cons(y0, y1))), y2)))) -> c13(MARK(s(cons(head(head(cons(y0, y1))), y2)))) MARK(incr(s(cons(head(head(s(y0))), y1)))) -> c13(MARK(s(cons(head(head(s(y0))), y1)))) MARK(incr(s(cons(head(head(y0)), y1)))) -> c13(MARK(s(cons(head(head(y0)), y1)))) MARK(incr(s(cons(head(tail(incr(y0))), y1)))) -> c13(MARK(s(cons(head(tail(incr(y0))), y1)))) MARK(incr(s(cons(head(tail(pairs)), y0)))) -> c13(MARK(s(cons(head(tail(pairs)), y0)))) MARK(incr(s(cons(head(tail(odds)), y0)))) -> c13(MARK(s(cons(head(tail(odds)), y0)))) MARK(incr(s(cons(head(tail(head(y0))), y1)))) -> c13(MARK(s(cons(head(tail(head(y0))), y1)))) MARK(incr(s(cons(head(tail(tail(y0))), y1)))) -> c13(MARK(s(cons(head(tail(tail(y0))), y1)))) MARK(incr(s(cons(head(tail(cons(y0, y1))), y2)))) -> c13(MARK(s(cons(head(tail(cons(y0, y1))), y2)))) MARK(incr(s(cons(head(tail(s(y0))), y1)))) -> c13(MARK(s(cons(head(tail(s(y0))), y1)))) MARK(incr(s(cons(head(tail(y0)), y1)))) -> c13(MARK(s(cons(head(tail(y0)), y1)))) MARK(incr(s(cons(tail(incr(incr(y0))), y1)))) -> c13(MARK(s(cons(tail(incr(incr(y0))), y1)))) MARK(incr(s(cons(tail(incr(odds)), y0)))) -> c13(MARK(s(cons(tail(incr(odds)), y0)))) MARK(incr(s(cons(tail(incr(head(y0))), y1)))) -> c13(MARK(s(cons(tail(incr(head(y0))), y1)))) MARK(incr(s(cons(tail(incr(tail(y0))), y1)))) -> c13(MARK(s(cons(tail(incr(tail(y0))), y1)))) MARK(incr(s(cons(tail(incr(cons(y0, y1))), y2)))) -> c13(MARK(s(cons(tail(incr(cons(y0, y1))), y2)))) MARK(incr(s(cons(tail(incr(s(y0))), y1)))) -> c13(MARK(s(cons(tail(incr(s(y0))), y1)))) MARK(incr(s(cons(tail(incr(y0)), y1)))) -> c13(MARK(s(cons(tail(incr(y0)), y1)))) MARK(incr(s(cons(tail(head(incr(y0))), y1)))) -> c13(MARK(s(cons(tail(head(incr(y0))), y1)))) MARK(incr(s(cons(tail(head(odds)), y0)))) -> c13(MARK(s(cons(tail(head(odds)), y0)))) MARK(incr(s(cons(tail(head(head(y0))), y1)))) -> c13(MARK(s(cons(tail(head(head(y0))), y1)))) MARK(incr(s(cons(tail(head(tail(y0))), y1)))) -> c13(MARK(s(cons(tail(head(tail(y0))), y1)))) MARK(incr(s(cons(tail(head(cons(y0, y1))), y2)))) -> c13(MARK(s(cons(tail(head(cons(y0, y1))), y2)))) MARK(incr(s(cons(tail(head(s(y0))), y1)))) -> c13(MARK(s(cons(tail(head(s(y0))), y1)))) MARK(incr(s(cons(tail(head(y0)), y1)))) -> c13(MARK(s(cons(tail(head(y0)), y1)))) MARK(incr(s(cons(tail(tail(incr(y0))), y1)))) -> c13(MARK(s(cons(tail(tail(incr(y0))), y1)))) MARK(incr(s(cons(tail(tail(pairs)), y0)))) -> c13(MARK(s(cons(tail(tail(pairs)), y0)))) MARK(incr(s(cons(tail(tail(odds)), y0)))) -> c13(MARK(s(cons(tail(tail(odds)), y0)))) MARK(incr(s(cons(tail(tail(head(y0))), y1)))) -> c13(MARK(s(cons(tail(tail(head(y0))), y1)))) MARK(incr(s(cons(tail(tail(tail(y0))), y1)))) -> c13(MARK(s(cons(tail(tail(tail(y0))), y1)))) MARK(incr(s(cons(tail(tail(cons(y0, y1))), y2)))) -> c13(MARK(s(cons(tail(tail(cons(y0, y1))), y2)))) MARK(incr(s(cons(tail(tail(s(y0))), y1)))) -> c13(MARK(s(cons(tail(tail(s(y0))), y1)))) MARK(incr(s(cons(tail(tail(y0)), y1)))) -> c13(MARK(s(cons(tail(tail(y0)), y1)))) MARK(incr(s(cons(tail(pairs), y0)))) -> c13(MARK(s(cons(tail(pairs), y0)))) MARK(incr(s(cons(incr(incr(nats)), y0)))) -> c13(MARK(s(cons(incr(incr(nats)), y0)))) MARK(incr(s(cons(incr(incr(pairs)), y0)))) -> c13(MARK(s(cons(incr(incr(pairs)), y0)))) MARK(incr(s(cons(incr(odds), y0)))) -> c13(MARK(s(cons(incr(odds), y0)))) MARK(incr(s(cons(head(incr(nats)), y0)))) -> c13(MARK(s(cons(head(incr(nats)), y0)))) MARK(incr(s(cons(head(incr(pairs)), y0)))) -> c13(MARK(s(cons(head(incr(pairs)), y0)))) MARK(incr(s(cons(head(odds), y0)))) -> c13(MARK(s(cons(head(odds), y0)))) MARK(incr(s(cons(tail(incr(nats)), y0)))) -> c13(MARK(s(cons(tail(incr(nats)), y0)))) MARK(incr(s(cons(tail(incr(pairs)), y0)))) -> c13(MARK(s(cons(tail(incr(pairs)), y0)))) MARK(incr(s(cons(tail(odds), y0)))) -> c13(MARK(s(cons(tail(odds), y0)))) MARK(incr(s(cons(incr(cons(nats, y0)), y1)))) -> c13(MARK(s(cons(incr(cons(nats, y0)), y1)))) MARK(incr(s(cons(incr(cons(incr(y0), y1)), y2)))) -> c13(MARK(s(cons(incr(cons(incr(y0), y1)), y2)))) MARK(incr(s(cons(incr(cons(pairs, y0)), y1)))) -> c13(MARK(s(cons(incr(cons(pairs, y0)), y1)))) MARK(incr(s(cons(incr(cons(odds, y0)), y1)))) -> c13(MARK(s(cons(incr(cons(odds, y0)), y1)))) MARK(incr(s(cons(incr(cons(head(y0), y1)), y2)))) -> c13(MARK(s(cons(incr(cons(head(y0), y1)), y2)))) MARK(incr(s(cons(incr(cons(tail(y0), y1)), y2)))) -> c13(MARK(s(cons(incr(cons(tail(y0), y1)), y2)))) MARK(incr(s(cons(incr(cons(cons(y0, y1), y2)), y3)))) -> c13(MARK(s(cons(incr(cons(cons(y0, y1), y2)), y3)))) MARK(incr(s(cons(incr(cons(s(y0), y1)), y2)))) -> c13(MARK(s(cons(incr(cons(s(y0), y1)), y2)))) MARK(incr(s(cons(incr(cons(y0, y1)), y2)))) -> c13(MARK(s(cons(incr(cons(y0, y1)), y2)))) MARK(incr(s(cons(incr(cons(0, y0)), y1)))) -> c13(MARK(s(cons(incr(cons(0, y0)), y1)))) MARK(incr(s(cons(incr(head(nats)), y0)))) -> c13(MARK(s(cons(incr(head(nats)), y0)))) MARK(incr(s(cons(incr(head(pairs)), y0)))) -> c13(MARK(s(cons(incr(head(pairs)), y0)))) MARK(incr(s(cons(head(head(nats)), y0)))) -> c13(MARK(s(cons(head(head(nats)), y0)))) MARK(incr(s(cons(head(head(pairs)), y0)))) -> c13(MARK(s(cons(head(head(pairs)), y0)))) MARK(incr(s(cons(tail(head(nats)), y0)))) -> c13(MARK(s(cons(tail(head(nats)), y0)))) MARK(incr(s(cons(tail(head(pairs)), y0)))) -> c13(MARK(s(cons(tail(head(pairs)), y0)))) MARK(incr(s(cons(head(cons(nats, y0)), y1)))) -> c13(MARK(s(cons(head(cons(nats, y0)), y1)))) MARK(incr(s(cons(head(cons(incr(y0), y1)), y2)))) -> c13(MARK(s(cons(head(cons(incr(y0), y1)), y2)))) MARK(incr(s(cons(head(cons(pairs, y0)), y1)))) -> c13(MARK(s(cons(head(cons(pairs, y0)), y1)))) MARK(incr(s(cons(head(cons(odds, y0)), y1)))) -> c13(MARK(s(cons(head(cons(odds, y0)), y1)))) MARK(incr(s(cons(head(cons(head(y0), y1)), y2)))) -> c13(MARK(s(cons(head(cons(head(y0), y1)), y2)))) MARK(incr(s(cons(head(cons(tail(y0), y1)), y2)))) -> c13(MARK(s(cons(head(cons(tail(y0), y1)), y2)))) MARK(incr(s(cons(head(cons(cons(y0, y1), y2)), y3)))) -> c13(MARK(s(cons(head(cons(cons(y0, y1), y2)), y3)))) MARK(incr(s(cons(head(cons(s(y0), y1)), y2)))) -> c13(MARK(s(cons(head(cons(s(y0), y1)), y2)))) MARK(incr(s(cons(head(cons(y0, y1)), y2)))) -> c13(MARK(s(cons(head(cons(y0, y1)), y2)))) MARK(incr(s(cons(head(cons(0, y0)), y1)))) -> c13(MARK(s(cons(head(cons(0, y0)), y1)))) MARK(incr(s(cons(incr(tail(nats)), y0)))) -> c13(MARK(s(cons(incr(tail(nats)), y0)))) MARK(incr(s(cons(head(tail(nats)), y0)))) -> c13(MARK(s(cons(head(tail(nats)), y0)))) MARK(incr(s(cons(tail(tail(nats)), y0)))) -> c13(MARK(s(cons(tail(tail(nats)), y0)))) MARK(incr(s(incr(cons(nats, y0))))) -> c13(MARK(s(incr(cons(nats, y0))))) MARK(incr(s(incr(cons(pairs, y0))))) -> c13(MARK(s(incr(cons(pairs, y0))))) MARK(incr(s(incr(cons(odds, y0))))) -> c13(MARK(s(incr(cons(odds, y0))))) MARK(incr(s(head(cons(nats, y0))))) -> c13(MARK(s(head(cons(nats, y0))))) MARK(incr(s(head(cons(pairs, y0))))) -> c13(MARK(s(head(cons(pairs, y0))))) MARK(incr(s(head(cons(odds, y0))))) -> c13(MARK(s(head(cons(odds, y0))))) MARK(head(s(s(y0)))) -> c16(MARK(s(s(y0)))) MARK(head(s(incr(s(y0))))) -> c16(MARK(s(incr(s(y0))))) MARK(head(s(head(s(y0))))) -> c16(MARK(s(head(s(y0))))) MARK(head(s(tail(cons(y0, y1))))) -> c16(MARK(s(tail(cons(y0, y1))))) MARK(head(s(tail(s(y0))))) -> c16(MARK(s(tail(s(y0))))) MARK(head(s(incr(incr(incr(y0)))))) -> c16(MARK(s(incr(incr(incr(y0)))))) MARK(head(s(incr(incr(odds))))) -> c16(MARK(s(incr(incr(odds))))) MARK(head(s(incr(incr(head(y0)))))) -> c16(MARK(s(incr(incr(head(y0)))))) MARK(head(s(incr(incr(tail(y0)))))) -> c16(MARK(s(incr(incr(tail(y0)))))) MARK(head(s(incr(incr(cons(y0, y1)))))) -> c16(MARK(s(incr(incr(cons(y0, y1)))))) MARK(head(s(incr(incr(s(y0)))))) -> c16(MARK(s(incr(incr(s(y0)))))) MARK(head(s(incr(incr(y0))))) -> c16(MARK(s(incr(incr(y0))))) MARK(head(s(incr(head(incr(y0)))))) -> c16(MARK(s(incr(head(incr(y0)))))) MARK(head(s(incr(head(odds))))) -> c16(MARK(s(incr(head(odds))))) MARK(head(s(incr(head(head(y0)))))) -> c16(MARK(s(incr(head(head(y0)))))) MARK(head(s(incr(head(tail(y0)))))) -> c16(MARK(s(incr(head(tail(y0)))))) MARK(head(s(incr(head(cons(y0, y1)))))) -> c16(MARK(s(incr(head(cons(y0, y1)))))) MARK(head(s(incr(head(s(y0)))))) -> c16(MARK(s(incr(head(s(y0)))))) MARK(head(s(incr(head(y0))))) -> c16(MARK(s(incr(head(y0))))) MARK(head(s(incr(tail(incr(y0)))))) -> c16(MARK(s(incr(tail(incr(y0)))))) MARK(head(s(incr(tail(pairs))))) -> c16(MARK(s(incr(tail(pairs))))) MARK(head(s(incr(tail(odds))))) -> c16(MARK(s(incr(tail(odds))))) MARK(head(s(incr(tail(head(y0)))))) -> c16(MARK(s(incr(tail(head(y0)))))) MARK(head(s(incr(tail(tail(y0)))))) -> c16(MARK(s(incr(tail(tail(y0)))))) MARK(head(s(incr(tail(cons(y0, y1)))))) -> c16(MARK(s(incr(tail(cons(y0, y1)))))) MARK(head(s(incr(tail(s(y0)))))) -> c16(MARK(s(incr(tail(s(y0)))))) MARK(head(s(incr(tail(y0))))) -> c16(MARK(s(incr(tail(y0))))) MARK(head(s(head(incr(incr(y0)))))) -> c16(MARK(s(head(incr(incr(y0)))))) MARK(head(s(head(incr(odds))))) -> c16(MARK(s(head(incr(odds))))) MARK(head(s(head(incr(head(y0)))))) -> c16(MARK(s(head(incr(head(y0)))))) MARK(head(s(head(incr(tail(y0)))))) -> c16(MARK(s(head(incr(tail(y0)))))) MARK(head(s(head(incr(cons(y0, y1)))))) -> c16(MARK(s(head(incr(cons(y0, y1)))))) MARK(head(s(head(incr(s(y0)))))) -> c16(MARK(s(head(incr(s(y0)))))) MARK(head(s(head(incr(y0))))) -> c16(MARK(s(head(incr(y0))))) MARK(head(s(head(head(incr(y0)))))) -> c16(MARK(s(head(head(incr(y0)))))) MARK(head(s(head(head(odds))))) -> c16(MARK(s(head(head(odds))))) MARK(head(s(head(head(head(y0)))))) -> c16(MARK(s(head(head(head(y0)))))) MARK(head(s(head(head(tail(y0)))))) -> c16(MARK(s(head(head(tail(y0)))))) MARK(head(s(head(head(cons(y0, y1)))))) -> c16(MARK(s(head(head(cons(y0, y1)))))) MARK(head(s(head(head(s(y0)))))) -> c16(MARK(s(head(head(s(y0)))))) MARK(head(s(head(head(y0))))) -> c16(MARK(s(head(head(y0))))) MARK(head(s(head(tail(incr(y0)))))) -> c16(MARK(s(head(tail(incr(y0)))))) MARK(head(s(head(tail(pairs))))) -> c16(MARK(s(head(tail(pairs))))) MARK(head(s(head(tail(odds))))) -> c16(MARK(s(head(tail(odds))))) MARK(head(s(head(tail(head(y0)))))) -> c16(MARK(s(head(tail(head(y0)))))) MARK(head(s(head(tail(tail(y0)))))) -> c16(MARK(s(head(tail(tail(y0)))))) MARK(head(s(head(tail(cons(y0, y1)))))) -> c16(MARK(s(head(tail(cons(y0, y1)))))) MARK(head(s(head(tail(s(y0)))))) -> c16(MARK(s(head(tail(s(y0)))))) MARK(head(s(head(tail(y0))))) -> c16(MARK(s(head(tail(y0))))) MARK(head(s(tail(incr(incr(y0)))))) -> c16(MARK(s(tail(incr(incr(y0)))))) MARK(head(s(tail(incr(odds))))) -> c16(MARK(s(tail(incr(odds))))) MARK(head(s(tail(incr(head(y0)))))) -> c16(MARK(s(tail(incr(head(y0)))))) MARK(head(s(tail(incr(tail(y0)))))) -> c16(MARK(s(tail(incr(tail(y0)))))) MARK(head(s(tail(incr(cons(y0, y1)))))) -> c16(MARK(s(tail(incr(cons(y0, y1)))))) MARK(head(s(tail(incr(s(y0)))))) -> c16(MARK(s(tail(incr(s(y0)))))) MARK(head(s(tail(incr(y0))))) -> c16(MARK(s(tail(incr(y0))))) MARK(head(s(tail(head(incr(y0)))))) -> c16(MARK(s(tail(head(incr(y0)))))) MARK(head(s(tail(head(odds))))) -> c16(MARK(s(tail(head(odds))))) MARK(head(s(tail(head(head(y0)))))) -> c16(MARK(s(tail(head(head(y0)))))) MARK(head(s(tail(head(tail(y0)))))) -> c16(MARK(s(tail(head(tail(y0)))))) MARK(head(s(tail(head(cons(y0, y1)))))) -> c16(MARK(s(tail(head(cons(y0, y1)))))) MARK(head(s(tail(head(s(y0)))))) -> c16(MARK(s(tail(head(s(y0)))))) MARK(head(s(tail(head(y0))))) -> c16(MARK(s(tail(head(y0))))) MARK(head(s(tail(tail(incr(y0)))))) -> c16(MARK(s(tail(tail(incr(y0)))))) MARK(head(s(tail(tail(pairs))))) -> c16(MARK(s(tail(tail(pairs))))) MARK(head(s(tail(tail(odds))))) -> c16(MARK(s(tail(tail(odds))))) MARK(head(s(tail(tail(head(y0)))))) -> c16(MARK(s(tail(tail(head(y0)))))) MARK(head(s(tail(tail(tail(y0)))))) -> c16(MARK(s(tail(tail(tail(y0)))))) MARK(head(s(tail(tail(cons(y0, y1)))))) -> c16(MARK(s(tail(tail(cons(y0, y1)))))) MARK(head(s(tail(tail(s(y0)))))) -> c16(MARK(s(tail(tail(s(y0)))))) MARK(head(s(tail(tail(y0))))) -> c16(MARK(s(tail(tail(y0))))) MARK(head(s(tail(pairs)))) -> c16(MARK(s(tail(pairs)))) MARK(head(s(incr(incr(nats))))) -> c16(MARK(s(incr(incr(nats))))) MARK(head(s(incr(incr(pairs))))) -> c16(MARK(s(incr(incr(pairs))))) MARK(head(s(incr(odds)))) -> c16(MARK(s(incr(odds)))) MARK(head(s(head(incr(nats))))) -> c16(MARK(s(head(incr(nats))))) MARK(head(s(head(incr(pairs))))) -> c16(MARK(s(head(incr(pairs))))) MARK(head(s(head(odds)))) -> c16(MARK(s(head(odds)))) MARK(head(s(tail(incr(nats))))) -> c16(MARK(s(tail(incr(nats))))) MARK(head(s(tail(incr(pairs))))) -> c16(MARK(s(tail(incr(pairs))))) MARK(head(s(tail(odds)))) -> c16(MARK(s(tail(odds)))) MARK(head(s(incr(cons(incr(y0), y1))))) -> c16(MARK(s(incr(cons(incr(y0), y1))))) MARK(head(s(incr(cons(head(y0), y1))))) -> c16(MARK(s(incr(cons(head(y0), y1))))) MARK(head(s(incr(cons(tail(y0), y1))))) -> c16(MARK(s(incr(cons(tail(y0), y1))))) MARK(head(s(incr(cons(cons(y0, y1), y2))))) -> c16(MARK(s(incr(cons(cons(y0, y1), y2))))) MARK(head(s(incr(cons(s(y0), y1))))) -> c16(MARK(s(incr(cons(s(y0), y1))))) MARK(head(s(incr(cons(y0, y1))))) -> c16(MARK(s(incr(cons(y0, y1))))) MARK(head(s(incr(head(nats))))) -> c16(MARK(s(incr(head(nats))))) MARK(head(s(incr(head(pairs))))) -> c16(MARK(s(incr(head(pairs))))) MARK(head(s(head(head(nats))))) -> c16(MARK(s(head(head(nats))))) MARK(head(s(head(head(pairs))))) -> c16(MARK(s(head(head(pairs))))) MARK(head(s(tail(head(nats))))) -> c16(MARK(s(tail(head(nats))))) MARK(head(s(tail(head(pairs))))) -> c16(MARK(s(tail(head(pairs))))) MARK(head(s(head(cons(incr(y0), y1))))) -> c16(MARK(s(head(cons(incr(y0), y1))))) MARK(head(s(head(cons(head(y0), y1))))) -> c16(MARK(s(head(cons(head(y0), y1))))) MARK(head(s(head(cons(tail(y0), y1))))) -> c16(MARK(s(head(cons(tail(y0), y1))))) MARK(head(s(head(cons(cons(y0, y1), y2))))) -> c16(MARK(s(head(cons(cons(y0, y1), y2))))) MARK(head(s(head(cons(s(y0), y1))))) -> c16(MARK(s(head(cons(s(y0), y1))))) MARK(head(s(head(cons(y0, y1))))) -> c16(MARK(s(head(cons(y0, y1))))) MARK(head(s(incr(tail(nats))))) -> c16(MARK(s(incr(tail(nats))))) MARK(head(s(head(tail(nats))))) -> c16(MARK(s(head(tail(nats))))) MARK(head(s(tail(tail(nats))))) -> c16(MARK(s(tail(tail(nats))))) MARK(head(s(cons(cons(y0, y1), y2)))) -> c16(MARK(s(cons(cons(y0, y1), y2)))) MARK(head(s(cons(s(y0), y1)))) -> c16(MARK(s(cons(s(y0), y1)))) MARK(head(s(cons(incr(s(y0)), y1)))) -> c16(MARK(s(cons(incr(s(y0)), y1)))) MARK(head(s(cons(head(s(y0)), y1)))) -> c16(MARK(s(cons(head(s(y0)), y1)))) MARK(head(s(cons(tail(cons(y0, y1)), y2)))) -> c16(MARK(s(cons(tail(cons(y0, y1)), y2)))) MARK(head(s(cons(tail(s(y0)), y1)))) -> c16(MARK(s(cons(tail(s(y0)), y1)))) MARK(head(s(cons(incr(incr(incr(y0))), y1)))) -> c16(MARK(s(cons(incr(incr(incr(y0))), y1)))) MARK(head(s(cons(incr(incr(odds)), y0)))) -> c16(MARK(s(cons(incr(incr(odds)), y0)))) MARK(head(s(cons(incr(incr(head(y0))), y1)))) -> c16(MARK(s(cons(incr(incr(head(y0))), y1)))) MARK(head(s(cons(incr(incr(tail(y0))), y1)))) -> c16(MARK(s(cons(incr(incr(tail(y0))), y1)))) MARK(head(s(cons(incr(incr(cons(y0, y1))), y2)))) -> c16(MARK(s(cons(incr(incr(cons(y0, y1))), y2)))) MARK(head(s(cons(incr(incr(s(y0))), y1)))) -> c16(MARK(s(cons(incr(incr(s(y0))), y1)))) MARK(head(s(cons(incr(incr(y0)), y1)))) -> c16(MARK(s(cons(incr(incr(y0)), y1)))) MARK(head(s(cons(incr(head(incr(y0))), y1)))) -> c16(MARK(s(cons(incr(head(incr(y0))), y1)))) MARK(head(s(cons(incr(head(odds)), y0)))) -> c16(MARK(s(cons(incr(head(odds)), y0)))) MARK(head(s(cons(incr(head(head(y0))), y1)))) -> c16(MARK(s(cons(incr(head(head(y0))), y1)))) MARK(head(s(cons(incr(head(tail(y0))), y1)))) -> c16(MARK(s(cons(incr(head(tail(y0))), y1)))) MARK(head(s(cons(incr(head(cons(y0, y1))), y2)))) -> c16(MARK(s(cons(incr(head(cons(y0, y1))), y2)))) MARK(head(s(cons(incr(head(s(y0))), y1)))) -> c16(MARK(s(cons(incr(head(s(y0))), y1)))) MARK(head(s(cons(incr(head(y0)), y1)))) -> c16(MARK(s(cons(incr(head(y0)), y1)))) MARK(head(s(cons(incr(tail(incr(y0))), y1)))) -> c16(MARK(s(cons(incr(tail(incr(y0))), y1)))) MARK(head(s(cons(incr(tail(pairs)), y0)))) -> c16(MARK(s(cons(incr(tail(pairs)), y0)))) MARK(head(s(cons(incr(tail(odds)), y0)))) -> c16(MARK(s(cons(incr(tail(odds)), y0)))) MARK(head(s(cons(incr(tail(head(y0))), y1)))) -> c16(MARK(s(cons(incr(tail(head(y0))), y1)))) MARK(head(s(cons(incr(tail(tail(y0))), y1)))) -> c16(MARK(s(cons(incr(tail(tail(y0))), y1)))) MARK(head(s(cons(incr(tail(cons(y0, y1))), y2)))) -> c16(MARK(s(cons(incr(tail(cons(y0, y1))), y2)))) MARK(head(s(cons(incr(tail(s(y0))), y1)))) -> c16(MARK(s(cons(incr(tail(s(y0))), y1)))) MARK(head(s(cons(incr(tail(y0)), y1)))) -> c16(MARK(s(cons(incr(tail(y0)), y1)))) MARK(head(s(cons(head(incr(incr(y0))), y1)))) -> c16(MARK(s(cons(head(incr(incr(y0))), y1)))) MARK(head(s(cons(head(incr(odds)), y0)))) -> c16(MARK(s(cons(head(incr(odds)), y0)))) MARK(head(s(cons(head(incr(head(y0))), y1)))) -> c16(MARK(s(cons(head(incr(head(y0))), y1)))) MARK(head(s(cons(head(incr(tail(y0))), y1)))) -> c16(MARK(s(cons(head(incr(tail(y0))), y1)))) MARK(head(s(cons(head(incr(cons(y0, y1))), y2)))) -> c16(MARK(s(cons(head(incr(cons(y0, y1))), y2)))) MARK(head(s(cons(head(incr(s(y0))), y1)))) -> c16(MARK(s(cons(head(incr(s(y0))), y1)))) MARK(head(s(cons(head(incr(y0)), y1)))) -> c16(MARK(s(cons(head(incr(y0)), y1)))) MARK(head(s(cons(head(head(incr(y0))), y1)))) -> c16(MARK(s(cons(head(head(incr(y0))), y1)))) MARK(head(s(cons(head(head(odds)), y0)))) -> c16(MARK(s(cons(head(head(odds)), y0)))) MARK(head(s(cons(head(head(head(y0))), y1)))) -> c16(MARK(s(cons(head(head(head(y0))), y1)))) MARK(head(s(cons(head(head(tail(y0))), y1)))) -> c16(MARK(s(cons(head(head(tail(y0))), y1)))) MARK(head(s(cons(head(head(cons(y0, y1))), y2)))) -> c16(MARK(s(cons(head(head(cons(y0, y1))), y2)))) MARK(head(s(cons(head(head(s(y0))), y1)))) -> c16(MARK(s(cons(head(head(s(y0))), y1)))) MARK(head(s(cons(head(head(y0)), y1)))) -> c16(MARK(s(cons(head(head(y0)), y1)))) MARK(head(s(cons(head(tail(incr(y0))), y1)))) -> c16(MARK(s(cons(head(tail(incr(y0))), y1)))) MARK(head(s(cons(head(tail(pairs)), y0)))) -> c16(MARK(s(cons(head(tail(pairs)), y0)))) MARK(head(s(cons(head(tail(odds)), y0)))) -> c16(MARK(s(cons(head(tail(odds)), y0)))) MARK(head(s(cons(head(tail(head(y0))), y1)))) -> c16(MARK(s(cons(head(tail(head(y0))), y1)))) MARK(head(s(cons(head(tail(tail(y0))), y1)))) -> c16(MARK(s(cons(head(tail(tail(y0))), y1)))) MARK(head(s(cons(head(tail(cons(y0, y1))), y2)))) -> c16(MARK(s(cons(head(tail(cons(y0, y1))), y2)))) MARK(head(s(cons(head(tail(s(y0))), y1)))) -> c16(MARK(s(cons(head(tail(s(y0))), y1)))) MARK(head(s(cons(head(tail(y0)), y1)))) -> c16(MARK(s(cons(head(tail(y0)), y1)))) MARK(head(s(cons(tail(incr(incr(y0))), y1)))) -> c16(MARK(s(cons(tail(incr(incr(y0))), y1)))) MARK(head(s(cons(tail(incr(odds)), y0)))) -> c16(MARK(s(cons(tail(incr(odds)), y0)))) MARK(head(s(cons(tail(incr(head(y0))), y1)))) -> c16(MARK(s(cons(tail(incr(head(y0))), y1)))) MARK(head(s(cons(tail(incr(tail(y0))), y1)))) -> c16(MARK(s(cons(tail(incr(tail(y0))), y1)))) MARK(head(s(cons(tail(incr(cons(y0, y1))), y2)))) -> c16(MARK(s(cons(tail(incr(cons(y0, y1))), y2)))) MARK(head(s(cons(tail(incr(s(y0))), y1)))) -> c16(MARK(s(cons(tail(incr(s(y0))), y1)))) MARK(head(s(cons(tail(incr(y0)), y1)))) -> c16(MARK(s(cons(tail(incr(y0)), y1)))) MARK(head(s(cons(tail(head(incr(y0))), y1)))) -> c16(MARK(s(cons(tail(head(incr(y0))), y1)))) MARK(head(s(cons(tail(head(odds)), y0)))) -> c16(MARK(s(cons(tail(head(odds)), y0)))) MARK(head(s(cons(tail(head(head(y0))), y1)))) -> c16(MARK(s(cons(tail(head(head(y0))), y1)))) MARK(head(s(cons(tail(head(tail(y0))), y1)))) -> c16(MARK(s(cons(tail(head(tail(y0))), y1)))) MARK(head(s(cons(tail(head(cons(y0, y1))), y2)))) -> c16(MARK(s(cons(tail(head(cons(y0, y1))), y2)))) MARK(head(s(cons(tail(head(s(y0))), y1)))) -> c16(MARK(s(cons(tail(head(s(y0))), y1)))) MARK(head(s(cons(tail(head(y0)), y1)))) -> c16(MARK(s(cons(tail(head(y0)), y1)))) MARK(head(s(cons(tail(tail(incr(y0))), y1)))) -> c16(MARK(s(cons(tail(tail(incr(y0))), y1)))) MARK(head(s(cons(tail(tail(pairs)), y0)))) -> c16(MARK(s(cons(tail(tail(pairs)), y0)))) MARK(head(s(cons(tail(tail(odds)), y0)))) -> c16(MARK(s(cons(tail(tail(odds)), y0)))) MARK(head(s(cons(tail(tail(head(y0))), y1)))) -> c16(MARK(s(cons(tail(tail(head(y0))), y1)))) MARK(head(s(cons(tail(tail(tail(y0))), y1)))) -> c16(MARK(s(cons(tail(tail(tail(y0))), y1)))) MARK(head(s(cons(tail(tail(cons(y0, y1))), y2)))) -> c16(MARK(s(cons(tail(tail(cons(y0, y1))), y2)))) MARK(head(s(cons(tail(tail(s(y0))), y1)))) -> c16(MARK(s(cons(tail(tail(s(y0))), y1)))) MARK(head(s(cons(tail(tail(y0)), y1)))) -> c16(MARK(s(cons(tail(tail(y0)), y1)))) MARK(head(s(cons(tail(pairs), y0)))) -> c16(MARK(s(cons(tail(pairs), y0)))) MARK(head(s(cons(incr(incr(nats)), y0)))) -> c16(MARK(s(cons(incr(incr(nats)), y0)))) MARK(head(s(cons(incr(incr(pairs)), y0)))) -> c16(MARK(s(cons(incr(incr(pairs)), y0)))) MARK(head(s(cons(incr(odds), y0)))) -> c16(MARK(s(cons(incr(odds), y0)))) MARK(head(s(cons(head(incr(nats)), y0)))) -> c16(MARK(s(cons(head(incr(nats)), y0)))) MARK(head(s(cons(head(incr(pairs)), y0)))) -> c16(MARK(s(cons(head(incr(pairs)), y0)))) MARK(head(s(cons(head(odds), y0)))) -> c16(MARK(s(cons(head(odds), y0)))) MARK(head(s(cons(tail(incr(nats)), y0)))) -> c16(MARK(s(cons(tail(incr(nats)), y0)))) MARK(head(s(cons(tail(incr(pairs)), y0)))) -> c16(MARK(s(cons(tail(incr(pairs)), y0)))) MARK(head(s(cons(tail(odds), y0)))) -> c16(MARK(s(cons(tail(odds), y0)))) MARK(head(s(cons(incr(cons(nats, y0)), y1)))) -> c16(MARK(s(cons(incr(cons(nats, y0)), y1)))) MARK(head(s(cons(incr(cons(incr(y0), y1)), y2)))) -> c16(MARK(s(cons(incr(cons(incr(y0), y1)), y2)))) MARK(head(s(cons(incr(cons(pairs, y0)), y1)))) -> c16(MARK(s(cons(incr(cons(pairs, y0)), y1)))) MARK(head(s(cons(incr(cons(odds, y0)), y1)))) -> c16(MARK(s(cons(incr(cons(odds, y0)), y1)))) MARK(head(s(cons(incr(cons(head(y0), y1)), y2)))) -> c16(MARK(s(cons(incr(cons(head(y0), y1)), y2)))) MARK(head(s(cons(incr(cons(tail(y0), y1)), y2)))) -> c16(MARK(s(cons(incr(cons(tail(y0), y1)), y2)))) MARK(head(s(cons(incr(cons(cons(y0, y1), y2)), y3)))) -> c16(MARK(s(cons(incr(cons(cons(y0, y1), y2)), y3)))) MARK(head(s(cons(incr(cons(s(y0), y1)), y2)))) -> c16(MARK(s(cons(incr(cons(s(y0), y1)), y2)))) MARK(head(s(cons(incr(cons(y0, y1)), y2)))) -> c16(MARK(s(cons(incr(cons(y0, y1)), y2)))) MARK(head(s(cons(incr(cons(0, y0)), y1)))) -> c16(MARK(s(cons(incr(cons(0, y0)), y1)))) MARK(head(s(cons(incr(head(nats)), y0)))) -> c16(MARK(s(cons(incr(head(nats)), y0)))) MARK(head(s(cons(incr(head(pairs)), y0)))) -> c16(MARK(s(cons(incr(head(pairs)), y0)))) MARK(head(s(cons(head(head(nats)), y0)))) -> c16(MARK(s(cons(head(head(nats)), y0)))) MARK(head(s(cons(head(head(pairs)), y0)))) -> c16(MARK(s(cons(head(head(pairs)), y0)))) MARK(head(s(cons(tail(head(nats)), y0)))) -> c16(MARK(s(cons(tail(head(nats)), y0)))) MARK(head(s(cons(tail(head(pairs)), y0)))) -> c16(MARK(s(cons(tail(head(pairs)), y0)))) MARK(head(s(cons(head(cons(nats, y0)), y1)))) -> c16(MARK(s(cons(head(cons(nats, y0)), y1)))) MARK(head(s(cons(head(cons(incr(y0), y1)), y2)))) -> c16(MARK(s(cons(head(cons(incr(y0), y1)), y2)))) MARK(head(s(cons(head(cons(pairs, y0)), y1)))) -> c16(MARK(s(cons(head(cons(pairs, y0)), y1)))) MARK(head(s(cons(head(cons(odds, y0)), y1)))) -> c16(MARK(s(cons(head(cons(odds, y0)), y1)))) MARK(head(s(cons(head(cons(head(y0), y1)), y2)))) -> c16(MARK(s(cons(head(cons(head(y0), y1)), y2)))) MARK(head(s(cons(head(cons(tail(y0), y1)), y2)))) -> c16(MARK(s(cons(head(cons(tail(y0), y1)), y2)))) MARK(head(s(cons(head(cons(cons(y0, y1), y2)), y3)))) -> c16(MARK(s(cons(head(cons(cons(y0, y1), y2)), y3)))) MARK(head(s(cons(head(cons(s(y0), y1)), y2)))) -> c16(MARK(s(cons(head(cons(s(y0), y1)), y2)))) MARK(head(s(cons(head(cons(y0, y1)), y2)))) -> c16(MARK(s(cons(head(cons(y0, y1)), y2)))) MARK(head(s(cons(head(cons(0, y0)), y1)))) -> c16(MARK(s(cons(head(cons(0, y0)), y1)))) MARK(head(s(cons(incr(tail(nats)), y0)))) -> c16(MARK(s(cons(incr(tail(nats)), y0)))) MARK(head(s(cons(head(tail(nats)), y0)))) -> c16(MARK(s(cons(head(tail(nats)), y0)))) MARK(head(s(cons(tail(tail(nats)), y0)))) -> c16(MARK(s(cons(tail(tail(nats)), y0)))) MARK(head(s(incr(cons(nats, y0))))) -> c16(MARK(s(incr(cons(nats, y0))))) MARK(head(s(incr(cons(pairs, y0))))) -> c16(MARK(s(incr(cons(pairs, y0))))) MARK(head(s(incr(cons(odds, y0))))) -> c16(MARK(s(incr(cons(odds, y0))))) MARK(head(s(head(cons(nats, y0))))) -> c16(MARK(s(head(cons(nats, y0))))) MARK(head(s(head(cons(pairs, y0))))) -> c16(MARK(s(head(cons(pairs, y0))))) MARK(head(s(head(cons(odds, y0))))) -> c16(MARK(s(head(cons(odds, y0))))) MARK(tail(s(s(y0)))) -> c17(MARK(s(s(y0)))) MARK(tail(s(incr(s(y0))))) -> c17(MARK(s(incr(s(y0))))) MARK(tail(s(head(s(y0))))) -> c17(MARK(s(head(s(y0))))) MARK(tail(s(tail(cons(y0, y1))))) -> c17(MARK(s(tail(cons(y0, y1))))) MARK(tail(s(tail(s(y0))))) -> c17(MARK(s(tail(s(y0))))) MARK(tail(s(incr(incr(incr(y0)))))) -> c17(MARK(s(incr(incr(incr(y0)))))) MARK(tail(s(incr(incr(odds))))) -> c17(MARK(s(incr(incr(odds))))) MARK(tail(s(incr(incr(head(y0)))))) -> c17(MARK(s(incr(incr(head(y0)))))) MARK(tail(s(incr(incr(tail(y0)))))) -> c17(MARK(s(incr(incr(tail(y0)))))) MARK(tail(s(incr(incr(cons(y0, y1)))))) -> c17(MARK(s(incr(incr(cons(y0, y1)))))) MARK(tail(s(incr(incr(s(y0)))))) -> c17(MARK(s(incr(incr(s(y0)))))) MARK(tail(s(incr(incr(y0))))) -> c17(MARK(s(incr(incr(y0))))) MARK(tail(s(incr(head(incr(y0)))))) -> c17(MARK(s(incr(head(incr(y0)))))) MARK(tail(s(incr(head(odds))))) -> c17(MARK(s(incr(head(odds))))) MARK(tail(s(incr(head(head(y0)))))) -> c17(MARK(s(incr(head(head(y0)))))) MARK(tail(s(incr(head(tail(y0)))))) -> c17(MARK(s(incr(head(tail(y0)))))) MARK(tail(s(incr(head(cons(y0, y1)))))) -> c17(MARK(s(incr(head(cons(y0, y1)))))) MARK(tail(s(incr(head(s(y0)))))) -> c17(MARK(s(incr(head(s(y0)))))) MARK(tail(s(incr(head(y0))))) -> c17(MARK(s(incr(head(y0))))) MARK(tail(s(incr(tail(incr(y0)))))) -> c17(MARK(s(incr(tail(incr(y0)))))) MARK(tail(s(incr(tail(pairs))))) -> c17(MARK(s(incr(tail(pairs))))) MARK(tail(s(incr(tail(odds))))) -> c17(MARK(s(incr(tail(odds))))) MARK(tail(s(incr(tail(head(y0)))))) -> c17(MARK(s(incr(tail(head(y0)))))) MARK(tail(s(incr(tail(tail(y0)))))) -> c17(MARK(s(incr(tail(tail(y0)))))) MARK(tail(s(incr(tail(cons(y0, y1)))))) -> c17(MARK(s(incr(tail(cons(y0, y1)))))) MARK(tail(s(incr(tail(s(y0)))))) -> c17(MARK(s(incr(tail(s(y0)))))) MARK(tail(s(incr(tail(y0))))) -> c17(MARK(s(incr(tail(y0))))) MARK(tail(s(head(incr(incr(y0)))))) -> c17(MARK(s(head(incr(incr(y0)))))) MARK(tail(s(head(incr(odds))))) -> c17(MARK(s(head(incr(odds))))) MARK(tail(s(head(incr(head(y0)))))) -> c17(MARK(s(head(incr(head(y0)))))) MARK(tail(s(head(incr(tail(y0)))))) -> c17(MARK(s(head(incr(tail(y0)))))) MARK(tail(s(head(incr(cons(y0, y1)))))) -> c17(MARK(s(head(incr(cons(y0, y1)))))) MARK(tail(s(head(incr(s(y0)))))) -> c17(MARK(s(head(incr(s(y0)))))) MARK(tail(s(head(incr(y0))))) -> c17(MARK(s(head(incr(y0))))) MARK(tail(s(head(head(incr(y0)))))) -> c17(MARK(s(head(head(incr(y0)))))) MARK(tail(s(head(head(odds))))) -> c17(MARK(s(head(head(odds))))) MARK(tail(s(head(head(head(y0)))))) -> c17(MARK(s(head(head(head(y0)))))) MARK(tail(s(head(head(tail(y0)))))) -> c17(MARK(s(head(head(tail(y0)))))) MARK(tail(s(head(head(cons(y0, y1)))))) -> c17(MARK(s(head(head(cons(y0, y1)))))) MARK(tail(s(head(head(s(y0)))))) -> c17(MARK(s(head(head(s(y0)))))) MARK(tail(s(head(head(y0))))) -> c17(MARK(s(head(head(y0))))) MARK(tail(s(head(tail(incr(y0)))))) -> c17(MARK(s(head(tail(incr(y0)))))) MARK(tail(s(head(tail(pairs))))) -> c17(MARK(s(head(tail(pairs))))) MARK(tail(s(head(tail(odds))))) -> c17(MARK(s(head(tail(odds))))) MARK(tail(s(head(tail(head(y0)))))) -> c17(MARK(s(head(tail(head(y0)))))) MARK(tail(s(head(tail(tail(y0)))))) -> c17(MARK(s(head(tail(tail(y0)))))) MARK(tail(s(head(tail(cons(y0, y1)))))) -> c17(MARK(s(head(tail(cons(y0, y1)))))) MARK(tail(s(head(tail(s(y0)))))) -> c17(MARK(s(head(tail(s(y0)))))) MARK(tail(s(head(tail(y0))))) -> c17(MARK(s(head(tail(y0))))) MARK(tail(s(tail(incr(incr(y0)))))) -> c17(MARK(s(tail(incr(incr(y0)))))) MARK(tail(s(tail(incr(odds))))) -> c17(MARK(s(tail(incr(odds))))) MARK(tail(s(tail(incr(head(y0)))))) -> c17(MARK(s(tail(incr(head(y0)))))) MARK(tail(s(tail(incr(tail(y0)))))) -> c17(MARK(s(tail(incr(tail(y0)))))) MARK(tail(s(tail(incr(cons(y0, y1)))))) -> c17(MARK(s(tail(incr(cons(y0, y1)))))) MARK(tail(s(tail(incr(s(y0)))))) -> c17(MARK(s(tail(incr(s(y0)))))) MARK(tail(s(tail(incr(y0))))) -> c17(MARK(s(tail(incr(y0))))) MARK(tail(s(tail(head(incr(y0)))))) -> c17(MARK(s(tail(head(incr(y0)))))) MARK(tail(s(tail(head(odds))))) -> c17(MARK(s(tail(head(odds))))) MARK(tail(s(tail(head(head(y0)))))) -> c17(MARK(s(tail(head(head(y0)))))) MARK(tail(s(tail(head(tail(y0)))))) -> c17(MARK(s(tail(head(tail(y0)))))) MARK(tail(s(tail(head(cons(y0, y1)))))) -> c17(MARK(s(tail(head(cons(y0, y1)))))) MARK(tail(s(tail(head(s(y0)))))) -> c17(MARK(s(tail(head(s(y0)))))) MARK(tail(s(tail(head(y0))))) -> c17(MARK(s(tail(head(y0))))) MARK(tail(s(tail(tail(incr(y0)))))) -> c17(MARK(s(tail(tail(incr(y0)))))) MARK(tail(s(tail(tail(pairs))))) -> c17(MARK(s(tail(tail(pairs))))) MARK(tail(s(tail(tail(odds))))) -> c17(MARK(s(tail(tail(odds))))) MARK(tail(s(tail(tail(head(y0)))))) -> c17(MARK(s(tail(tail(head(y0)))))) MARK(tail(s(tail(tail(tail(y0)))))) -> c17(MARK(s(tail(tail(tail(y0)))))) MARK(tail(s(tail(tail(cons(y0, y1)))))) -> c17(MARK(s(tail(tail(cons(y0, y1)))))) MARK(tail(s(tail(tail(s(y0)))))) -> c17(MARK(s(tail(tail(s(y0)))))) MARK(tail(s(tail(tail(y0))))) -> c17(MARK(s(tail(tail(y0))))) MARK(tail(s(tail(pairs)))) -> c17(MARK(s(tail(pairs)))) MARK(tail(s(incr(incr(nats))))) -> c17(MARK(s(incr(incr(nats))))) MARK(tail(s(incr(incr(pairs))))) -> c17(MARK(s(incr(incr(pairs))))) MARK(tail(s(incr(odds)))) -> c17(MARK(s(incr(odds)))) MARK(tail(s(head(incr(nats))))) -> c17(MARK(s(head(incr(nats))))) MARK(tail(s(head(incr(pairs))))) -> c17(MARK(s(head(incr(pairs))))) MARK(tail(s(head(odds)))) -> c17(MARK(s(head(odds)))) MARK(tail(s(tail(incr(nats))))) -> c17(MARK(s(tail(incr(nats))))) MARK(tail(s(tail(incr(pairs))))) -> c17(MARK(s(tail(incr(pairs))))) MARK(tail(s(tail(odds)))) -> c17(MARK(s(tail(odds)))) MARK(tail(s(incr(cons(incr(y0), y1))))) -> c17(MARK(s(incr(cons(incr(y0), y1))))) MARK(tail(s(incr(cons(head(y0), y1))))) -> c17(MARK(s(incr(cons(head(y0), y1))))) MARK(tail(s(incr(cons(tail(y0), y1))))) -> c17(MARK(s(incr(cons(tail(y0), y1))))) MARK(tail(s(incr(cons(cons(y0, y1), y2))))) -> c17(MARK(s(incr(cons(cons(y0, y1), y2))))) MARK(tail(s(incr(cons(s(y0), y1))))) -> c17(MARK(s(incr(cons(s(y0), y1))))) MARK(tail(s(incr(cons(y0, y1))))) -> c17(MARK(s(incr(cons(y0, y1))))) MARK(tail(s(incr(head(nats))))) -> c17(MARK(s(incr(head(nats))))) MARK(tail(s(incr(head(pairs))))) -> c17(MARK(s(incr(head(pairs))))) MARK(tail(s(head(head(nats))))) -> c17(MARK(s(head(head(nats))))) MARK(tail(s(head(head(pairs))))) -> c17(MARK(s(head(head(pairs))))) MARK(tail(s(tail(head(nats))))) -> c17(MARK(s(tail(head(nats))))) MARK(tail(s(tail(head(pairs))))) -> c17(MARK(s(tail(head(pairs))))) MARK(tail(s(head(cons(incr(y0), y1))))) -> c17(MARK(s(head(cons(incr(y0), y1))))) MARK(tail(s(head(cons(head(y0), y1))))) -> c17(MARK(s(head(cons(head(y0), y1))))) MARK(tail(s(head(cons(tail(y0), y1))))) -> c17(MARK(s(head(cons(tail(y0), y1))))) MARK(tail(s(head(cons(cons(y0, y1), y2))))) -> c17(MARK(s(head(cons(cons(y0, y1), y2))))) MARK(tail(s(head(cons(s(y0), y1))))) -> c17(MARK(s(head(cons(s(y0), y1))))) MARK(tail(s(head(cons(y0, y1))))) -> c17(MARK(s(head(cons(y0, y1))))) MARK(tail(s(incr(tail(nats))))) -> c17(MARK(s(incr(tail(nats))))) MARK(tail(s(head(tail(nats))))) -> c17(MARK(s(head(tail(nats))))) MARK(tail(s(tail(tail(nats))))) -> c17(MARK(s(tail(tail(nats))))) MARK(tail(s(cons(cons(y0, y1), y2)))) -> c17(MARK(s(cons(cons(y0, y1), y2)))) MARK(tail(s(cons(s(y0), y1)))) -> c17(MARK(s(cons(s(y0), y1)))) MARK(tail(s(cons(incr(s(y0)), y1)))) -> c17(MARK(s(cons(incr(s(y0)), y1)))) MARK(tail(s(cons(head(s(y0)), y1)))) -> c17(MARK(s(cons(head(s(y0)), y1)))) MARK(tail(s(cons(tail(cons(y0, y1)), y2)))) -> c17(MARK(s(cons(tail(cons(y0, y1)), y2)))) MARK(tail(s(cons(tail(s(y0)), y1)))) -> c17(MARK(s(cons(tail(s(y0)), y1)))) MARK(tail(s(cons(incr(incr(incr(y0))), y1)))) -> c17(MARK(s(cons(incr(incr(incr(y0))), y1)))) MARK(tail(s(cons(incr(incr(odds)), y0)))) -> c17(MARK(s(cons(incr(incr(odds)), y0)))) MARK(tail(s(cons(incr(incr(head(y0))), y1)))) -> c17(MARK(s(cons(incr(incr(head(y0))), y1)))) MARK(tail(s(cons(incr(incr(tail(y0))), y1)))) -> c17(MARK(s(cons(incr(incr(tail(y0))), y1)))) MARK(tail(s(cons(incr(incr(cons(y0, y1))), y2)))) -> c17(MARK(s(cons(incr(incr(cons(y0, y1))), y2)))) MARK(tail(s(cons(incr(incr(s(y0))), y1)))) -> c17(MARK(s(cons(incr(incr(s(y0))), y1)))) MARK(tail(s(cons(incr(incr(y0)), y1)))) -> c17(MARK(s(cons(incr(incr(y0)), y1)))) MARK(tail(s(cons(incr(head(incr(y0))), y1)))) -> c17(MARK(s(cons(incr(head(incr(y0))), y1)))) MARK(tail(s(cons(incr(head(odds)), y0)))) -> c17(MARK(s(cons(incr(head(odds)), y0)))) MARK(tail(s(cons(incr(head(head(y0))), y1)))) -> c17(MARK(s(cons(incr(head(head(y0))), y1)))) MARK(tail(s(cons(incr(head(tail(y0))), y1)))) -> c17(MARK(s(cons(incr(head(tail(y0))), y1)))) MARK(tail(s(cons(incr(head(cons(y0, y1))), y2)))) -> c17(MARK(s(cons(incr(head(cons(y0, y1))), y2)))) MARK(tail(s(cons(incr(head(s(y0))), y1)))) -> c17(MARK(s(cons(incr(head(s(y0))), y1)))) MARK(tail(s(cons(incr(head(y0)), y1)))) -> c17(MARK(s(cons(incr(head(y0)), y1)))) MARK(tail(s(cons(incr(tail(incr(y0))), y1)))) -> c17(MARK(s(cons(incr(tail(incr(y0))), y1)))) MARK(tail(s(cons(incr(tail(pairs)), y0)))) -> c17(MARK(s(cons(incr(tail(pairs)), y0)))) MARK(tail(s(cons(incr(tail(odds)), y0)))) -> c17(MARK(s(cons(incr(tail(odds)), y0)))) MARK(tail(s(cons(incr(tail(head(y0))), y1)))) -> c17(MARK(s(cons(incr(tail(head(y0))), y1)))) MARK(tail(s(cons(incr(tail(tail(y0))), y1)))) -> c17(MARK(s(cons(incr(tail(tail(y0))), y1)))) MARK(tail(s(cons(incr(tail(cons(y0, y1))), y2)))) -> c17(MARK(s(cons(incr(tail(cons(y0, y1))), y2)))) MARK(tail(s(cons(incr(tail(s(y0))), y1)))) -> c17(MARK(s(cons(incr(tail(s(y0))), y1)))) MARK(tail(s(cons(incr(tail(y0)), y1)))) -> c17(MARK(s(cons(incr(tail(y0)), y1)))) MARK(tail(s(cons(head(incr(incr(y0))), y1)))) -> c17(MARK(s(cons(head(incr(incr(y0))), y1)))) MARK(tail(s(cons(head(incr(odds)), y0)))) -> c17(MARK(s(cons(head(incr(odds)), y0)))) MARK(tail(s(cons(head(incr(head(y0))), y1)))) -> c17(MARK(s(cons(head(incr(head(y0))), y1)))) MARK(tail(s(cons(head(incr(tail(y0))), y1)))) -> c17(MARK(s(cons(head(incr(tail(y0))), y1)))) MARK(tail(s(cons(head(incr(cons(y0, y1))), y2)))) -> c17(MARK(s(cons(head(incr(cons(y0, y1))), y2)))) MARK(tail(s(cons(head(incr(s(y0))), y1)))) -> c17(MARK(s(cons(head(incr(s(y0))), y1)))) MARK(tail(s(cons(head(incr(y0)), y1)))) -> c17(MARK(s(cons(head(incr(y0)), y1)))) MARK(tail(s(cons(head(head(incr(y0))), y1)))) -> c17(MARK(s(cons(head(head(incr(y0))), y1)))) MARK(tail(s(cons(head(head(odds)), y0)))) -> c17(MARK(s(cons(head(head(odds)), y0)))) MARK(tail(s(cons(head(head(head(y0))), y1)))) -> c17(MARK(s(cons(head(head(head(y0))), y1)))) MARK(tail(s(cons(head(head(tail(y0))), y1)))) -> c17(MARK(s(cons(head(head(tail(y0))), y1)))) MARK(tail(s(cons(head(head(cons(y0, y1))), y2)))) -> c17(MARK(s(cons(head(head(cons(y0, y1))), y2)))) MARK(tail(s(cons(head(head(s(y0))), y1)))) -> c17(MARK(s(cons(head(head(s(y0))), y1)))) MARK(tail(s(cons(head(head(y0)), y1)))) -> c17(MARK(s(cons(head(head(y0)), y1)))) MARK(tail(s(cons(head(tail(incr(y0))), y1)))) -> c17(MARK(s(cons(head(tail(incr(y0))), y1)))) MARK(tail(s(cons(head(tail(pairs)), y0)))) -> c17(MARK(s(cons(head(tail(pairs)), y0)))) MARK(tail(s(cons(head(tail(odds)), y0)))) -> c17(MARK(s(cons(head(tail(odds)), y0)))) MARK(tail(s(cons(head(tail(head(y0))), y1)))) -> c17(MARK(s(cons(head(tail(head(y0))), y1)))) MARK(tail(s(cons(head(tail(tail(y0))), y1)))) -> c17(MARK(s(cons(head(tail(tail(y0))), y1)))) MARK(tail(s(cons(head(tail(cons(y0, y1))), y2)))) -> c17(MARK(s(cons(head(tail(cons(y0, y1))), y2)))) MARK(tail(s(cons(head(tail(s(y0))), y1)))) -> c17(MARK(s(cons(head(tail(s(y0))), y1)))) MARK(tail(s(cons(head(tail(y0)), y1)))) -> c17(MARK(s(cons(head(tail(y0)), y1)))) MARK(tail(s(cons(tail(incr(incr(y0))), y1)))) -> c17(MARK(s(cons(tail(incr(incr(y0))), y1)))) MARK(tail(s(cons(tail(incr(odds)), y0)))) -> c17(MARK(s(cons(tail(incr(odds)), y0)))) MARK(tail(s(cons(tail(incr(head(y0))), y1)))) -> c17(MARK(s(cons(tail(incr(head(y0))), y1)))) MARK(tail(s(cons(tail(incr(tail(y0))), y1)))) -> c17(MARK(s(cons(tail(incr(tail(y0))), y1)))) MARK(tail(s(cons(tail(incr(cons(y0, y1))), y2)))) -> c17(MARK(s(cons(tail(incr(cons(y0, y1))), y2)))) MARK(tail(s(cons(tail(incr(s(y0))), y1)))) -> c17(MARK(s(cons(tail(incr(s(y0))), y1)))) MARK(tail(s(cons(tail(incr(y0)), y1)))) -> c17(MARK(s(cons(tail(incr(y0)), y1)))) MARK(tail(s(cons(tail(head(incr(y0))), y1)))) -> c17(MARK(s(cons(tail(head(incr(y0))), y1)))) MARK(tail(s(cons(tail(head(odds)), y0)))) -> c17(MARK(s(cons(tail(head(odds)), y0)))) MARK(tail(s(cons(tail(head(head(y0))), y1)))) -> c17(MARK(s(cons(tail(head(head(y0))), y1)))) MARK(tail(s(cons(tail(head(tail(y0))), y1)))) -> c17(MARK(s(cons(tail(head(tail(y0))), y1)))) MARK(tail(s(cons(tail(head(cons(y0, y1))), y2)))) -> c17(MARK(s(cons(tail(head(cons(y0, y1))), y2)))) MARK(tail(s(cons(tail(head(s(y0))), y1)))) -> c17(MARK(s(cons(tail(head(s(y0))), y1)))) MARK(tail(s(cons(tail(head(y0)), y1)))) -> c17(MARK(s(cons(tail(head(y0)), y1)))) MARK(tail(s(cons(tail(tail(incr(y0))), y1)))) -> c17(MARK(s(cons(tail(tail(incr(y0))), y1)))) MARK(tail(s(cons(tail(tail(pairs)), y0)))) -> c17(MARK(s(cons(tail(tail(pairs)), y0)))) MARK(tail(s(cons(tail(tail(odds)), y0)))) -> c17(MARK(s(cons(tail(tail(odds)), y0)))) MARK(tail(s(cons(tail(tail(head(y0))), y1)))) -> c17(MARK(s(cons(tail(tail(head(y0))), y1)))) MARK(tail(s(cons(tail(tail(tail(y0))), y1)))) -> c17(MARK(s(cons(tail(tail(tail(y0))), y1)))) MARK(tail(s(cons(tail(tail(cons(y0, y1))), y2)))) -> c17(MARK(s(cons(tail(tail(cons(y0, y1))), y2)))) MARK(tail(s(cons(tail(tail(s(y0))), y1)))) -> c17(MARK(s(cons(tail(tail(s(y0))), y1)))) MARK(tail(s(cons(tail(tail(y0)), y1)))) -> c17(MARK(s(cons(tail(tail(y0)), y1)))) MARK(tail(s(cons(tail(pairs), y0)))) -> c17(MARK(s(cons(tail(pairs), y0)))) MARK(tail(s(cons(incr(incr(nats)), y0)))) -> c17(MARK(s(cons(incr(incr(nats)), y0)))) MARK(tail(s(cons(incr(incr(pairs)), y0)))) -> c17(MARK(s(cons(incr(incr(pairs)), y0)))) MARK(tail(s(cons(incr(odds), y0)))) -> c17(MARK(s(cons(incr(odds), y0)))) MARK(tail(s(cons(head(incr(nats)), y0)))) -> c17(MARK(s(cons(head(incr(nats)), y0)))) MARK(tail(s(cons(head(incr(pairs)), y0)))) -> c17(MARK(s(cons(head(incr(pairs)), y0)))) MARK(tail(s(cons(head(odds), y0)))) -> c17(MARK(s(cons(head(odds), y0)))) MARK(tail(s(cons(tail(incr(nats)), y0)))) -> c17(MARK(s(cons(tail(incr(nats)), y0)))) MARK(tail(s(cons(tail(incr(pairs)), y0)))) -> c17(MARK(s(cons(tail(incr(pairs)), y0)))) MARK(tail(s(cons(tail(odds), y0)))) -> c17(MARK(s(cons(tail(odds), y0)))) MARK(tail(s(cons(incr(cons(nats, y0)), y1)))) -> c17(MARK(s(cons(incr(cons(nats, y0)), y1)))) MARK(tail(s(cons(incr(cons(incr(y0), y1)), y2)))) -> c17(MARK(s(cons(incr(cons(incr(y0), y1)), y2)))) MARK(tail(s(cons(incr(cons(pairs, y0)), y1)))) -> c17(MARK(s(cons(incr(cons(pairs, y0)), y1)))) MARK(tail(s(cons(incr(cons(odds, y0)), y1)))) -> c17(MARK(s(cons(incr(cons(odds, y0)), y1)))) MARK(tail(s(cons(incr(cons(head(y0), y1)), y2)))) -> c17(MARK(s(cons(incr(cons(head(y0), y1)), y2)))) MARK(tail(s(cons(incr(cons(tail(y0), y1)), y2)))) -> c17(MARK(s(cons(incr(cons(tail(y0), y1)), y2)))) MARK(tail(s(cons(incr(cons(cons(y0, y1), y2)), y3)))) -> c17(MARK(s(cons(incr(cons(cons(y0, y1), y2)), y3)))) MARK(tail(s(cons(incr(cons(s(y0), y1)), y2)))) -> c17(MARK(s(cons(incr(cons(s(y0), y1)), y2)))) MARK(tail(s(cons(incr(cons(y0, y1)), y2)))) -> c17(MARK(s(cons(incr(cons(y0, y1)), y2)))) MARK(tail(s(cons(incr(cons(0, y0)), y1)))) -> c17(MARK(s(cons(incr(cons(0, y0)), y1)))) MARK(tail(s(cons(incr(head(nats)), y0)))) -> c17(MARK(s(cons(incr(head(nats)), y0)))) MARK(tail(s(cons(incr(head(pairs)), y0)))) -> c17(MARK(s(cons(incr(head(pairs)), y0)))) MARK(tail(s(cons(head(head(nats)), y0)))) -> c17(MARK(s(cons(head(head(nats)), y0)))) MARK(tail(s(cons(head(head(pairs)), y0)))) -> c17(MARK(s(cons(head(head(pairs)), y0)))) MARK(tail(s(cons(tail(head(nats)), y0)))) -> c17(MARK(s(cons(tail(head(nats)), y0)))) MARK(tail(s(cons(tail(head(pairs)), y0)))) -> c17(MARK(s(cons(tail(head(pairs)), y0)))) MARK(tail(s(cons(head(cons(nats, y0)), y1)))) -> c17(MARK(s(cons(head(cons(nats, y0)), y1)))) MARK(tail(s(cons(head(cons(incr(y0), y1)), y2)))) -> c17(MARK(s(cons(head(cons(incr(y0), y1)), y2)))) MARK(tail(s(cons(head(cons(pairs, y0)), y1)))) -> c17(MARK(s(cons(head(cons(pairs, y0)), y1)))) MARK(tail(s(cons(head(cons(odds, y0)), y1)))) -> c17(MARK(s(cons(head(cons(odds, y0)), y1)))) MARK(tail(s(cons(head(cons(head(y0), y1)), y2)))) -> c17(MARK(s(cons(head(cons(head(y0), y1)), y2)))) MARK(tail(s(cons(head(cons(tail(y0), y1)), y2)))) -> c17(MARK(s(cons(head(cons(tail(y0), y1)), y2)))) MARK(tail(s(cons(head(cons(cons(y0, y1), y2)), y3)))) -> c17(MARK(s(cons(head(cons(cons(y0, y1), y2)), y3)))) MARK(tail(s(cons(head(cons(s(y0), y1)), y2)))) -> c17(MARK(s(cons(head(cons(s(y0), y1)), y2)))) MARK(tail(s(cons(head(cons(y0, y1)), y2)))) -> c17(MARK(s(cons(head(cons(y0, y1)), y2)))) MARK(tail(s(cons(head(cons(0, y0)), y1)))) -> c17(MARK(s(cons(head(cons(0, y0)), y1)))) MARK(tail(s(cons(incr(tail(nats)), y0)))) -> c17(MARK(s(cons(incr(tail(nats)), y0)))) MARK(tail(s(cons(head(tail(nats)), y0)))) -> c17(MARK(s(cons(head(tail(nats)), y0)))) MARK(tail(s(cons(tail(tail(nats)), y0)))) -> c17(MARK(s(cons(tail(tail(nats)), y0)))) MARK(tail(s(incr(cons(nats, y0))))) -> c17(MARK(s(incr(cons(nats, y0))))) MARK(tail(s(incr(cons(pairs, y0))))) -> c17(MARK(s(incr(cons(pairs, y0))))) MARK(tail(s(incr(cons(odds, y0))))) -> c17(MARK(s(incr(cons(odds, y0))))) MARK(tail(s(head(cons(nats, y0))))) -> c17(MARK(s(head(cons(nats, y0))))) MARK(tail(s(head(cons(pairs, y0))))) -> c17(MARK(s(head(cons(pairs, y0))))) MARK(tail(s(head(cons(odds, y0))))) -> c17(MARK(s(head(cons(odds, y0))))) S tuples: MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(tail(incr(incr(z0)))) -> c17(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(odds))) -> c17(A__TAIL(a__incr(a__odds)), MARK(incr(odds))) MARK(tail(incr(head(z0)))) -> c17(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c17(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c17(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c17(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c17(MARK(incr(x0))) MARK(tail(head(incr(z0)))) -> c17(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(odds))) -> c17(A__TAIL(a__head(a__odds)), MARK(head(odds))) MARK(tail(head(head(z0)))) -> c17(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c17(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c17(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c17(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c17(MARK(head(x0))) MARK(tail(tail(incr(z0)))) -> c17(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(pairs))) -> c17(A__TAIL(a__tail(a__pairs)), MARK(tail(pairs))) MARK(tail(tail(odds))) -> c17(A__TAIL(a__tail(a__odds)), MARK(tail(odds))) MARK(tail(tail(head(z0)))) -> c17(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c17(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c17(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c17(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c17(MARK(tail(x0))) MARK(tail(pairs)) -> c17(A__TAIL(cons(0, incr(odds)))) A__INCR(cons(cons(y0, y1), z1)) -> c6(MARK(cons(y0, y1))) A__INCR(cons(s(y0), z1)) -> c6(MARK(s(y0))) A__INCR(cons(incr(cons(y0, y1)), z1)) -> c6(MARK(incr(cons(y0, y1)))) A__INCR(cons(incr(s(y0)), z1)) -> c6(MARK(incr(s(y0)))) A__INCR(cons(head(cons(y0, y1)), z1)) -> c6(MARK(head(cons(y0, y1)))) A__INCR(cons(head(s(y0)), z1)) -> c6(MARK(head(s(y0)))) A__INCR(cons(tail(cons(y0, y1)), z1)) -> c6(MARK(tail(cons(y0, y1)))) A__INCR(cons(tail(s(y0)), z1)) -> c6(MARK(tail(s(y0)))) A__INCR(cons(incr(incr(nats)), z1)) -> c6(MARK(incr(incr(nats)))) A__INCR(cons(incr(incr(incr(y0))), z1)) -> c6(MARK(incr(incr(incr(y0))))) A__INCR(cons(incr(incr(pairs)), z1)) -> c6(MARK(incr(incr(pairs)))) A__INCR(cons(incr(incr(odds)), z1)) -> c6(MARK(incr(incr(odds)))) A__INCR(cons(incr(incr(head(y0))), z1)) -> c6(MARK(incr(incr(head(y0))))) A__INCR(cons(incr(incr(tail(y0))), z1)) -> c6(MARK(incr(incr(tail(y0))))) A__INCR(cons(incr(incr(cons(y0, y1))), z1)) -> c6(MARK(incr(incr(cons(y0, y1))))) A__INCR(cons(incr(incr(s(y0))), z1)) -> c6(MARK(incr(incr(s(y0))))) A__INCR(cons(incr(incr(y0)), z1)) -> c6(MARK(incr(incr(y0)))) A__INCR(cons(incr(odds), z1)) -> c6(MARK(incr(odds))) A__INCR(cons(incr(head(nats)), z1)) -> c6(MARK(incr(head(nats)))) A__INCR(cons(incr(head(incr(y0))), z1)) -> c6(MARK(incr(head(incr(y0))))) A__INCR(cons(incr(head(pairs)), z1)) -> c6(MARK(incr(head(pairs)))) A__INCR(cons(incr(head(odds)), z1)) -> c6(MARK(incr(head(odds)))) A__INCR(cons(incr(head(head(y0))), z1)) -> c6(MARK(incr(head(head(y0))))) A__INCR(cons(incr(head(tail(y0))), z1)) -> c6(MARK(incr(head(tail(y0))))) A__INCR(cons(incr(head(cons(y0, y1))), z1)) -> c6(MARK(incr(head(cons(y0, y1))))) A__INCR(cons(incr(head(s(y0))), z1)) -> c6(MARK(incr(head(s(y0))))) A__INCR(cons(incr(head(y0)), z1)) -> c6(MARK(incr(head(y0)))) A__INCR(cons(incr(tail(nats)), z1)) -> c6(MARK(incr(tail(nats)))) A__INCR(cons(incr(tail(incr(y0))), z1)) -> c6(MARK(incr(tail(incr(y0))))) A__INCR(cons(incr(tail(pairs)), z1)) -> c6(MARK(incr(tail(pairs)))) A__INCR(cons(incr(tail(odds)), z1)) -> c6(MARK(incr(tail(odds)))) A__INCR(cons(incr(tail(head(y0))), z1)) -> c6(MARK(incr(tail(head(y0))))) A__INCR(cons(incr(tail(tail(y0))), z1)) -> c6(MARK(incr(tail(tail(y0))))) A__INCR(cons(incr(tail(cons(y0, y1))), z1)) -> c6(MARK(incr(tail(cons(y0, y1))))) A__INCR(cons(incr(tail(s(y0))), z1)) -> c6(MARK(incr(tail(s(y0))))) A__INCR(cons(incr(tail(y0)), z1)) -> c6(MARK(incr(tail(y0)))) A__INCR(cons(head(incr(nats)), z1)) -> c6(MARK(head(incr(nats)))) A__INCR(cons(head(incr(incr(y0))), z1)) -> c6(MARK(head(incr(incr(y0))))) A__INCR(cons(head(incr(pairs)), z1)) -> c6(MARK(head(incr(pairs)))) A__INCR(cons(head(incr(odds)), z1)) -> c6(MARK(head(incr(odds)))) A__INCR(cons(head(incr(head(y0))), z1)) -> c6(MARK(head(incr(head(y0))))) A__INCR(cons(head(incr(tail(y0))), z1)) -> c6(MARK(head(incr(tail(y0))))) A__INCR(cons(head(incr(cons(y0, y1))), z1)) -> c6(MARK(head(incr(cons(y0, y1))))) A__INCR(cons(head(incr(s(y0))), z1)) -> c6(MARK(head(incr(s(y0))))) A__INCR(cons(head(incr(y0)), z1)) -> c6(MARK(head(incr(y0)))) A__INCR(cons(head(odds), z1)) -> c6(MARK(head(odds))) A__INCR(cons(head(head(nats)), z1)) -> c6(MARK(head(head(nats)))) A__INCR(cons(head(head(incr(y0))), z1)) -> c6(MARK(head(head(incr(y0))))) A__INCR(cons(head(head(pairs)), z1)) -> c6(MARK(head(head(pairs)))) A__INCR(cons(head(head(odds)), z1)) -> c6(MARK(head(head(odds)))) A__INCR(cons(head(head(head(y0))), z1)) -> c6(MARK(head(head(head(y0))))) A__INCR(cons(head(head(tail(y0))), z1)) -> c6(MARK(head(head(tail(y0))))) A__INCR(cons(head(head(cons(y0, y1))), z1)) -> c6(MARK(head(head(cons(y0, y1))))) A__INCR(cons(head(head(s(y0))), z1)) -> c6(MARK(head(head(s(y0))))) 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A__TAIL(cons(z0, head(incr(odds)))) -> c10(MARK(head(incr(odds)))) A__TAIL(cons(z0, head(incr(head(y0))))) -> c10(MARK(head(incr(head(y0))))) A__TAIL(cons(z0, head(incr(tail(y0))))) -> c10(MARK(head(incr(tail(y0))))) A__TAIL(cons(z0, head(incr(cons(y0, y1))))) -> c10(MARK(head(incr(cons(y0, y1))))) A__TAIL(cons(z0, head(incr(s(y0))))) -> c10(MARK(head(incr(s(y0))))) A__TAIL(cons(z0, head(incr(y0)))) -> c10(MARK(head(incr(y0)))) A__TAIL(cons(z0, head(head(incr(y0))))) -> c10(MARK(head(head(incr(y0))))) A__TAIL(cons(z0, head(head(odds)))) -> c10(MARK(head(head(odds)))) A__TAIL(cons(z0, head(head(head(y0))))) -> c10(MARK(head(head(head(y0))))) A__TAIL(cons(z0, head(head(tail(y0))))) -> c10(MARK(head(head(tail(y0))))) A__TAIL(cons(z0, head(head(cons(y0, y1))))) -> c10(MARK(head(head(cons(y0, y1))))) A__TAIL(cons(z0, head(head(s(y0))))) -> c10(MARK(head(head(s(y0))))) A__TAIL(cons(z0, head(head(y0)))) -> c10(MARK(head(head(y0)))) A__TAIL(cons(z0, head(tail(nats)))) -> c10(MARK(head(tail(nats)))) A__TAIL(cons(z0, head(tail(incr(y0))))) -> c10(MARK(head(tail(incr(y0))))) A__TAIL(cons(z0, head(tail(pairs)))) -> c10(MARK(head(tail(pairs)))) A__TAIL(cons(z0, head(tail(odds)))) -> c10(MARK(head(tail(odds)))) A__TAIL(cons(z0, head(tail(head(y0))))) -> c10(MARK(head(tail(head(y0))))) A__TAIL(cons(z0, head(tail(tail(y0))))) -> c10(MARK(head(tail(tail(y0))))) A__TAIL(cons(z0, head(tail(cons(y0, y1))))) -> c10(MARK(head(tail(cons(y0, y1))))) A__TAIL(cons(z0, head(tail(s(y0))))) -> c10(MARK(head(tail(s(y0))))) A__TAIL(cons(z0, head(tail(y0)))) -> c10(MARK(head(tail(y0)))) A__TAIL(cons(z0, tail(incr(incr(y0))))) -> c10(MARK(tail(incr(incr(y0))))) A__TAIL(cons(z0, tail(incr(odds)))) -> c10(MARK(tail(incr(odds)))) A__TAIL(cons(z0, tail(incr(head(y0))))) -> c10(MARK(tail(incr(head(y0))))) A__TAIL(cons(z0, tail(incr(tail(y0))))) -> c10(MARK(tail(incr(tail(y0))))) A__TAIL(cons(z0, tail(incr(cons(y0, y1))))) -> c10(MARK(tail(incr(cons(y0, y1))))) A__TAIL(cons(z0, tail(incr(s(y0))))) -> c10(MARK(tail(incr(s(y0))))) A__TAIL(cons(z0, tail(incr(y0)))) -> c10(MARK(tail(incr(y0)))) A__TAIL(cons(z0, tail(head(incr(y0))))) -> c10(MARK(tail(head(incr(y0))))) A__TAIL(cons(z0, tail(head(odds)))) -> c10(MARK(tail(head(odds)))) A__TAIL(cons(z0, tail(head(head(y0))))) -> c10(MARK(tail(head(head(y0))))) A__TAIL(cons(z0, tail(head(tail(y0))))) -> c10(MARK(tail(head(tail(y0))))) A__TAIL(cons(z0, tail(head(cons(y0, y1))))) -> c10(MARK(tail(head(cons(y0, y1))))) A__TAIL(cons(z0, tail(head(s(y0))))) -> c10(MARK(tail(head(s(y0))))) A__TAIL(cons(z0, tail(head(y0)))) -> c10(MARK(tail(head(y0)))) A__TAIL(cons(z0, tail(tail(nats)))) -> c10(MARK(tail(tail(nats)))) A__TAIL(cons(z0, tail(tail(incr(y0))))) -> c10(MARK(tail(tail(incr(y0))))) A__TAIL(cons(z0, tail(tail(pairs)))) -> c10(MARK(tail(tail(pairs)))) A__TAIL(cons(z0, tail(tail(odds)))) -> c10(MARK(tail(tail(odds)))) A__TAIL(cons(z0, tail(tail(head(y0))))) -> c10(MARK(tail(tail(head(y0))))) A__TAIL(cons(z0, tail(tail(tail(y0))))) -> c10(MARK(tail(tail(tail(y0))))) A__TAIL(cons(z0, tail(tail(cons(y0, y1))))) -> c10(MARK(tail(tail(cons(y0, y1))))) A__TAIL(cons(z0, tail(tail(s(y0))))) -> c10(MARK(tail(tail(s(y0))))) A__TAIL(cons(z0, tail(tail(y0)))) -> c10(MARK(tail(tail(y0)))) A__TAIL(cons(z0, tail(pairs))) -> c10(MARK(tail(pairs))) A__TAIL(cons(z0, incr(incr(nats)))) -> c10(MARK(incr(incr(nats)))) A__TAIL(cons(z0, incr(incr(pairs)))) -> c10(MARK(incr(incr(pairs)))) A__TAIL(cons(z0, incr(odds))) -> c10(MARK(incr(odds))) A__TAIL(cons(z0, head(incr(nats)))) -> c10(MARK(head(incr(nats)))) A__TAIL(cons(z0, head(incr(pairs)))) -> c10(MARK(head(incr(pairs)))) A__TAIL(cons(z0, head(odds))) -> c10(MARK(head(odds))) A__TAIL(cons(z0, tail(incr(nats)))) -> c10(MARK(tail(incr(nats)))) A__TAIL(cons(z0, tail(incr(pairs)))) -> c10(MARK(tail(incr(pairs)))) A__TAIL(cons(z0, tail(odds))) -> c10(MARK(tail(odds))) A__TAIL(cons(z0, incr(cons(nats, y0)))) -> c10(MARK(incr(cons(nats, y0)))) A__TAIL(cons(z0, incr(cons(incr(y0), y1)))) -> c10(MARK(incr(cons(incr(y0), y1)))) A__TAIL(cons(z0, incr(cons(pairs, y0)))) -> c10(MARK(incr(cons(pairs, y0)))) A__TAIL(cons(z0, incr(cons(odds, y0)))) -> c10(MARK(incr(cons(odds, y0)))) A__TAIL(cons(z0, incr(cons(head(y0), y1)))) -> c10(MARK(incr(cons(head(y0), y1)))) A__TAIL(cons(z0, incr(cons(tail(y0), y1)))) -> c10(MARK(incr(cons(tail(y0), y1)))) A__TAIL(cons(z0, incr(cons(cons(y0, y1), y2)))) -> c10(MARK(incr(cons(cons(y0, y1), y2)))) A__TAIL(cons(z0, incr(cons(s(y0), y1)))) -> c10(MARK(incr(cons(s(y0), y1)))) A__TAIL(cons(z0, incr(cons(y0, y1)))) -> c10(MARK(incr(cons(y0, y1)))) A__TAIL(cons(z0, incr(cons(0, y0)))) -> c10(MARK(incr(cons(0, y0)))) A__TAIL(cons(z0, incr(head(nats)))) -> c10(MARK(incr(head(nats)))) A__TAIL(cons(z0, incr(head(pairs)))) -> c10(MARK(incr(head(pairs)))) A__TAIL(cons(z0, head(head(nats)))) -> c10(MARK(head(head(nats)))) A__TAIL(cons(z0, head(head(pairs)))) -> c10(MARK(head(head(pairs)))) A__TAIL(cons(z0, tail(head(nats)))) -> c10(MARK(tail(head(nats)))) A__TAIL(cons(z0, tail(head(pairs)))) -> c10(MARK(tail(head(pairs)))) A__TAIL(cons(z0, head(cons(nats, y0)))) -> c10(MARK(head(cons(nats, y0)))) A__TAIL(cons(z0, head(cons(incr(y0), y1)))) -> c10(MARK(head(cons(incr(y0), y1)))) A__TAIL(cons(z0, head(cons(pairs, y0)))) -> c10(MARK(head(cons(pairs, y0)))) A__TAIL(cons(z0, head(cons(odds, y0)))) -> c10(MARK(head(cons(odds, y0)))) A__TAIL(cons(z0, head(cons(head(y0), y1)))) -> c10(MARK(head(cons(head(y0), y1)))) A__TAIL(cons(z0, head(cons(tail(y0), y1)))) -> c10(MARK(head(cons(tail(y0), y1)))) A__TAIL(cons(z0, head(cons(cons(y0, y1), y2)))) -> c10(MARK(head(cons(cons(y0, y1), y2)))) A__TAIL(cons(z0, head(cons(s(y0), y1)))) -> c10(MARK(head(cons(s(y0), y1)))) A__TAIL(cons(z0, head(cons(y0, y1)))) -> c10(MARK(head(cons(y0, y1)))) A__TAIL(cons(z0, head(cons(0, y0)))) -> c10(MARK(head(cons(0, y0)))) MARK(incr(tail(nats))) -> c13(A__INCR(a__tail(a__nats))) MARK(head(tail(nats))) -> c16(A__HEAD(a__tail(a__nats))) MARK(tail(tail(nats))) -> c17(A__TAIL(a__tail(a__nats))) MARK(cons(cons(y0, y1), z1)) -> c18(MARK(cons(y0, y1))) MARK(cons(s(y0), z1)) -> c18(MARK(s(y0))) MARK(cons(incr(s(y0)), z1)) -> c18(MARK(incr(s(y0)))) MARK(cons(head(s(y0)), z1)) -> c18(MARK(head(s(y0)))) MARK(cons(tail(cons(y0, y1)), z1)) -> c18(MARK(tail(cons(y0, y1)))) MARK(cons(tail(s(y0)), z1)) -> c18(MARK(tail(s(y0)))) MARK(cons(incr(incr(incr(y0))), z1)) -> c18(MARK(incr(incr(incr(y0))))) MARK(cons(incr(incr(odds)), z1)) -> c18(MARK(incr(incr(odds)))) MARK(cons(incr(incr(head(y0))), z1)) -> c18(MARK(incr(incr(head(y0))))) MARK(cons(incr(incr(tail(y0))), z1)) -> c18(MARK(incr(incr(tail(y0))))) MARK(cons(incr(incr(cons(y0, y1))), z1)) -> c18(MARK(incr(incr(cons(y0, y1))))) MARK(cons(incr(incr(s(y0))), z1)) -> c18(MARK(incr(incr(s(y0))))) MARK(cons(incr(incr(y0)), z1)) -> c18(MARK(incr(incr(y0)))) MARK(cons(incr(head(incr(y0))), z1)) -> c18(MARK(incr(head(incr(y0))))) MARK(cons(incr(head(odds)), z1)) -> c18(MARK(incr(head(odds)))) MARK(cons(incr(head(head(y0))), z1)) -> c18(MARK(incr(head(head(y0))))) MARK(cons(incr(head(tail(y0))), z1)) -> c18(MARK(incr(head(tail(y0))))) MARK(cons(incr(head(cons(y0, y1))), z1)) -> c18(MARK(incr(head(cons(y0, y1))))) MARK(cons(incr(head(s(y0))), z1)) -> c18(MARK(incr(head(s(y0))))) MARK(cons(incr(head(y0)), z1)) -> c18(MARK(incr(head(y0)))) MARK(cons(incr(tail(incr(y0))), z1)) -> c18(MARK(incr(tail(incr(y0))))) MARK(cons(incr(tail(pairs)), z1)) -> c18(MARK(incr(tail(pairs)))) MARK(cons(incr(tail(odds)), z1)) -> c18(MARK(incr(tail(odds)))) MARK(cons(incr(tail(head(y0))), z1)) -> c18(MARK(incr(tail(head(y0))))) MARK(cons(incr(tail(tail(y0))), z1)) -> c18(MARK(incr(tail(tail(y0))))) MARK(cons(incr(tail(cons(y0, y1))), z1)) -> c18(MARK(incr(tail(cons(y0, y1))))) MARK(cons(incr(tail(s(y0))), z1)) -> c18(MARK(incr(tail(s(y0))))) MARK(cons(incr(tail(y0)), z1)) -> c18(MARK(incr(tail(y0)))) MARK(cons(head(incr(incr(y0))), z1)) -> c18(MARK(head(incr(incr(y0))))) MARK(cons(head(incr(odds)), z1)) -> c18(MARK(head(incr(odds)))) MARK(cons(head(incr(head(y0))), z1)) -> c18(MARK(head(incr(head(y0))))) MARK(cons(head(incr(tail(y0))), z1)) -> c18(MARK(head(incr(tail(y0))))) MARK(cons(head(incr(cons(y0, y1))), z1)) -> c18(MARK(head(incr(cons(y0, y1))))) MARK(cons(head(incr(s(y0))), z1)) -> c18(MARK(head(incr(s(y0))))) MARK(cons(head(incr(y0)), z1)) -> c18(MARK(head(incr(y0)))) MARK(cons(head(head(incr(y0))), z1)) -> c18(MARK(head(head(incr(y0))))) MARK(cons(head(head(odds)), z1)) -> c18(MARK(head(head(odds)))) MARK(cons(head(head(head(y0))), z1)) -> c18(MARK(head(head(head(y0))))) MARK(cons(head(head(tail(y0))), z1)) -> c18(MARK(head(head(tail(y0))))) MARK(cons(head(head(cons(y0, y1))), z1)) -> c18(MARK(head(head(cons(y0, y1))))) MARK(cons(head(head(s(y0))), z1)) -> c18(MARK(head(head(s(y0))))) MARK(cons(head(head(y0)), z1)) -> c18(MARK(head(head(y0)))) MARK(cons(head(tail(incr(y0))), z1)) -> c18(MARK(head(tail(incr(y0))))) MARK(cons(head(tail(pairs)), z1)) -> c18(MARK(head(tail(pairs)))) MARK(cons(head(tail(odds)), z1)) -> c18(MARK(head(tail(odds)))) MARK(cons(head(tail(head(y0))), z1)) -> c18(MARK(head(tail(head(y0))))) MARK(cons(head(tail(tail(y0))), z1)) -> c18(MARK(head(tail(tail(y0))))) MARK(cons(head(tail(cons(y0, y1))), z1)) -> c18(MARK(head(tail(cons(y0, y1))))) MARK(cons(head(tail(s(y0))), z1)) -> c18(MARK(head(tail(s(y0))))) MARK(cons(head(tail(y0)), z1)) -> c18(MARK(head(tail(y0)))) MARK(cons(tail(incr(incr(y0))), z1)) -> c18(MARK(tail(incr(incr(y0))))) MARK(cons(tail(incr(odds)), z1)) -> c18(MARK(tail(incr(odds)))) MARK(cons(tail(incr(head(y0))), z1)) -> c18(MARK(tail(incr(head(y0))))) MARK(cons(tail(incr(tail(y0))), z1)) -> c18(MARK(tail(incr(tail(y0))))) MARK(cons(tail(incr(cons(y0, y1))), z1)) -> c18(MARK(tail(incr(cons(y0, y1))))) MARK(cons(tail(incr(s(y0))), z1)) -> c18(MARK(tail(incr(s(y0))))) MARK(cons(tail(incr(y0)), z1)) -> c18(MARK(tail(incr(y0)))) MARK(cons(tail(head(incr(y0))), z1)) -> c18(MARK(tail(head(incr(y0))))) MARK(cons(tail(head(odds)), z1)) -> c18(MARK(tail(head(odds)))) MARK(cons(tail(head(head(y0))), z1)) -> c18(MARK(tail(head(head(y0))))) MARK(cons(tail(head(tail(y0))), z1)) -> c18(MARK(tail(head(tail(y0))))) MARK(cons(tail(head(cons(y0, y1))), z1)) -> c18(MARK(tail(head(cons(y0, y1))))) MARK(cons(tail(head(s(y0))), z1)) -> c18(MARK(tail(head(s(y0))))) MARK(cons(tail(head(y0)), z1)) -> c18(MARK(tail(head(y0)))) MARK(cons(tail(tail(incr(y0))), z1)) -> c18(MARK(tail(tail(incr(y0))))) MARK(cons(tail(tail(pairs)), z1)) -> c18(MARK(tail(tail(pairs)))) MARK(cons(tail(tail(odds)), z1)) -> c18(MARK(tail(tail(odds)))) MARK(cons(tail(tail(head(y0))), z1)) -> c18(MARK(tail(tail(head(y0))))) MARK(cons(tail(tail(tail(y0))), z1)) -> c18(MARK(tail(tail(tail(y0))))) MARK(cons(tail(tail(cons(y0, y1))), z1)) -> c18(MARK(tail(tail(cons(y0, y1))))) MARK(cons(tail(tail(s(y0))), z1)) -> c18(MARK(tail(tail(s(y0))))) MARK(cons(tail(tail(y0)), z1)) -> c18(MARK(tail(tail(y0)))) MARK(cons(tail(pairs), z1)) -> c18(MARK(tail(pairs))) MARK(cons(incr(incr(nats)), z1)) -> c18(MARK(incr(incr(nats)))) MARK(cons(incr(incr(pairs)), z1)) -> c18(MARK(incr(incr(pairs)))) MARK(cons(incr(odds), z1)) -> c18(MARK(incr(odds))) MARK(cons(head(incr(nats)), z1)) -> c18(MARK(head(incr(nats)))) MARK(cons(head(incr(pairs)), z1)) -> c18(MARK(head(incr(pairs)))) MARK(cons(head(odds), z1)) -> c18(MARK(head(odds))) MARK(cons(tail(incr(nats)), z1)) -> c18(MARK(tail(incr(nats)))) MARK(cons(tail(incr(pairs)), z1)) -> c18(MARK(tail(incr(pairs)))) MARK(cons(tail(odds), z1)) -> c18(MARK(tail(odds))) MARK(cons(incr(cons(nats, y0)), z1)) -> c18(MARK(incr(cons(nats, y0)))) MARK(cons(incr(cons(incr(y0), y1)), z1)) -> c18(MARK(incr(cons(incr(y0), y1)))) MARK(cons(incr(cons(pairs, y0)), z1)) -> c18(MARK(incr(cons(pairs, y0)))) MARK(cons(incr(cons(odds, y0)), z1)) -> c18(MARK(incr(cons(odds, y0)))) MARK(cons(incr(cons(head(y0), y1)), z1)) -> c18(MARK(incr(cons(head(y0), y1)))) MARK(cons(incr(cons(tail(y0), y1)), z1)) -> c18(MARK(incr(cons(tail(y0), y1)))) MARK(cons(incr(cons(cons(y0, y1), y2)), z1)) -> c18(MARK(incr(cons(cons(y0, y1), y2)))) MARK(cons(incr(cons(s(y0), y1)), z1)) -> c18(MARK(incr(cons(s(y0), y1)))) MARK(cons(incr(cons(y0, y1)), z1)) -> c18(MARK(incr(cons(y0, y1)))) MARK(cons(incr(cons(0, y0)), z1)) -> c18(MARK(incr(cons(0, y0)))) MARK(cons(incr(head(nats)), z1)) -> c18(MARK(incr(head(nats)))) MARK(cons(incr(head(pairs)), z1)) -> c18(MARK(incr(head(pairs)))) MARK(cons(head(head(nats)), z1)) -> c18(MARK(head(head(nats)))) MARK(cons(head(head(pairs)), z1)) -> c18(MARK(head(head(pairs)))) MARK(cons(tail(head(nats)), z1)) -> c18(MARK(tail(head(nats)))) MARK(cons(tail(head(pairs)), z1)) -> c18(MARK(tail(head(pairs)))) MARK(cons(head(cons(nats, y0)), z1)) -> c18(MARK(head(cons(nats, y0)))) MARK(cons(head(cons(incr(y0), y1)), z1)) -> c18(MARK(head(cons(incr(y0), y1)))) MARK(cons(head(cons(pairs, y0)), z1)) -> c18(MARK(head(cons(pairs, y0)))) MARK(cons(head(cons(odds, y0)), z1)) -> c18(MARK(head(cons(odds, y0)))) MARK(cons(head(cons(head(y0), y1)), z1)) -> c18(MARK(head(cons(head(y0), y1)))) MARK(cons(head(cons(tail(y0), y1)), z1)) -> c18(MARK(head(cons(tail(y0), y1)))) MARK(cons(head(cons(cons(y0, y1), y2)), z1)) -> c18(MARK(head(cons(cons(y0, y1), y2)))) MARK(cons(head(cons(s(y0), y1)), z1)) -> c18(MARK(head(cons(s(y0), y1)))) MARK(cons(head(cons(y0, y1)), z1)) -> c18(MARK(head(cons(y0, y1)))) MARK(cons(head(cons(0, y0)), z1)) -> c18(MARK(head(cons(0, y0)))) MARK(cons(incr(tail(nats)), z1)) -> c18(MARK(incr(tail(nats)))) MARK(cons(head(tail(nats)), z1)) -> c18(MARK(head(tail(nats)))) MARK(cons(tail(tail(nats)), z1)) -> c18(MARK(tail(tail(nats)))) MARK(incr(cons(nats, x1))) -> c13(A__INCR(cons(a__nats, x1))) MARK(incr(cons(pairs, x1))) -> c13(A__INCR(cons(a__pairs, x1))) MARK(incr(cons(odds, x1))) -> c13(A__INCR(cons(a__odds, x1))) MARK(head(cons(nats, x1))) -> c16(A__HEAD(cons(a__nats, x1))) MARK(head(cons(pairs, x1))) -> c16(A__HEAD(cons(a__pairs, x1))) MARK(head(cons(odds, x1))) -> c16(A__HEAD(cons(a__odds, x1))) MARK(s(s(y0))) -> c20(MARK(s(y0))) MARK(s(incr(s(y0)))) -> c20(MARK(incr(s(y0)))) MARK(s(head(s(y0)))) -> c20(MARK(head(s(y0)))) MARK(s(tail(cons(y0, y1)))) -> c20(MARK(tail(cons(y0, y1)))) MARK(s(tail(s(y0)))) -> c20(MARK(tail(s(y0)))) MARK(s(incr(incr(incr(y0))))) -> c20(MARK(incr(incr(incr(y0))))) MARK(s(incr(incr(odds)))) -> c20(MARK(incr(incr(odds)))) MARK(s(incr(incr(head(y0))))) -> c20(MARK(incr(incr(head(y0))))) MARK(s(incr(incr(tail(y0))))) -> c20(MARK(incr(incr(tail(y0))))) MARK(s(incr(incr(cons(y0, y1))))) -> c20(MARK(incr(incr(cons(y0, y1))))) MARK(s(incr(incr(s(y0))))) -> c20(MARK(incr(incr(s(y0))))) MARK(s(incr(incr(y0)))) -> c20(MARK(incr(incr(y0)))) MARK(s(incr(head(incr(y0))))) -> c20(MARK(incr(head(incr(y0))))) MARK(s(incr(head(odds)))) -> c20(MARK(incr(head(odds)))) MARK(s(incr(head(head(y0))))) -> c20(MARK(incr(head(head(y0))))) MARK(s(incr(head(tail(y0))))) -> c20(MARK(incr(head(tail(y0))))) MARK(s(incr(head(cons(y0, y1))))) -> c20(MARK(incr(head(cons(y0, y1))))) MARK(s(incr(head(s(y0))))) -> c20(MARK(incr(head(s(y0))))) MARK(s(incr(head(y0)))) -> c20(MARK(incr(head(y0)))) MARK(s(incr(tail(incr(y0))))) -> c20(MARK(incr(tail(incr(y0))))) MARK(s(incr(tail(pairs)))) -> c20(MARK(incr(tail(pairs)))) MARK(s(incr(tail(odds)))) -> c20(MARK(incr(tail(odds)))) MARK(s(incr(tail(head(y0))))) -> c20(MARK(incr(tail(head(y0))))) MARK(s(incr(tail(tail(y0))))) -> c20(MARK(incr(tail(tail(y0))))) MARK(s(incr(tail(cons(y0, y1))))) -> c20(MARK(incr(tail(cons(y0, y1))))) MARK(s(incr(tail(s(y0))))) -> c20(MARK(incr(tail(s(y0))))) MARK(s(incr(tail(y0)))) -> c20(MARK(incr(tail(y0)))) MARK(s(head(incr(incr(y0))))) -> c20(MARK(head(incr(incr(y0))))) MARK(s(head(incr(odds)))) -> c20(MARK(head(incr(odds)))) MARK(s(head(incr(head(y0))))) -> c20(MARK(head(incr(head(y0))))) MARK(s(head(incr(tail(y0))))) -> c20(MARK(head(incr(tail(y0))))) MARK(s(head(incr(cons(y0, y1))))) -> c20(MARK(head(incr(cons(y0, y1))))) MARK(s(head(incr(s(y0))))) -> c20(MARK(head(incr(s(y0))))) MARK(s(head(incr(y0)))) -> c20(MARK(head(incr(y0)))) MARK(s(head(head(incr(y0))))) -> c20(MARK(head(head(incr(y0))))) MARK(s(head(head(odds)))) -> c20(MARK(head(head(odds)))) MARK(s(head(head(head(y0))))) -> c20(MARK(head(head(head(y0))))) MARK(s(head(head(tail(y0))))) -> c20(MARK(head(head(tail(y0))))) MARK(s(head(head(cons(y0, y1))))) -> c20(MARK(head(head(cons(y0, y1))))) MARK(s(head(head(s(y0))))) -> c20(MARK(head(head(s(y0))))) MARK(s(head(head(y0)))) -> c20(MARK(head(head(y0)))) MARK(s(head(tail(incr(y0))))) -> c20(MARK(head(tail(incr(y0))))) MARK(s(head(tail(pairs)))) -> c20(MARK(head(tail(pairs)))) MARK(s(head(tail(odds)))) -> c20(MARK(head(tail(odds)))) MARK(s(head(tail(head(y0))))) -> c20(MARK(head(tail(head(y0))))) MARK(s(head(tail(tail(y0))))) -> c20(MARK(head(tail(tail(y0))))) MARK(s(head(tail(cons(y0, y1))))) -> c20(MARK(head(tail(cons(y0, y1))))) MARK(s(head(tail(s(y0))))) -> c20(MARK(head(tail(s(y0))))) MARK(s(head(tail(y0)))) -> c20(MARK(head(tail(y0)))) MARK(s(tail(incr(incr(y0))))) -> c20(MARK(tail(incr(incr(y0))))) MARK(s(tail(incr(odds)))) -> c20(MARK(tail(incr(odds)))) MARK(s(tail(incr(head(y0))))) -> c20(MARK(tail(incr(head(y0))))) MARK(s(tail(incr(tail(y0))))) -> c20(MARK(tail(incr(tail(y0))))) MARK(s(tail(incr(cons(y0, y1))))) -> c20(MARK(tail(incr(cons(y0, y1))))) MARK(s(tail(incr(s(y0))))) -> c20(MARK(tail(incr(s(y0))))) MARK(s(tail(incr(y0)))) -> c20(MARK(tail(incr(y0)))) MARK(s(tail(head(incr(y0))))) -> c20(MARK(tail(head(incr(y0))))) MARK(s(tail(head(odds)))) -> c20(MARK(tail(head(odds)))) MARK(s(tail(head(head(y0))))) -> c20(MARK(tail(head(head(y0))))) MARK(s(tail(head(tail(y0))))) -> c20(MARK(tail(head(tail(y0))))) MARK(s(tail(head(cons(y0, y1))))) -> c20(MARK(tail(head(cons(y0, y1))))) MARK(s(tail(head(s(y0))))) -> c20(MARK(tail(head(s(y0))))) MARK(s(tail(head(y0)))) -> c20(MARK(tail(head(y0)))) MARK(s(tail(tail(incr(y0))))) -> c20(MARK(tail(tail(incr(y0))))) MARK(s(tail(tail(pairs)))) -> c20(MARK(tail(tail(pairs)))) MARK(s(tail(tail(odds)))) -> c20(MARK(tail(tail(odds)))) MARK(s(tail(tail(head(y0))))) -> c20(MARK(tail(tail(head(y0))))) MARK(s(tail(tail(tail(y0))))) -> c20(MARK(tail(tail(tail(y0))))) MARK(s(tail(tail(cons(y0, y1))))) -> c20(MARK(tail(tail(cons(y0, y1))))) MARK(s(tail(tail(s(y0))))) -> c20(MARK(tail(tail(s(y0))))) MARK(s(tail(tail(y0)))) -> c20(MARK(tail(tail(y0)))) MARK(s(tail(pairs))) -> c20(MARK(tail(pairs))) MARK(s(incr(incr(nats)))) -> c20(MARK(incr(incr(nats)))) MARK(s(incr(incr(pairs)))) -> c20(MARK(incr(incr(pairs)))) MARK(s(incr(odds))) -> c20(MARK(incr(odds))) MARK(s(head(incr(nats)))) -> c20(MARK(head(incr(nats)))) MARK(s(head(incr(pairs)))) -> c20(MARK(head(incr(pairs)))) MARK(s(head(odds))) -> c20(MARK(head(odds))) MARK(s(tail(incr(nats)))) -> c20(MARK(tail(incr(nats)))) MARK(s(tail(incr(pairs)))) -> c20(MARK(tail(incr(pairs)))) MARK(s(tail(odds))) -> c20(MARK(tail(odds))) MARK(s(incr(cons(incr(y0), y1)))) -> c20(MARK(incr(cons(incr(y0), y1)))) MARK(s(incr(cons(head(y0), y1)))) -> c20(MARK(incr(cons(head(y0), y1)))) MARK(s(incr(cons(tail(y0), y1)))) -> c20(MARK(incr(cons(tail(y0), y1)))) MARK(s(incr(cons(cons(y0, y1), y2)))) -> c20(MARK(incr(cons(cons(y0, y1), y2)))) MARK(s(incr(cons(s(y0), y1)))) -> c20(MARK(incr(cons(s(y0), y1)))) MARK(s(incr(cons(y0, y1)))) -> c20(MARK(incr(cons(y0, y1)))) MARK(s(incr(head(nats)))) -> c20(MARK(incr(head(nats)))) MARK(s(incr(head(pairs)))) -> c20(MARK(incr(head(pairs)))) MARK(s(head(head(nats)))) -> c20(MARK(head(head(nats)))) MARK(s(head(head(pairs)))) -> c20(MARK(head(head(pairs)))) MARK(s(tail(head(nats)))) -> c20(MARK(tail(head(nats)))) MARK(s(tail(head(pairs)))) -> c20(MARK(tail(head(pairs)))) MARK(s(head(cons(incr(y0), y1)))) -> c20(MARK(head(cons(incr(y0), y1)))) MARK(s(head(cons(head(y0), y1)))) -> c20(MARK(head(cons(head(y0), y1)))) MARK(s(head(cons(tail(y0), y1)))) -> c20(MARK(head(cons(tail(y0), y1)))) MARK(s(head(cons(cons(y0, y1), y2)))) -> c20(MARK(head(cons(cons(y0, y1), y2)))) MARK(s(head(cons(s(y0), y1)))) -> c20(MARK(head(cons(s(y0), y1)))) MARK(s(head(cons(y0, y1)))) -> c20(MARK(head(cons(y0, y1)))) MARK(s(incr(tail(nats)))) -> c20(MARK(incr(tail(nats)))) MARK(s(head(tail(nats)))) -> c20(MARK(head(tail(nats)))) MARK(s(tail(tail(nats)))) -> c20(MARK(tail(tail(nats)))) MARK(s(cons(cons(y0, y1), y2))) -> c20(MARK(cons(cons(y0, y1), y2))) MARK(s(cons(s(y0), y1))) -> c20(MARK(cons(s(y0), y1))) MARK(s(cons(incr(s(y0)), y1))) -> c20(MARK(cons(incr(s(y0)), y1))) MARK(s(cons(head(s(y0)), y1))) -> c20(MARK(cons(head(s(y0)), y1))) MARK(s(cons(tail(cons(y0, y1)), y2))) -> c20(MARK(cons(tail(cons(y0, y1)), y2))) MARK(s(cons(tail(s(y0)), y1))) -> c20(MARK(cons(tail(s(y0)), y1))) MARK(s(cons(incr(incr(incr(y0))), y1))) -> c20(MARK(cons(incr(incr(incr(y0))), y1))) MARK(s(cons(incr(incr(odds)), y0))) -> c20(MARK(cons(incr(incr(odds)), y0))) MARK(s(cons(incr(incr(head(y0))), y1))) -> c20(MARK(cons(incr(incr(head(y0))), y1))) MARK(s(cons(incr(incr(tail(y0))), y1))) -> c20(MARK(cons(incr(incr(tail(y0))), y1))) MARK(s(cons(incr(incr(cons(y0, y1))), y2))) -> c20(MARK(cons(incr(incr(cons(y0, y1))), y2))) MARK(s(cons(incr(incr(s(y0))), y1))) -> c20(MARK(cons(incr(incr(s(y0))), y1))) MARK(s(cons(incr(incr(y0)), y1))) -> c20(MARK(cons(incr(incr(y0)), y1))) MARK(s(cons(incr(head(incr(y0))), y1))) -> c20(MARK(cons(incr(head(incr(y0))), y1))) MARK(s(cons(incr(head(odds)), y0))) -> c20(MARK(cons(incr(head(odds)), y0))) MARK(s(cons(incr(head(head(y0))), y1))) -> c20(MARK(cons(incr(head(head(y0))), y1))) MARK(s(cons(incr(head(tail(y0))), y1))) -> c20(MARK(cons(incr(head(tail(y0))), y1))) MARK(s(cons(incr(head(cons(y0, y1))), y2))) -> c20(MARK(cons(incr(head(cons(y0, y1))), y2))) MARK(s(cons(incr(head(s(y0))), y1))) -> c20(MARK(cons(incr(head(s(y0))), y1))) MARK(s(cons(incr(head(y0)), y1))) -> c20(MARK(cons(incr(head(y0)), y1))) MARK(s(cons(incr(tail(incr(y0))), y1))) -> c20(MARK(cons(incr(tail(incr(y0))), y1))) MARK(s(cons(incr(tail(pairs)), y0))) -> c20(MARK(cons(incr(tail(pairs)), y0))) MARK(s(cons(incr(tail(odds)), y0))) -> c20(MARK(cons(incr(tail(odds)), y0))) MARK(s(cons(incr(tail(head(y0))), y1))) -> c20(MARK(cons(incr(tail(head(y0))), y1))) MARK(s(cons(incr(tail(tail(y0))), y1))) -> c20(MARK(cons(incr(tail(tail(y0))), y1))) MARK(s(cons(incr(tail(cons(y0, y1))), y2))) -> c20(MARK(cons(incr(tail(cons(y0, y1))), y2))) MARK(s(cons(incr(tail(s(y0))), y1))) -> c20(MARK(cons(incr(tail(s(y0))), y1))) MARK(s(cons(incr(tail(y0)), y1))) -> c20(MARK(cons(incr(tail(y0)), y1))) MARK(s(cons(head(incr(incr(y0))), y1))) -> c20(MARK(cons(head(incr(incr(y0))), y1))) MARK(s(cons(head(incr(odds)), y0))) -> c20(MARK(cons(head(incr(odds)), y0))) MARK(s(cons(head(incr(head(y0))), y1))) -> c20(MARK(cons(head(incr(head(y0))), y1))) MARK(s(cons(head(incr(tail(y0))), y1))) -> c20(MARK(cons(head(incr(tail(y0))), y1))) MARK(s(cons(head(incr(cons(y0, y1))), y2))) -> c20(MARK(cons(head(incr(cons(y0, y1))), y2))) MARK(s(cons(head(incr(s(y0))), y1))) -> c20(MARK(cons(head(incr(s(y0))), y1))) MARK(s(cons(head(incr(y0)), y1))) -> c20(MARK(cons(head(incr(y0)), y1))) MARK(s(cons(head(head(incr(y0))), y1))) -> c20(MARK(cons(head(head(incr(y0))), y1))) MARK(s(cons(head(head(odds)), y0))) -> c20(MARK(cons(head(head(odds)), y0))) MARK(s(cons(head(head(head(y0))), y1))) -> c20(MARK(cons(head(head(head(y0))), y1))) MARK(s(cons(head(head(tail(y0))), y1))) -> c20(MARK(cons(head(head(tail(y0))), y1))) MARK(s(cons(head(head(cons(y0, y1))), y2))) -> c20(MARK(cons(head(head(cons(y0, y1))), y2))) MARK(s(cons(head(head(s(y0))), y1))) -> c20(MARK(cons(head(head(s(y0))), y1))) MARK(s(cons(head(head(y0)), y1))) -> c20(MARK(cons(head(head(y0)), y1))) MARK(s(cons(head(tail(incr(y0))), y1))) -> c20(MARK(cons(head(tail(incr(y0))), y1))) MARK(s(cons(head(tail(pairs)), y0))) -> c20(MARK(cons(head(tail(pairs)), y0))) MARK(s(cons(head(tail(odds)), y0))) -> c20(MARK(cons(head(tail(odds)), y0))) MARK(s(cons(head(tail(head(y0))), y1))) -> c20(MARK(cons(head(tail(head(y0))), y1))) MARK(s(cons(head(tail(tail(y0))), y1))) -> c20(MARK(cons(head(tail(tail(y0))), y1))) MARK(s(cons(head(tail(cons(y0, y1))), y2))) -> c20(MARK(cons(head(tail(cons(y0, y1))), y2))) MARK(s(cons(head(tail(s(y0))), y1))) -> c20(MARK(cons(head(tail(s(y0))), y1))) MARK(s(cons(head(tail(y0)), y1))) -> c20(MARK(cons(head(tail(y0)), y1))) MARK(s(cons(tail(incr(incr(y0))), y1))) -> c20(MARK(cons(tail(incr(incr(y0))), y1))) MARK(s(cons(tail(incr(odds)), y0))) -> c20(MARK(cons(tail(incr(odds)), y0))) MARK(s(cons(tail(incr(head(y0))), y1))) -> c20(MARK(cons(tail(incr(head(y0))), y1))) MARK(s(cons(tail(incr(tail(y0))), y1))) -> c20(MARK(cons(tail(incr(tail(y0))), y1))) MARK(s(cons(tail(incr(cons(y0, y1))), y2))) -> c20(MARK(cons(tail(incr(cons(y0, y1))), y2))) MARK(s(cons(tail(incr(s(y0))), y1))) -> c20(MARK(cons(tail(incr(s(y0))), y1))) MARK(s(cons(tail(incr(y0)), y1))) -> c20(MARK(cons(tail(incr(y0)), y1))) MARK(s(cons(tail(head(incr(y0))), y1))) -> c20(MARK(cons(tail(head(incr(y0))), y1))) MARK(s(cons(tail(head(odds)), y0))) -> c20(MARK(cons(tail(head(odds)), y0))) MARK(s(cons(tail(head(head(y0))), y1))) -> c20(MARK(cons(tail(head(head(y0))), y1))) MARK(s(cons(tail(head(tail(y0))), y1))) -> c20(MARK(cons(tail(head(tail(y0))), y1))) MARK(s(cons(tail(head(cons(y0, y1))), y2))) -> c20(MARK(cons(tail(head(cons(y0, y1))), y2))) MARK(s(cons(tail(head(s(y0))), y1))) -> c20(MARK(cons(tail(head(s(y0))), y1))) MARK(s(cons(tail(head(y0)), y1))) -> c20(MARK(cons(tail(head(y0)), y1))) MARK(s(cons(tail(tail(incr(y0))), y1))) -> c20(MARK(cons(tail(tail(incr(y0))), y1))) MARK(s(cons(tail(tail(pairs)), y0))) -> c20(MARK(cons(tail(tail(pairs)), y0))) MARK(s(cons(tail(tail(odds)), y0))) -> c20(MARK(cons(tail(tail(odds)), y0))) MARK(s(cons(tail(tail(head(y0))), y1))) -> c20(MARK(cons(tail(tail(head(y0))), y1))) MARK(s(cons(tail(tail(tail(y0))), y1))) -> c20(MARK(cons(tail(tail(tail(y0))), y1))) MARK(s(cons(tail(tail(cons(y0, y1))), y2))) -> c20(MARK(cons(tail(tail(cons(y0, y1))), y2))) MARK(s(cons(tail(tail(s(y0))), y1))) -> c20(MARK(cons(tail(tail(s(y0))), y1))) MARK(s(cons(tail(tail(y0)), y1))) -> c20(MARK(cons(tail(tail(y0)), y1))) MARK(s(cons(tail(pairs), y0))) -> c20(MARK(cons(tail(pairs), y0))) MARK(s(cons(incr(incr(nats)), y0))) -> c20(MARK(cons(incr(incr(nats)), y0))) MARK(s(cons(incr(incr(pairs)), y0))) -> c20(MARK(cons(incr(incr(pairs)), y0))) MARK(s(cons(incr(odds), y0))) -> c20(MARK(cons(incr(odds), y0))) MARK(s(cons(head(incr(nats)), y0))) -> c20(MARK(cons(head(incr(nats)), y0))) MARK(s(cons(head(incr(pairs)), y0))) -> c20(MARK(cons(head(incr(pairs)), y0))) MARK(s(cons(head(odds), y0))) -> c20(MARK(cons(head(odds), y0))) MARK(s(cons(tail(incr(nats)), y0))) -> c20(MARK(cons(tail(incr(nats)), y0))) MARK(s(cons(tail(incr(pairs)), y0))) -> c20(MARK(cons(tail(incr(pairs)), y0))) MARK(s(cons(tail(odds), y0))) -> c20(MARK(cons(tail(odds), y0))) MARK(s(cons(incr(cons(nats, y0)), y1))) -> c20(MARK(cons(incr(cons(nats, y0)), y1))) MARK(s(cons(incr(cons(incr(y0), y1)), y2))) -> c20(MARK(cons(incr(cons(incr(y0), y1)), y2))) MARK(s(cons(incr(cons(pairs, y0)), y1))) -> c20(MARK(cons(incr(cons(pairs, y0)), y1))) MARK(s(cons(incr(cons(odds, y0)), y1))) -> c20(MARK(cons(incr(cons(odds, y0)), y1))) MARK(s(cons(incr(cons(head(y0), y1)), y2))) -> c20(MARK(cons(incr(cons(head(y0), y1)), y2))) MARK(s(cons(incr(cons(tail(y0), y1)), y2))) -> c20(MARK(cons(incr(cons(tail(y0), y1)), y2))) MARK(s(cons(incr(cons(cons(y0, y1), y2)), y3))) -> c20(MARK(cons(incr(cons(cons(y0, y1), y2)), y3))) MARK(s(cons(incr(cons(s(y0), y1)), y2))) -> c20(MARK(cons(incr(cons(s(y0), y1)), y2))) MARK(s(cons(incr(cons(y0, y1)), y2))) -> c20(MARK(cons(incr(cons(y0, y1)), y2))) MARK(s(cons(incr(cons(0, y0)), y1))) -> c20(MARK(cons(incr(cons(0, y0)), y1))) MARK(s(cons(incr(head(nats)), y0))) -> c20(MARK(cons(incr(head(nats)), y0))) MARK(s(cons(incr(head(pairs)), y0))) -> c20(MARK(cons(incr(head(pairs)), y0))) MARK(s(cons(head(head(nats)), y0))) -> c20(MARK(cons(head(head(nats)), y0))) MARK(s(cons(head(head(pairs)), y0))) -> c20(MARK(cons(head(head(pairs)), y0))) MARK(s(cons(tail(head(nats)), y0))) -> c20(MARK(cons(tail(head(nats)), y0))) MARK(s(cons(tail(head(pairs)), y0))) -> c20(MARK(cons(tail(head(pairs)), y0))) MARK(s(cons(head(cons(nats, y0)), y1))) -> c20(MARK(cons(head(cons(nats, y0)), y1))) MARK(s(cons(head(cons(incr(y0), y1)), y2))) -> c20(MARK(cons(head(cons(incr(y0), y1)), y2))) MARK(s(cons(head(cons(pairs, y0)), y1))) -> c20(MARK(cons(head(cons(pairs, y0)), y1))) MARK(s(cons(head(cons(odds, y0)), y1))) -> c20(MARK(cons(head(cons(odds, y0)), y1))) MARK(s(cons(head(cons(head(y0), y1)), y2))) -> c20(MARK(cons(head(cons(head(y0), y1)), y2))) MARK(s(cons(head(cons(tail(y0), y1)), y2))) -> c20(MARK(cons(head(cons(tail(y0), y1)), y2))) MARK(s(cons(head(cons(cons(y0, y1), y2)), y3))) -> c20(MARK(cons(head(cons(cons(y0, y1), y2)), y3))) MARK(s(cons(head(cons(s(y0), y1)), y2))) -> c20(MARK(cons(head(cons(s(y0), y1)), y2))) MARK(s(cons(head(cons(y0, y1)), y2))) -> c20(MARK(cons(head(cons(y0, y1)), y2))) MARK(s(cons(head(cons(0, y0)), y1))) -> c20(MARK(cons(head(cons(0, y0)), y1))) MARK(s(cons(incr(tail(nats)), y0))) -> c20(MARK(cons(incr(tail(nats)), y0))) MARK(s(cons(head(tail(nats)), y0))) -> c20(MARK(cons(head(tail(nats)), y0))) MARK(s(cons(tail(tail(nats)), y0))) -> c20(MARK(cons(tail(tail(nats)), y0))) MARK(s(incr(cons(nats, y0)))) -> c20(MARK(incr(cons(nats, y0)))) MARK(s(incr(cons(pairs, y0)))) -> c20(MARK(incr(cons(pairs, y0)))) MARK(s(incr(cons(odds, y0)))) -> c20(MARK(incr(cons(odds, y0)))) MARK(s(head(cons(nats, y0)))) -> c20(MARK(head(cons(nats, y0)))) MARK(s(head(cons(pairs, y0)))) -> c20(MARK(head(cons(pairs, y0)))) MARK(s(head(cons(odds, y0)))) -> c20(MARK(head(cons(odds, y0)))) MARK(incr(s(s(y0)))) -> c13(MARK(s(s(y0)))) MARK(incr(s(incr(s(y0))))) -> c13(MARK(s(incr(s(y0))))) MARK(incr(s(head(s(y0))))) -> c13(MARK(s(head(s(y0))))) MARK(incr(s(tail(cons(y0, y1))))) -> c13(MARK(s(tail(cons(y0, y1))))) MARK(incr(s(tail(s(y0))))) -> c13(MARK(s(tail(s(y0))))) MARK(incr(s(incr(incr(incr(y0)))))) -> c13(MARK(s(incr(incr(incr(y0)))))) MARK(incr(s(incr(incr(odds))))) -> c13(MARK(s(incr(incr(odds))))) MARK(incr(s(incr(incr(head(y0)))))) -> c13(MARK(s(incr(incr(head(y0)))))) MARK(incr(s(incr(incr(tail(y0)))))) -> c13(MARK(s(incr(incr(tail(y0)))))) MARK(incr(s(incr(incr(cons(y0, y1)))))) -> c13(MARK(s(incr(incr(cons(y0, y1)))))) MARK(incr(s(incr(incr(s(y0)))))) -> c13(MARK(s(incr(incr(s(y0)))))) MARK(incr(s(incr(incr(y0))))) -> c13(MARK(s(incr(incr(y0))))) MARK(incr(s(incr(head(incr(y0)))))) -> c13(MARK(s(incr(head(incr(y0)))))) MARK(incr(s(incr(head(odds))))) -> c13(MARK(s(incr(head(odds))))) MARK(incr(s(incr(head(head(y0)))))) -> c13(MARK(s(incr(head(head(y0)))))) MARK(incr(s(incr(head(tail(y0)))))) -> c13(MARK(s(incr(head(tail(y0)))))) MARK(incr(s(incr(head(cons(y0, y1)))))) -> c13(MARK(s(incr(head(cons(y0, y1)))))) MARK(incr(s(incr(head(s(y0)))))) -> c13(MARK(s(incr(head(s(y0)))))) MARK(incr(s(incr(head(y0))))) -> c13(MARK(s(incr(head(y0))))) MARK(incr(s(incr(tail(incr(y0)))))) -> c13(MARK(s(incr(tail(incr(y0)))))) MARK(incr(s(incr(tail(pairs))))) -> c13(MARK(s(incr(tail(pairs))))) MARK(incr(s(incr(tail(odds))))) -> c13(MARK(s(incr(tail(odds))))) MARK(incr(s(incr(tail(head(y0)))))) -> c13(MARK(s(incr(tail(head(y0)))))) MARK(incr(s(incr(tail(tail(y0)))))) -> c13(MARK(s(incr(tail(tail(y0)))))) MARK(incr(s(incr(tail(cons(y0, y1)))))) -> c13(MARK(s(incr(tail(cons(y0, y1)))))) MARK(incr(s(incr(tail(s(y0)))))) -> c13(MARK(s(incr(tail(s(y0)))))) MARK(incr(s(incr(tail(y0))))) -> c13(MARK(s(incr(tail(y0))))) MARK(incr(s(head(incr(incr(y0)))))) -> c13(MARK(s(head(incr(incr(y0)))))) MARK(incr(s(head(incr(odds))))) -> c13(MARK(s(head(incr(odds))))) MARK(incr(s(head(incr(head(y0)))))) -> c13(MARK(s(head(incr(head(y0)))))) MARK(incr(s(head(incr(tail(y0)))))) -> c13(MARK(s(head(incr(tail(y0)))))) MARK(incr(s(head(incr(cons(y0, y1)))))) -> c13(MARK(s(head(incr(cons(y0, y1)))))) MARK(incr(s(head(incr(s(y0)))))) -> c13(MARK(s(head(incr(s(y0)))))) MARK(incr(s(head(incr(y0))))) -> c13(MARK(s(head(incr(y0))))) MARK(incr(s(head(head(incr(y0)))))) -> c13(MARK(s(head(head(incr(y0)))))) MARK(incr(s(head(head(odds))))) -> c13(MARK(s(head(head(odds))))) MARK(incr(s(head(head(head(y0)))))) -> c13(MARK(s(head(head(head(y0)))))) MARK(incr(s(head(head(tail(y0)))))) -> c13(MARK(s(head(head(tail(y0)))))) MARK(incr(s(head(head(cons(y0, y1)))))) -> c13(MARK(s(head(head(cons(y0, y1)))))) MARK(incr(s(head(head(s(y0)))))) -> c13(MARK(s(head(head(s(y0)))))) MARK(incr(s(head(head(y0))))) -> c13(MARK(s(head(head(y0))))) MARK(incr(s(head(tail(incr(y0)))))) -> c13(MARK(s(head(tail(incr(y0)))))) MARK(incr(s(head(tail(pairs))))) -> c13(MARK(s(head(tail(pairs))))) MARK(incr(s(head(tail(odds))))) -> c13(MARK(s(head(tail(odds))))) MARK(incr(s(head(tail(head(y0)))))) -> c13(MARK(s(head(tail(head(y0)))))) MARK(incr(s(head(tail(tail(y0)))))) -> c13(MARK(s(head(tail(tail(y0)))))) MARK(incr(s(head(tail(cons(y0, y1)))))) -> c13(MARK(s(head(tail(cons(y0, y1)))))) MARK(incr(s(head(tail(s(y0)))))) -> c13(MARK(s(head(tail(s(y0)))))) MARK(incr(s(head(tail(y0))))) -> c13(MARK(s(head(tail(y0))))) MARK(incr(s(tail(incr(incr(y0)))))) -> c13(MARK(s(tail(incr(incr(y0)))))) MARK(incr(s(tail(incr(odds))))) -> c13(MARK(s(tail(incr(odds))))) MARK(incr(s(tail(incr(head(y0)))))) -> c13(MARK(s(tail(incr(head(y0)))))) MARK(incr(s(tail(incr(tail(y0)))))) -> c13(MARK(s(tail(incr(tail(y0)))))) MARK(incr(s(tail(incr(cons(y0, y1)))))) -> c13(MARK(s(tail(incr(cons(y0, y1)))))) MARK(incr(s(tail(incr(s(y0)))))) -> c13(MARK(s(tail(incr(s(y0)))))) MARK(incr(s(tail(incr(y0))))) -> c13(MARK(s(tail(incr(y0))))) MARK(incr(s(tail(head(incr(y0)))))) -> c13(MARK(s(tail(head(incr(y0)))))) MARK(incr(s(tail(head(odds))))) -> c13(MARK(s(tail(head(odds))))) MARK(incr(s(tail(head(head(y0)))))) -> c13(MARK(s(tail(head(head(y0)))))) MARK(incr(s(tail(head(tail(y0)))))) -> c13(MARK(s(tail(head(tail(y0)))))) MARK(incr(s(tail(head(cons(y0, y1)))))) -> c13(MARK(s(tail(head(cons(y0, y1)))))) MARK(incr(s(tail(head(s(y0)))))) -> c13(MARK(s(tail(head(s(y0)))))) MARK(incr(s(tail(head(y0))))) -> c13(MARK(s(tail(head(y0))))) MARK(incr(s(tail(tail(incr(y0)))))) -> c13(MARK(s(tail(tail(incr(y0)))))) MARK(incr(s(tail(tail(pairs))))) -> c13(MARK(s(tail(tail(pairs))))) MARK(incr(s(tail(tail(odds))))) -> c13(MARK(s(tail(tail(odds))))) MARK(incr(s(tail(tail(head(y0)))))) -> c13(MARK(s(tail(tail(head(y0)))))) MARK(incr(s(tail(tail(tail(y0)))))) -> c13(MARK(s(tail(tail(tail(y0)))))) MARK(incr(s(tail(tail(cons(y0, y1)))))) -> c13(MARK(s(tail(tail(cons(y0, y1)))))) MARK(incr(s(tail(tail(s(y0)))))) -> c13(MARK(s(tail(tail(s(y0)))))) MARK(incr(s(tail(tail(y0))))) -> c13(MARK(s(tail(tail(y0))))) MARK(incr(s(tail(pairs)))) -> c13(MARK(s(tail(pairs)))) MARK(incr(s(incr(incr(nats))))) -> c13(MARK(s(incr(incr(nats))))) MARK(incr(s(incr(incr(pairs))))) -> c13(MARK(s(incr(incr(pairs))))) MARK(incr(s(incr(odds)))) -> c13(MARK(s(incr(odds)))) MARK(incr(s(head(incr(nats))))) -> c13(MARK(s(head(incr(nats))))) MARK(incr(s(head(incr(pairs))))) -> c13(MARK(s(head(incr(pairs))))) MARK(incr(s(head(odds)))) -> c13(MARK(s(head(odds)))) MARK(incr(s(tail(incr(nats))))) -> c13(MARK(s(tail(incr(nats))))) MARK(incr(s(tail(incr(pairs))))) -> c13(MARK(s(tail(incr(pairs))))) MARK(incr(s(tail(odds)))) -> c13(MARK(s(tail(odds)))) MARK(incr(s(incr(cons(incr(y0), y1))))) -> c13(MARK(s(incr(cons(incr(y0), y1))))) MARK(incr(s(incr(cons(head(y0), y1))))) -> c13(MARK(s(incr(cons(head(y0), y1))))) MARK(incr(s(incr(cons(tail(y0), y1))))) -> c13(MARK(s(incr(cons(tail(y0), y1))))) MARK(incr(s(incr(cons(cons(y0, y1), y2))))) -> c13(MARK(s(incr(cons(cons(y0, y1), y2))))) MARK(incr(s(incr(cons(s(y0), y1))))) -> c13(MARK(s(incr(cons(s(y0), y1))))) MARK(incr(s(incr(cons(y0, y1))))) -> c13(MARK(s(incr(cons(y0, y1))))) MARK(incr(s(incr(head(nats))))) -> c13(MARK(s(incr(head(nats))))) MARK(incr(s(incr(head(pairs))))) -> c13(MARK(s(incr(head(pairs))))) MARK(incr(s(head(head(nats))))) -> c13(MARK(s(head(head(nats))))) MARK(incr(s(head(head(pairs))))) -> c13(MARK(s(head(head(pairs))))) MARK(incr(s(tail(head(nats))))) -> c13(MARK(s(tail(head(nats))))) MARK(incr(s(tail(head(pairs))))) -> c13(MARK(s(tail(head(pairs))))) MARK(incr(s(head(cons(incr(y0), y1))))) -> c13(MARK(s(head(cons(incr(y0), y1))))) MARK(incr(s(head(cons(head(y0), y1))))) -> c13(MARK(s(head(cons(head(y0), y1))))) MARK(incr(s(head(cons(tail(y0), y1))))) -> c13(MARK(s(head(cons(tail(y0), y1))))) MARK(incr(s(head(cons(cons(y0, y1), y2))))) -> c13(MARK(s(head(cons(cons(y0, y1), y2))))) MARK(incr(s(head(cons(s(y0), y1))))) -> c13(MARK(s(head(cons(s(y0), y1))))) MARK(incr(s(head(cons(y0, y1))))) -> c13(MARK(s(head(cons(y0, y1))))) MARK(incr(s(incr(tail(nats))))) -> c13(MARK(s(incr(tail(nats))))) MARK(incr(s(head(tail(nats))))) -> c13(MARK(s(head(tail(nats))))) MARK(incr(s(tail(tail(nats))))) -> c13(MARK(s(tail(tail(nats))))) MARK(incr(s(cons(cons(y0, y1), y2)))) -> c13(MARK(s(cons(cons(y0, y1), y2)))) MARK(incr(s(cons(s(y0), y1)))) -> c13(MARK(s(cons(s(y0), y1)))) MARK(incr(s(cons(incr(s(y0)), y1)))) -> c13(MARK(s(cons(incr(s(y0)), y1)))) MARK(incr(s(cons(head(s(y0)), y1)))) -> c13(MARK(s(cons(head(s(y0)), y1)))) MARK(incr(s(cons(tail(cons(y0, y1)), y2)))) -> c13(MARK(s(cons(tail(cons(y0, y1)), y2)))) MARK(incr(s(cons(tail(s(y0)), y1)))) -> c13(MARK(s(cons(tail(s(y0)), y1)))) MARK(incr(s(cons(incr(incr(incr(y0))), y1)))) -> c13(MARK(s(cons(incr(incr(incr(y0))), y1)))) MARK(incr(s(cons(incr(incr(odds)), y0)))) -> c13(MARK(s(cons(incr(incr(odds)), y0)))) MARK(incr(s(cons(incr(incr(head(y0))), y1)))) -> c13(MARK(s(cons(incr(incr(head(y0))), y1)))) MARK(incr(s(cons(incr(incr(tail(y0))), y1)))) -> c13(MARK(s(cons(incr(incr(tail(y0))), y1)))) MARK(incr(s(cons(incr(incr(cons(y0, y1))), y2)))) -> c13(MARK(s(cons(incr(incr(cons(y0, y1))), y2)))) MARK(incr(s(cons(incr(incr(s(y0))), y1)))) -> c13(MARK(s(cons(incr(incr(s(y0))), y1)))) MARK(incr(s(cons(incr(incr(y0)), y1)))) -> c13(MARK(s(cons(incr(incr(y0)), y1)))) MARK(incr(s(cons(incr(head(incr(y0))), y1)))) -> c13(MARK(s(cons(incr(head(incr(y0))), y1)))) MARK(incr(s(cons(incr(head(odds)), y0)))) -> c13(MARK(s(cons(incr(head(odds)), y0)))) MARK(incr(s(cons(incr(head(head(y0))), y1)))) -> c13(MARK(s(cons(incr(head(head(y0))), y1)))) MARK(incr(s(cons(incr(head(tail(y0))), y1)))) -> c13(MARK(s(cons(incr(head(tail(y0))), y1)))) MARK(incr(s(cons(incr(head(cons(y0, y1))), y2)))) -> c13(MARK(s(cons(incr(head(cons(y0, y1))), y2)))) MARK(incr(s(cons(incr(head(s(y0))), y1)))) -> c13(MARK(s(cons(incr(head(s(y0))), y1)))) MARK(incr(s(cons(incr(head(y0)), y1)))) -> c13(MARK(s(cons(incr(head(y0)), y1)))) MARK(incr(s(cons(incr(tail(incr(y0))), y1)))) -> c13(MARK(s(cons(incr(tail(incr(y0))), y1)))) MARK(incr(s(cons(incr(tail(pairs)), y0)))) -> c13(MARK(s(cons(incr(tail(pairs)), y0)))) MARK(incr(s(cons(incr(tail(odds)), y0)))) -> c13(MARK(s(cons(incr(tail(odds)), y0)))) MARK(incr(s(cons(incr(tail(head(y0))), y1)))) -> c13(MARK(s(cons(incr(tail(head(y0))), y1)))) MARK(incr(s(cons(incr(tail(tail(y0))), y1)))) -> c13(MARK(s(cons(incr(tail(tail(y0))), y1)))) MARK(incr(s(cons(incr(tail(cons(y0, y1))), y2)))) -> c13(MARK(s(cons(incr(tail(cons(y0, y1))), y2)))) MARK(incr(s(cons(incr(tail(s(y0))), y1)))) -> c13(MARK(s(cons(incr(tail(s(y0))), y1)))) MARK(incr(s(cons(incr(tail(y0)), y1)))) -> c13(MARK(s(cons(incr(tail(y0)), y1)))) MARK(incr(s(cons(head(incr(incr(y0))), y1)))) -> c13(MARK(s(cons(head(incr(incr(y0))), y1)))) MARK(incr(s(cons(head(incr(odds)), y0)))) -> c13(MARK(s(cons(head(incr(odds)), y0)))) MARK(incr(s(cons(head(incr(head(y0))), y1)))) -> c13(MARK(s(cons(head(incr(head(y0))), y1)))) MARK(incr(s(cons(head(incr(tail(y0))), y1)))) -> c13(MARK(s(cons(head(incr(tail(y0))), y1)))) MARK(incr(s(cons(head(incr(cons(y0, y1))), y2)))) -> c13(MARK(s(cons(head(incr(cons(y0, y1))), y2)))) MARK(incr(s(cons(head(incr(s(y0))), y1)))) -> c13(MARK(s(cons(head(incr(s(y0))), y1)))) MARK(incr(s(cons(head(incr(y0)), y1)))) -> c13(MARK(s(cons(head(incr(y0)), y1)))) MARK(incr(s(cons(head(head(incr(y0))), y1)))) -> c13(MARK(s(cons(head(head(incr(y0))), y1)))) MARK(incr(s(cons(head(head(odds)), y0)))) -> c13(MARK(s(cons(head(head(odds)), y0)))) MARK(incr(s(cons(head(head(head(y0))), y1)))) -> c13(MARK(s(cons(head(head(head(y0))), y1)))) MARK(incr(s(cons(head(head(tail(y0))), y1)))) -> c13(MARK(s(cons(head(head(tail(y0))), y1)))) MARK(incr(s(cons(head(head(cons(y0, y1))), y2)))) -> c13(MARK(s(cons(head(head(cons(y0, y1))), y2)))) MARK(incr(s(cons(head(head(s(y0))), y1)))) -> c13(MARK(s(cons(head(head(s(y0))), y1)))) MARK(incr(s(cons(head(head(y0)), y1)))) -> c13(MARK(s(cons(head(head(y0)), y1)))) MARK(incr(s(cons(head(tail(incr(y0))), y1)))) -> c13(MARK(s(cons(head(tail(incr(y0))), y1)))) MARK(incr(s(cons(head(tail(pairs)), y0)))) -> c13(MARK(s(cons(head(tail(pairs)), y0)))) MARK(incr(s(cons(head(tail(odds)), y0)))) -> c13(MARK(s(cons(head(tail(odds)), y0)))) MARK(incr(s(cons(head(tail(head(y0))), y1)))) -> c13(MARK(s(cons(head(tail(head(y0))), y1)))) MARK(incr(s(cons(head(tail(tail(y0))), y1)))) -> c13(MARK(s(cons(head(tail(tail(y0))), y1)))) MARK(incr(s(cons(head(tail(cons(y0, y1))), y2)))) -> c13(MARK(s(cons(head(tail(cons(y0, y1))), y2)))) MARK(incr(s(cons(head(tail(s(y0))), y1)))) -> c13(MARK(s(cons(head(tail(s(y0))), y1)))) MARK(incr(s(cons(head(tail(y0)), y1)))) -> c13(MARK(s(cons(head(tail(y0)), y1)))) MARK(incr(s(cons(tail(incr(incr(y0))), y1)))) -> c13(MARK(s(cons(tail(incr(incr(y0))), y1)))) MARK(incr(s(cons(tail(incr(odds)), y0)))) -> c13(MARK(s(cons(tail(incr(odds)), y0)))) MARK(incr(s(cons(tail(incr(head(y0))), y1)))) -> c13(MARK(s(cons(tail(incr(head(y0))), y1)))) MARK(incr(s(cons(tail(incr(tail(y0))), y1)))) -> c13(MARK(s(cons(tail(incr(tail(y0))), y1)))) MARK(incr(s(cons(tail(incr(cons(y0, y1))), y2)))) -> c13(MARK(s(cons(tail(incr(cons(y0, y1))), y2)))) MARK(incr(s(cons(tail(incr(s(y0))), y1)))) -> c13(MARK(s(cons(tail(incr(s(y0))), y1)))) MARK(incr(s(cons(tail(incr(y0)), y1)))) -> c13(MARK(s(cons(tail(incr(y0)), y1)))) MARK(incr(s(cons(tail(head(incr(y0))), y1)))) -> c13(MARK(s(cons(tail(head(incr(y0))), y1)))) MARK(incr(s(cons(tail(head(odds)), y0)))) -> c13(MARK(s(cons(tail(head(odds)), y0)))) MARK(incr(s(cons(tail(head(head(y0))), y1)))) -> c13(MARK(s(cons(tail(head(head(y0))), y1)))) MARK(incr(s(cons(tail(head(tail(y0))), y1)))) -> c13(MARK(s(cons(tail(head(tail(y0))), y1)))) MARK(incr(s(cons(tail(head(cons(y0, y1))), y2)))) -> c13(MARK(s(cons(tail(head(cons(y0, y1))), y2)))) MARK(incr(s(cons(tail(head(s(y0))), y1)))) -> c13(MARK(s(cons(tail(head(s(y0))), y1)))) MARK(incr(s(cons(tail(head(y0)), y1)))) -> c13(MARK(s(cons(tail(head(y0)), y1)))) MARK(incr(s(cons(tail(tail(incr(y0))), y1)))) -> c13(MARK(s(cons(tail(tail(incr(y0))), y1)))) MARK(incr(s(cons(tail(tail(pairs)), y0)))) -> c13(MARK(s(cons(tail(tail(pairs)), y0)))) MARK(incr(s(cons(tail(tail(odds)), y0)))) -> c13(MARK(s(cons(tail(tail(odds)), y0)))) MARK(incr(s(cons(tail(tail(head(y0))), y1)))) -> c13(MARK(s(cons(tail(tail(head(y0))), y1)))) MARK(incr(s(cons(tail(tail(tail(y0))), y1)))) -> c13(MARK(s(cons(tail(tail(tail(y0))), y1)))) MARK(incr(s(cons(tail(tail(cons(y0, y1))), y2)))) -> c13(MARK(s(cons(tail(tail(cons(y0, y1))), y2)))) MARK(incr(s(cons(tail(tail(s(y0))), y1)))) -> c13(MARK(s(cons(tail(tail(s(y0))), y1)))) MARK(incr(s(cons(tail(tail(y0)), y1)))) -> c13(MARK(s(cons(tail(tail(y0)), y1)))) MARK(incr(s(cons(tail(pairs), y0)))) -> c13(MARK(s(cons(tail(pairs), y0)))) MARK(incr(s(cons(incr(incr(nats)), y0)))) -> c13(MARK(s(cons(incr(incr(nats)), y0)))) MARK(incr(s(cons(incr(incr(pairs)), y0)))) -> c13(MARK(s(cons(incr(incr(pairs)), y0)))) MARK(incr(s(cons(incr(odds), y0)))) -> c13(MARK(s(cons(incr(odds), y0)))) MARK(incr(s(cons(head(incr(nats)), y0)))) -> c13(MARK(s(cons(head(incr(nats)), y0)))) MARK(incr(s(cons(head(incr(pairs)), y0)))) -> c13(MARK(s(cons(head(incr(pairs)), y0)))) MARK(incr(s(cons(head(odds), y0)))) -> c13(MARK(s(cons(head(odds), y0)))) MARK(incr(s(cons(tail(incr(nats)), y0)))) -> c13(MARK(s(cons(tail(incr(nats)), y0)))) MARK(incr(s(cons(tail(incr(pairs)), y0)))) -> c13(MARK(s(cons(tail(incr(pairs)), y0)))) MARK(incr(s(cons(tail(odds), y0)))) -> c13(MARK(s(cons(tail(odds), y0)))) MARK(incr(s(cons(incr(cons(nats, y0)), y1)))) -> c13(MARK(s(cons(incr(cons(nats, y0)), y1)))) MARK(incr(s(cons(incr(cons(incr(y0), y1)), y2)))) -> c13(MARK(s(cons(incr(cons(incr(y0), y1)), y2)))) MARK(incr(s(cons(incr(cons(pairs, y0)), y1)))) -> c13(MARK(s(cons(incr(cons(pairs, y0)), y1)))) MARK(incr(s(cons(incr(cons(odds, y0)), y1)))) -> c13(MARK(s(cons(incr(cons(odds, y0)), y1)))) MARK(incr(s(cons(incr(cons(head(y0), y1)), y2)))) -> c13(MARK(s(cons(incr(cons(head(y0), y1)), y2)))) MARK(incr(s(cons(incr(cons(tail(y0), y1)), y2)))) -> c13(MARK(s(cons(incr(cons(tail(y0), y1)), y2)))) MARK(incr(s(cons(incr(cons(cons(y0, y1), y2)), y3)))) -> c13(MARK(s(cons(incr(cons(cons(y0, y1), y2)), y3)))) MARK(incr(s(cons(incr(cons(s(y0), y1)), y2)))) -> c13(MARK(s(cons(incr(cons(s(y0), y1)), y2)))) MARK(incr(s(cons(incr(cons(y0, y1)), y2)))) -> c13(MARK(s(cons(incr(cons(y0, y1)), y2)))) MARK(incr(s(cons(incr(cons(0, y0)), y1)))) -> c13(MARK(s(cons(incr(cons(0, y0)), y1)))) MARK(incr(s(cons(incr(head(nats)), y0)))) -> c13(MARK(s(cons(incr(head(nats)), y0)))) MARK(incr(s(cons(incr(head(pairs)), y0)))) -> c13(MARK(s(cons(incr(head(pairs)), y0)))) MARK(incr(s(cons(head(head(nats)), y0)))) -> c13(MARK(s(cons(head(head(nats)), y0)))) MARK(incr(s(cons(head(head(pairs)), y0)))) -> c13(MARK(s(cons(head(head(pairs)), y0)))) MARK(incr(s(cons(tail(head(nats)), y0)))) -> c13(MARK(s(cons(tail(head(nats)), y0)))) MARK(incr(s(cons(tail(head(pairs)), y0)))) -> c13(MARK(s(cons(tail(head(pairs)), y0)))) MARK(incr(s(cons(head(cons(nats, y0)), y1)))) -> c13(MARK(s(cons(head(cons(nats, y0)), y1)))) MARK(incr(s(cons(head(cons(incr(y0), y1)), y2)))) -> c13(MARK(s(cons(head(cons(incr(y0), y1)), y2)))) MARK(incr(s(cons(head(cons(pairs, y0)), y1)))) -> c13(MARK(s(cons(head(cons(pairs, y0)), y1)))) MARK(incr(s(cons(head(cons(odds, y0)), y1)))) -> c13(MARK(s(cons(head(cons(odds, y0)), y1)))) MARK(incr(s(cons(head(cons(head(y0), y1)), y2)))) -> c13(MARK(s(cons(head(cons(head(y0), y1)), y2)))) MARK(incr(s(cons(head(cons(tail(y0), y1)), y2)))) -> c13(MARK(s(cons(head(cons(tail(y0), y1)), y2)))) MARK(incr(s(cons(head(cons(cons(y0, y1), y2)), y3)))) -> c13(MARK(s(cons(head(cons(cons(y0, y1), y2)), y3)))) MARK(incr(s(cons(head(cons(s(y0), y1)), y2)))) -> c13(MARK(s(cons(head(cons(s(y0), y1)), y2)))) MARK(incr(s(cons(head(cons(y0, y1)), y2)))) -> c13(MARK(s(cons(head(cons(y0, y1)), y2)))) MARK(incr(s(cons(head(cons(0, y0)), y1)))) -> c13(MARK(s(cons(head(cons(0, y0)), y1)))) MARK(incr(s(cons(incr(tail(nats)), y0)))) -> c13(MARK(s(cons(incr(tail(nats)), y0)))) MARK(incr(s(cons(head(tail(nats)), y0)))) -> c13(MARK(s(cons(head(tail(nats)), y0)))) MARK(incr(s(cons(tail(tail(nats)), y0)))) -> c13(MARK(s(cons(tail(tail(nats)), y0)))) MARK(incr(s(incr(cons(nats, y0))))) -> c13(MARK(s(incr(cons(nats, y0))))) MARK(incr(s(incr(cons(pairs, y0))))) -> c13(MARK(s(incr(cons(pairs, y0))))) MARK(incr(s(incr(cons(odds, y0))))) -> c13(MARK(s(incr(cons(odds, y0))))) MARK(incr(s(head(cons(nats, y0))))) -> c13(MARK(s(head(cons(nats, y0))))) MARK(incr(s(head(cons(pairs, y0))))) -> c13(MARK(s(head(cons(pairs, y0))))) MARK(incr(s(head(cons(odds, y0))))) -> c13(MARK(s(head(cons(odds, y0))))) MARK(head(s(s(y0)))) -> c16(MARK(s(s(y0)))) MARK(head(s(incr(s(y0))))) -> c16(MARK(s(incr(s(y0))))) MARK(head(s(head(s(y0))))) -> c16(MARK(s(head(s(y0))))) MARK(head(s(tail(cons(y0, y1))))) -> c16(MARK(s(tail(cons(y0, y1))))) MARK(head(s(tail(s(y0))))) -> c16(MARK(s(tail(s(y0))))) MARK(head(s(incr(incr(incr(y0)))))) -> c16(MARK(s(incr(incr(incr(y0)))))) MARK(head(s(incr(incr(odds))))) -> c16(MARK(s(incr(incr(odds))))) MARK(head(s(incr(incr(head(y0)))))) -> c16(MARK(s(incr(incr(head(y0)))))) MARK(head(s(incr(incr(tail(y0)))))) -> c16(MARK(s(incr(incr(tail(y0)))))) MARK(head(s(incr(incr(cons(y0, y1)))))) -> c16(MARK(s(incr(incr(cons(y0, y1)))))) MARK(head(s(incr(incr(s(y0)))))) -> c16(MARK(s(incr(incr(s(y0)))))) MARK(head(s(incr(incr(y0))))) -> c16(MARK(s(incr(incr(y0))))) MARK(head(s(incr(head(incr(y0)))))) -> c16(MARK(s(incr(head(incr(y0)))))) MARK(head(s(incr(head(odds))))) -> c16(MARK(s(incr(head(odds))))) MARK(head(s(incr(head(head(y0)))))) -> c16(MARK(s(incr(head(head(y0)))))) MARK(head(s(incr(head(tail(y0)))))) -> c16(MARK(s(incr(head(tail(y0)))))) MARK(head(s(incr(head(cons(y0, y1)))))) -> c16(MARK(s(incr(head(cons(y0, y1)))))) MARK(head(s(incr(head(s(y0)))))) -> c16(MARK(s(incr(head(s(y0)))))) MARK(head(s(incr(head(y0))))) -> c16(MARK(s(incr(head(y0))))) MARK(head(s(incr(tail(incr(y0)))))) -> c16(MARK(s(incr(tail(incr(y0)))))) MARK(head(s(incr(tail(pairs))))) -> c16(MARK(s(incr(tail(pairs))))) MARK(head(s(incr(tail(odds))))) -> c16(MARK(s(incr(tail(odds))))) MARK(head(s(incr(tail(head(y0)))))) -> c16(MARK(s(incr(tail(head(y0)))))) MARK(head(s(incr(tail(tail(y0)))))) -> c16(MARK(s(incr(tail(tail(y0)))))) MARK(head(s(incr(tail(cons(y0, y1)))))) -> c16(MARK(s(incr(tail(cons(y0, y1)))))) MARK(head(s(incr(tail(s(y0)))))) -> c16(MARK(s(incr(tail(s(y0)))))) MARK(head(s(incr(tail(y0))))) -> c16(MARK(s(incr(tail(y0))))) MARK(head(s(head(incr(incr(y0)))))) -> c16(MARK(s(head(incr(incr(y0)))))) MARK(head(s(head(incr(odds))))) -> c16(MARK(s(head(incr(odds))))) MARK(head(s(head(incr(head(y0)))))) -> c16(MARK(s(head(incr(head(y0)))))) MARK(head(s(head(incr(tail(y0)))))) -> c16(MARK(s(head(incr(tail(y0)))))) MARK(head(s(head(incr(cons(y0, y1)))))) -> c16(MARK(s(head(incr(cons(y0, y1)))))) MARK(head(s(head(incr(s(y0)))))) -> c16(MARK(s(head(incr(s(y0)))))) MARK(head(s(head(incr(y0))))) -> c16(MARK(s(head(incr(y0))))) MARK(head(s(head(head(incr(y0)))))) -> c16(MARK(s(head(head(incr(y0)))))) MARK(head(s(head(head(odds))))) -> c16(MARK(s(head(head(odds))))) MARK(head(s(head(head(head(y0)))))) -> c16(MARK(s(head(head(head(y0)))))) MARK(head(s(head(head(tail(y0)))))) -> c16(MARK(s(head(head(tail(y0)))))) MARK(head(s(head(head(cons(y0, y1)))))) -> c16(MARK(s(head(head(cons(y0, y1)))))) MARK(head(s(head(head(s(y0)))))) -> c16(MARK(s(head(head(s(y0)))))) MARK(head(s(head(head(y0))))) -> c16(MARK(s(head(head(y0))))) MARK(head(s(head(tail(incr(y0)))))) -> c16(MARK(s(head(tail(incr(y0)))))) MARK(head(s(head(tail(pairs))))) -> c16(MARK(s(head(tail(pairs))))) MARK(head(s(head(tail(odds))))) -> c16(MARK(s(head(tail(odds))))) MARK(head(s(head(tail(head(y0)))))) -> c16(MARK(s(head(tail(head(y0)))))) MARK(head(s(head(tail(tail(y0)))))) -> c16(MARK(s(head(tail(tail(y0)))))) MARK(head(s(head(tail(cons(y0, y1)))))) -> c16(MARK(s(head(tail(cons(y0, y1)))))) MARK(head(s(head(tail(s(y0)))))) -> c16(MARK(s(head(tail(s(y0)))))) MARK(head(s(head(tail(y0))))) -> c16(MARK(s(head(tail(y0))))) MARK(head(s(tail(incr(incr(y0)))))) -> c16(MARK(s(tail(incr(incr(y0)))))) MARK(head(s(tail(incr(odds))))) -> c16(MARK(s(tail(incr(odds))))) MARK(head(s(tail(incr(head(y0)))))) -> c16(MARK(s(tail(incr(head(y0)))))) MARK(head(s(tail(incr(tail(y0)))))) -> c16(MARK(s(tail(incr(tail(y0)))))) MARK(head(s(tail(incr(cons(y0, y1)))))) -> c16(MARK(s(tail(incr(cons(y0, y1)))))) MARK(head(s(tail(incr(s(y0)))))) -> c16(MARK(s(tail(incr(s(y0)))))) MARK(head(s(tail(incr(y0))))) -> c16(MARK(s(tail(incr(y0))))) MARK(head(s(tail(head(incr(y0)))))) -> c16(MARK(s(tail(head(incr(y0)))))) MARK(head(s(tail(head(odds))))) -> c16(MARK(s(tail(head(odds))))) MARK(head(s(tail(head(head(y0)))))) -> c16(MARK(s(tail(head(head(y0)))))) MARK(head(s(tail(head(tail(y0)))))) -> c16(MARK(s(tail(head(tail(y0)))))) MARK(head(s(tail(head(cons(y0, y1)))))) -> c16(MARK(s(tail(head(cons(y0, y1)))))) MARK(head(s(tail(head(s(y0)))))) -> c16(MARK(s(tail(head(s(y0)))))) MARK(head(s(tail(head(y0))))) -> c16(MARK(s(tail(head(y0))))) MARK(head(s(tail(tail(incr(y0)))))) -> c16(MARK(s(tail(tail(incr(y0)))))) MARK(head(s(tail(tail(pairs))))) -> c16(MARK(s(tail(tail(pairs))))) MARK(head(s(tail(tail(odds))))) -> c16(MARK(s(tail(tail(odds))))) MARK(head(s(tail(tail(head(y0)))))) -> c16(MARK(s(tail(tail(head(y0)))))) MARK(head(s(tail(tail(tail(y0)))))) -> c16(MARK(s(tail(tail(tail(y0)))))) MARK(head(s(tail(tail(cons(y0, y1)))))) -> c16(MARK(s(tail(tail(cons(y0, y1)))))) MARK(head(s(tail(tail(s(y0)))))) -> c16(MARK(s(tail(tail(s(y0)))))) MARK(head(s(tail(tail(y0))))) -> c16(MARK(s(tail(tail(y0))))) MARK(head(s(tail(pairs)))) -> c16(MARK(s(tail(pairs)))) MARK(head(s(incr(incr(nats))))) -> c16(MARK(s(incr(incr(nats))))) MARK(head(s(incr(incr(pairs))))) -> c16(MARK(s(incr(incr(pairs))))) MARK(head(s(incr(odds)))) -> c16(MARK(s(incr(odds)))) MARK(head(s(head(incr(nats))))) -> c16(MARK(s(head(incr(nats))))) MARK(head(s(head(incr(pairs))))) -> c16(MARK(s(head(incr(pairs))))) MARK(head(s(head(odds)))) -> c16(MARK(s(head(odds)))) MARK(head(s(tail(incr(nats))))) -> c16(MARK(s(tail(incr(nats))))) MARK(head(s(tail(incr(pairs))))) -> c16(MARK(s(tail(incr(pairs))))) MARK(head(s(tail(odds)))) -> c16(MARK(s(tail(odds)))) MARK(head(s(incr(cons(incr(y0), y1))))) -> c16(MARK(s(incr(cons(incr(y0), y1))))) MARK(head(s(incr(cons(head(y0), y1))))) -> c16(MARK(s(incr(cons(head(y0), y1))))) MARK(head(s(incr(cons(tail(y0), y1))))) -> c16(MARK(s(incr(cons(tail(y0), y1))))) MARK(head(s(incr(cons(cons(y0, y1), y2))))) -> c16(MARK(s(incr(cons(cons(y0, y1), y2))))) MARK(head(s(incr(cons(s(y0), y1))))) -> c16(MARK(s(incr(cons(s(y0), y1))))) MARK(head(s(incr(cons(y0, y1))))) -> c16(MARK(s(incr(cons(y0, y1))))) MARK(head(s(incr(head(nats))))) -> c16(MARK(s(incr(head(nats))))) MARK(head(s(incr(head(pairs))))) -> c16(MARK(s(incr(head(pairs))))) MARK(head(s(head(head(nats))))) -> c16(MARK(s(head(head(nats))))) MARK(head(s(head(head(pairs))))) -> c16(MARK(s(head(head(pairs))))) MARK(head(s(tail(head(nats))))) -> c16(MARK(s(tail(head(nats))))) MARK(head(s(tail(head(pairs))))) -> c16(MARK(s(tail(head(pairs))))) MARK(head(s(head(cons(incr(y0), y1))))) -> c16(MARK(s(head(cons(incr(y0), y1))))) MARK(head(s(head(cons(head(y0), y1))))) -> c16(MARK(s(head(cons(head(y0), y1))))) MARK(head(s(head(cons(tail(y0), y1))))) -> c16(MARK(s(head(cons(tail(y0), y1))))) MARK(head(s(head(cons(cons(y0, y1), y2))))) -> c16(MARK(s(head(cons(cons(y0, y1), y2))))) MARK(head(s(head(cons(s(y0), y1))))) -> c16(MARK(s(head(cons(s(y0), y1))))) MARK(head(s(head(cons(y0, y1))))) -> c16(MARK(s(head(cons(y0, y1))))) MARK(head(s(incr(tail(nats))))) -> c16(MARK(s(incr(tail(nats))))) MARK(head(s(head(tail(nats))))) -> c16(MARK(s(head(tail(nats))))) MARK(head(s(tail(tail(nats))))) -> c16(MARK(s(tail(tail(nats))))) MARK(head(s(cons(cons(y0, y1), y2)))) -> c16(MARK(s(cons(cons(y0, y1), y2)))) MARK(head(s(cons(s(y0), y1)))) -> c16(MARK(s(cons(s(y0), y1)))) MARK(head(s(cons(incr(s(y0)), y1)))) -> c16(MARK(s(cons(incr(s(y0)), y1)))) MARK(head(s(cons(head(s(y0)), y1)))) -> c16(MARK(s(cons(head(s(y0)), y1)))) MARK(head(s(cons(tail(cons(y0, y1)), y2)))) -> c16(MARK(s(cons(tail(cons(y0, y1)), y2)))) MARK(head(s(cons(tail(s(y0)), y1)))) -> c16(MARK(s(cons(tail(s(y0)), y1)))) MARK(head(s(cons(incr(incr(incr(y0))), y1)))) -> c16(MARK(s(cons(incr(incr(incr(y0))), y1)))) MARK(head(s(cons(incr(incr(odds)), y0)))) -> c16(MARK(s(cons(incr(incr(odds)), y0)))) MARK(head(s(cons(incr(incr(head(y0))), y1)))) -> c16(MARK(s(cons(incr(incr(head(y0))), y1)))) MARK(head(s(cons(incr(incr(tail(y0))), y1)))) -> c16(MARK(s(cons(incr(incr(tail(y0))), y1)))) MARK(head(s(cons(incr(incr(cons(y0, y1))), y2)))) -> c16(MARK(s(cons(incr(incr(cons(y0, y1))), y2)))) MARK(head(s(cons(incr(incr(s(y0))), y1)))) -> c16(MARK(s(cons(incr(incr(s(y0))), y1)))) MARK(head(s(cons(incr(incr(y0)), y1)))) -> c16(MARK(s(cons(incr(incr(y0)), y1)))) MARK(head(s(cons(incr(head(incr(y0))), y1)))) -> c16(MARK(s(cons(incr(head(incr(y0))), y1)))) MARK(head(s(cons(incr(head(odds)), y0)))) -> c16(MARK(s(cons(incr(head(odds)), y0)))) MARK(head(s(cons(incr(head(head(y0))), y1)))) -> c16(MARK(s(cons(incr(head(head(y0))), y1)))) MARK(head(s(cons(incr(head(tail(y0))), y1)))) -> c16(MARK(s(cons(incr(head(tail(y0))), y1)))) MARK(head(s(cons(incr(head(cons(y0, y1))), y2)))) -> c16(MARK(s(cons(incr(head(cons(y0, y1))), y2)))) MARK(head(s(cons(incr(head(s(y0))), y1)))) -> c16(MARK(s(cons(incr(head(s(y0))), y1)))) MARK(head(s(cons(incr(head(y0)), y1)))) -> c16(MARK(s(cons(incr(head(y0)), y1)))) MARK(head(s(cons(incr(tail(incr(y0))), y1)))) -> c16(MARK(s(cons(incr(tail(incr(y0))), y1)))) MARK(head(s(cons(incr(tail(pairs)), y0)))) -> c16(MARK(s(cons(incr(tail(pairs)), y0)))) MARK(head(s(cons(incr(tail(odds)), y0)))) -> c16(MARK(s(cons(incr(tail(odds)), y0)))) MARK(head(s(cons(incr(tail(head(y0))), y1)))) -> c16(MARK(s(cons(incr(tail(head(y0))), y1)))) MARK(head(s(cons(incr(tail(tail(y0))), y1)))) -> c16(MARK(s(cons(incr(tail(tail(y0))), y1)))) MARK(head(s(cons(incr(tail(cons(y0, y1))), y2)))) -> c16(MARK(s(cons(incr(tail(cons(y0, y1))), y2)))) MARK(head(s(cons(incr(tail(s(y0))), y1)))) -> c16(MARK(s(cons(incr(tail(s(y0))), y1)))) MARK(head(s(cons(incr(tail(y0)), y1)))) -> c16(MARK(s(cons(incr(tail(y0)), y1)))) MARK(head(s(cons(head(incr(incr(y0))), y1)))) -> c16(MARK(s(cons(head(incr(incr(y0))), y1)))) MARK(head(s(cons(head(incr(odds)), y0)))) -> c16(MARK(s(cons(head(incr(odds)), y0)))) MARK(head(s(cons(head(incr(head(y0))), y1)))) -> c16(MARK(s(cons(head(incr(head(y0))), y1)))) MARK(head(s(cons(head(incr(tail(y0))), y1)))) -> c16(MARK(s(cons(head(incr(tail(y0))), y1)))) MARK(head(s(cons(head(incr(cons(y0, y1))), y2)))) -> c16(MARK(s(cons(head(incr(cons(y0, y1))), y2)))) MARK(head(s(cons(head(incr(s(y0))), y1)))) -> c16(MARK(s(cons(head(incr(s(y0))), y1)))) MARK(head(s(cons(head(incr(y0)), y1)))) -> c16(MARK(s(cons(head(incr(y0)), y1)))) MARK(head(s(cons(head(head(incr(y0))), y1)))) -> c16(MARK(s(cons(head(head(incr(y0))), y1)))) MARK(head(s(cons(head(head(odds)), y0)))) -> c16(MARK(s(cons(head(head(odds)), y0)))) MARK(head(s(cons(head(head(head(y0))), y1)))) -> c16(MARK(s(cons(head(head(head(y0))), y1)))) MARK(head(s(cons(head(head(tail(y0))), y1)))) -> c16(MARK(s(cons(head(head(tail(y0))), y1)))) MARK(head(s(cons(head(head(cons(y0, y1))), y2)))) -> c16(MARK(s(cons(head(head(cons(y0, y1))), y2)))) MARK(head(s(cons(head(head(s(y0))), y1)))) -> c16(MARK(s(cons(head(head(s(y0))), y1)))) MARK(head(s(cons(head(head(y0)), y1)))) -> c16(MARK(s(cons(head(head(y0)), y1)))) MARK(head(s(cons(head(tail(incr(y0))), y1)))) -> c16(MARK(s(cons(head(tail(incr(y0))), y1)))) MARK(head(s(cons(head(tail(pairs)), y0)))) -> c16(MARK(s(cons(head(tail(pairs)), y0)))) MARK(head(s(cons(head(tail(odds)), y0)))) -> c16(MARK(s(cons(head(tail(odds)), y0)))) MARK(head(s(cons(head(tail(head(y0))), y1)))) -> c16(MARK(s(cons(head(tail(head(y0))), y1)))) MARK(head(s(cons(head(tail(tail(y0))), y1)))) -> c16(MARK(s(cons(head(tail(tail(y0))), y1)))) MARK(head(s(cons(head(tail(cons(y0, y1))), y2)))) -> c16(MARK(s(cons(head(tail(cons(y0, y1))), y2)))) MARK(head(s(cons(head(tail(s(y0))), y1)))) -> c16(MARK(s(cons(head(tail(s(y0))), y1)))) MARK(head(s(cons(head(tail(y0)), y1)))) -> c16(MARK(s(cons(head(tail(y0)), y1)))) MARK(head(s(cons(tail(incr(incr(y0))), y1)))) -> c16(MARK(s(cons(tail(incr(incr(y0))), y1)))) MARK(head(s(cons(tail(incr(odds)), y0)))) -> c16(MARK(s(cons(tail(incr(odds)), y0)))) MARK(head(s(cons(tail(incr(head(y0))), y1)))) -> c16(MARK(s(cons(tail(incr(head(y0))), y1)))) MARK(head(s(cons(tail(incr(tail(y0))), y1)))) -> c16(MARK(s(cons(tail(incr(tail(y0))), y1)))) MARK(head(s(cons(tail(incr(cons(y0, y1))), y2)))) -> c16(MARK(s(cons(tail(incr(cons(y0, y1))), y2)))) MARK(head(s(cons(tail(incr(s(y0))), y1)))) -> c16(MARK(s(cons(tail(incr(s(y0))), y1)))) MARK(head(s(cons(tail(incr(y0)), y1)))) -> c16(MARK(s(cons(tail(incr(y0)), y1)))) MARK(head(s(cons(tail(head(incr(y0))), y1)))) -> c16(MARK(s(cons(tail(head(incr(y0))), y1)))) MARK(head(s(cons(tail(head(odds)), y0)))) -> c16(MARK(s(cons(tail(head(odds)), y0)))) MARK(head(s(cons(tail(head(head(y0))), y1)))) -> c16(MARK(s(cons(tail(head(head(y0))), y1)))) MARK(head(s(cons(tail(head(tail(y0))), y1)))) -> c16(MARK(s(cons(tail(head(tail(y0))), y1)))) MARK(head(s(cons(tail(head(cons(y0, y1))), y2)))) -> c16(MARK(s(cons(tail(head(cons(y0, y1))), y2)))) MARK(head(s(cons(tail(head(s(y0))), y1)))) -> c16(MARK(s(cons(tail(head(s(y0))), y1)))) MARK(head(s(cons(tail(head(y0)), y1)))) -> c16(MARK(s(cons(tail(head(y0)), y1)))) MARK(head(s(cons(tail(tail(incr(y0))), y1)))) -> c16(MARK(s(cons(tail(tail(incr(y0))), y1)))) MARK(head(s(cons(tail(tail(pairs)), y0)))) -> c16(MARK(s(cons(tail(tail(pairs)), y0)))) MARK(head(s(cons(tail(tail(odds)), y0)))) -> c16(MARK(s(cons(tail(tail(odds)), y0)))) MARK(head(s(cons(tail(tail(head(y0))), y1)))) -> c16(MARK(s(cons(tail(tail(head(y0))), y1)))) MARK(head(s(cons(tail(tail(tail(y0))), y1)))) -> c16(MARK(s(cons(tail(tail(tail(y0))), y1)))) MARK(head(s(cons(tail(tail(cons(y0, y1))), y2)))) -> c16(MARK(s(cons(tail(tail(cons(y0, y1))), y2)))) MARK(head(s(cons(tail(tail(s(y0))), y1)))) -> c16(MARK(s(cons(tail(tail(s(y0))), y1)))) MARK(head(s(cons(tail(tail(y0)), y1)))) -> c16(MARK(s(cons(tail(tail(y0)), y1)))) MARK(head(s(cons(tail(pairs), y0)))) -> c16(MARK(s(cons(tail(pairs), y0)))) MARK(head(s(cons(incr(incr(nats)), y0)))) -> c16(MARK(s(cons(incr(incr(nats)), y0)))) MARK(head(s(cons(incr(incr(pairs)), y0)))) -> c16(MARK(s(cons(incr(incr(pairs)), y0)))) MARK(head(s(cons(incr(odds), y0)))) -> c16(MARK(s(cons(incr(odds), y0)))) MARK(head(s(cons(head(incr(nats)), y0)))) -> c16(MARK(s(cons(head(incr(nats)), y0)))) MARK(head(s(cons(head(incr(pairs)), y0)))) -> c16(MARK(s(cons(head(incr(pairs)), y0)))) MARK(head(s(cons(head(odds), y0)))) -> c16(MARK(s(cons(head(odds), y0)))) MARK(head(s(cons(tail(incr(nats)), y0)))) -> c16(MARK(s(cons(tail(incr(nats)), y0)))) MARK(head(s(cons(tail(incr(pairs)), y0)))) -> c16(MARK(s(cons(tail(incr(pairs)), y0)))) MARK(head(s(cons(tail(odds), y0)))) -> c16(MARK(s(cons(tail(odds), y0)))) MARK(head(s(cons(incr(cons(nats, y0)), y1)))) -> c16(MARK(s(cons(incr(cons(nats, y0)), y1)))) MARK(head(s(cons(incr(cons(incr(y0), y1)), y2)))) -> c16(MARK(s(cons(incr(cons(incr(y0), y1)), y2)))) MARK(head(s(cons(incr(cons(pairs, y0)), y1)))) -> c16(MARK(s(cons(incr(cons(pairs, y0)), y1)))) MARK(head(s(cons(incr(cons(odds, y0)), y1)))) -> c16(MARK(s(cons(incr(cons(odds, y0)), y1)))) MARK(head(s(cons(incr(cons(head(y0), y1)), y2)))) -> c16(MARK(s(cons(incr(cons(head(y0), y1)), y2)))) MARK(head(s(cons(incr(cons(tail(y0), y1)), y2)))) -> c16(MARK(s(cons(incr(cons(tail(y0), y1)), y2)))) MARK(head(s(cons(incr(cons(cons(y0, y1), y2)), y3)))) -> c16(MARK(s(cons(incr(cons(cons(y0, y1), y2)), y3)))) MARK(head(s(cons(incr(cons(s(y0), y1)), y2)))) -> c16(MARK(s(cons(incr(cons(s(y0), y1)), y2)))) MARK(head(s(cons(incr(cons(y0, y1)), y2)))) -> c16(MARK(s(cons(incr(cons(y0, y1)), y2)))) MARK(head(s(cons(incr(cons(0, y0)), y1)))) -> c16(MARK(s(cons(incr(cons(0, y0)), y1)))) MARK(head(s(cons(incr(head(nats)), y0)))) -> c16(MARK(s(cons(incr(head(nats)), y0)))) MARK(head(s(cons(incr(head(pairs)), y0)))) -> c16(MARK(s(cons(incr(head(pairs)), y0)))) MARK(head(s(cons(head(head(nats)), y0)))) -> c16(MARK(s(cons(head(head(nats)), y0)))) MARK(head(s(cons(head(head(pairs)), y0)))) -> c16(MARK(s(cons(head(head(pairs)), y0)))) MARK(head(s(cons(tail(head(nats)), y0)))) -> c16(MARK(s(cons(tail(head(nats)), y0)))) MARK(head(s(cons(tail(head(pairs)), y0)))) -> c16(MARK(s(cons(tail(head(pairs)), y0)))) MARK(head(s(cons(head(cons(nats, y0)), y1)))) -> c16(MARK(s(cons(head(cons(nats, y0)), y1)))) MARK(head(s(cons(head(cons(incr(y0), y1)), y2)))) -> c16(MARK(s(cons(head(cons(incr(y0), y1)), y2)))) MARK(head(s(cons(head(cons(pairs, y0)), y1)))) -> c16(MARK(s(cons(head(cons(pairs, y0)), y1)))) MARK(head(s(cons(head(cons(odds, y0)), y1)))) -> c16(MARK(s(cons(head(cons(odds, y0)), y1)))) MARK(head(s(cons(head(cons(head(y0), y1)), y2)))) -> c16(MARK(s(cons(head(cons(head(y0), y1)), y2)))) MARK(head(s(cons(head(cons(tail(y0), y1)), y2)))) -> c16(MARK(s(cons(head(cons(tail(y0), y1)), y2)))) MARK(head(s(cons(head(cons(cons(y0, y1), y2)), y3)))) -> c16(MARK(s(cons(head(cons(cons(y0, y1), y2)), y3)))) MARK(head(s(cons(head(cons(s(y0), y1)), y2)))) -> c16(MARK(s(cons(head(cons(s(y0), y1)), y2)))) MARK(head(s(cons(head(cons(y0, y1)), y2)))) -> c16(MARK(s(cons(head(cons(y0, y1)), y2)))) MARK(head(s(cons(head(cons(0, y0)), y1)))) -> c16(MARK(s(cons(head(cons(0, y0)), y1)))) MARK(head(s(cons(incr(tail(nats)), y0)))) -> c16(MARK(s(cons(incr(tail(nats)), y0)))) MARK(head(s(cons(head(tail(nats)), y0)))) -> c16(MARK(s(cons(head(tail(nats)), y0)))) MARK(head(s(cons(tail(tail(nats)), y0)))) -> c16(MARK(s(cons(tail(tail(nats)), y0)))) MARK(head(s(incr(cons(nats, y0))))) -> c16(MARK(s(incr(cons(nats, y0))))) MARK(head(s(incr(cons(pairs, y0))))) -> c16(MARK(s(incr(cons(pairs, y0))))) MARK(head(s(incr(cons(odds, y0))))) -> c16(MARK(s(incr(cons(odds, y0))))) MARK(head(s(head(cons(nats, y0))))) -> c16(MARK(s(head(cons(nats, y0))))) MARK(head(s(head(cons(pairs, y0))))) -> c16(MARK(s(head(cons(pairs, y0))))) MARK(head(s(head(cons(odds, y0))))) -> c16(MARK(s(head(cons(odds, y0))))) MARK(tail(s(s(y0)))) -> c17(MARK(s(s(y0)))) MARK(tail(s(incr(s(y0))))) -> c17(MARK(s(incr(s(y0))))) MARK(tail(s(head(s(y0))))) -> c17(MARK(s(head(s(y0))))) MARK(tail(s(tail(cons(y0, y1))))) -> c17(MARK(s(tail(cons(y0, y1))))) MARK(tail(s(tail(s(y0))))) -> c17(MARK(s(tail(s(y0))))) MARK(tail(s(incr(incr(incr(y0)))))) -> c17(MARK(s(incr(incr(incr(y0)))))) MARK(tail(s(incr(incr(odds))))) -> c17(MARK(s(incr(incr(odds))))) MARK(tail(s(incr(incr(head(y0)))))) -> c17(MARK(s(incr(incr(head(y0)))))) MARK(tail(s(incr(incr(tail(y0)))))) -> c17(MARK(s(incr(incr(tail(y0)))))) MARK(tail(s(incr(incr(cons(y0, y1)))))) -> c17(MARK(s(incr(incr(cons(y0, y1)))))) MARK(tail(s(incr(incr(s(y0)))))) -> c17(MARK(s(incr(incr(s(y0)))))) MARK(tail(s(incr(incr(y0))))) -> c17(MARK(s(incr(incr(y0))))) MARK(tail(s(incr(head(incr(y0)))))) -> c17(MARK(s(incr(head(incr(y0)))))) MARK(tail(s(incr(head(odds))))) -> c17(MARK(s(incr(head(odds))))) MARK(tail(s(incr(head(head(y0)))))) -> c17(MARK(s(incr(head(head(y0)))))) MARK(tail(s(incr(head(tail(y0)))))) -> c17(MARK(s(incr(head(tail(y0)))))) MARK(tail(s(incr(head(cons(y0, y1)))))) -> c17(MARK(s(incr(head(cons(y0, y1)))))) MARK(tail(s(incr(head(s(y0)))))) -> c17(MARK(s(incr(head(s(y0)))))) MARK(tail(s(incr(head(y0))))) -> c17(MARK(s(incr(head(y0))))) MARK(tail(s(incr(tail(incr(y0)))))) -> c17(MARK(s(incr(tail(incr(y0)))))) MARK(tail(s(incr(tail(pairs))))) -> c17(MARK(s(incr(tail(pairs))))) MARK(tail(s(incr(tail(odds))))) -> c17(MARK(s(incr(tail(odds))))) MARK(tail(s(incr(tail(head(y0)))))) -> c17(MARK(s(incr(tail(head(y0)))))) MARK(tail(s(incr(tail(tail(y0)))))) -> c17(MARK(s(incr(tail(tail(y0)))))) MARK(tail(s(incr(tail(cons(y0, y1)))))) -> c17(MARK(s(incr(tail(cons(y0, y1)))))) MARK(tail(s(incr(tail(s(y0)))))) -> c17(MARK(s(incr(tail(s(y0)))))) MARK(tail(s(incr(tail(y0))))) -> c17(MARK(s(incr(tail(y0))))) MARK(tail(s(head(incr(incr(y0)))))) -> c17(MARK(s(head(incr(incr(y0)))))) MARK(tail(s(head(incr(odds))))) -> c17(MARK(s(head(incr(odds))))) MARK(tail(s(head(incr(head(y0)))))) -> c17(MARK(s(head(incr(head(y0)))))) MARK(tail(s(head(incr(tail(y0)))))) -> c17(MARK(s(head(incr(tail(y0)))))) MARK(tail(s(head(incr(cons(y0, y1)))))) -> c17(MARK(s(head(incr(cons(y0, y1)))))) MARK(tail(s(head(incr(s(y0)))))) -> c17(MARK(s(head(incr(s(y0)))))) MARK(tail(s(head(incr(y0))))) -> c17(MARK(s(head(incr(y0))))) MARK(tail(s(head(head(incr(y0)))))) -> c17(MARK(s(head(head(incr(y0)))))) MARK(tail(s(head(head(odds))))) -> c17(MARK(s(head(head(odds))))) MARK(tail(s(head(head(head(y0)))))) -> c17(MARK(s(head(head(head(y0)))))) MARK(tail(s(head(head(tail(y0)))))) -> c17(MARK(s(head(head(tail(y0)))))) MARK(tail(s(head(head(cons(y0, y1)))))) -> c17(MARK(s(head(head(cons(y0, y1)))))) MARK(tail(s(head(head(s(y0)))))) -> c17(MARK(s(head(head(s(y0)))))) MARK(tail(s(head(head(y0))))) -> c17(MARK(s(head(head(y0))))) MARK(tail(s(head(tail(incr(y0)))))) -> c17(MARK(s(head(tail(incr(y0)))))) MARK(tail(s(head(tail(pairs))))) -> c17(MARK(s(head(tail(pairs))))) MARK(tail(s(head(tail(odds))))) -> c17(MARK(s(head(tail(odds))))) MARK(tail(s(head(tail(head(y0)))))) -> c17(MARK(s(head(tail(head(y0)))))) MARK(tail(s(head(tail(tail(y0)))))) -> c17(MARK(s(head(tail(tail(y0)))))) MARK(tail(s(head(tail(cons(y0, y1)))))) -> c17(MARK(s(head(tail(cons(y0, y1)))))) MARK(tail(s(head(tail(s(y0)))))) -> c17(MARK(s(head(tail(s(y0)))))) MARK(tail(s(head(tail(y0))))) -> c17(MARK(s(head(tail(y0))))) MARK(tail(s(tail(incr(incr(y0)))))) -> c17(MARK(s(tail(incr(incr(y0)))))) MARK(tail(s(tail(incr(odds))))) -> c17(MARK(s(tail(incr(odds))))) MARK(tail(s(tail(incr(head(y0)))))) -> c17(MARK(s(tail(incr(head(y0)))))) MARK(tail(s(tail(incr(tail(y0)))))) -> c17(MARK(s(tail(incr(tail(y0)))))) MARK(tail(s(tail(incr(cons(y0, y1)))))) -> c17(MARK(s(tail(incr(cons(y0, y1)))))) MARK(tail(s(tail(incr(s(y0)))))) -> c17(MARK(s(tail(incr(s(y0)))))) MARK(tail(s(tail(incr(y0))))) -> c17(MARK(s(tail(incr(y0))))) MARK(tail(s(tail(head(incr(y0)))))) -> c17(MARK(s(tail(head(incr(y0)))))) MARK(tail(s(tail(head(odds))))) -> c17(MARK(s(tail(head(odds))))) MARK(tail(s(tail(head(head(y0)))))) -> c17(MARK(s(tail(head(head(y0)))))) MARK(tail(s(tail(head(tail(y0)))))) -> c17(MARK(s(tail(head(tail(y0)))))) MARK(tail(s(tail(head(cons(y0, y1)))))) -> c17(MARK(s(tail(head(cons(y0, y1)))))) MARK(tail(s(tail(head(s(y0)))))) -> c17(MARK(s(tail(head(s(y0)))))) MARK(tail(s(tail(head(y0))))) -> c17(MARK(s(tail(head(y0))))) MARK(tail(s(tail(tail(incr(y0)))))) -> c17(MARK(s(tail(tail(incr(y0)))))) MARK(tail(s(tail(tail(pairs))))) -> c17(MARK(s(tail(tail(pairs))))) MARK(tail(s(tail(tail(odds))))) -> c17(MARK(s(tail(tail(odds))))) MARK(tail(s(tail(tail(head(y0)))))) -> c17(MARK(s(tail(tail(head(y0)))))) MARK(tail(s(tail(tail(tail(y0)))))) -> c17(MARK(s(tail(tail(tail(y0)))))) MARK(tail(s(tail(tail(cons(y0, y1)))))) -> c17(MARK(s(tail(tail(cons(y0, y1)))))) MARK(tail(s(tail(tail(s(y0)))))) -> c17(MARK(s(tail(tail(s(y0)))))) MARK(tail(s(tail(tail(y0))))) -> c17(MARK(s(tail(tail(y0))))) MARK(tail(s(tail(pairs)))) -> c17(MARK(s(tail(pairs)))) MARK(tail(s(incr(incr(nats))))) -> c17(MARK(s(incr(incr(nats))))) MARK(tail(s(incr(incr(pairs))))) -> c17(MARK(s(incr(incr(pairs))))) MARK(tail(s(incr(odds)))) -> c17(MARK(s(incr(odds)))) MARK(tail(s(head(incr(nats))))) -> c17(MARK(s(head(incr(nats))))) MARK(tail(s(head(incr(pairs))))) -> c17(MARK(s(head(incr(pairs))))) MARK(tail(s(head(odds)))) -> c17(MARK(s(head(odds)))) MARK(tail(s(tail(incr(nats))))) -> c17(MARK(s(tail(incr(nats))))) MARK(tail(s(tail(incr(pairs))))) -> c17(MARK(s(tail(incr(pairs))))) MARK(tail(s(tail(odds)))) -> c17(MARK(s(tail(odds)))) MARK(tail(s(incr(cons(incr(y0), y1))))) -> c17(MARK(s(incr(cons(incr(y0), y1))))) MARK(tail(s(incr(cons(head(y0), y1))))) -> c17(MARK(s(incr(cons(head(y0), y1))))) MARK(tail(s(incr(cons(tail(y0), y1))))) -> c17(MARK(s(incr(cons(tail(y0), y1))))) MARK(tail(s(incr(cons(cons(y0, y1), y2))))) -> c17(MARK(s(incr(cons(cons(y0, y1), y2))))) MARK(tail(s(incr(cons(s(y0), y1))))) -> c17(MARK(s(incr(cons(s(y0), y1))))) MARK(tail(s(incr(cons(y0, y1))))) -> c17(MARK(s(incr(cons(y0, y1))))) MARK(tail(s(incr(head(nats))))) -> c17(MARK(s(incr(head(nats))))) MARK(tail(s(incr(head(pairs))))) -> c17(MARK(s(incr(head(pairs))))) MARK(tail(s(head(head(nats))))) -> c17(MARK(s(head(head(nats))))) MARK(tail(s(head(head(pairs))))) -> c17(MARK(s(head(head(pairs))))) MARK(tail(s(tail(head(nats))))) -> c17(MARK(s(tail(head(nats))))) MARK(tail(s(tail(head(pairs))))) -> c17(MARK(s(tail(head(pairs))))) MARK(tail(s(head(cons(incr(y0), y1))))) -> c17(MARK(s(head(cons(incr(y0), y1))))) MARK(tail(s(head(cons(head(y0), y1))))) -> c17(MARK(s(head(cons(head(y0), y1))))) MARK(tail(s(head(cons(tail(y0), y1))))) -> c17(MARK(s(head(cons(tail(y0), y1))))) MARK(tail(s(head(cons(cons(y0, y1), y2))))) -> c17(MARK(s(head(cons(cons(y0, y1), y2))))) MARK(tail(s(head(cons(s(y0), y1))))) -> c17(MARK(s(head(cons(s(y0), y1))))) MARK(tail(s(head(cons(y0, y1))))) -> c17(MARK(s(head(cons(y0, y1))))) MARK(tail(s(incr(tail(nats))))) -> c17(MARK(s(incr(tail(nats))))) MARK(tail(s(head(tail(nats))))) -> c17(MARK(s(head(tail(nats))))) MARK(tail(s(tail(tail(nats))))) -> c17(MARK(s(tail(tail(nats))))) MARK(tail(s(cons(cons(y0, y1), y2)))) -> c17(MARK(s(cons(cons(y0, y1), y2)))) MARK(tail(s(cons(s(y0), y1)))) -> c17(MARK(s(cons(s(y0), y1)))) MARK(tail(s(cons(incr(s(y0)), y1)))) -> c17(MARK(s(cons(incr(s(y0)), y1)))) MARK(tail(s(cons(head(s(y0)), y1)))) -> c17(MARK(s(cons(head(s(y0)), y1)))) MARK(tail(s(cons(tail(cons(y0, y1)), y2)))) -> c17(MARK(s(cons(tail(cons(y0, y1)), y2)))) MARK(tail(s(cons(tail(s(y0)), y1)))) -> c17(MARK(s(cons(tail(s(y0)), y1)))) MARK(tail(s(cons(incr(incr(incr(y0))), y1)))) -> c17(MARK(s(cons(incr(incr(incr(y0))), y1)))) MARK(tail(s(cons(incr(incr(odds)), y0)))) -> c17(MARK(s(cons(incr(incr(odds)), y0)))) MARK(tail(s(cons(incr(incr(head(y0))), y1)))) -> c17(MARK(s(cons(incr(incr(head(y0))), y1)))) MARK(tail(s(cons(incr(incr(tail(y0))), y1)))) -> c17(MARK(s(cons(incr(incr(tail(y0))), y1)))) MARK(tail(s(cons(incr(incr(cons(y0, y1))), y2)))) -> c17(MARK(s(cons(incr(incr(cons(y0, y1))), y2)))) MARK(tail(s(cons(incr(incr(s(y0))), y1)))) -> c17(MARK(s(cons(incr(incr(s(y0))), y1)))) MARK(tail(s(cons(incr(incr(y0)), y1)))) -> c17(MARK(s(cons(incr(incr(y0)), y1)))) MARK(tail(s(cons(incr(head(incr(y0))), y1)))) -> c17(MARK(s(cons(incr(head(incr(y0))), y1)))) MARK(tail(s(cons(incr(head(odds)), y0)))) -> c17(MARK(s(cons(incr(head(odds)), y0)))) MARK(tail(s(cons(incr(head(head(y0))), y1)))) -> c17(MARK(s(cons(incr(head(head(y0))), y1)))) MARK(tail(s(cons(incr(head(tail(y0))), y1)))) -> c17(MARK(s(cons(incr(head(tail(y0))), y1)))) MARK(tail(s(cons(incr(head(cons(y0, y1))), y2)))) -> c17(MARK(s(cons(incr(head(cons(y0, y1))), y2)))) MARK(tail(s(cons(incr(head(s(y0))), y1)))) -> c17(MARK(s(cons(incr(head(s(y0))), y1)))) MARK(tail(s(cons(incr(head(y0)), y1)))) -> c17(MARK(s(cons(incr(head(y0)), y1)))) MARK(tail(s(cons(incr(tail(incr(y0))), y1)))) -> c17(MARK(s(cons(incr(tail(incr(y0))), y1)))) MARK(tail(s(cons(incr(tail(pairs)), y0)))) -> c17(MARK(s(cons(incr(tail(pairs)), y0)))) MARK(tail(s(cons(incr(tail(odds)), y0)))) -> c17(MARK(s(cons(incr(tail(odds)), y0)))) MARK(tail(s(cons(incr(tail(head(y0))), y1)))) -> c17(MARK(s(cons(incr(tail(head(y0))), y1)))) MARK(tail(s(cons(incr(tail(tail(y0))), y1)))) -> c17(MARK(s(cons(incr(tail(tail(y0))), y1)))) MARK(tail(s(cons(incr(tail(cons(y0, y1))), y2)))) -> c17(MARK(s(cons(incr(tail(cons(y0, y1))), y2)))) MARK(tail(s(cons(incr(tail(s(y0))), y1)))) -> c17(MARK(s(cons(incr(tail(s(y0))), y1)))) MARK(tail(s(cons(incr(tail(y0)), y1)))) -> c17(MARK(s(cons(incr(tail(y0)), y1)))) MARK(tail(s(cons(head(incr(incr(y0))), y1)))) -> c17(MARK(s(cons(head(incr(incr(y0))), y1)))) MARK(tail(s(cons(head(incr(odds)), y0)))) -> c17(MARK(s(cons(head(incr(odds)), y0)))) MARK(tail(s(cons(head(incr(head(y0))), y1)))) -> c17(MARK(s(cons(head(incr(head(y0))), y1)))) MARK(tail(s(cons(head(incr(tail(y0))), y1)))) -> c17(MARK(s(cons(head(incr(tail(y0))), y1)))) MARK(tail(s(cons(head(incr(cons(y0, y1))), y2)))) -> c17(MARK(s(cons(head(incr(cons(y0, y1))), y2)))) MARK(tail(s(cons(head(incr(s(y0))), y1)))) -> c17(MARK(s(cons(head(incr(s(y0))), y1)))) MARK(tail(s(cons(head(incr(y0)), y1)))) -> c17(MARK(s(cons(head(incr(y0)), y1)))) MARK(tail(s(cons(head(head(incr(y0))), y1)))) -> c17(MARK(s(cons(head(head(incr(y0))), y1)))) MARK(tail(s(cons(head(head(odds)), y0)))) -> c17(MARK(s(cons(head(head(odds)), y0)))) MARK(tail(s(cons(head(head(head(y0))), y1)))) -> c17(MARK(s(cons(head(head(head(y0))), y1)))) MARK(tail(s(cons(head(head(tail(y0))), y1)))) -> c17(MARK(s(cons(head(head(tail(y0))), y1)))) MARK(tail(s(cons(head(head(cons(y0, y1))), y2)))) -> c17(MARK(s(cons(head(head(cons(y0, y1))), y2)))) MARK(tail(s(cons(head(head(s(y0))), y1)))) -> c17(MARK(s(cons(head(head(s(y0))), y1)))) MARK(tail(s(cons(head(head(y0)), y1)))) -> c17(MARK(s(cons(head(head(y0)), y1)))) MARK(tail(s(cons(head(tail(incr(y0))), y1)))) -> c17(MARK(s(cons(head(tail(incr(y0))), y1)))) MARK(tail(s(cons(head(tail(pairs)), y0)))) -> c17(MARK(s(cons(head(tail(pairs)), y0)))) MARK(tail(s(cons(head(tail(odds)), y0)))) -> c17(MARK(s(cons(head(tail(odds)), y0)))) MARK(tail(s(cons(head(tail(head(y0))), y1)))) -> c17(MARK(s(cons(head(tail(head(y0))), y1)))) MARK(tail(s(cons(head(tail(tail(y0))), y1)))) -> c17(MARK(s(cons(head(tail(tail(y0))), y1)))) MARK(tail(s(cons(head(tail(cons(y0, y1))), y2)))) -> c17(MARK(s(cons(head(tail(cons(y0, y1))), y2)))) MARK(tail(s(cons(head(tail(s(y0))), y1)))) -> c17(MARK(s(cons(head(tail(s(y0))), y1)))) MARK(tail(s(cons(head(tail(y0)), y1)))) -> c17(MARK(s(cons(head(tail(y0)), y1)))) MARK(tail(s(cons(tail(incr(incr(y0))), y1)))) -> c17(MARK(s(cons(tail(incr(incr(y0))), y1)))) MARK(tail(s(cons(tail(incr(odds)), y0)))) -> c17(MARK(s(cons(tail(incr(odds)), y0)))) MARK(tail(s(cons(tail(incr(head(y0))), y1)))) -> c17(MARK(s(cons(tail(incr(head(y0))), y1)))) MARK(tail(s(cons(tail(incr(tail(y0))), y1)))) -> c17(MARK(s(cons(tail(incr(tail(y0))), y1)))) MARK(tail(s(cons(tail(incr(cons(y0, y1))), y2)))) -> c17(MARK(s(cons(tail(incr(cons(y0, y1))), y2)))) MARK(tail(s(cons(tail(incr(s(y0))), y1)))) -> c17(MARK(s(cons(tail(incr(s(y0))), y1)))) MARK(tail(s(cons(tail(incr(y0)), y1)))) -> c17(MARK(s(cons(tail(incr(y0)), y1)))) MARK(tail(s(cons(tail(head(incr(y0))), y1)))) -> c17(MARK(s(cons(tail(head(incr(y0))), y1)))) MARK(tail(s(cons(tail(head(odds)), y0)))) -> c17(MARK(s(cons(tail(head(odds)), y0)))) MARK(tail(s(cons(tail(head(head(y0))), y1)))) -> c17(MARK(s(cons(tail(head(head(y0))), y1)))) MARK(tail(s(cons(tail(head(tail(y0))), y1)))) -> c17(MARK(s(cons(tail(head(tail(y0))), y1)))) MARK(tail(s(cons(tail(head(cons(y0, y1))), y2)))) -> c17(MARK(s(cons(tail(head(cons(y0, y1))), y2)))) MARK(tail(s(cons(tail(head(s(y0))), y1)))) -> c17(MARK(s(cons(tail(head(s(y0))), y1)))) MARK(tail(s(cons(tail(head(y0)), y1)))) -> c17(MARK(s(cons(tail(head(y0)), y1)))) MARK(tail(s(cons(tail(tail(incr(y0))), y1)))) -> c17(MARK(s(cons(tail(tail(incr(y0))), y1)))) MARK(tail(s(cons(tail(tail(pairs)), y0)))) -> c17(MARK(s(cons(tail(tail(pairs)), y0)))) MARK(tail(s(cons(tail(tail(odds)), y0)))) -> c17(MARK(s(cons(tail(tail(odds)), y0)))) MARK(tail(s(cons(tail(tail(head(y0))), y1)))) -> c17(MARK(s(cons(tail(tail(head(y0))), y1)))) MARK(tail(s(cons(tail(tail(tail(y0))), y1)))) -> c17(MARK(s(cons(tail(tail(tail(y0))), y1)))) MARK(tail(s(cons(tail(tail(cons(y0, y1))), y2)))) -> c17(MARK(s(cons(tail(tail(cons(y0, y1))), y2)))) MARK(tail(s(cons(tail(tail(s(y0))), y1)))) -> c17(MARK(s(cons(tail(tail(s(y0))), y1)))) MARK(tail(s(cons(tail(tail(y0)), y1)))) -> c17(MARK(s(cons(tail(tail(y0)), y1)))) MARK(tail(s(cons(tail(pairs), y0)))) -> c17(MARK(s(cons(tail(pairs), y0)))) MARK(tail(s(cons(incr(incr(nats)), y0)))) -> c17(MARK(s(cons(incr(incr(nats)), y0)))) MARK(tail(s(cons(incr(incr(pairs)), y0)))) -> c17(MARK(s(cons(incr(incr(pairs)), y0)))) MARK(tail(s(cons(incr(odds), y0)))) -> c17(MARK(s(cons(incr(odds), y0)))) MARK(tail(s(cons(head(incr(nats)), y0)))) -> c17(MARK(s(cons(head(incr(nats)), y0)))) MARK(tail(s(cons(head(incr(pairs)), y0)))) -> c17(MARK(s(cons(head(incr(pairs)), y0)))) MARK(tail(s(cons(head(odds), y0)))) -> c17(MARK(s(cons(head(odds), y0)))) MARK(tail(s(cons(tail(incr(nats)), y0)))) -> c17(MARK(s(cons(tail(incr(nats)), y0)))) MARK(tail(s(cons(tail(incr(pairs)), y0)))) -> c17(MARK(s(cons(tail(incr(pairs)), y0)))) MARK(tail(s(cons(tail(odds), y0)))) -> c17(MARK(s(cons(tail(odds), y0)))) MARK(tail(s(cons(incr(cons(nats, y0)), y1)))) -> c17(MARK(s(cons(incr(cons(nats, y0)), y1)))) MARK(tail(s(cons(incr(cons(incr(y0), y1)), y2)))) -> c17(MARK(s(cons(incr(cons(incr(y0), y1)), y2)))) MARK(tail(s(cons(incr(cons(pairs, y0)), y1)))) -> c17(MARK(s(cons(incr(cons(pairs, y0)), y1)))) MARK(tail(s(cons(incr(cons(odds, y0)), y1)))) -> c17(MARK(s(cons(incr(cons(odds, y0)), y1)))) MARK(tail(s(cons(incr(cons(head(y0), y1)), y2)))) -> c17(MARK(s(cons(incr(cons(head(y0), y1)), y2)))) MARK(tail(s(cons(incr(cons(tail(y0), y1)), y2)))) -> c17(MARK(s(cons(incr(cons(tail(y0), y1)), y2)))) MARK(tail(s(cons(incr(cons(cons(y0, y1), y2)), y3)))) -> c17(MARK(s(cons(incr(cons(cons(y0, y1), y2)), y3)))) MARK(tail(s(cons(incr(cons(s(y0), y1)), y2)))) -> c17(MARK(s(cons(incr(cons(s(y0), y1)), y2)))) MARK(tail(s(cons(incr(cons(y0, y1)), y2)))) -> c17(MARK(s(cons(incr(cons(y0, y1)), y2)))) MARK(tail(s(cons(incr(cons(0, y0)), y1)))) -> c17(MARK(s(cons(incr(cons(0, y0)), y1)))) MARK(tail(s(cons(incr(head(nats)), y0)))) -> c17(MARK(s(cons(incr(head(nats)), y0)))) MARK(tail(s(cons(incr(head(pairs)), y0)))) -> c17(MARK(s(cons(incr(head(pairs)), y0)))) MARK(tail(s(cons(head(head(nats)), y0)))) -> c17(MARK(s(cons(head(head(nats)), y0)))) MARK(tail(s(cons(head(head(pairs)), y0)))) -> c17(MARK(s(cons(head(head(pairs)), y0)))) MARK(tail(s(cons(tail(head(nats)), y0)))) -> c17(MARK(s(cons(tail(head(nats)), y0)))) MARK(tail(s(cons(tail(head(pairs)), y0)))) -> c17(MARK(s(cons(tail(head(pairs)), y0)))) MARK(tail(s(cons(head(cons(nats, y0)), y1)))) -> c17(MARK(s(cons(head(cons(nats, y0)), y1)))) MARK(tail(s(cons(head(cons(incr(y0), y1)), y2)))) -> c17(MARK(s(cons(head(cons(incr(y0), y1)), y2)))) MARK(tail(s(cons(head(cons(pairs, y0)), y1)))) -> c17(MARK(s(cons(head(cons(pairs, y0)), y1)))) MARK(tail(s(cons(head(cons(odds, y0)), y1)))) -> c17(MARK(s(cons(head(cons(odds, y0)), y1)))) MARK(tail(s(cons(head(cons(head(y0), y1)), y2)))) -> c17(MARK(s(cons(head(cons(head(y0), y1)), y2)))) MARK(tail(s(cons(head(cons(tail(y0), y1)), y2)))) -> c17(MARK(s(cons(head(cons(tail(y0), y1)), y2)))) MARK(tail(s(cons(head(cons(cons(y0, y1), y2)), y3)))) -> c17(MARK(s(cons(head(cons(cons(y0, y1), y2)), y3)))) MARK(tail(s(cons(head(cons(s(y0), y1)), y2)))) -> c17(MARK(s(cons(head(cons(s(y0), y1)), y2)))) MARK(tail(s(cons(head(cons(y0, y1)), y2)))) -> c17(MARK(s(cons(head(cons(y0, y1)), y2)))) MARK(tail(s(cons(head(cons(0, y0)), y1)))) -> c17(MARK(s(cons(head(cons(0, y0)), y1)))) MARK(tail(s(cons(incr(tail(nats)), y0)))) -> c17(MARK(s(cons(incr(tail(nats)), y0)))) MARK(tail(s(cons(head(tail(nats)), y0)))) -> c17(MARK(s(cons(head(tail(nats)), y0)))) MARK(tail(s(cons(tail(tail(nats)), y0)))) -> c17(MARK(s(cons(tail(tail(nats)), y0)))) MARK(tail(s(incr(cons(nats, y0))))) -> c17(MARK(s(incr(cons(nats, y0))))) MARK(tail(s(incr(cons(pairs, y0))))) -> c17(MARK(s(incr(cons(pairs, y0))))) MARK(tail(s(incr(cons(odds, y0))))) -> c17(MARK(s(incr(cons(odds, y0))))) MARK(tail(s(head(cons(nats, y0))))) -> c17(MARK(s(head(cons(nats, y0))))) MARK(tail(s(head(cons(pairs, y0))))) -> c17(MARK(s(head(cons(pairs, y0))))) MARK(tail(s(head(cons(odds, y0))))) -> c17(MARK(s(head(cons(odds, y0))))) K tuples:none Defined Rule Symbols: a__nats, a__pairs, a__odds, a__incr_1, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: MARK_1, A__INCR_1, A__HEAD_1, A__TAIL_1 Compound Symbols: c17_2, c13_2, c13_1, c16_2, c16_1, c17_1, c6_1, c8_1, c10_1, c18_1, c20_1 ---------------------------------------- (199) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID)) Use forward instantiation to replace MARK(incr(incr(x0))) -> c13(MARK(incr(x0))) by MARK(incr(incr(incr(incr(y0))))) -> c13(MARK(incr(incr(incr(y0))))) MARK(incr(incr(incr(odds)))) -> c13(MARK(incr(incr(odds)))) MARK(incr(incr(incr(head(y0))))) -> c13(MARK(incr(incr(head(y0))))) MARK(incr(incr(incr(tail(y0))))) -> c13(MARK(incr(incr(tail(y0))))) MARK(incr(incr(incr(cons(y0, y1))))) -> c13(MARK(incr(incr(cons(y0, y1))))) MARK(incr(incr(incr(s(y0))))) -> c13(MARK(incr(incr(s(y0))))) MARK(incr(incr(incr(y0)))) -> c13(MARK(incr(incr(y0)))) MARK(incr(incr(head(incr(y0))))) -> c13(MARK(incr(head(incr(y0))))) MARK(incr(incr(head(odds)))) -> c13(MARK(incr(head(odds)))) MARK(incr(incr(head(head(y0))))) -> c13(MARK(incr(head(head(y0))))) MARK(incr(incr(head(tail(y0))))) -> c13(MARK(incr(head(tail(y0))))) MARK(incr(incr(head(cons(y0, y1))))) -> c13(MARK(incr(head(cons(y0, y1))))) MARK(incr(incr(head(s(y0))))) -> c13(MARK(incr(head(s(y0))))) MARK(incr(incr(head(y0)))) -> c13(MARK(incr(head(y0)))) MARK(incr(incr(tail(incr(y0))))) -> c13(MARK(incr(tail(incr(y0))))) MARK(incr(incr(tail(pairs)))) -> c13(MARK(incr(tail(pairs)))) MARK(incr(incr(tail(odds)))) -> c13(MARK(incr(tail(odds)))) MARK(incr(incr(tail(head(y0))))) -> c13(MARK(incr(tail(head(y0))))) MARK(incr(incr(tail(tail(y0))))) -> c13(MARK(incr(tail(tail(y0))))) MARK(incr(incr(tail(cons(y0, y1))))) -> c13(MARK(incr(tail(cons(y0, y1))))) MARK(incr(incr(tail(s(y0))))) -> c13(MARK(incr(tail(s(y0))))) MARK(incr(incr(tail(y0)))) -> c13(MARK(incr(tail(y0)))) MARK(incr(incr(incr(nats)))) -> c13(MARK(incr(incr(nats)))) MARK(incr(incr(incr(pairs)))) -> c13(MARK(incr(incr(pairs)))) MARK(incr(incr(odds))) -> c13(MARK(incr(odds))) MARK(incr(incr(cons(incr(y0), y1)))) -> c13(MARK(incr(cons(incr(y0), y1)))) MARK(incr(incr(cons(head(y0), y1)))) -> c13(MARK(incr(cons(head(y0), y1)))) MARK(incr(incr(cons(tail(y0), y1)))) -> c13(MARK(incr(cons(tail(y0), y1)))) MARK(incr(incr(cons(cons(y0, y1), y2)))) -> c13(MARK(incr(cons(cons(y0, y1), y2)))) MARK(incr(incr(cons(s(y0), y1)))) -> c13(MARK(incr(cons(s(y0), y1)))) MARK(incr(incr(cons(y0, y1)))) -> c13(MARK(incr(cons(y0, y1)))) MARK(incr(incr(head(nats)))) -> c13(MARK(incr(head(nats)))) MARK(incr(incr(head(pairs)))) -> c13(MARK(incr(head(pairs)))) MARK(incr(incr(tail(nats)))) -> c13(MARK(incr(tail(nats)))) MARK(incr(incr(cons(nats, y0)))) -> c13(MARK(incr(cons(nats, y0)))) MARK(incr(incr(cons(pairs, y0)))) -> c13(MARK(incr(cons(pairs, y0)))) MARK(incr(incr(cons(odds, y0)))) -> c13(MARK(incr(cons(odds, y0)))) MARK(incr(incr(s(s(y0))))) -> c13(MARK(incr(s(s(y0))))) MARK(incr(incr(s(incr(s(y0)))))) -> c13(MARK(incr(s(incr(s(y0)))))) MARK(incr(incr(s(head(s(y0)))))) -> c13(MARK(incr(s(head(s(y0)))))) MARK(incr(incr(s(tail(cons(y0, y1)))))) -> c13(MARK(incr(s(tail(cons(y0, y1)))))) MARK(incr(incr(s(tail(s(y0)))))) -> c13(MARK(incr(s(tail(s(y0)))))) MARK(incr(incr(s(incr(incr(incr(y0))))))) -> c13(MARK(incr(s(incr(incr(incr(y0))))))) MARK(incr(incr(s(incr(incr(odds)))))) -> c13(MARK(incr(s(incr(incr(odds)))))) MARK(incr(incr(s(incr(incr(head(y0))))))) -> c13(MARK(incr(s(incr(incr(head(y0))))))) MARK(incr(incr(s(incr(incr(tail(y0))))))) -> c13(MARK(incr(s(incr(incr(tail(y0))))))) MARK(incr(incr(s(incr(incr(cons(y0, y1))))))) -> c13(MARK(incr(s(incr(incr(cons(y0, y1))))))) MARK(incr(incr(s(incr(incr(s(y0))))))) -> c13(MARK(incr(s(incr(incr(s(y0))))))) MARK(incr(incr(s(incr(incr(y0)))))) -> c13(MARK(incr(s(incr(incr(y0)))))) MARK(incr(incr(s(incr(head(incr(y0))))))) -> c13(MARK(incr(s(incr(head(incr(y0))))))) MARK(incr(incr(s(incr(head(odds)))))) -> c13(MARK(incr(s(incr(head(odds)))))) MARK(incr(incr(s(incr(head(head(y0))))))) -> c13(MARK(incr(s(incr(head(head(y0))))))) MARK(incr(incr(s(incr(head(tail(y0))))))) -> c13(MARK(incr(s(incr(head(tail(y0))))))) MARK(incr(incr(s(incr(head(cons(y0, y1))))))) -> c13(MARK(incr(s(incr(head(cons(y0, y1))))))) MARK(incr(incr(s(incr(head(s(y0))))))) -> c13(MARK(incr(s(incr(head(s(y0))))))) MARK(incr(incr(s(incr(head(y0)))))) -> c13(MARK(incr(s(incr(head(y0)))))) MARK(incr(incr(s(incr(tail(incr(y0))))))) -> c13(MARK(incr(s(incr(tail(incr(y0))))))) MARK(incr(incr(s(incr(tail(pairs)))))) -> c13(MARK(incr(s(incr(tail(pairs)))))) MARK(incr(incr(s(incr(tail(odds)))))) -> c13(MARK(incr(s(incr(tail(odds)))))) MARK(incr(incr(s(incr(tail(head(y0))))))) -> c13(MARK(incr(s(incr(tail(head(y0))))))) MARK(incr(incr(s(incr(tail(tail(y0))))))) -> c13(MARK(incr(s(incr(tail(tail(y0))))))) MARK(incr(incr(s(incr(tail(cons(y0, y1))))))) -> c13(MARK(incr(s(incr(tail(cons(y0, y1))))))) MARK(incr(incr(s(incr(tail(s(y0))))))) -> c13(MARK(incr(s(incr(tail(s(y0))))))) MARK(incr(incr(s(incr(tail(y0)))))) -> c13(MARK(incr(s(incr(tail(y0)))))) MARK(incr(incr(s(head(incr(incr(y0))))))) -> c13(MARK(incr(s(head(incr(incr(y0))))))) MARK(incr(incr(s(head(incr(odds)))))) -> c13(MARK(incr(s(head(incr(odds)))))) MARK(incr(incr(s(head(incr(head(y0))))))) -> c13(MARK(incr(s(head(incr(head(y0))))))) MARK(incr(incr(s(head(incr(tail(y0))))))) -> c13(MARK(incr(s(head(incr(tail(y0))))))) MARK(incr(incr(s(head(incr(cons(y0, y1))))))) -> c13(MARK(incr(s(head(incr(cons(y0, y1))))))) MARK(incr(incr(s(head(incr(s(y0))))))) -> c13(MARK(incr(s(head(incr(s(y0))))))) MARK(incr(incr(s(head(incr(y0)))))) -> c13(MARK(incr(s(head(incr(y0)))))) MARK(incr(incr(s(head(head(incr(y0))))))) -> c13(MARK(incr(s(head(head(incr(y0))))))) MARK(incr(incr(s(head(head(odds)))))) -> c13(MARK(incr(s(head(head(odds)))))) MARK(incr(incr(s(head(head(head(y0))))))) -> c13(MARK(incr(s(head(head(head(y0))))))) MARK(incr(incr(s(head(head(tail(y0))))))) -> c13(MARK(incr(s(head(head(tail(y0))))))) MARK(incr(incr(s(head(head(cons(y0, y1))))))) -> c13(MARK(incr(s(head(head(cons(y0, y1))))))) MARK(incr(incr(s(head(head(s(y0))))))) -> c13(MARK(incr(s(head(head(s(y0))))))) MARK(incr(incr(s(head(head(y0)))))) -> c13(MARK(incr(s(head(head(y0)))))) MARK(incr(incr(s(head(tail(incr(y0))))))) -> c13(MARK(incr(s(head(tail(incr(y0))))))) MARK(incr(incr(s(head(tail(pairs)))))) -> c13(MARK(incr(s(head(tail(pairs)))))) MARK(incr(incr(s(head(tail(odds)))))) -> c13(MARK(incr(s(head(tail(odds)))))) MARK(incr(incr(s(head(tail(head(y0))))))) -> c13(MARK(incr(s(head(tail(head(y0))))))) MARK(incr(incr(s(head(tail(tail(y0))))))) -> c13(MARK(incr(s(head(tail(tail(y0))))))) MARK(incr(incr(s(head(tail(cons(y0, y1))))))) -> c13(MARK(incr(s(head(tail(cons(y0, y1))))))) MARK(incr(incr(s(head(tail(s(y0))))))) -> c13(MARK(incr(s(head(tail(s(y0))))))) MARK(incr(incr(s(head(tail(y0)))))) -> c13(MARK(incr(s(head(tail(y0)))))) MARK(incr(incr(s(tail(incr(incr(y0))))))) -> c13(MARK(incr(s(tail(incr(incr(y0))))))) MARK(incr(incr(s(tail(incr(odds)))))) -> c13(MARK(incr(s(tail(incr(odds)))))) MARK(incr(incr(s(tail(incr(head(y0))))))) -> c13(MARK(incr(s(tail(incr(head(y0))))))) MARK(incr(incr(s(tail(incr(tail(y0))))))) -> c13(MARK(incr(s(tail(incr(tail(y0))))))) MARK(incr(incr(s(tail(incr(cons(y0, y1))))))) -> c13(MARK(incr(s(tail(incr(cons(y0, y1))))))) MARK(incr(incr(s(tail(incr(s(y0))))))) -> c13(MARK(incr(s(tail(incr(s(y0))))))) MARK(incr(incr(s(tail(incr(y0)))))) -> c13(MARK(incr(s(tail(incr(y0)))))) MARK(incr(incr(s(tail(head(incr(y0))))))) -> c13(MARK(incr(s(tail(head(incr(y0))))))) MARK(incr(incr(s(tail(head(odds)))))) -> c13(MARK(incr(s(tail(head(odds)))))) MARK(incr(incr(s(tail(head(head(y0))))))) -> c13(MARK(incr(s(tail(head(head(y0))))))) MARK(incr(incr(s(tail(head(tail(y0))))))) -> c13(MARK(incr(s(tail(head(tail(y0))))))) MARK(incr(incr(s(tail(head(cons(y0, y1))))))) -> c13(MARK(incr(s(tail(head(cons(y0, y1))))))) MARK(incr(incr(s(tail(head(s(y0))))))) -> c13(MARK(incr(s(tail(head(s(y0))))))) MARK(incr(incr(s(tail(head(y0)))))) -> c13(MARK(incr(s(tail(head(y0)))))) MARK(incr(incr(s(tail(tail(incr(y0))))))) -> c13(MARK(incr(s(tail(tail(incr(y0))))))) MARK(incr(incr(s(tail(tail(pairs)))))) -> c13(MARK(incr(s(tail(tail(pairs)))))) MARK(incr(incr(s(tail(tail(odds)))))) -> c13(MARK(incr(s(tail(tail(odds)))))) MARK(incr(incr(s(tail(tail(head(y0))))))) -> c13(MARK(incr(s(tail(tail(head(y0))))))) MARK(incr(incr(s(tail(tail(tail(y0))))))) -> c13(MARK(incr(s(tail(tail(tail(y0))))))) MARK(incr(incr(s(tail(tail(cons(y0, y1))))))) -> c13(MARK(incr(s(tail(tail(cons(y0, y1))))))) MARK(incr(incr(s(tail(tail(s(y0))))))) -> c13(MARK(incr(s(tail(tail(s(y0))))))) MARK(incr(incr(s(tail(tail(y0)))))) -> c13(MARK(incr(s(tail(tail(y0)))))) MARK(incr(incr(s(tail(pairs))))) -> c13(MARK(incr(s(tail(pairs))))) MARK(incr(incr(s(incr(incr(nats)))))) -> c13(MARK(incr(s(incr(incr(nats)))))) MARK(incr(incr(s(incr(incr(pairs)))))) -> c13(MARK(incr(s(incr(incr(pairs)))))) MARK(incr(incr(s(incr(odds))))) -> c13(MARK(incr(s(incr(odds))))) MARK(incr(incr(s(head(incr(nats)))))) -> c13(MARK(incr(s(head(incr(nats)))))) MARK(incr(incr(s(head(incr(pairs)))))) -> c13(MARK(incr(s(head(incr(pairs)))))) MARK(incr(incr(s(head(odds))))) -> c13(MARK(incr(s(head(odds))))) MARK(incr(incr(s(tail(incr(nats)))))) -> c13(MARK(incr(s(tail(incr(nats)))))) MARK(incr(incr(s(tail(incr(pairs)))))) -> c13(MARK(incr(s(tail(incr(pairs)))))) MARK(incr(incr(s(tail(odds))))) -> c13(MARK(incr(s(tail(odds))))) MARK(incr(incr(s(incr(cons(incr(y0), y1)))))) -> c13(MARK(incr(s(incr(cons(incr(y0), y1)))))) MARK(incr(incr(s(incr(cons(head(y0), y1)))))) -> c13(MARK(incr(s(incr(cons(head(y0), y1)))))) MARK(incr(incr(s(incr(cons(tail(y0), y1)))))) -> c13(MARK(incr(s(incr(cons(tail(y0), y1)))))) MARK(incr(incr(s(incr(cons(cons(y0, y1), y2)))))) -> c13(MARK(incr(s(incr(cons(cons(y0, y1), y2)))))) MARK(incr(incr(s(incr(cons(s(y0), y1)))))) -> c13(MARK(incr(s(incr(cons(s(y0), y1)))))) MARK(incr(incr(s(incr(cons(y0, y1)))))) -> c13(MARK(incr(s(incr(cons(y0, y1)))))) MARK(incr(incr(s(incr(head(nats)))))) -> c13(MARK(incr(s(incr(head(nats)))))) MARK(incr(incr(s(incr(head(pairs)))))) -> c13(MARK(incr(s(incr(head(pairs)))))) MARK(incr(incr(s(head(head(nats)))))) -> c13(MARK(incr(s(head(head(nats)))))) MARK(incr(incr(s(head(head(pairs)))))) -> c13(MARK(incr(s(head(head(pairs)))))) MARK(incr(incr(s(tail(head(nats)))))) -> c13(MARK(incr(s(tail(head(nats)))))) MARK(incr(incr(s(tail(head(pairs)))))) -> c13(MARK(incr(s(tail(head(pairs)))))) MARK(incr(incr(s(head(cons(incr(y0), y1)))))) -> c13(MARK(incr(s(head(cons(incr(y0), y1)))))) MARK(incr(incr(s(head(cons(head(y0), y1)))))) -> c13(MARK(incr(s(head(cons(head(y0), y1)))))) MARK(incr(incr(s(head(cons(tail(y0), y1)))))) -> c13(MARK(incr(s(head(cons(tail(y0), y1)))))) MARK(incr(incr(s(head(cons(cons(y0, y1), y2)))))) -> c13(MARK(incr(s(head(cons(cons(y0, y1), y2)))))) MARK(incr(incr(s(head(cons(s(y0), y1)))))) -> c13(MARK(incr(s(head(cons(s(y0), y1)))))) MARK(incr(incr(s(head(cons(y0, y1)))))) -> c13(MARK(incr(s(head(cons(y0, y1)))))) MARK(incr(incr(s(incr(tail(nats)))))) -> c13(MARK(incr(s(incr(tail(nats)))))) MARK(incr(incr(s(head(tail(nats)))))) -> c13(MARK(incr(s(head(tail(nats)))))) MARK(incr(incr(s(tail(tail(nats)))))) -> c13(MARK(incr(s(tail(tail(nats)))))) MARK(incr(incr(s(cons(cons(y0, y1), y2))))) -> c13(MARK(incr(s(cons(cons(y0, y1), y2))))) MARK(incr(incr(s(cons(s(y0), y1))))) -> c13(MARK(incr(s(cons(s(y0), y1))))) MARK(incr(incr(s(cons(incr(s(y0)), y1))))) -> c13(MARK(incr(s(cons(incr(s(y0)), y1))))) MARK(incr(incr(s(cons(head(s(y0)), y1))))) -> c13(MARK(incr(s(cons(head(s(y0)), y1))))) MARK(incr(incr(s(cons(tail(cons(y0, y1)), y2))))) -> c13(MARK(incr(s(cons(tail(cons(y0, y1)), y2))))) MARK(incr(incr(s(cons(tail(s(y0)), y1))))) -> c13(MARK(incr(s(cons(tail(s(y0)), y1))))) MARK(incr(incr(s(cons(incr(incr(incr(y0))), y1))))) -> c13(MARK(incr(s(cons(incr(incr(incr(y0))), y1))))) MARK(incr(incr(s(cons(incr(incr(odds)), y0))))) -> c13(MARK(incr(s(cons(incr(incr(odds)), y0))))) MARK(incr(incr(s(cons(incr(incr(head(y0))), y1))))) -> c13(MARK(incr(s(cons(incr(incr(head(y0))), y1))))) MARK(incr(incr(s(cons(incr(incr(tail(y0))), y1))))) -> c13(MARK(incr(s(cons(incr(incr(tail(y0))), y1))))) MARK(incr(incr(s(cons(incr(incr(cons(y0, y1))), y2))))) -> c13(MARK(incr(s(cons(incr(incr(cons(y0, y1))), y2))))) MARK(incr(incr(s(cons(incr(incr(s(y0))), y1))))) -> c13(MARK(incr(s(cons(incr(incr(s(y0))), y1))))) MARK(incr(incr(s(cons(incr(incr(y0)), y1))))) -> c13(MARK(incr(s(cons(incr(incr(y0)), y1))))) MARK(incr(incr(s(cons(incr(head(incr(y0))), y1))))) -> c13(MARK(incr(s(cons(incr(head(incr(y0))), y1))))) MARK(incr(incr(s(cons(incr(head(odds)), y0))))) -> c13(MARK(incr(s(cons(incr(head(odds)), y0))))) MARK(incr(incr(s(cons(incr(head(head(y0))), y1))))) -> c13(MARK(incr(s(cons(incr(head(head(y0))), y1))))) MARK(incr(incr(s(cons(incr(head(tail(y0))), y1))))) -> c13(MARK(incr(s(cons(incr(head(tail(y0))), y1))))) MARK(incr(incr(s(cons(incr(head(cons(y0, y1))), y2))))) -> c13(MARK(incr(s(cons(incr(head(cons(y0, y1))), y2))))) MARK(incr(incr(s(cons(incr(head(s(y0))), y1))))) -> c13(MARK(incr(s(cons(incr(head(s(y0))), y1))))) MARK(incr(incr(s(cons(incr(head(y0)), y1))))) -> c13(MARK(incr(s(cons(incr(head(y0)), y1))))) MARK(incr(incr(s(cons(incr(tail(incr(y0))), y1))))) -> c13(MARK(incr(s(cons(incr(tail(incr(y0))), y1))))) MARK(incr(incr(s(cons(incr(tail(pairs)), y0))))) -> c13(MARK(incr(s(cons(incr(tail(pairs)), y0))))) MARK(incr(incr(s(cons(incr(tail(odds)), y0))))) -> c13(MARK(incr(s(cons(incr(tail(odds)), y0))))) MARK(incr(incr(s(cons(incr(tail(head(y0))), y1))))) -> c13(MARK(incr(s(cons(incr(tail(head(y0))), y1))))) MARK(incr(incr(s(cons(incr(tail(tail(y0))), y1))))) -> c13(MARK(incr(s(cons(incr(tail(tail(y0))), y1))))) MARK(incr(incr(s(cons(incr(tail(cons(y0, y1))), y2))))) -> c13(MARK(incr(s(cons(incr(tail(cons(y0, y1))), y2))))) MARK(incr(incr(s(cons(incr(tail(s(y0))), y1))))) -> c13(MARK(incr(s(cons(incr(tail(s(y0))), y1))))) MARK(incr(incr(s(cons(incr(tail(y0)), y1))))) -> c13(MARK(incr(s(cons(incr(tail(y0)), y1))))) MARK(incr(incr(s(cons(head(incr(incr(y0))), y1))))) -> c13(MARK(incr(s(cons(head(incr(incr(y0))), y1))))) MARK(incr(incr(s(cons(head(incr(odds)), y0))))) -> c13(MARK(incr(s(cons(head(incr(odds)), y0))))) MARK(incr(incr(s(cons(head(incr(head(y0))), y1))))) -> c13(MARK(incr(s(cons(head(incr(head(y0))), y1))))) MARK(incr(incr(s(cons(head(incr(tail(y0))), y1))))) -> c13(MARK(incr(s(cons(head(incr(tail(y0))), y1))))) MARK(incr(incr(s(cons(head(incr(cons(y0, y1))), y2))))) -> c13(MARK(incr(s(cons(head(incr(cons(y0, y1))), y2))))) MARK(incr(incr(s(cons(head(incr(s(y0))), y1))))) -> c13(MARK(incr(s(cons(head(incr(s(y0))), y1))))) MARK(incr(incr(s(cons(head(incr(y0)), y1))))) -> c13(MARK(incr(s(cons(head(incr(y0)), y1))))) MARK(incr(incr(s(cons(head(head(incr(y0))), y1))))) -> c13(MARK(incr(s(cons(head(head(incr(y0))), y1))))) MARK(incr(incr(s(cons(head(head(odds)), y0))))) -> c13(MARK(incr(s(cons(head(head(odds)), y0))))) MARK(incr(incr(s(cons(head(head(head(y0))), y1))))) -> c13(MARK(incr(s(cons(head(head(head(y0))), y1))))) MARK(incr(incr(s(cons(head(head(tail(y0))), y1))))) -> c13(MARK(incr(s(cons(head(head(tail(y0))), y1))))) MARK(incr(incr(s(cons(head(head(cons(y0, y1))), y2))))) -> c13(MARK(incr(s(cons(head(head(cons(y0, y1))), y2))))) MARK(incr(incr(s(cons(head(head(s(y0))), y1))))) -> c13(MARK(incr(s(cons(head(head(s(y0))), y1))))) MARK(incr(incr(s(cons(head(head(y0)), y1))))) -> c13(MARK(incr(s(cons(head(head(y0)), y1))))) MARK(incr(incr(s(cons(head(tail(incr(y0))), y1))))) -> c13(MARK(incr(s(cons(head(tail(incr(y0))), y1))))) MARK(incr(incr(s(cons(head(tail(pairs)), y0))))) -> c13(MARK(incr(s(cons(head(tail(pairs)), y0))))) MARK(incr(incr(s(cons(head(tail(odds)), y0))))) -> c13(MARK(incr(s(cons(head(tail(odds)), y0))))) MARK(incr(incr(s(cons(head(tail(head(y0))), y1))))) -> c13(MARK(incr(s(cons(head(tail(head(y0))), y1))))) MARK(incr(incr(s(cons(head(tail(tail(y0))), y1))))) -> c13(MARK(incr(s(cons(head(tail(tail(y0))), y1))))) MARK(incr(incr(s(cons(head(tail(cons(y0, y1))), y2))))) -> c13(MARK(incr(s(cons(head(tail(cons(y0, y1))), y2))))) MARK(incr(incr(s(cons(head(tail(s(y0))), y1))))) -> c13(MARK(incr(s(cons(head(tail(s(y0))), y1))))) MARK(incr(incr(s(cons(head(tail(y0)), y1))))) -> c13(MARK(incr(s(cons(head(tail(y0)), y1))))) MARK(incr(incr(s(cons(tail(incr(incr(y0))), y1))))) -> c13(MARK(incr(s(cons(tail(incr(incr(y0))), y1))))) MARK(incr(incr(s(cons(tail(incr(odds)), y0))))) -> c13(MARK(incr(s(cons(tail(incr(odds)), y0))))) MARK(incr(incr(s(cons(tail(incr(head(y0))), y1))))) -> c13(MARK(incr(s(cons(tail(incr(head(y0))), y1))))) MARK(incr(incr(s(cons(tail(incr(tail(y0))), y1))))) -> c13(MARK(incr(s(cons(tail(incr(tail(y0))), y1))))) MARK(incr(incr(s(cons(tail(incr(cons(y0, y1))), y2))))) -> c13(MARK(incr(s(cons(tail(incr(cons(y0, y1))), y2))))) MARK(incr(incr(s(cons(tail(incr(s(y0))), y1))))) -> c13(MARK(incr(s(cons(tail(incr(s(y0))), y1))))) MARK(incr(incr(s(cons(tail(incr(y0)), y1))))) -> c13(MARK(incr(s(cons(tail(incr(y0)), y1))))) MARK(incr(incr(s(cons(tail(head(incr(y0))), y1))))) -> c13(MARK(incr(s(cons(tail(head(incr(y0))), y1))))) MARK(incr(incr(s(cons(tail(head(odds)), y0))))) -> c13(MARK(incr(s(cons(tail(head(odds)), y0))))) MARK(incr(incr(s(cons(tail(head(head(y0))), y1))))) -> c13(MARK(incr(s(cons(tail(head(head(y0))), y1))))) MARK(incr(incr(s(cons(tail(head(tail(y0))), y1))))) -> c13(MARK(incr(s(cons(tail(head(tail(y0))), y1))))) MARK(incr(incr(s(cons(tail(head(cons(y0, y1))), y2))))) -> c13(MARK(incr(s(cons(tail(head(cons(y0, y1))), y2))))) MARK(incr(incr(s(cons(tail(head(s(y0))), y1))))) -> c13(MARK(incr(s(cons(tail(head(s(y0))), y1))))) MARK(incr(incr(s(cons(tail(head(y0)), y1))))) -> c13(MARK(incr(s(cons(tail(head(y0)), y1))))) MARK(incr(incr(s(cons(tail(tail(incr(y0))), y1))))) -> c13(MARK(incr(s(cons(tail(tail(incr(y0))), y1))))) MARK(incr(incr(s(cons(tail(tail(pairs)), y0))))) -> c13(MARK(incr(s(cons(tail(tail(pairs)), y0))))) MARK(incr(incr(s(cons(tail(tail(odds)), y0))))) -> c13(MARK(incr(s(cons(tail(tail(odds)), y0))))) MARK(incr(incr(s(cons(tail(tail(head(y0))), y1))))) -> c13(MARK(incr(s(cons(tail(tail(head(y0))), y1))))) MARK(incr(incr(s(cons(tail(tail(tail(y0))), y1))))) -> c13(MARK(incr(s(cons(tail(tail(tail(y0))), y1))))) MARK(incr(incr(s(cons(tail(tail(cons(y0, y1))), y2))))) -> c13(MARK(incr(s(cons(tail(tail(cons(y0, y1))), y2))))) MARK(incr(incr(s(cons(tail(tail(s(y0))), y1))))) -> c13(MARK(incr(s(cons(tail(tail(s(y0))), y1))))) MARK(incr(incr(s(cons(tail(tail(y0)), y1))))) -> c13(MARK(incr(s(cons(tail(tail(y0)), y1))))) MARK(incr(incr(s(cons(tail(pairs), y0))))) -> c13(MARK(incr(s(cons(tail(pairs), y0))))) MARK(incr(incr(s(cons(incr(incr(nats)), y0))))) -> c13(MARK(incr(s(cons(incr(incr(nats)), y0))))) MARK(incr(incr(s(cons(incr(incr(pairs)), y0))))) -> c13(MARK(incr(s(cons(incr(incr(pairs)), y0))))) MARK(incr(incr(s(cons(incr(odds), y0))))) -> c13(MARK(incr(s(cons(incr(odds), y0))))) MARK(incr(incr(s(cons(head(incr(nats)), y0))))) -> c13(MARK(incr(s(cons(head(incr(nats)), y0))))) MARK(incr(incr(s(cons(head(incr(pairs)), y0))))) -> c13(MARK(incr(s(cons(head(incr(pairs)), y0))))) MARK(incr(incr(s(cons(head(odds), y0))))) -> c13(MARK(incr(s(cons(head(odds), y0))))) MARK(incr(incr(s(cons(tail(incr(nats)), y0))))) -> c13(MARK(incr(s(cons(tail(incr(nats)), y0))))) MARK(incr(incr(s(cons(tail(incr(pairs)), y0))))) -> c13(MARK(incr(s(cons(tail(incr(pairs)), y0))))) MARK(incr(incr(s(cons(tail(odds), y0))))) -> c13(MARK(incr(s(cons(tail(odds), y0))))) MARK(incr(incr(s(cons(incr(cons(nats, y0)), y1))))) -> c13(MARK(incr(s(cons(incr(cons(nats, y0)), y1))))) MARK(incr(incr(s(cons(incr(cons(incr(y0), y1)), y2))))) -> c13(MARK(incr(s(cons(incr(cons(incr(y0), y1)), y2))))) MARK(incr(incr(s(cons(incr(cons(pairs, y0)), y1))))) -> c13(MARK(incr(s(cons(incr(cons(pairs, y0)), y1))))) MARK(incr(incr(s(cons(incr(cons(odds, y0)), y1))))) -> c13(MARK(incr(s(cons(incr(cons(odds, y0)), y1))))) MARK(incr(incr(s(cons(incr(cons(head(y0), y1)), y2))))) -> c13(MARK(incr(s(cons(incr(cons(head(y0), y1)), y2))))) MARK(incr(incr(s(cons(incr(cons(tail(y0), y1)), y2))))) -> c13(MARK(incr(s(cons(incr(cons(tail(y0), y1)), y2))))) MARK(incr(incr(s(cons(incr(cons(cons(y0, y1), y2)), y3))))) -> c13(MARK(incr(s(cons(incr(cons(cons(y0, y1), y2)), y3))))) MARK(incr(incr(s(cons(incr(cons(s(y0), y1)), y2))))) -> c13(MARK(incr(s(cons(incr(cons(s(y0), y1)), y2))))) MARK(incr(incr(s(cons(incr(cons(y0, y1)), y2))))) -> c13(MARK(incr(s(cons(incr(cons(y0, y1)), y2))))) MARK(incr(incr(s(cons(incr(cons(0, y0)), y1))))) -> c13(MARK(incr(s(cons(incr(cons(0, y0)), y1))))) MARK(incr(incr(s(cons(incr(head(nats)), y0))))) -> c13(MARK(incr(s(cons(incr(head(nats)), y0))))) MARK(incr(incr(s(cons(incr(head(pairs)), y0))))) -> c13(MARK(incr(s(cons(incr(head(pairs)), y0))))) MARK(incr(incr(s(cons(head(head(nats)), y0))))) -> c13(MARK(incr(s(cons(head(head(nats)), y0))))) MARK(incr(incr(s(cons(head(head(pairs)), y0))))) -> c13(MARK(incr(s(cons(head(head(pairs)), y0))))) MARK(incr(incr(s(cons(tail(head(nats)), y0))))) -> c13(MARK(incr(s(cons(tail(head(nats)), y0))))) MARK(incr(incr(s(cons(tail(head(pairs)), y0))))) -> c13(MARK(incr(s(cons(tail(head(pairs)), y0))))) MARK(incr(incr(s(cons(head(cons(nats, y0)), y1))))) -> c13(MARK(incr(s(cons(head(cons(nats, y0)), y1))))) MARK(incr(incr(s(cons(head(cons(incr(y0), y1)), y2))))) -> c13(MARK(incr(s(cons(head(cons(incr(y0), y1)), y2))))) MARK(incr(incr(s(cons(head(cons(pairs, y0)), y1))))) -> c13(MARK(incr(s(cons(head(cons(pairs, y0)), y1))))) MARK(incr(incr(s(cons(head(cons(odds, y0)), y1))))) -> c13(MARK(incr(s(cons(head(cons(odds, y0)), y1))))) MARK(incr(incr(s(cons(head(cons(head(y0), y1)), y2))))) -> c13(MARK(incr(s(cons(head(cons(head(y0), y1)), y2))))) MARK(incr(incr(s(cons(head(cons(tail(y0), y1)), y2))))) -> c13(MARK(incr(s(cons(head(cons(tail(y0), y1)), y2))))) MARK(incr(incr(s(cons(head(cons(cons(y0, y1), y2)), y3))))) -> c13(MARK(incr(s(cons(head(cons(cons(y0, y1), y2)), y3))))) MARK(incr(incr(s(cons(head(cons(s(y0), y1)), y2))))) -> c13(MARK(incr(s(cons(head(cons(s(y0), y1)), y2))))) MARK(incr(incr(s(cons(head(cons(y0, y1)), y2))))) -> c13(MARK(incr(s(cons(head(cons(y0, y1)), y2))))) MARK(incr(incr(s(cons(head(cons(0, y0)), y1))))) -> c13(MARK(incr(s(cons(head(cons(0, y0)), y1))))) MARK(incr(incr(s(cons(incr(tail(nats)), y0))))) -> c13(MARK(incr(s(cons(incr(tail(nats)), y0))))) MARK(incr(incr(s(cons(head(tail(nats)), y0))))) -> c13(MARK(incr(s(cons(head(tail(nats)), y0))))) MARK(incr(incr(s(cons(tail(tail(nats)), y0))))) -> c13(MARK(incr(s(cons(tail(tail(nats)), y0))))) MARK(incr(incr(s(incr(cons(nats, y0)))))) -> c13(MARK(incr(s(incr(cons(nats, y0)))))) MARK(incr(incr(s(incr(cons(pairs, y0)))))) -> c13(MARK(incr(s(incr(cons(pairs, y0)))))) MARK(incr(incr(s(incr(cons(odds, y0)))))) -> c13(MARK(incr(s(incr(cons(odds, y0)))))) MARK(incr(incr(s(head(cons(nats, y0)))))) -> c13(MARK(incr(s(head(cons(nats, y0)))))) MARK(incr(incr(s(head(cons(pairs, y0)))))) -> c13(MARK(incr(s(head(cons(pairs, y0)))))) MARK(incr(incr(s(head(cons(odds, y0)))))) -> c13(MARK(incr(s(head(cons(odds, y0)))))) ---------------------------------------- (200) Obligation: Complexity Dependency Tuples Problem Rules: a__nats -> cons(0, incr(nats)) a__nats -> nats a__pairs -> cons(0, incr(odds)) a__pairs -> pairs a__odds -> a__incr(a__pairs) a__odds -> odds a__incr(cons(z0, z1)) -> cons(s(mark(z0)), incr(z1)) a__incr(z0) -> incr(z0) 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(nats) -> a__nats mark(incr(z0)) -> a__incr(mark(z0)) mark(pairs) -> a__pairs mark(odds) -> a__odds mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(tail(incr(incr(z0)))) -> c17(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(odds))) -> c17(A__TAIL(a__incr(a__odds)), MARK(incr(odds))) MARK(tail(incr(head(z0)))) -> c17(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c17(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c17(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c17(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c17(MARK(incr(x0))) MARK(tail(head(incr(z0)))) -> c17(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(odds))) -> c17(A__TAIL(a__head(a__odds)), MARK(head(odds))) MARK(tail(head(head(z0)))) -> c17(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c17(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c17(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c17(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c17(MARK(head(x0))) MARK(tail(tail(incr(z0)))) -> c17(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(pairs))) -> c17(A__TAIL(a__tail(a__pairs)), MARK(tail(pairs))) MARK(tail(tail(odds))) -> c17(A__TAIL(a__tail(a__odds)), MARK(tail(odds))) MARK(tail(tail(head(z0)))) -> c17(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c17(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c17(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(s(z0)))) -> c17(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(x0))) -> c17(MARK(tail(x0))) MARK(tail(pairs)) -> c17(A__TAIL(cons(0, incr(odds)))) A__INCR(cons(cons(y0, y1), z1)) -> c6(MARK(cons(y0, y1))) A__INCR(cons(s(y0), z1)) -> c6(MARK(s(y0))) A__INCR(cons(incr(cons(y0, y1)), z1)) -> c6(MARK(incr(cons(y0, y1)))) A__INCR(cons(incr(s(y0)), z1)) -> c6(MARK(incr(s(y0)))) A__INCR(cons(head(cons(y0, y1)), z1)) -> c6(MARK(head(cons(y0, y1)))) A__INCR(cons(head(s(y0)), z1)) -> c6(MARK(head(s(y0)))) A__INCR(cons(tail(cons(y0, y1)), z1)) -> c6(MARK(tail(cons(y0, y1)))) A__INCR(cons(tail(s(y0)), z1)) -> c6(MARK(tail(s(y0)))) A__INCR(cons(incr(incr(nats)), z1)) -> c6(MARK(incr(incr(nats)))) A__INCR(cons(incr(incr(incr(y0))), z1)) -> c6(MARK(incr(incr(incr(y0))))) A__INCR(cons(incr(incr(pairs)), z1)) -> c6(MARK(incr(incr(pairs)))) A__INCR(cons(incr(incr(odds)), z1)) -> c6(MARK(incr(incr(odds)))) A__INCR(cons(incr(incr(head(y0))), z1)) -> c6(MARK(incr(incr(head(y0))))) A__INCR(cons(incr(incr(tail(y0))), z1)) -> c6(MARK(incr(incr(tail(y0))))) A__INCR(cons(incr(incr(cons(y0, y1))), z1)) -> c6(MARK(incr(incr(cons(y0, y1))))) A__INCR(cons(incr(incr(s(y0))), z1)) -> c6(MARK(incr(incr(s(y0))))) A__INCR(cons(incr(incr(y0)), z1)) -> c6(MARK(incr(incr(y0)))) A__INCR(cons(incr(odds), z1)) -> c6(MARK(incr(odds))) A__INCR(cons(incr(head(nats)), z1)) -> c6(MARK(incr(head(nats)))) A__INCR(cons(incr(head(incr(y0))), z1)) -> c6(MARK(incr(head(incr(y0))))) A__INCR(cons(incr(head(pairs)), z1)) -> c6(MARK(incr(head(pairs)))) A__INCR(cons(incr(head(odds)), z1)) -> c6(MARK(incr(head(odds)))) A__INCR(cons(incr(head(head(y0))), z1)) -> c6(MARK(incr(head(head(y0))))) A__INCR(cons(incr(head(tail(y0))), z1)) -> c6(MARK(incr(head(tail(y0))))) A__INCR(cons(incr(head(cons(y0, y1))), z1)) -> c6(MARK(incr(head(cons(y0, y1))))) A__INCR(cons(incr(head(s(y0))), z1)) -> c6(MARK(incr(head(s(y0))))) A__INCR(cons(incr(head(y0)), z1)) -> c6(MARK(incr(head(y0)))) A__INCR(cons(incr(tail(nats)), z1)) -> c6(MARK(incr(tail(nats)))) A__INCR(cons(incr(tail(incr(y0))), z1)) -> c6(MARK(incr(tail(incr(y0))))) A__INCR(cons(incr(tail(pairs)), z1)) -> c6(MARK(incr(tail(pairs)))) A__INCR(cons(incr(tail(odds)), z1)) -> c6(MARK(incr(tail(odds)))) A__INCR(cons(incr(tail(head(y0))), z1)) -> c6(MARK(incr(tail(head(y0))))) A__INCR(cons(incr(tail(tail(y0))), z1)) -> c6(MARK(incr(tail(tail(y0))))) A__INCR(cons(incr(tail(cons(y0, y1))), z1)) -> c6(MARK(incr(tail(cons(y0, y1))))) A__INCR(cons(incr(tail(s(y0))), z1)) -> c6(MARK(incr(tail(s(y0))))) A__INCR(cons(incr(tail(y0)), z1)) -> c6(MARK(incr(tail(y0)))) A__INCR(cons(head(incr(nats)), z1)) -> c6(MARK(head(incr(nats)))) A__INCR(cons(head(incr(incr(y0))), z1)) -> c6(MARK(head(incr(incr(y0))))) 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MARK(cons(head(head(y0)), z1)) -> c18(MARK(head(head(y0)))) MARK(cons(head(tail(incr(y0))), z1)) -> c18(MARK(head(tail(incr(y0))))) MARK(cons(head(tail(pairs)), z1)) -> c18(MARK(head(tail(pairs)))) MARK(cons(head(tail(odds)), z1)) -> c18(MARK(head(tail(odds)))) MARK(cons(head(tail(head(y0))), z1)) -> c18(MARK(head(tail(head(y0))))) MARK(cons(head(tail(tail(y0))), z1)) -> c18(MARK(head(tail(tail(y0))))) MARK(cons(head(tail(cons(y0, y1))), z1)) -> c18(MARK(head(tail(cons(y0, y1))))) MARK(cons(head(tail(s(y0))), z1)) -> c18(MARK(head(tail(s(y0))))) MARK(cons(head(tail(y0)), z1)) -> c18(MARK(head(tail(y0)))) MARK(cons(tail(incr(incr(y0))), z1)) -> c18(MARK(tail(incr(incr(y0))))) MARK(cons(tail(incr(odds)), z1)) -> c18(MARK(tail(incr(odds)))) MARK(cons(tail(incr(head(y0))), z1)) -> c18(MARK(tail(incr(head(y0))))) MARK(cons(tail(incr(tail(y0))), z1)) -> c18(MARK(tail(incr(tail(y0))))) MARK(cons(tail(incr(cons(y0, y1))), z1)) -> c18(MARK(tail(incr(cons(y0, y1))))) MARK(cons(tail(incr(s(y0))), 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c20(MARK(incr(head(odds)))) MARK(s(incr(head(head(y0))))) -> c20(MARK(incr(head(head(y0))))) MARK(s(incr(head(tail(y0))))) -> c20(MARK(incr(head(tail(y0))))) MARK(s(incr(head(cons(y0, y1))))) -> c20(MARK(incr(head(cons(y0, y1))))) MARK(s(incr(head(s(y0))))) -> c20(MARK(incr(head(s(y0))))) MARK(s(incr(head(y0)))) -> c20(MARK(incr(head(y0)))) MARK(s(incr(tail(incr(y0))))) -> c20(MARK(incr(tail(incr(y0))))) MARK(s(incr(tail(pairs)))) -> c20(MARK(incr(tail(pairs)))) MARK(s(incr(tail(odds)))) -> c20(MARK(incr(tail(odds)))) MARK(s(incr(tail(head(y0))))) -> c20(MARK(incr(tail(head(y0))))) MARK(s(incr(tail(tail(y0))))) -> c20(MARK(incr(tail(tail(y0))))) MARK(s(incr(tail(cons(y0, y1))))) -> c20(MARK(incr(tail(cons(y0, y1))))) MARK(s(incr(tail(s(y0))))) -> c20(MARK(incr(tail(s(y0))))) MARK(s(incr(tail(y0)))) -> c20(MARK(incr(tail(y0)))) MARK(s(head(incr(incr(y0))))) -> c20(MARK(head(incr(incr(y0))))) MARK(s(head(incr(odds)))) -> c20(MARK(head(incr(odds)))) MARK(s(head(incr(head(y0))))) -> c20(MARK(head(incr(head(y0))))) MARK(s(head(incr(tail(y0))))) -> c20(MARK(head(incr(tail(y0))))) MARK(s(head(incr(cons(y0, y1))))) -> c20(MARK(head(incr(cons(y0, y1))))) MARK(s(head(incr(s(y0))))) -> c20(MARK(head(incr(s(y0))))) MARK(s(head(incr(y0)))) -> c20(MARK(head(incr(y0)))) MARK(s(head(head(incr(y0))))) -> c20(MARK(head(head(incr(y0))))) MARK(s(head(head(odds)))) -> c20(MARK(head(head(odds)))) MARK(s(head(head(head(y0))))) -> c20(MARK(head(head(head(y0))))) MARK(s(head(head(tail(y0))))) -> c20(MARK(head(head(tail(y0))))) MARK(s(head(head(cons(y0, y1))))) -> c20(MARK(head(head(cons(y0, y1))))) MARK(s(head(head(s(y0))))) -> c20(MARK(head(head(s(y0))))) MARK(s(head(head(y0)))) -> c20(MARK(head(head(y0)))) MARK(s(head(tail(incr(y0))))) -> c20(MARK(head(tail(incr(y0))))) MARK(s(head(tail(pairs)))) -> c20(MARK(head(tail(pairs)))) MARK(s(head(tail(odds)))) -> c20(MARK(head(tail(odds)))) MARK(s(head(tail(head(y0))))) -> c20(MARK(head(tail(head(y0))))) MARK(s(head(tail(tail(y0))))) -> c20(MARK(head(tail(tail(y0))))) MARK(s(head(tail(cons(y0, y1))))) -> c20(MARK(head(tail(cons(y0, y1))))) MARK(s(head(tail(s(y0))))) -> c20(MARK(head(tail(s(y0))))) MARK(s(head(tail(y0)))) -> c20(MARK(head(tail(y0)))) MARK(s(tail(incr(incr(y0))))) -> c20(MARK(tail(incr(incr(y0))))) MARK(s(tail(incr(odds)))) -> c20(MARK(tail(incr(odds)))) MARK(s(tail(incr(head(y0))))) -> c20(MARK(tail(incr(head(y0))))) MARK(s(tail(incr(tail(y0))))) -> c20(MARK(tail(incr(tail(y0))))) MARK(s(tail(incr(cons(y0, y1))))) -> c20(MARK(tail(incr(cons(y0, y1))))) MARK(s(tail(incr(s(y0))))) -> c20(MARK(tail(incr(s(y0))))) MARK(s(tail(incr(y0)))) -> c20(MARK(tail(incr(y0)))) MARK(s(tail(head(incr(y0))))) -> c20(MARK(tail(head(incr(y0))))) MARK(s(tail(head(odds)))) -> c20(MARK(tail(head(odds)))) MARK(s(tail(head(head(y0))))) -> c20(MARK(tail(head(head(y0))))) MARK(s(tail(head(tail(y0))))) -> c20(MARK(tail(head(tail(y0))))) MARK(s(tail(head(cons(y0, y1))))) -> c20(MARK(tail(head(cons(y0, y1))))) MARK(s(tail(head(s(y0))))) -> c20(MARK(tail(head(s(y0))))) MARK(s(tail(head(y0)))) -> c20(MARK(tail(head(y0)))) MARK(s(tail(tail(incr(y0))))) -> c20(MARK(tail(tail(incr(y0))))) MARK(s(tail(tail(pairs)))) -> c20(MARK(tail(tail(pairs)))) MARK(s(tail(tail(odds)))) -> c20(MARK(tail(tail(odds)))) MARK(s(tail(tail(head(y0))))) -> c20(MARK(tail(tail(head(y0))))) MARK(s(tail(tail(tail(y0))))) -> c20(MARK(tail(tail(tail(y0))))) MARK(s(tail(tail(cons(y0, y1))))) -> c20(MARK(tail(tail(cons(y0, y1))))) MARK(s(tail(tail(s(y0))))) -> c20(MARK(tail(tail(s(y0))))) MARK(s(tail(tail(y0)))) -> c20(MARK(tail(tail(y0)))) MARK(s(tail(pairs))) -> c20(MARK(tail(pairs))) MARK(s(incr(incr(nats)))) -> c20(MARK(incr(incr(nats)))) MARK(s(incr(incr(pairs)))) -> c20(MARK(incr(incr(pairs)))) MARK(s(incr(odds))) -> c20(MARK(incr(odds))) MARK(s(head(incr(nats)))) -> c20(MARK(head(incr(nats)))) MARK(s(head(incr(pairs)))) -> c20(MARK(head(incr(pairs)))) MARK(s(head(odds))) -> c20(MARK(head(odds))) MARK(s(tail(incr(nats)))) -> c20(MARK(tail(incr(nats)))) MARK(s(tail(incr(pairs)))) -> c20(MARK(tail(incr(pairs)))) MARK(s(tail(odds))) -> c20(MARK(tail(odds))) MARK(s(incr(cons(incr(y0), y1)))) -> c20(MARK(incr(cons(incr(y0), y1)))) MARK(s(incr(cons(head(y0), y1)))) -> c20(MARK(incr(cons(head(y0), y1)))) MARK(s(incr(cons(tail(y0), y1)))) -> c20(MARK(incr(cons(tail(y0), y1)))) MARK(s(incr(cons(cons(y0, y1), y2)))) -> c20(MARK(incr(cons(cons(y0, y1), y2)))) MARK(s(incr(cons(s(y0), y1)))) -> c20(MARK(incr(cons(s(y0), y1)))) MARK(s(incr(cons(y0, y1)))) -> c20(MARK(incr(cons(y0, y1)))) MARK(s(incr(head(nats)))) -> c20(MARK(incr(head(nats)))) MARK(s(incr(head(pairs)))) -> c20(MARK(incr(head(pairs)))) MARK(s(head(head(nats)))) -> c20(MARK(head(head(nats)))) MARK(s(head(head(pairs)))) -> c20(MARK(head(head(pairs)))) MARK(s(tail(head(nats)))) -> c20(MARK(tail(head(nats)))) MARK(s(tail(head(pairs)))) -> c20(MARK(tail(head(pairs)))) MARK(s(head(cons(incr(y0), y1)))) -> c20(MARK(head(cons(incr(y0), y1)))) MARK(s(head(cons(head(y0), y1)))) -> c20(MARK(head(cons(head(y0), y1)))) MARK(s(head(cons(tail(y0), y1)))) -> c20(MARK(head(cons(tail(y0), y1)))) MARK(s(head(cons(cons(y0, y1), y2)))) -> c20(MARK(head(cons(cons(y0, y1), y2)))) MARK(s(head(cons(s(y0), y1)))) -> c20(MARK(head(cons(s(y0), y1)))) MARK(s(head(cons(y0, y1)))) -> c20(MARK(head(cons(y0, y1)))) MARK(s(incr(tail(nats)))) -> c20(MARK(incr(tail(nats)))) MARK(s(head(tail(nats)))) -> c20(MARK(head(tail(nats)))) MARK(s(tail(tail(nats)))) -> c20(MARK(tail(tail(nats)))) MARK(s(cons(cons(y0, y1), y2))) -> c20(MARK(cons(cons(y0, y1), y2))) MARK(s(cons(s(y0), y1))) -> c20(MARK(cons(s(y0), y1))) MARK(s(cons(incr(s(y0)), y1))) -> c20(MARK(cons(incr(s(y0)), y1))) MARK(s(cons(head(s(y0)), y1))) -> c20(MARK(cons(head(s(y0)), y1))) MARK(s(cons(tail(cons(y0, y1)), y2))) -> c20(MARK(cons(tail(cons(y0, y1)), y2))) MARK(s(cons(tail(s(y0)), y1))) -> c20(MARK(cons(tail(s(y0)), y1))) MARK(s(cons(incr(incr(incr(y0))), y1))) -> c20(MARK(cons(incr(incr(incr(y0))), y1))) MARK(s(cons(incr(incr(odds)), y0))) -> c20(MARK(cons(incr(incr(odds)), y0))) MARK(s(cons(incr(incr(head(y0))), y1))) -> c20(MARK(cons(incr(incr(head(y0))), y1))) MARK(s(cons(incr(incr(tail(y0))), y1))) -> c20(MARK(cons(incr(incr(tail(y0))), y1))) MARK(s(cons(incr(incr(cons(y0, y1))), y2))) -> c20(MARK(cons(incr(incr(cons(y0, y1))), y2))) MARK(s(cons(incr(incr(s(y0))), y1))) -> c20(MARK(cons(incr(incr(s(y0))), y1))) MARK(s(cons(incr(incr(y0)), y1))) -> c20(MARK(cons(incr(incr(y0)), y1))) MARK(s(cons(incr(head(incr(y0))), y1))) -> c20(MARK(cons(incr(head(incr(y0))), y1))) MARK(s(cons(incr(head(odds)), y0))) -> c20(MARK(cons(incr(head(odds)), y0))) MARK(s(cons(incr(head(head(y0))), y1))) -> c20(MARK(cons(incr(head(head(y0))), y1))) MARK(s(cons(incr(head(tail(y0))), y1))) -> c20(MARK(cons(incr(head(tail(y0))), y1))) MARK(s(cons(incr(head(cons(y0, y1))), y2))) -> c20(MARK(cons(incr(head(cons(y0, y1))), y2))) MARK(s(cons(incr(head(s(y0))), y1))) -> c20(MARK(cons(incr(head(s(y0))), y1))) MARK(s(cons(incr(head(y0)), y1))) -> c20(MARK(cons(incr(head(y0)), y1))) MARK(s(cons(incr(tail(incr(y0))), y1))) -> c20(MARK(cons(incr(tail(incr(y0))), y1))) MARK(s(cons(incr(tail(pairs)), y0))) -> c20(MARK(cons(incr(tail(pairs)), y0))) MARK(s(cons(incr(tail(odds)), y0))) -> c20(MARK(cons(incr(tail(odds)), y0))) MARK(s(cons(incr(tail(head(y0))), y1))) -> c20(MARK(cons(incr(tail(head(y0))), y1))) MARK(s(cons(incr(tail(tail(y0))), y1))) -> c20(MARK(cons(incr(tail(tail(y0))), y1))) MARK(s(cons(incr(tail(cons(y0, y1))), y2))) -> c20(MARK(cons(incr(tail(cons(y0, y1))), y2))) MARK(s(cons(incr(tail(s(y0))), y1))) -> c20(MARK(cons(incr(tail(s(y0))), y1))) MARK(s(cons(incr(tail(y0)), y1))) -> c20(MARK(cons(incr(tail(y0)), y1))) MARK(s(cons(head(incr(incr(y0))), y1))) -> c20(MARK(cons(head(incr(incr(y0))), y1))) MARK(s(cons(head(incr(odds)), y0))) -> c20(MARK(cons(head(incr(odds)), y0))) MARK(s(cons(head(incr(head(y0))), y1))) -> c20(MARK(cons(head(incr(head(y0))), y1))) MARK(s(cons(head(incr(tail(y0))), y1))) -> c20(MARK(cons(head(incr(tail(y0))), y1))) MARK(s(cons(head(incr(cons(y0, y1))), y2))) -> c20(MARK(cons(head(incr(cons(y0, y1))), y2))) MARK(s(cons(head(incr(s(y0))), y1))) -> c20(MARK(cons(head(incr(s(y0))), y1))) MARK(s(cons(head(incr(y0)), y1))) -> c20(MARK(cons(head(incr(y0)), y1))) MARK(s(cons(head(head(incr(y0))), y1))) -> c20(MARK(cons(head(head(incr(y0))), y1))) MARK(s(cons(head(head(odds)), y0))) -> c20(MARK(cons(head(head(odds)), y0))) MARK(s(cons(head(head(head(y0))), y1))) -> c20(MARK(cons(head(head(head(y0))), y1))) MARK(s(cons(head(head(tail(y0))), y1))) -> c20(MARK(cons(head(head(tail(y0))), y1))) MARK(s(cons(head(head(cons(y0, y1))), y2))) -> c20(MARK(cons(head(head(cons(y0, y1))), y2))) MARK(s(cons(head(head(s(y0))), y1))) -> c20(MARK(cons(head(head(s(y0))), y1))) MARK(s(cons(head(head(y0)), y1))) -> c20(MARK(cons(head(head(y0)), y1))) MARK(s(cons(head(tail(incr(y0))), y1))) -> c20(MARK(cons(head(tail(incr(y0))), y1))) MARK(s(cons(head(tail(pairs)), y0))) -> c20(MARK(cons(head(tail(pairs)), y0))) MARK(s(cons(head(tail(odds)), y0))) -> c20(MARK(cons(head(tail(odds)), y0))) MARK(s(cons(head(tail(head(y0))), y1))) -> c20(MARK(cons(head(tail(head(y0))), y1))) MARK(s(cons(head(tail(tail(y0))), y1))) -> c20(MARK(cons(head(tail(tail(y0))), y1))) MARK(s(cons(head(tail(cons(y0, y1))), y2))) -> c20(MARK(cons(head(tail(cons(y0, y1))), y2))) MARK(s(cons(head(tail(s(y0))), y1))) -> c20(MARK(cons(head(tail(s(y0))), y1))) MARK(s(cons(head(tail(y0)), y1))) -> c20(MARK(cons(head(tail(y0)), y1))) MARK(s(cons(tail(incr(incr(y0))), y1))) -> c20(MARK(cons(tail(incr(incr(y0))), y1))) MARK(s(cons(tail(incr(odds)), y0))) -> c20(MARK(cons(tail(incr(odds)), y0))) MARK(s(cons(tail(incr(head(y0))), y1))) -> c20(MARK(cons(tail(incr(head(y0))), y1))) MARK(s(cons(tail(incr(tail(y0))), y1))) -> c20(MARK(cons(tail(incr(tail(y0))), y1))) MARK(s(cons(tail(incr(cons(y0, y1))), y2))) -> c20(MARK(cons(tail(incr(cons(y0, y1))), y2))) MARK(s(cons(tail(incr(s(y0))), y1))) -> c20(MARK(cons(tail(incr(s(y0))), y1))) MARK(s(cons(tail(incr(y0)), y1))) -> c20(MARK(cons(tail(incr(y0)), y1))) MARK(s(cons(tail(head(incr(y0))), y1))) -> c20(MARK(cons(tail(head(incr(y0))), y1))) MARK(s(cons(tail(head(odds)), y0))) -> c20(MARK(cons(tail(head(odds)), y0))) MARK(s(cons(tail(head(head(y0))), y1))) -> c20(MARK(cons(tail(head(head(y0))), y1))) MARK(s(cons(tail(head(tail(y0))), y1))) -> c20(MARK(cons(tail(head(tail(y0))), y1))) MARK(s(cons(tail(head(cons(y0, y1))), y2))) -> c20(MARK(cons(tail(head(cons(y0, y1))), y2))) MARK(s(cons(tail(head(s(y0))), y1))) -> c20(MARK(cons(tail(head(s(y0))), y1))) MARK(s(cons(tail(head(y0)), y1))) -> c20(MARK(cons(tail(head(y0)), y1))) MARK(s(cons(tail(tail(incr(y0))), y1))) -> c20(MARK(cons(tail(tail(incr(y0))), y1))) MARK(s(cons(tail(tail(pairs)), y0))) -> c20(MARK(cons(tail(tail(pairs)), y0))) MARK(s(cons(tail(tail(odds)), y0))) -> c20(MARK(cons(tail(tail(odds)), y0))) MARK(s(cons(tail(tail(head(y0))), y1))) -> c20(MARK(cons(tail(tail(head(y0))), y1))) MARK(s(cons(tail(tail(tail(y0))), y1))) -> c20(MARK(cons(tail(tail(tail(y0))), y1))) MARK(s(cons(tail(tail(cons(y0, y1))), y2))) -> c20(MARK(cons(tail(tail(cons(y0, y1))), y2))) MARK(s(cons(tail(tail(s(y0))), y1))) -> c20(MARK(cons(tail(tail(s(y0))), y1))) MARK(s(cons(tail(tail(y0)), y1))) -> c20(MARK(cons(tail(tail(y0)), y1))) MARK(s(cons(tail(pairs), y0))) -> c20(MARK(cons(tail(pairs), y0))) MARK(s(cons(incr(incr(nats)), y0))) -> c20(MARK(cons(incr(incr(nats)), y0))) MARK(s(cons(incr(incr(pairs)), y0))) -> c20(MARK(cons(incr(incr(pairs)), y0))) MARK(s(cons(incr(odds), y0))) -> c20(MARK(cons(incr(odds), y0))) MARK(s(cons(head(incr(nats)), y0))) -> c20(MARK(cons(head(incr(nats)), y0))) MARK(s(cons(head(incr(pairs)), y0))) -> c20(MARK(cons(head(incr(pairs)), y0))) MARK(s(cons(head(odds), y0))) -> c20(MARK(cons(head(odds), y0))) MARK(s(cons(tail(incr(nats)), y0))) -> c20(MARK(cons(tail(incr(nats)), y0))) MARK(s(cons(tail(incr(pairs)), y0))) -> c20(MARK(cons(tail(incr(pairs)), y0))) MARK(s(cons(tail(odds), y0))) -> c20(MARK(cons(tail(odds), y0))) MARK(s(cons(incr(cons(nats, y0)), y1))) -> c20(MARK(cons(incr(cons(nats, y0)), y1))) MARK(s(cons(incr(cons(incr(y0), y1)), y2))) -> c20(MARK(cons(incr(cons(incr(y0), y1)), y2))) MARK(s(cons(incr(cons(pairs, y0)), y1))) -> c20(MARK(cons(incr(cons(pairs, y0)), y1))) MARK(s(cons(incr(cons(odds, y0)), y1))) -> c20(MARK(cons(incr(cons(odds, y0)), y1))) MARK(s(cons(incr(cons(head(y0), y1)), y2))) -> c20(MARK(cons(incr(cons(head(y0), y1)), y2))) MARK(s(cons(incr(cons(tail(y0), y1)), y2))) -> c20(MARK(cons(incr(cons(tail(y0), y1)), y2))) MARK(s(cons(incr(cons(cons(y0, y1), y2)), y3))) -> c20(MARK(cons(incr(cons(cons(y0, y1), y2)), y3))) MARK(s(cons(incr(cons(s(y0), y1)), y2))) -> c20(MARK(cons(incr(cons(s(y0), y1)), y2))) MARK(s(cons(incr(cons(y0, y1)), y2))) -> c20(MARK(cons(incr(cons(y0, y1)), y2))) MARK(s(cons(incr(cons(0, y0)), y1))) -> c20(MARK(cons(incr(cons(0, y0)), y1))) MARK(s(cons(incr(head(nats)), y0))) -> c20(MARK(cons(incr(head(nats)), y0))) MARK(s(cons(incr(head(pairs)), y0))) -> c20(MARK(cons(incr(head(pairs)), y0))) MARK(s(cons(head(head(nats)), y0))) -> c20(MARK(cons(head(head(nats)), y0))) MARK(s(cons(head(head(pairs)), y0))) -> c20(MARK(cons(head(head(pairs)), y0))) MARK(s(cons(tail(head(nats)), y0))) -> c20(MARK(cons(tail(head(nats)), y0))) MARK(s(cons(tail(head(pairs)), y0))) -> c20(MARK(cons(tail(head(pairs)), y0))) MARK(s(cons(head(cons(nats, y0)), y1))) -> c20(MARK(cons(head(cons(nats, y0)), y1))) MARK(s(cons(head(cons(incr(y0), y1)), y2))) -> c20(MARK(cons(head(cons(incr(y0), y1)), y2))) MARK(s(cons(head(cons(pairs, y0)), y1))) -> c20(MARK(cons(head(cons(pairs, y0)), y1))) MARK(s(cons(head(cons(odds, y0)), y1))) -> c20(MARK(cons(head(cons(odds, y0)), y1))) MARK(s(cons(head(cons(head(y0), y1)), y2))) -> c20(MARK(cons(head(cons(head(y0), y1)), y2))) MARK(s(cons(head(cons(tail(y0), y1)), y2))) -> c20(MARK(cons(head(cons(tail(y0), y1)), y2))) MARK(s(cons(head(cons(cons(y0, y1), y2)), y3))) -> c20(MARK(cons(head(cons(cons(y0, y1), y2)), y3))) MARK(s(cons(head(cons(s(y0), y1)), y2))) -> c20(MARK(cons(head(cons(s(y0), y1)), y2))) MARK(s(cons(head(cons(y0, y1)), y2))) -> c20(MARK(cons(head(cons(y0, y1)), y2))) MARK(s(cons(head(cons(0, y0)), y1))) -> c20(MARK(cons(head(cons(0, y0)), y1))) MARK(s(cons(incr(tail(nats)), y0))) -> c20(MARK(cons(incr(tail(nats)), y0))) MARK(s(cons(head(tail(nats)), y0))) -> c20(MARK(cons(head(tail(nats)), y0))) MARK(s(cons(tail(tail(nats)), y0))) -> c20(MARK(cons(tail(tail(nats)), y0))) MARK(s(incr(cons(nats, y0)))) -> c20(MARK(incr(cons(nats, y0)))) MARK(s(incr(cons(pairs, y0)))) -> c20(MARK(incr(cons(pairs, y0)))) MARK(s(incr(cons(odds, y0)))) -> c20(MARK(incr(cons(odds, y0)))) MARK(s(head(cons(nats, y0)))) -> c20(MARK(head(cons(nats, y0)))) MARK(s(head(cons(pairs, y0)))) -> c20(MARK(head(cons(pairs, y0)))) MARK(s(head(cons(odds, y0)))) -> c20(MARK(head(cons(odds, y0)))) MARK(incr(s(s(y0)))) -> c13(MARK(s(s(y0)))) MARK(incr(s(incr(s(y0))))) -> c13(MARK(s(incr(s(y0))))) MARK(incr(s(head(s(y0))))) -> c13(MARK(s(head(s(y0))))) MARK(incr(s(tail(cons(y0, y1))))) -> c13(MARK(s(tail(cons(y0, y1))))) MARK(incr(s(tail(s(y0))))) -> c13(MARK(s(tail(s(y0))))) MARK(incr(s(incr(incr(incr(y0)))))) -> c13(MARK(s(incr(incr(incr(y0)))))) MARK(incr(s(incr(incr(odds))))) -> c13(MARK(s(incr(incr(odds))))) MARK(incr(s(incr(incr(head(y0)))))) -> c13(MARK(s(incr(incr(head(y0)))))) MARK(incr(s(incr(incr(tail(y0)))))) -> c13(MARK(s(incr(incr(tail(y0)))))) MARK(incr(s(incr(incr(cons(y0, y1)))))) -> c13(MARK(s(incr(incr(cons(y0, y1)))))) MARK(incr(s(incr(incr(s(y0)))))) -> c13(MARK(s(incr(incr(s(y0)))))) MARK(incr(s(incr(incr(y0))))) -> c13(MARK(s(incr(incr(y0))))) MARK(incr(s(incr(head(incr(y0)))))) -> c13(MARK(s(incr(head(incr(y0)))))) MARK(incr(s(incr(head(odds))))) -> c13(MARK(s(incr(head(odds))))) MARK(incr(s(incr(head(head(y0)))))) -> c13(MARK(s(incr(head(head(y0)))))) MARK(incr(s(incr(head(tail(y0)))))) -> c13(MARK(s(incr(head(tail(y0)))))) MARK(incr(s(incr(head(cons(y0, y1)))))) -> c13(MARK(s(incr(head(cons(y0, y1)))))) MARK(incr(s(incr(head(s(y0)))))) -> c13(MARK(s(incr(head(s(y0)))))) MARK(incr(s(incr(head(y0))))) -> c13(MARK(s(incr(head(y0))))) MARK(incr(s(incr(tail(incr(y0)))))) -> c13(MARK(s(incr(tail(incr(y0)))))) MARK(incr(s(incr(tail(pairs))))) -> c13(MARK(s(incr(tail(pairs))))) MARK(incr(s(incr(tail(odds))))) -> c13(MARK(s(incr(tail(odds))))) MARK(incr(s(incr(tail(head(y0)))))) -> c13(MARK(s(incr(tail(head(y0)))))) MARK(incr(s(incr(tail(tail(y0)))))) -> c13(MARK(s(incr(tail(tail(y0)))))) MARK(incr(s(incr(tail(cons(y0, y1)))))) -> c13(MARK(s(incr(tail(cons(y0, y1)))))) MARK(incr(s(incr(tail(s(y0)))))) -> c13(MARK(s(incr(tail(s(y0)))))) MARK(incr(s(incr(tail(y0))))) -> c13(MARK(s(incr(tail(y0))))) MARK(incr(s(head(incr(incr(y0)))))) -> c13(MARK(s(head(incr(incr(y0)))))) MARK(incr(s(head(incr(odds))))) -> c13(MARK(s(head(incr(odds))))) MARK(incr(s(head(incr(head(y0)))))) -> c13(MARK(s(head(incr(head(y0)))))) MARK(incr(s(head(incr(tail(y0)))))) -> c13(MARK(s(head(incr(tail(y0)))))) MARK(incr(s(head(incr(cons(y0, y1)))))) -> c13(MARK(s(head(incr(cons(y0, y1)))))) MARK(incr(s(head(incr(s(y0)))))) -> c13(MARK(s(head(incr(s(y0)))))) MARK(incr(s(head(incr(y0))))) -> c13(MARK(s(head(incr(y0))))) MARK(incr(s(head(head(incr(y0)))))) -> c13(MARK(s(head(head(incr(y0)))))) MARK(incr(s(head(head(odds))))) -> c13(MARK(s(head(head(odds))))) MARK(incr(s(head(head(head(y0)))))) -> c13(MARK(s(head(head(head(y0)))))) MARK(incr(s(head(head(tail(y0)))))) -> c13(MARK(s(head(head(tail(y0)))))) MARK(incr(s(head(head(cons(y0, y1)))))) -> c13(MARK(s(head(head(cons(y0, y1)))))) MARK(incr(s(head(head(s(y0)))))) -> c13(MARK(s(head(head(s(y0)))))) MARK(incr(s(head(head(y0))))) -> c13(MARK(s(head(head(y0))))) MARK(incr(s(head(tail(incr(y0)))))) -> c13(MARK(s(head(tail(incr(y0)))))) MARK(incr(s(head(tail(pairs))))) -> c13(MARK(s(head(tail(pairs))))) MARK(incr(s(head(tail(odds))))) -> c13(MARK(s(head(tail(odds))))) MARK(incr(s(head(tail(head(y0)))))) -> c13(MARK(s(head(tail(head(y0)))))) MARK(incr(s(head(tail(tail(y0)))))) -> c13(MARK(s(head(tail(tail(y0)))))) MARK(incr(s(head(tail(cons(y0, y1)))))) -> c13(MARK(s(head(tail(cons(y0, y1)))))) MARK(incr(s(head(tail(s(y0)))))) -> c13(MARK(s(head(tail(s(y0)))))) MARK(incr(s(head(tail(y0))))) -> c13(MARK(s(head(tail(y0))))) MARK(incr(s(tail(incr(incr(y0)))))) -> c13(MARK(s(tail(incr(incr(y0)))))) MARK(incr(s(tail(incr(odds))))) -> c13(MARK(s(tail(incr(odds))))) MARK(incr(s(tail(incr(head(y0)))))) -> c13(MARK(s(tail(incr(head(y0)))))) MARK(incr(s(tail(incr(tail(y0)))))) -> c13(MARK(s(tail(incr(tail(y0)))))) MARK(incr(s(tail(incr(cons(y0, y1)))))) -> c13(MARK(s(tail(incr(cons(y0, y1)))))) MARK(incr(s(tail(incr(s(y0)))))) -> c13(MARK(s(tail(incr(s(y0)))))) MARK(incr(s(tail(incr(y0))))) -> c13(MARK(s(tail(incr(y0))))) MARK(incr(s(tail(head(incr(y0)))))) -> c13(MARK(s(tail(head(incr(y0)))))) MARK(incr(s(tail(head(odds))))) -> c13(MARK(s(tail(head(odds))))) MARK(incr(s(tail(head(head(y0)))))) -> c13(MARK(s(tail(head(head(y0)))))) MARK(incr(s(tail(head(tail(y0)))))) -> c13(MARK(s(tail(head(tail(y0)))))) MARK(incr(s(tail(head(cons(y0, y1)))))) -> c13(MARK(s(tail(head(cons(y0, y1)))))) MARK(incr(s(tail(head(s(y0)))))) -> c13(MARK(s(tail(head(s(y0)))))) MARK(incr(s(tail(head(y0))))) -> c13(MARK(s(tail(head(y0))))) MARK(incr(s(tail(tail(incr(y0)))))) -> c13(MARK(s(tail(tail(incr(y0)))))) MARK(incr(s(tail(tail(pairs))))) -> c13(MARK(s(tail(tail(pairs))))) MARK(incr(s(tail(tail(odds))))) -> c13(MARK(s(tail(tail(odds))))) MARK(incr(s(tail(tail(head(y0)))))) -> c13(MARK(s(tail(tail(head(y0)))))) MARK(incr(s(tail(tail(tail(y0)))))) -> c13(MARK(s(tail(tail(tail(y0)))))) MARK(incr(s(tail(tail(cons(y0, y1)))))) -> c13(MARK(s(tail(tail(cons(y0, y1)))))) MARK(incr(s(tail(tail(s(y0)))))) -> c13(MARK(s(tail(tail(s(y0)))))) MARK(incr(s(tail(tail(y0))))) -> c13(MARK(s(tail(tail(y0))))) MARK(incr(s(tail(pairs)))) -> c13(MARK(s(tail(pairs)))) MARK(incr(s(incr(incr(nats))))) -> c13(MARK(s(incr(incr(nats))))) MARK(incr(s(incr(incr(pairs))))) -> c13(MARK(s(incr(incr(pairs))))) MARK(incr(s(incr(odds)))) -> c13(MARK(s(incr(odds)))) MARK(incr(s(head(incr(nats))))) -> c13(MARK(s(head(incr(nats))))) MARK(incr(s(head(incr(pairs))))) -> c13(MARK(s(head(incr(pairs))))) MARK(incr(s(head(odds)))) -> c13(MARK(s(head(odds)))) MARK(incr(s(tail(incr(nats))))) -> c13(MARK(s(tail(incr(nats))))) MARK(incr(s(tail(incr(pairs))))) -> c13(MARK(s(tail(incr(pairs))))) MARK(incr(s(tail(odds)))) -> c13(MARK(s(tail(odds)))) MARK(incr(s(incr(cons(incr(y0), y1))))) -> c13(MARK(s(incr(cons(incr(y0), y1))))) MARK(incr(s(incr(cons(head(y0), y1))))) -> c13(MARK(s(incr(cons(head(y0), y1))))) MARK(incr(s(incr(cons(tail(y0), y1))))) -> c13(MARK(s(incr(cons(tail(y0), y1))))) MARK(incr(s(incr(cons(cons(y0, y1), y2))))) -> c13(MARK(s(incr(cons(cons(y0, y1), y2))))) MARK(incr(s(incr(cons(s(y0), y1))))) -> c13(MARK(s(incr(cons(s(y0), y1))))) MARK(incr(s(incr(cons(y0, y1))))) -> c13(MARK(s(incr(cons(y0, y1))))) MARK(incr(s(incr(head(nats))))) -> c13(MARK(s(incr(head(nats))))) MARK(incr(s(incr(head(pairs))))) -> c13(MARK(s(incr(head(pairs))))) MARK(incr(s(head(head(nats))))) -> c13(MARK(s(head(head(nats))))) MARK(incr(s(head(head(pairs))))) -> c13(MARK(s(head(head(pairs))))) MARK(incr(s(tail(head(nats))))) -> c13(MARK(s(tail(head(nats))))) MARK(incr(s(tail(head(pairs))))) -> c13(MARK(s(tail(head(pairs))))) MARK(incr(s(head(cons(incr(y0), y1))))) -> c13(MARK(s(head(cons(incr(y0), y1))))) MARK(incr(s(head(cons(head(y0), y1))))) -> c13(MARK(s(head(cons(head(y0), y1))))) MARK(incr(s(head(cons(tail(y0), y1))))) -> c13(MARK(s(head(cons(tail(y0), y1))))) MARK(incr(s(head(cons(cons(y0, y1), y2))))) -> c13(MARK(s(head(cons(cons(y0, y1), y2))))) MARK(incr(s(head(cons(s(y0), y1))))) -> c13(MARK(s(head(cons(s(y0), y1))))) MARK(incr(s(head(cons(y0, y1))))) -> c13(MARK(s(head(cons(y0, y1))))) MARK(incr(s(incr(tail(nats))))) -> c13(MARK(s(incr(tail(nats))))) MARK(incr(s(head(tail(nats))))) -> c13(MARK(s(head(tail(nats))))) MARK(incr(s(tail(tail(nats))))) -> c13(MARK(s(tail(tail(nats))))) MARK(incr(s(cons(cons(y0, y1), y2)))) -> c13(MARK(s(cons(cons(y0, y1), y2)))) MARK(incr(s(cons(s(y0), y1)))) -> c13(MARK(s(cons(s(y0), y1)))) MARK(incr(s(cons(incr(s(y0)), y1)))) -> c13(MARK(s(cons(incr(s(y0)), y1)))) MARK(incr(s(cons(head(s(y0)), y1)))) -> c13(MARK(s(cons(head(s(y0)), y1)))) MARK(incr(s(cons(tail(cons(y0, y1)), y2)))) -> c13(MARK(s(cons(tail(cons(y0, y1)), y2)))) MARK(incr(s(cons(tail(s(y0)), y1)))) -> c13(MARK(s(cons(tail(s(y0)), y1)))) MARK(incr(s(cons(incr(incr(incr(y0))), y1)))) -> c13(MARK(s(cons(incr(incr(incr(y0))), y1)))) MARK(incr(s(cons(incr(incr(odds)), y0)))) -> c13(MARK(s(cons(incr(incr(odds)), y0)))) MARK(incr(s(cons(incr(incr(head(y0))), y1)))) -> c13(MARK(s(cons(incr(incr(head(y0))), y1)))) MARK(incr(s(cons(incr(incr(tail(y0))), y1)))) -> c13(MARK(s(cons(incr(incr(tail(y0))), y1)))) MARK(incr(s(cons(incr(incr(cons(y0, y1))), y2)))) -> c13(MARK(s(cons(incr(incr(cons(y0, y1))), y2)))) MARK(incr(s(cons(incr(incr(s(y0))), y1)))) -> c13(MARK(s(cons(incr(incr(s(y0))), y1)))) MARK(incr(s(cons(incr(incr(y0)), y1)))) -> c13(MARK(s(cons(incr(incr(y0)), y1)))) MARK(incr(s(cons(incr(head(incr(y0))), y1)))) -> c13(MARK(s(cons(incr(head(incr(y0))), y1)))) MARK(incr(s(cons(incr(head(odds)), y0)))) -> c13(MARK(s(cons(incr(head(odds)), y0)))) MARK(incr(s(cons(incr(head(head(y0))), y1)))) -> c13(MARK(s(cons(incr(head(head(y0))), y1)))) MARK(incr(s(cons(incr(head(tail(y0))), y1)))) -> c13(MARK(s(cons(incr(head(tail(y0))), y1)))) MARK(incr(s(cons(incr(head(cons(y0, y1))), y2)))) -> c13(MARK(s(cons(incr(head(cons(y0, y1))), y2)))) MARK(incr(s(cons(incr(head(s(y0))), y1)))) -> c13(MARK(s(cons(incr(head(s(y0))), y1)))) MARK(incr(s(cons(incr(head(y0)), y1)))) -> c13(MARK(s(cons(incr(head(y0)), y1)))) MARK(incr(s(cons(incr(tail(incr(y0))), y1)))) -> c13(MARK(s(cons(incr(tail(incr(y0))), y1)))) MARK(incr(s(cons(incr(tail(pairs)), y0)))) -> c13(MARK(s(cons(incr(tail(pairs)), y0)))) MARK(incr(s(cons(incr(tail(odds)), y0)))) -> c13(MARK(s(cons(incr(tail(odds)), y0)))) MARK(incr(s(cons(incr(tail(head(y0))), y1)))) -> c13(MARK(s(cons(incr(tail(head(y0))), y1)))) MARK(incr(s(cons(incr(tail(tail(y0))), y1)))) -> c13(MARK(s(cons(incr(tail(tail(y0))), y1)))) MARK(incr(s(cons(incr(tail(cons(y0, y1))), y2)))) -> c13(MARK(s(cons(incr(tail(cons(y0, y1))), y2)))) MARK(incr(s(cons(incr(tail(s(y0))), y1)))) -> c13(MARK(s(cons(incr(tail(s(y0))), y1)))) MARK(incr(s(cons(incr(tail(y0)), y1)))) -> c13(MARK(s(cons(incr(tail(y0)), y1)))) MARK(incr(s(cons(head(incr(incr(y0))), y1)))) -> c13(MARK(s(cons(head(incr(incr(y0))), y1)))) MARK(incr(s(cons(head(incr(odds)), y0)))) -> c13(MARK(s(cons(head(incr(odds)), y0)))) MARK(incr(s(cons(head(incr(head(y0))), y1)))) -> c13(MARK(s(cons(head(incr(head(y0))), y1)))) MARK(incr(s(cons(head(incr(tail(y0))), y1)))) -> c13(MARK(s(cons(head(incr(tail(y0))), y1)))) MARK(incr(s(cons(head(incr(cons(y0, y1))), y2)))) -> c13(MARK(s(cons(head(incr(cons(y0, y1))), y2)))) MARK(incr(s(cons(head(incr(s(y0))), y1)))) -> c13(MARK(s(cons(head(incr(s(y0))), y1)))) MARK(incr(s(cons(head(incr(y0)), y1)))) -> c13(MARK(s(cons(head(incr(y0)), y1)))) MARK(incr(s(cons(head(head(incr(y0))), y1)))) -> c13(MARK(s(cons(head(head(incr(y0))), y1)))) MARK(incr(s(cons(head(head(odds)), y0)))) -> c13(MARK(s(cons(head(head(odds)), y0)))) MARK(incr(s(cons(head(head(head(y0))), y1)))) -> c13(MARK(s(cons(head(head(head(y0))), y1)))) MARK(incr(s(cons(head(head(tail(y0))), y1)))) -> c13(MARK(s(cons(head(head(tail(y0))), y1)))) MARK(incr(s(cons(head(head(cons(y0, y1))), y2)))) -> c13(MARK(s(cons(head(head(cons(y0, y1))), y2)))) MARK(incr(s(cons(head(head(s(y0))), y1)))) -> c13(MARK(s(cons(head(head(s(y0))), y1)))) MARK(incr(s(cons(head(head(y0)), y1)))) -> c13(MARK(s(cons(head(head(y0)), y1)))) MARK(incr(s(cons(head(tail(incr(y0))), y1)))) -> c13(MARK(s(cons(head(tail(incr(y0))), y1)))) MARK(incr(s(cons(head(tail(pairs)), y0)))) -> c13(MARK(s(cons(head(tail(pairs)), y0)))) MARK(incr(s(cons(head(tail(odds)), y0)))) -> c13(MARK(s(cons(head(tail(odds)), y0)))) MARK(incr(s(cons(head(tail(head(y0))), y1)))) -> c13(MARK(s(cons(head(tail(head(y0))), y1)))) MARK(incr(s(cons(head(tail(tail(y0))), y1)))) -> c13(MARK(s(cons(head(tail(tail(y0))), y1)))) MARK(incr(s(cons(head(tail(cons(y0, y1))), y2)))) -> c13(MARK(s(cons(head(tail(cons(y0, y1))), y2)))) MARK(incr(s(cons(head(tail(s(y0))), y1)))) -> c13(MARK(s(cons(head(tail(s(y0))), y1)))) MARK(incr(s(cons(head(tail(y0)), y1)))) -> c13(MARK(s(cons(head(tail(y0)), y1)))) MARK(incr(s(cons(tail(incr(incr(y0))), y1)))) -> c13(MARK(s(cons(tail(incr(incr(y0))), y1)))) MARK(incr(s(cons(tail(incr(odds)), y0)))) -> c13(MARK(s(cons(tail(incr(odds)), y0)))) MARK(incr(s(cons(tail(incr(head(y0))), y1)))) -> c13(MARK(s(cons(tail(incr(head(y0))), y1)))) MARK(incr(s(cons(tail(incr(tail(y0))), y1)))) -> c13(MARK(s(cons(tail(incr(tail(y0))), y1)))) MARK(incr(s(cons(tail(incr(cons(y0, y1))), y2)))) -> c13(MARK(s(cons(tail(incr(cons(y0, y1))), y2)))) MARK(incr(s(cons(tail(incr(s(y0))), y1)))) -> c13(MARK(s(cons(tail(incr(s(y0))), y1)))) MARK(incr(s(cons(tail(incr(y0)), y1)))) -> c13(MARK(s(cons(tail(incr(y0)), y1)))) MARK(incr(s(cons(tail(head(incr(y0))), y1)))) -> c13(MARK(s(cons(tail(head(incr(y0))), y1)))) MARK(incr(s(cons(tail(head(odds)), y0)))) -> c13(MARK(s(cons(tail(head(odds)), y0)))) MARK(incr(s(cons(tail(head(head(y0))), y1)))) -> c13(MARK(s(cons(tail(head(head(y0))), y1)))) MARK(incr(s(cons(tail(head(tail(y0))), y1)))) -> c13(MARK(s(cons(tail(head(tail(y0))), y1)))) MARK(incr(s(cons(tail(head(cons(y0, y1))), y2)))) -> c13(MARK(s(cons(tail(head(cons(y0, y1))), y2)))) MARK(incr(s(cons(tail(head(s(y0))), y1)))) -> c13(MARK(s(cons(tail(head(s(y0))), y1)))) MARK(incr(s(cons(tail(head(y0)), y1)))) -> c13(MARK(s(cons(tail(head(y0)), y1)))) MARK(incr(s(cons(tail(tail(incr(y0))), y1)))) -> c13(MARK(s(cons(tail(tail(incr(y0))), y1)))) MARK(incr(s(cons(tail(tail(pairs)), y0)))) -> c13(MARK(s(cons(tail(tail(pairs)), y0)))) MARK(incr(s(cons(tail(tail(odds)), y0)))) -> c13(MARK(s(cons(tail(tail(odds)), y0)))) MARK(incr(s(cons(tail(tail(head(y0))), y1)))) -> c13(MARK(s(cons(tail(tail(head(y0))), y1)))) MARK(incr(s(cons(tail(tail(tail(y0))), y1)))) -> c13(MARK(s(cons(tail(tail(tail(y0))), y1)))) MARK(incr(s(cons(tail(tail(cons(y0, y1))), y2)))) -> c13(MARK(s(cons(tail(tail(cons(y0, y1))), y2)))) MARK(incr(s(cons(tail(tail(s(y0))), y1)))) -> c13(MARK(s(cons(tail(tail(s(y0))), y1)))) MARK(incr(s(cons(tail(tail(y0)), y1)))) -> c13(MARK(s(cons(tail(tail(y0)), y1)))) MARK(incr(s(cons(tail(pairs), y0)))) -> c13(MARK(s(cons(tail(pairs), y0)))) MARK(incr(s(cons(incr(incr(nats)), y0)))) -> c13(MARK(s(cons(incr(incr(nats)), y0)))) MARK(incr(s(cons(incr(incr(pairs)), y0)))) -> c13(MARK(s(cons(incr(incr(pairs)), y0)))) MARK(incr(s(cons(incr(odds), y0)))) -> c13(MARK(s(cons(incr(odds), y0)))) MARK(incr(s(cons(head(incr(nats)), y0)))) -> c13(MARK(s(cons(head(incr(nats)), y0)))) MARK(incr(s(cons(head(incr(pairs)), y0)))) -> c13(MARK(s(cons(head(incr(pairs)), y0)))) MARK(incr(s(cons(head(odds), y0)))) -> c13(MARK(s(cons(head(odds), y0)))) MARK(incr(s(cons(tail(incr(nats)), y0)))) -> c13(MARK(s(cons(tail(incr(nats)), y0)))) MARK(incr(s(cons(tail(incr(pairs)), y0)))) -> c13(MARK(s(cons(tail(incr(pairs)), y0)))) MARK(incr(s(cons(tail(odds), y0)))) -> c13(MARK(s(cons(tail(odds), y0)))) MARK(incr(s(cons(incr(cons(nats, y0)), y1)))) -> c13(MARK(s(cons(incr(cons(nats, y0)), y1)))) MARK(incr(s(cons(incr(cons(incr(y0), y1)), y2)))) -> c13(MARK(s(cons(incr(cons(incr(y0), y1)), y2)))) MARK(incr(s(cons(incr(cons(pairs, y0)), y1)))) -> c13(MARK(s(cons(incr(cons(pairs, y0)), y1)))) MARK(incr(s(cons(incr(cons(odds, y0)), y1)))) -> c13(MARK(s(cons(incr(cons(odds, y0)), y1)))) MARK(incr(s(cons(incr(cons(head(y0), y1)), y2)))) -> c13(MARK(s(cons(incr(cons(head(y0), y1)), y2)))) MARK(incr(s(cons(incr(cons(tail(y0), y1)), y2)))) -> c13(MARK(s(cons(incr(cons(tail(y0), y1)), y2)))) MARK(incr(s(cons(incr(cons(cons(y0, y1), y2)), y3)))) -> c13(MARK(s(cons(incr(cons(cons(y0, y1), y2)), y3)))) MARK(incr(s(cons(incr(cons(s(y0), y1)), y2)))) -> c13(MARK(s(cons(incr(cons(s(y0), y1)), y2)))) MARK(incr(s(cons(incr(cons(y0, y1)), y2)))) -> c13(MARK(s(cons(incr(cons(y0, y1)), y2)))) MARK(incr(s(cons(incr(cons(0, y0)), y1)))) -> c13(MARK(s(cons(incr(cons(0, y0)), y1)))) MARK(incr(s(cons(incr(head(nats)), y0)))) -> c13(MARK(s(cons(incr(head(nats)), y0)))) MARK(incr(s(cons(incr(head(pairs)), y0)))) -> c13(MARK(s(cons(incr(head(pairs)), y0)))) MARK(incr(s(cons(head(head(nats)), y0)))) -> c13(MARK(s(cons(head(head(nats)), y0)))) MARK(incr(s(cons(head(head(pairs)), y0)))) -> c13(MARK(s(cons(head(head(pairs)), y0)))) MARK(incr(s(cons(tail(head(nats)), y0)))) -> c13(MARK(s(cons(tail(head(nats)), y0)))) MARK(incr(s(cons(tail(head(pairs)), y0)))) -> c13(MARK(s(cons(tail(head(pairs)), y0)))) MARK(incr(s(cons(head(cons(nats, y0)), y1)))) -> c13(MARK(s(cons(head(cons(nats, y0)), y1)))) MARK(incr(s(cons(head(cons(incr(y0), y1)), y2)))) -> c13(MARK(s(cons(head(cons(incr(y0), y1)), y2)))) MARK(incr(s(cons(head(cons(pairs, y0)), y1)))) -> c13(MARK(s(cons(head(cons(pairs, y0)), y1)))) MARK(incr(s(cons(head(cons(odds, y0)), y1)))) -> c13(MARK(s(cons(head(cons(odds, y0)), y1)))) MARK(incr(s(cons(head(cons(head(y0), y1)), y2)))) -> c13(MARK(s(cons(head(cons(head(y0), y1)), y2)))) MARK(incr(s(cons(head(cons(tail(y0), y1)), y2)))) -> c13(MARK(s(cons(head(cons(tail(y0), y1)), y2)))) MARK(incr(s(cons(head(cons(cons(y0, y1), y2)), y3)))) -> c13(MARK(s(cons(head(cons(cons(y0, y1), y2)), y3)))) MARK(incr(s(cons(head(cons(s(y0), y1)), y2)))) -> c13(MARK(s(cons(head(cons(s(y0), y1)), y2)))) MARK(incr(s(cons(head(cons(y0, y1)), y2)))) -> c13(MARK(s(cons(head(cons(y0, y1)), y2)))) MARK(incr(s(cons(head(cons(0, y0)), y1)))) -> c13(MARK(s(cons(head(cons(0, y0)), y1)))) MARK(incr(s(cons(incr(tail(nats)), y0)))) -> c13(MARK(s(cons(incr(tail(nats)), y0)))) MARK(incr(s(cons(head(tail(nats)), y0)))) -> c13(MARK(s(cons(head(tail(nats)), y0)))) MARK(incr(s(cons(tail(tail(nats)), y0)))) -> c13(MARK(s(cons(tail(tail(nats)), y0)))) MARK(incr(s(incr(cons(nats, y0))))) -> c13(MARK(s(incr(cons(nats, y0))))) MARK(incr(s(incr(cons(pairs, y0))))) -> c13(MARK(s(incr(cons(pairs, y0))))) MARK(incr(s(incr(cons(odds, y0))))) -> c13(MARK(s(incr(cons(odds, y0))))) MARK(incr(s(head(cons(nats, y0))))) -> c13(MARK(s(head(cons(nats, y0))))) MARK(incr(s(head(cons(pairs, y0))))) -> c13(MARK(s(head(cons(pairs, y0))))) MARK(incr(s(head(cons(odds, y0))))) -> c13(MARK(s(head(cons(odds, y0))))) MARK(head(s(s(y0)))) -> c16(MARK(s(s(y0)))) MARK(head(s(incr(s(y0))))) -> c16(MARK(s(incr(s(y0))))) MARK(head(s(head(s(y0))))) -> c16(MARK(s(head(s(y0))))) MARK(head(s(tail(cons(y0, y1))))) -> c16(MARK(s(tail(cons(y0, y1))))) MARK(head(s(tail(s(y0))))) -> c16(MARK(s(tail(s(y0))))) MARK(head(s(incr(incr(incr(y0)))))) -> c16(MARK(s(incr(incr(incr(y0)))))) MARK(head(s(incr(incr(odds))))) -> c16(MARK(s(incr(incr(odds))))) MARK(head(s(incr(incr(head(y0)))))) -> c16(MARK(s(incr(incr(head(y0)))))) MARK(head(s(incr(incr(tail(y0)))))) -> c16(MARK(s(incr(incr(tail(y0)))))) MARK(head(s(incr(incr(cons(y0, y1)))))) -> c16(MARK(s(incr(incr(cons(y0, y1)))))) MARK(head(s(incr(incr(s(y0)))))) -> c16(MARK(s(incr(incr(s(y0)))))) MARK(head(s(incr(incr(y0))))) -> c16(MARK(s(incr(incr(y0))))) MARK(head(s(incr(head(incr(y0)))))) -> c16(MARK(s(incr(head(incr(y0)))))) MARK(head(s(incr(head(odds))))) -> c16(MARK(s(incr(head(odds))))) MARK(head(s(incr(head(head(y0)))))) -> c16(MARK(s(incr(head(head(y0)))))) MARK(head(s(incr(head(tail(y0)))))) -> c16(MARK(s(incr(head(tail(y0)))))) MARK(head(s(incr(head(cons(y0, y1)))))) -> c16(MARK(s(incr(head(cons(y0, y1)))))) MARK(head(s(incr(head(s(y0)))))) -> c16(MARK(s(incr(head(s(y0)))))) MARK(head(s(incr(head(y0))))) -> c16(MARK(s(incr(head(y0))))) MARK(head(s(incr(tail(incr(y0)))))) -> c16(MARK(s(incr(tail(incr(y0)))))) MARK(head(s(incr(tail(pairs))))) -> c16(MARK(s(incr(tail(pairs))))) MARK(head(s(incr(tail(odds))))) -> c16(MARK(s(incr(tail(odds))))) MARK(head(s(incr(tail(head(y0)))))) -> c16(MARK(s(incr(tail(head(y0)))))) MARK(head(s(incr(tail(tail(y0)))))) -> c16(MARK(s(incr(tail(tail(y0)))))) MARK(head(s(incr(tail(cons(y0, y1)))))) -> c16(MARK(s(incr(tail(cons(y0, y1)))))) MARK(head(s(incr(tail(s(y0)))))) -> c16(MARK(s(incr(tail(s(y0)))))) MARK(head(s(incr(tail(y0))))) -> c16(MARK(s(incr(tail(y0))))) MARK(head(s(head(incr(incr(y0)))))) -> c16(MARK(s(head(incr(incr(y0)))))) MARK(head(s(head(incr(odds))))) -> c16(MARK(s(head(incr(odds))))) MARK(head(s(head(incr(head(y0)))))) -> c16(MARK(s(head(incr(head(y0)))))) MARK(head(s(head(incr(tail(y0)))))) -> c16(MARK(s(head(incr(tail(y0)))))) MARK(head(s(head(incr(cons(y0, y1)))))) -> c16(MARK(s(head(incr(cons(y0, y1)))))) MARK(head(s(head(incr(s(y0)))))) -> c16(MARK(s(head(incr(s(y0)))))) MARK(head(s(head(incr(y0))))) -> c16(MARK(s(head(incr(y0))))) MARK(head(s(head(head(incr(y0)))))) -> c16(MARK(s(head(head(incr(y0)))))) MARK(head(s(head(head(odds))))) -> c16(MARK(s(head(head(odds))))) MARK(head(s(head(head(head(y0)))))) -> c16(MARK(s(head(head(head(y0)))))) MARK(head(s(head(head(tail(y0)))))) -> c16(MARK(s(head(head(tail(y0)))))) MARK(head(s(head(head(cons(y0, y1)))))) -> c16(MARK(s(head(head(cons(y0, y1)))))) MARK(head(s(head(head(s(y0)))))) -> c16(MARK(s(head(head(s(y0)))))) MARK(head(s(head(head(y0))))) -> c16(MARK(s(head(head(y0))))) MARK(head(s(head(tail(incr(y0)))))) -> c16(MARK(s(head(tail(incr(y0)))))) MARK(head(s(head(tail(pairs))))) -> c16(MARK(s(head(tail(pairs))))) MARK(head(s(head(tail(odds))))) -> c16(MARK(s(head(tail(odds))))) MARK(head(s(head(tail(head(y0)))))) -> c16(MARK(s(head(tail(head(y0)))))) MARK(head(s(head(tail(tail(y0)))))) -> c16(MARK(s(head(tail(tail(y0)))))) MARK(head(s(head(tail(cons(y0, y1)))))) -> c16(MARK(s(head(tail(cons(y0, y1)))))) MARK(head(s(head(tail(s(y0)))))) -> c16(MARK(s(head(tail(s(y0)))))) MARK(head(s(head(tail(y0))))) -> c16(MARK(s(head(tail(y0))))) MARK(head(s(tail(incr(incr(y0)))))) -> c16(MARK(s(tail(incr(incr(y0)))))) MARK(head(s(tail(incr(odds))))) -> c16(MARK(s(tail(incr(odds))))) MARK(head(s(tail(incr(head(y0)))))) -> c16(MARK(s(tail(incr(head(y0)))))) MARK(head(s(tail(incr(tail(y0)))))) -> c16(MARK(s(tail(incr(tail(y0)))))) MARK(head(s(tail(incr(cons(y0, y1)))))) -> c16(MARK(s(tail(incr(cons(y0, y1)))))) MARK(head(s(tail(incr(s(y0)))))) -> c16(MARK(s(tail(incr(s(y0)))))) MARK(head(s(tail(incr(y0))))) -> c16(MARK(s(tail(incr(y0))))) MARK(head(s(tail(head(incr(y0)))))) -> c16(MARK(s(tail(head(incr(y0)))))) MARK(head(s(tail(head(odds))))) -> c16(MARK(s(tail(head(odds))))) MARK(head(s(tail(head(head(y0)))))) -> c16(MARK(s(tail(head(head(y0)))))) MARK(head(s(tail(head(tail(y0)))))) -> c16(MARK(s(tail(head(tail(y0)))))) MARK(head(s(tail(head(cons(y0, y1)))))) -> c16(MARK(s(tail(head(cons(y0, y1)))))) MARK(head(s(tail(head(s(y0)))))) -> c16(MARK(s(tail(head(s(y0)))))) MARK(head(s(tail(head(y0))))) -> c16(MARK(s(tail(head(y0))))) MARK(head(s(tail(tail(incr(y0)))))) -> c16(MARK(s(tail(tail(incr(y0)))))) MARK(head(s(tail(tail(pairs))))) -> c16(MARK(s(tail(tail(pairs))))) MARK(head(s(tail(tail(odds))))) -> c16(MARK(s(tail(tail(odds))))) MARK(head(s(tail(tail(head(y0)))))) -> c16(MARK(s(tail(tail(head(y0)))))) MARK(head(s(tail(tail(tail(y0)))))) -> c16(MARK(s(tail(tail(tail(y0)))))) MARK(head(s(tail(tail(cons(y0, y1)))))) -> c16(MARK(s(tail(tail(cons(y0, y1)))))) MARK(head(s(tail(tail(s(y0)))))) -> c16(MARK(s(tail(tail(s(y0)))))) MARK(head(s(tail(tail(y0))))) -> c16(MARK(s(tail(tail(y0))))) MARK(head(s(tail(pairs)))) -> c16(MARK(s(tail(pairs)))) MARK(head(s(incr(incr(nats))))) -> c16(MARK(s(incr(incr(nats))))) MARK(head(s(incr(incr(pairs))))) -> c16(MARK(s(incr(incr(pairs))))) MARK(head(s(incr(odds)))) -> c16(MARK(s(incr(odds)))) MARK(head(s(head(incr(nats))))) -> c16(MARK(s(head(incr(nats))))) MARK(head(s(head(incr(pairs))))) -> c16(MARK(s(head(incr(pairs))))) MARK(head(s(head(odds)))) -> c16(MARK(s(head(odds)))) MARK(head(s(tail(incr(nats))))) -> c16(MARK(s(tail(incr(nats))))) MARK(head(s(tail(incr(pairs))))) -> c16(MARK(s(tail(incr(pairs))))) MARK(head(s(tail(odds)))) -> c16(MARK(s(tail(odds)))) MARK(head(s(incr(cons(incr(y0), y1))))) -> c16(MARK(s(incr(cons(incr(y0), y1))))) MARK(head(s(incr(cons(head(y0), y1))))) -> c16(MARK(s(incr(cons(head(y0), y1))))) MARK(head(s(incr(cons(tail(y0), y1))))) -> c16(MARK(s(incr(cons(tail(y0), y1))))) MARK(head(s(incr(cons(cons(y0, y1), y2))))) -> c16(MARK(s(incr(cons(cons(y0, y1), y2))))) MARK(head(s(incr(cons(s(y0), y1))))) -> c16(MARK(s(incr(cons(s(y0), y1))))) MARK(head(s(incr(cons(y0, y1))))) -> c16(MARK(s(incr(cons(y0, y1))))) MARK(head(s(incr(head(nats))))) -> c16(MARK(s(incr(head(nats))))) MARK(head(s(incr(head(pairs))))) -> c16(MARK(s(incr(head(pairs))))) MARK(head(s(head(head(nats))))) -> c16(MARK(s(head(head(nats))))) MARK(head(s(head(head(pairs))))) -> c16(MARK(s(head(head(pairs))))) MARK(head(s(tail(head(nats))))) -> c16(MARK(s(tail(head(nats))))) MARK(head(s(tail(head(pairs))))) -> c16(MARK(s(tail(head(pairs))))) MARK(head(s(head(cons(incr(y0), y1))))) -> c16(MARK(s(head(cons(incr(y0), y1))))) MARK(head(s(head(cons(head(y0), y1))))) -> c16(MARK(s(head(cons(head(y0), y1))))) MARK(head(s(head(cons(tail(y0), y1))))) -> c16(MARK(s(head(cons(tail(y0), y1))))) MARK(head(s(head(cons(cons(y0, y1), y2))))) -> c16(MARK(s(head(cons(cons(y0, y1), y2))))) MARK(head(s(head(cons(s(y0), y1))))) -> c16(MARK(s(head(cons(s(y0), y1))))) MARK(head(s(head(cons(y0, y1))))) -> c16(MARK(s(head(cons(y0, y1))))) MARK(head(s(incr(tail(nats))))) -> c16(MARK(s(incr(tail(nats))))) MARK(head(s(head(tail(nats))))) -> c16(MARK(s(head(tail(nats))))) MARK(head(s(tail(tail(nats))))) -> c16(MARK(s(tail(tail(nats))))) MARK(head(s(cons(cons(y0, y1), y2)))) -> c16(MARK(s(cons(cons(y0, y1), y2)))) MARK(head(s(cons(s(y0), y1)))) -> c16(MARK(s(cons(s(y0), y1)))) MARK(head(s(cons(incr(s(y0)), y1)))) -> c16(MARK(s(cons(incr(s(y0)), y1)))) MARK(head(s(cons(head(s(y0)), y1)))) -> c16(MARK(s(cons(head(s(y0)), y1)))) MARK(head(s(cons(tail(cons(y0, y1)), y2)))) -> c16(MARK(s(cons(tail(cons(y0, y1)), y2)))) MARK(head(s(cons(tail(s(y0)), y1)))) -> c16(MARK(s(cons(tail(s(y0)), y1)))) MARK(head(s(cons(incr(incr(incr(y0))), y1)))) -> c16(MARK(s(cons(incr(incr(incr(y0))), y1)))) MARK(head(s(cons(incr(incr(odds)), y0)))) -> c16(MARK(s(cons(incr(incr(odds)), y0)))) MARK(head(s(cons(incr(incr(head(y0))), y1)))) -> c16(MARK(s(cons(incr(incr(head(y0))), y1)))) MARK(head(s(cons(incr(incr(tail(y0))), y1)))) -> c16(MARK(s(cons(incr(incr(tail(y0))), y1)))) MARK(head(s(cons(incr(incr(cons(y0, y1))), y2)))) -> c16(MARK(s(cons(incr(incr(cons(y0, y1))), y2)))) MARK(head(s(cons(incr(incr(s(y0))), y1)))) -> c16(MARK(s(cons(incr(incr(s(y0))), y1)))) MARK(head(s(cons(incr(incr(y0)), y1)))) -> c16(MARK(s(cons(incr(incr(y0)), y1)))) MARK(head(s(cons(incr(head(incr(y0))), y1)))) -> c16(MARK(s(cons(incr(head(incr(y0))), y1)))) MARK(head(s(cons(incr(head(odds)), y0)))) -> c16(MARK(s(cons(incr(head(odds)), y0)))) MARK(head(s(cons(incr(head(head(y0))), y1)))) -> c16(MARK(s(cons(incr(head(head(y0))), y1)))) MARK(head(s(cons(incr(head(tail(y0))), y1)))) -> c16(MARK(s(cons(incr(head(tail(y0))), y1)))) MARK(head(s(cons(incr(head(cons(y0, y1))), y2)))) -> c16(MARK(s(cons(incr(head(cons(y0, y1))), y2)))) MARK(head(s(cons(incr(head(s(y0))), y1)))) -> c16(MARK(s(cons(incr(head(s(y0))), y1)))) MARK(head(s(cons(incr(head(y0)), y1)))) -> c16(MARK(s(cons(incr(head(y0)), y1)))) MARK(head(s(cons(incr(tail(incr(y0))), y1)))) -> c16(MARK(s(cons(incr(tail(incr(y0))), y1)))) MARK(head(s(cons(incr(tail(pairs)), y0)))) -> c16(MARK(s(cons(incr(tail(pairs)), y0)))) MARK(head(s(cons(incr(tail(odds)), y0)))) -> c16(MARK(s(cons(incr(tail(odds)), y0)))) MARK(head(s(cons(incr(tail(head(y0))), y1)))) -> c16(MARK(s(cons(incr(tail(head(y0))), y1)))) MARK(head(s(cons(incr(tail(tail(y0))), y1)))) -> c16(MARK(s(cons(incr(tail(tail(y0))), y1)))) MARK(head(s(cons(incr(tail(cons(y0, y1))), y2)))) -> c16(MARK(s(cons(incr(tail(cons(y0, y1))), y2)))) MARK(head(s(cons(incr(tail(s(y0))), y1)))) -> c16(MARK(s(cons(incr(tail(s(y0))), y1)))) MARK(head(s(cons(incr(tail(y0)), y1)))) -> c16(MARK(s(cons(incr(tail(y0)), y1)))) MARK(head(s(cons(head(incr(incr(y0))), y1)))) -> c16(MARK(s(cons(head(incr(incr(y0))), y1)))) MARK(head(s(cons(head(incr(odds)), y0)))) -> c16(MARK(s(cons(head(incr(odds)), y0)))) MARK(head(s(cons(head(incr(head(y0))), y1)))) -> c16(MARK(s(cons(head(incr(head(y0))), y1)))) MARK(head(s(cons(head(incr(tail(y0))), y1)))) -> c16(MARK(s(cons(head(incr(tail(y0))), y1)))) MARK(head(s(cons(head(incr(cons(y0, y1))), y2)))) -> c16(MARK(s(cons(head(incr(cons(y0, y1))), y2)))) MARK(head(s(cons(head(incr(s(y0))), y1)))) -> c16(MARK(s(cons(head(incr(s(y0))), y1)))) MARK(head(s(cons(head(incr(y0)), y1)))) -> c16(MARK(s(cons(head(incr(y0)), y1)))) MARK(head(s(cons(head(head(incr(y0))), y1)))) -> c16(MARK(s(cons(head(head(incr(y0))), y1)))) MARK(head(s(cons(head(head(odds)), y0)))) -> c16(MARK(s(cons(head(head(odds)), y0)))) MARK(head(s(cons(head(head(head(y0))), y1)))) -> c16(MARK(s(cons(head(head(head(y0))), y1)))) MARK(head(s(cons(head(head(tail(y0))), y1)))) -> c16(MARK(s(cons(head(head(tail(y0))), y1)))) MARK(head(s(cons(head(head(cons(y0, y1))), y2)))) -> c16(MARK(s(cons(head(head(cons(y0, y1))), y2)))) MARK(head(s(cons(head(head(s(y0))), y1)))) -> c16(MARK(s(cons(head(head(s(y0))), y1)))) MARK(head(s(cons(head(head(y0)), y1)))) -> c16(MARK(s(cons(head(head(y0)), y1)))) MARK(head(s(cons(head(tail(incr(y0))), y1)))) -> c16(MARK(s(cons(head(tail(incr(y0))), y1)))) MARK(head(s(cons(head(tail(pairs)), y0)))) -> c16(MARK(s(cons(head(tail(pairs)), y0)))) MARK(head(s(cons(head(tail(odds)), y0)))) -> c16(MARK(s(cons(head(tail(odds)), y0)))) MARK(head(s(cons(head(tail(head(y0))), y1)))) -> c16(MARK(s(cons(head(tail(head(y0))), y1)))) MARK(head(s(cons(head(tail(tail(y0))), y1)))) -> c16(MARK(s(cons(head(tail(tail(y0))), y1)))) MARK(head(s(cons(head(tail(cons(y0, y1))), y2)))) -> c16(MARK(s(cons(head(tail(cons(y0, y1))), y2)))) MARK(head(s(cons(head(tail(s(y0))), y1)))) -> c16(MARK(s(cons(head(tail(s(y0))), y1)))) MARK(head(s(cons(head(tail(y0)), y1)))) -> c16(MARK(s(cons(head(tail(y0)), y1)))) MARK(head(s(cons(tail(incr(incr(y0))), y1)))) -> c16(MARK(s(cons(tail(incr(incr(y0))), y1)))) MARK(head(s(cons(tail(incr(odds)), y0)))) -> c16(MARK(s(cons(tail(incr(odds)), y0)))) MARK(head(s(cons(tail(incr(head(y0))), y1)))) -> c16(MARK(s(cons(tail(incr(head(y0))), y1)))) MARK(head(s(cons(tail(incr(tail(y0))), y1)))) -> c16(MARK(s(cons(tail(incr(tail(y0))), y1)))) MARK(head(s(cons(tail(incr(cons(y0, y1))), y2)))) -> c16(MARK(s(cons(tail(incr(cons(y0, y1))), y2)))) MARK(head(s(cons(tail(incr(s(y0))), y1)))) -> c16(MARK(s(cons(tail(incr(s(y0))), y1)))) MARK(head(s(cons(tail(incr(y0)), y1)))) -> c16(MARK(s(cons(tail(incr(y0)), y1)))) MARK(head(s(cons(tail(head(incr(y0))), y1)))) -> c16(MARK(s(cons(tail(head(incr(y0))), y1)))) MARK(head(s(cons(tail(head(odds)), y0)))) -> c16(MARK(s(cons(tail(head(odds)), y0)))) MARK(head(s(cons(tail(head(head(y0))), y1)))) -> c16(MARK(s(cons(tail(head(head(y0))), y1)))) MARK(head(s(cons(tail(head(tail(y0))), y1)))) -> c16(MARK(s(cons(tail(head(tail(y0))), y1)))) MARK(head(s(cons(tail(head(cons(y0, y1))), y2)))) -> c16(MARK(s(cons(tail(head(cons(y0, y1))), y2)))) MARK(head(s(cons(tail(head(s(y0))), y1)))) -> c16(MARK(s(cons(tail(head(s(y0))), y1)))) MARK(head(s(cons(tail(head(y0)), y1)))) -> c16(MARK(s(cons(tail(head(y0)), y1)))) MARK(head(s(cons(tail(tail(incr(y0))), y1)))) -> c16(MARK(s(cons(tail(tail(incr(y0))), y1)))) MARK(head(s(cons(tail(tail(pairs)), y0)))) -> c16(MARK(s(cons(tail(tail(pairs)), y0)))) MARK(head(s(cons(tail(tail(odds)), y0)))) -> c16(MARK(s(cons(tail(tail(odds)), y0)))) MARK(head(s(cons(tail(tail(head(y0))), y1)))) -> c16(MARK(s(cons(tail(tail(head(y0))), y1)))) MARK(head(s(cons(tail(tail(tail(y0))), y1)))) -> c16(MARK(s(cons(tail(tail(tail(y0))), y1)))) MARK(head(s(cons(tail(tail(cons(y0, y1))), y2)))) -> c16(MARK(s(cons(tail(tail(cons(y0, y1))), y2)))) MARK(head(s(cons(tail(tail(s(y0))), y1)))) -> c16(MARK(s(cons(tail(tail(s(y0))), y1)))) MARK(head(s(cons(tail(tail(y0)), y1)))) -> c16(MARK(s(cons(tail(tail(y0)), y1)))) MARK(head(s(cons(tail(pairs), y0)))) -> c16(MARK(s(cons(tail(pairs), y0)))) MARK(head(s(cons(incr(incr(nats)), y0)))) -> c16(MARK(s(cons(incr(incr(nats)), y0)))) MARK(head(s(cons(incr(incr(pairs)), y0)))) -> c16(MARK(s(cons(incr(incr(pairs)), y0)))) MARK(head(s(cons(incr(odds), y0)))) -> c16(MARK(s(cons(incr(odds), y0)))) MARK(head(s(cons(head(incr(nats)), y0)))) -> c16(MARK(s(cons(head(incr(nats)), y0)))) MARK(head(s(cons(head(incr(pairs)), y0)))) -> c16(MARK(s(cons(head(incr(pairs)), y0)))) MARK(head(s(cons(head(odds), y0)))) -> c16(MARK(s(cons(head(odds), y0)))) MARK(head(s(cons(tail(incr(nats)), y0)))) -> c16(MARK(s(cons(tail(incr(nats)), y0)))) MARK(head(s(cons(tail(incr(pairs)), y0)))) -> c16(MARK(s(cons(tail(incr(pairs)), y0)))) MARK(head(s(cons(tail(odds), y0)))) -> c16(MARK(s(cons(tail(odds), y0)))) MARK(head(s(cons(incr(cons(nats, y0)), y1)))) -> c16(MARK(s(cons(incr(cons(nats, y0)), y1)))) MARK(head(s(cons(incr(cons(incr(y0), y1)), y2)))) -> c16(MARK(s(cons(incr(cons(incr(y0), y1)), y2)))) MARK(head(s(cons(incr(cons(pairs, y0)), y1)))) -> c16(MARK(s(cons(incr(cons(pairs, y0)), y1)))) MARK(head(s(cons(incr(cons(odds, y0)), y1)))) -> c16(MARK(s(cons(incr(cons(odds, y0)), y1)))) MARK(head(s(cons(incr(cons(head(y0), y1)), y2)))) -> c16(MARK(s(cons(incr(cons(head(y0), y1)), y2)))) MARK(head(s(cons(incr(cons(tail(y0), y1)), y2)))) -> c16(MARK(s(cons(incr(cons(tail(y0), y1)), y2)))) MARK(head(s(cons(incr(cons(cons(y0, y1), y2)), y3)))) -> c16(MARK(s(cons(incr(cons(cons(y0, y1), y2)), y3)))) MARK(head(s(cons(incr(cons(s(y0), y1)), y2)))) -> c16(MARK(s(cons(incr(cons(s(y0), y1)), y2)))) MARK(head(s(cons(incr(cons(y0, y1)), y2)))) -> c16(MARK(s(cons(incr(cons(y0, y1)), y2)))) MARK(head(s(cons(incr(cons(0, y0)), y1)))) -> c16(MARK(s(cons(incr(cons(0, y0)), y1)))) MARK(head(s(cons(incr(head(nats)), y0)))) -> c16(MARK(s(cons(incr(head(nats)), y0)))) MARK(head(s(cons(incr(head(pairs)), y0)))) -> c16(MARK(s(cons(incr(head(pairs)), y0)))) MARK(head(s(cons(head(head(nats)), y0)))) -> c16(MARK(s(cons(head(head(nats)), y0)))) MARK(head(s(cons(head(head(pairs)), y0)))) -> c16(MARK(s(cons(head(head(pairs)), y0)))) MARK(head(s(cons(tail(head(nats)), y0)))) -> c16(MARK(s(cons(tail(head(nats)), y0)))) MARK(head(s(cons(tail(head(pairs)), y0)))) -> c16(MARK(s(cons(tail(head(pairs)), y0)))) MARK(head(s(cons(head(cons(nats, y0)), y1)))) -> c16(MARK(s(cons(head(cons(nats, y0)), y1)))) MARK(head(s(cons(head(cons(incr(y0), y1)), y2)))) -> c16(MARK(s(cons(head(cons(incr(y0), y1)), y2)))) MARK(head(s(cons(head(cons(pairs, y0)), y1)))) -> c16(MARK(s(cons(head(cons(pairs, y0)), y1)))) MARK(head(s(cons(head(cons(odds, y0)), y1)))) -> c16(MARK(s(cons(head(cons(odds, y0)), y1)))) MARK(head(s(cons(head(cons(head(y0), y1)), y2)))) -> c16(MARK(s(cons(head(cons(head(y0), y1)), y2)))) MARK(head(s(cons(head(cons(tail(y0), y1)), y2)))) -> c16(MARK(s(cons(head(cons(tail(y0), y1)), y2)))) MARK(head(s(cons(head(cons(cons(y0, y1), y2)), y3)))) -> c16(MARK(s(cons(head(cons(cons(y0, y1), y2)), y3)))) MARK(head(s(cons(head(cons(s(y0), y1)), y2)))) -> c16(MARK(s(cons(head(cons(s(y0), y1)), y2)))) MARK(head(s(cons(head(cons(y0, y1)), y2)))) -> c16(MARK(s(cons(head(cons(y0, y1)), y2)))) MARK(head(s(cons(head(cons(0, y0)), y1)))) -> c16(MARK(s(cons(head(cons(0, y0)), y1)))) MARK(head(s(cons(incr(tail(nats)), y0)))) -> c16(MARK(s(cons(incr(tail(nats)), y0)))) MARK(head(s(cons(head(tail(nats)), y0)))) -> c16(MARK(s(cons(head(tail(nats)), y0)))) MARK(head(s(cons(tail(tail(nats)), y0)))) -> c16(MARK(s(cons(tail(tail(nats)), y0)))) MARK(head(s(incr(cons(nats, y0))))) -> c16(MARK(s(incr(cons(nats, y0))))) MARK(head(s(incr(cons(pairs, y0))))) -> c16(MARK(s(incr(cons(pairs, y0))))) MARK(head(s(incr(cons(odds, y0))))) -> c16(MARK(s(incr(cons(odds, y0))))) MARK(head(s(head(cons(nats, y0))))) -> c16(MARK(s(head(cons(nats, y0))))) MARK(head(s(head(cons(pairs, y0))))) -> c16(MARK(s(head(cons(pairs, y0))))) MARK(head(s(head(cons(odds, y0))))) -> c16(MARK(s(head(cons(odds, y0))))) MARK(tail(s(s(y0)))) -> c17(MARK(s(s(y0)))) MARK(tail(s(incr(s(y0))))) -> c17(MARK(s(incr(s(y0))))) MARK(tail(s(head(s(y0))))) -> c17(MARK(s(head(s(y0))))) MARK(tail(s(tail(cons(y0, y1))))) -> c17(MARK(s(tail(cons(y0, y1))))) MARK(tail(s(tail(s(y0))))) -> c17(MARK(s(tail(s(y0))))) MARK(tail(s(incr(incr(incr(y0)))))) -> c17(MARK(s(incr(incr(incr(y0)))))) MARK(tail(s(incr(incr(odds))))) -> c17(MARK(s(incr(incr(odds))))) MARK(tail(s(incr(incr(head(y0)))))) -> c17(MARK(s(incr(incr(head(y0)))))) MARK(tail(s(incr(incr(tail(y0)))))) -> c17(MARK(s(incr(incr(tail(y0)))))) MARK(tail(s(incr(incr(cons(y0, y1)))))) -> c17(MARK(s(incr(incr(cons(y0, y1)))))) MARK(tail(s(incr(incr(s(y0)))))) -> c17(MARK(s(incr(incr(s(y0)))))) MARK(tail(s(incr(incr(y0))))) -> c17(MARK(s(incr(incr(y0))))) MARK(tail(s(incr(head(incr(y0)))))) -> c17(MARK(s(incr(head(incr(y0)))))) MARK(tail(s(incr(head(odds))))) -> c17(MARK(s(incr(head(odds))))) MARK(tail(s(incr(head(head(y0)))))) -> c17(MARK(s(incr(head(head(y0)))))) MARK(tail(s(incr(head(tail(y0)))))) -> c17(MARK(s(incr(head(tail(y0)))))) MARK(tail(s(incr(head(cons(y0, y1)))))) -> c17(MARK(s(incr(head(cons(y0, y1)))))) MARK(tail(s(incr(head(s(y0)))))) -> c17(MARK(s(incr(head(s(y0)))))) MARK(tail(s(incr(head(y0))))) -> c17(MARK(s(incr(head(y0))))) MARK(tail(s(incr(tail(incr(y0)))))) -> c17(MARK(s(incr(tail(incr(y0)))))) MARK(tail(s(incr(tail(pairs))))) -> c17(MARK(s(incr(tail(pairs))))) MARK(tail(s(incr(tail(odds))))) -> c17(MARK(s(incr(tail(odds))))) MARK(tail(s(incr(tail(head(y0)))))) -> c17(MARK(s(incr(tail(head(y0)))))) MARK(tail(s(incr(tail(tail(y0)))))) -> c17(MARK(s(incr(tail(tail(y0)))))) MARK(tail(s(incr(tail(cons(y0, y1)))))) -> c17(MARK(s(incr(tail(cons(y0, y1)))))) MARK(tail(s(incr(tail(s(y0)))))) -> c17(MARK(s(incr(tail(s(y0)))))) MARK(tail(s(incr(tail(y0))))) -> c17(MARK(s(incr(tail(y0))))) MARK(tail(s(head(incr(incr(y0)))))) -> c17(MARK(s(head(incr(incr(y0)))))) MARK(tail(s(head(incr(odds))))) -> c17(MARK(s(head(incr(odds))))) MARK(tail(s(head(incr(head(y0)))))) -> c17(MARK(s(head(incr(head(y0)))))) MARK(tail(s(head(incr(tail(y0)))))) -> c17(MARK(s(head(incr(tail(y0)))))) MARK(tail(s(head(incr(cons(y0, y1)))))) -> c17(MARK(s(head(incr(cons(y0, y1)))))) MARK(tail(s(head(incr(s(y0)))))) -> c17(MARK(s(head(incr(s(y0)))))) MARK(tail(s(head(incr(y0))))) -> c17(MARK(s(head(incr(y0))))) MARK(tail(s(head(head(incr(y0)))))) -> c17(MARK(s(head(head(incr(y0)))))) MARK(tail(s(head(head(odds))))) -> c17(MARK(s(head(head(odds))))) MARK(tail(s(head(head(head(y0)))))) -> c17(MARK(s(head(head(head(y0)))))) MARK(tail(s(head(head(tail(y0)))))) -> c17(MARK(s(head(head(tail(y0)))))) MARK(tail(s(head(head(cons(y0, y1)))))) -> c17(MARK(s(head(head(cons(y0, y1)))))) MARK(tail(s(head(head(s(y0)))))) -> c17(MARK(s(head(head(s(y0)))))) MARK(tail(s(head(head(y0))))) -> c17(MARK(s(head(head(y0))))) MARK(tail(s(head(tail(incr(y0)))))) -> c17(MARK(s(head(tail(incr(y0)))))) MARK(tail(s(head(tail(pairs))))) -> c17(MARK(s(head(tail(pairs))))) MARK(tail(s(head(tail(odds))))) -> c17(MARK(s(head(tail(odds))))) MARK(tail(s(head(tail(head(y0)))))) -> c17(MARK(s(head(tail(head(y0)))))) MARK(tail(s(head(tail(tail(y0)))))) -> c17(MARK(s(head(tail(tail(y0)))))) MARK(tail(s(head(tail(cons(y0, y1)))))) -> c17(MARK(s(head(tail(cons(y0, y1)))))) MARK(tail(s(head(tail(s(y0)))))) -> c17(MARK(s(head(tail(s(y0)))))) MARK(tail(s(head(tail(y0))))) -> c17(MARK(s(head(tail(y0))))) MARK(tail(s(tail(incr(incr(y0)))))) -> c17(MARK(s(tail(incr(incr(y0)))))) MARK(tail(s(tail(incr(odds))))) -> c17(MARK(s(tail(incr(odds))))) MARK(tail(s(tail(incr(head(y0)))))) -> c17(MARK(s(tail(incr(head(y0)))))) MARK(tail(s(tail(incr(tail(y0)))))) -> c17(MARK(s(tail(incr(tail(y0)))))) MARK(tail(s(tail(incr(cons(y0, y1)))))) -> c17(MARK(s(tail(incr(cons(y0, y1)))))) MARK(tail(s(tail(incr(s(y0)))))) -> c17(MARK(s(tail(incr(s(y0)))))) MARK(tail(s(tail(incr(y0))))) -> c17(MARK(s(tail(incr(y0))))) MARK(tail(s(tail(head(incr(y0)))))) -> c17(MARK(s(tail(head(incr(y0)))))) MARK(tail(s(tail(head(odds))))) -> c17(MARK(s(tail(head(odds))))) MARK(tail(s(tail(head(head(y0)))))) -> c17(MARK(s(tail(head(head(y0)))))) MARK(tail(s(tail(head(tail(y0)))))) -> c17(MARK(s(tail(head(tail(y0)))))) MARK(tail(s(tail(head(cons(y0, y1)))))) -> c17(MARK(s(tail(head(cons(y0, y1)))))) MARK(tail(s(tail(head(s(y0)))))) -> c17(MARK(s(tail(head(s(y0)))))) MARK(tail(s(tail(head(y0))))) -> c17(MARK(s(tail(head(y0))))) MARK(tail(s(tail(tail(incr(y0)))))) -> c17(MARK(s(tail(tail(incr(y0)))))) MARK(tail(s(tail(tail(pairs))))) -> c17(MARK(s(tail(tail(pairs))))) MARK(tail(s(tail(tail(odds))))) -> c17(MARK(s(tail(tail(odds))))) MARK(tail(s(tail(tail(head(y0)))))) -> c17(MARK(s(tail(tail(head(y0)))))) MARK(tail(s(tail(tail(tail(y0)))))) -> c17(MARK(s(tail(tail(tail(y0)))))) MARK(tail(s(tail(tail(cons(y0, y1)))))) -> c17(MARK(s(tail(tail(cons(y0, y1)))))) MARK(tail(s(tail(tail(s(y0)))))) -> c17(MARK(s(tail(tail(s(y0)))))) MARK(tail(s(tail(tail(y0))))) -> c17(MARK(s(tail(tail(y0))))) MARK(tail(s(tail(pairs)))) -> c17(MARK(s(tail(pairs)))) MARK(tail(s(incr(incr(nats))))) -> c17(MARK(s(incr(incr(nats))))) MARK(tail(s(incr(incr(pairs))))) -> c17(MARK(s(incr(incr(pairs))))) MARK(tail(s(incr(odds)))) -> c17(MARK(s(incr(odds)))) MARK(tail(s(head(incr(nats))))) -> c17(MARK(s(head(incr(nats))))) MARK(tail(s(head(incr(pairs))))) -> c17(MARK(s(head(incr(pairs))))) MARK(tail(s(head(odds)))) -> c17(MARK(s(head(odds)))) MARK(tail(s(tail(incr(nats))))) -> c17(MARK(s(tail(incr(nats))))) MARK(tail(s(tail(incr(pairs))))) -> c17(MARK(s(tail(incr(pairs))))) MARK(tail(s(tail(odds)))) -> c17(MARK(s(tail(odds)))) MARK(tail(s(incr(cons(incr(y0), y1))))) -> c17(MARK(s(incr(cons(incr(y0), y1))))) MARK(tail(s(incr(cons(head(y0), y1))))) -> c17(MARK(s(incr(cons(head(y0), y1))))) MARK(tail(s(incr(cons(tail(y0), y1))))) -> c17(MARK(s(incr(cons(tail(y0), y1))))) MARK(tail(s(incr(cons(cons(y0, y1), y2))))) -> c17(MARK(s(incr(cons(cons(y0, y1), y2))))) MARK(tail(s(incr(cons(s(y0), y1))))) -> c17(MARK(s(incr(cons(s(y0), y1))))) MARK(tail(s(incr(cons(y0, y1))))) -> c17(MARK(s(incr(cons(y0, y1))))) MARK(tail(s(incr(head(nats))))) -> c17(MARK(s(incr(head(nats))))) MARK(tail(s(incr(head(pairs))))) -> c17(MARK(s(incr(head(pairs))))) MARK(tail(s(head(head(nats))))) -> c17(MARK(s(head(head(nats))))) MARK(tail(s(head(head(pairs))))) -> c17(MARK(s(head(head(pairs))))) MARK(tail(s(tail(head(nats))))) -> c17(MARK(s(tail(head(nats))))) MARK(tail(s(tail(head(pairs))))) -> c17(MARK(s(tail(head(pairs))))) MARK(tail(s(head(cons(incr(y0), y1))))) -> c17(MARK(s(head(cons(incr(y0), y1))))) MARK(tail(s(head(cons(head(y0), y1))))) -> c17(MARK(s(head(cons(head(y0), y1))))) MARK(tail(s(head(cons(tail(y0), y1))))) -> c17(MARK(s(head(cons(tail(y0), y1))))) MARK(tail(s(head(cons(cons(y0, y1), y2))))) -> c17(MARK(s(head(cons(cons(y0, y1), y2))))) MARK(tail(s(head(cons(s(y0), y1))))) -> c17(MARK(s(head(cons(s(y0), y1))))) MARK(tail(s(head(cons(y0, y1))))) -> c17(MARK(s(head(cons(y0, y1))))) MARK(tail(s(incr(tail(nats))))) -> c17(MARK(s(incr(tail(nats))))) MARK(tail(s(head(tail(nats))))) -> c17(MARK(s(head(tail(nats))))) MARK(tail(s(tail(tail(nats))))) -> c17(MARK(s(tail(tail(nats))))) MARK(tail(s(cons(cons(y0, y1), y2)))) -> c17(MARK(s(cons(cons(y0, y1), y2)))) MARK(tail(s(cons(s(y0), y1)))) -> c17(MARK(s(cons(s(y0), y1)))) MARK(tail(s(cons(incr(s(y0)), y1)))) -> c17(MARK(s(cons(incr(s(y0)), y1)))) MARK(tail(s(cons(head(s(y0)), y1)))) -> c17(MARK(s(cons(head(s(y0)), y1)))) MARK(tail(s(cons(tail(cons(y0, y1)), y2)))) -> c17(MARK(s(cons(tail(cons(y0, y1)), y2)))) MARK(tail(s(cons(tail(s(y0)), y1)))) -> c17(MARK(s(cons(tail(s(y0)), y1)))) MARK(tail(s(cons(incr(incr(incr(y0))), y1)))) -> c17(MARK(s(cons(incr(incr(incr(y0))), y1)))) MARK(tail(s(cons(incr(incr(odds)), y0)))) -> c17(MARK(s(cons(incr(incr(odds)), y0)))) MARK(tail(s(cons(incr(incr(head(y0))), y1)))) -> c17(MARK(s(cons(incr(incr(head(y0))), y1)))) MARK(tail(s(cons(incr(incr(tail(y0))), y1)))) -> c17(MARK(s(cons(incr(incr(tail(y0))), y1)))) MARK(tail(s(cons(incr(incr(cons(y0, y1))), y2)))) -> c17(MARK(s(cons(incr(incr(cons(y0, y1))), y2)))) MARK(tail(s(cons(incr(incr(s(y0))), y1)))) -> c17(MARK(s(cons(incr(incr(s(y0))), y1)))) MARK(tail(s(cons(incr(incr(y0)), y1)))) -> c17(MARK(s(cons(incr(incr(y0)), y1)))) MARK(tail(s(cons(incr(head(incr(y0))), y1)))) -> c17(MARK(s(cons(incr(head(incr(y0))), y1)))) MARK(tail(s(cons(incr(head(odds)), y0)))) -> c17(MARK(s(cons(incr(head(odds)), y0)))) MARK(tail(s(cons(incr(head(head(y0))), y1)))) -> c17(MARK(s(cons(incr(head(head(y0))), y1)))) MARK(tail(s(cons(incr(head(tail(y0))), y1)))) -> c17(MARK(s(cons(incr(head(tail(y0))), y1)))) MARK(tail(s(cons(incr(head(cons(y0, y1))), y2)))) -> c17(MARK(s(cons(incr(head(cons(y0, y1))), y2)))) MARK(tail(s(cons(incr(head(s(y0))), y1)))) -> c17(MARK(s(cons(incr(head(s(y0))), y1)))) MARK(tail(s(cons(incr(head(y0)), y1)))) -> c17(MARK(s(cons(incr(head(y0)), y1)))) MARK(tail(s(cons(incr(tail(incr(y0))), y1)))) -> c17(MARK(s(cons(incr(tail(incr(y0))), y1)))) MARK(tail(s(cons(incr(tail(pairs)), y0)))) -> c17(MARK(s(cons(incr(tail(pairs)), y0)))) MARK(tail(s(cons(incr(tail(odds)), y0)))) -> c17(MARK(s(cons(incr(tail(odds)), y0)))) MARK(tail(s(cons(incr(tail(head(y0))), y1)))) -> c17(MARK(s(cons(incr(tail(head(y0))), y1)))) MARK(tail(s(cons(incr(tail(tail(y0))), y1)))) -> c17(MARK(s(cons(incr(tail(tail(y0))), y1)))) MARK(tail(s(cons(incr(tail(cons(y0, y1))), y2)))) -> c17(MARK(s(cons(incr(tail(cons(y0, y1))), y2)))) MARK(tail(s(cons(incr(tail(s(y0))), y1)))) -> c17(MARK(s(cons(incr(tail(s(y0))), y1)))) MARK(tail(s(cons(incr(tail(y0)), y1)))) -> c17(MARK(s(cons(incr(tail(y0)), y1)))) MARK(tail(s(cons(head(incr(incr(y0))), y1)))) -> c17(MARK(s(cons(head(incr(incr(y0))), y1)))) MARK(tail(s(cons(head(incr(odds)), y0)))) -> c17(MARK(s(cons(head(incr(odds)), y0)))) MARK(tail(s(cons(head(incr(head(y0))), y1)))) -> c17(MARK(s(cons(head(incr(head(y0))), y1)))) MARK(tail(s(cons(head(incr(tail(y0))), y1)))) -> c17(MARK(s(cons(head(incr(tail(y0))), y1)))) MARK(tail(s(cons(head(incr(cons(y0, y1))), y2)))) -> c17(MARK(s(cons(head(incr(cons(y0, y1))), y2)))) MARK(tail(s(cons(head(incr(s(y0))), y1)))) -> c17(MARK(s(cons(head(incr(s(y0))), y1)))) MARK(tail(s(cons(head(incr(y0)), y1)))) -> c17(MARK(s(cons(head(incr(y0)), y1)))) MARK(tail(s(cons(head(head(incr(y0))), y1)))) -> c17(MARK(s(cons(head(head(incr(y0))), y1)))) MARK(tail(s(cons(head(head(odds)), y0)))) -> c17(MARK(s(cons(head(head(odds)), y0)))) MARK(tail(s(cons(head(head(head(y0))), y1)))) -> c17(MARK(s(cons(head(head(head(y0))), y1)))) MARK(tail(s(cons(head(head(tail(y0))), y1)))) -> c17(MARK(s(cons(head(head(tail(y0))), y1)))) MARK(tail(s(cons(head(head(cons(y0, y1))), y2)))) -> c17(MARK(s(cons(head(head(cons(y0, y1))), y2)))) MARK(tail(s(cons(head(head(s(y0))), y1)))) -> c17(MARK(s(cons(head(head(s(y0))), y1)))) MARK(tail(s(cons(head(head(y0)), y1)))) -> c17(MARK(s(cons(head(head(y0)), y1)))) MARK(tail(s(cons(head(tail(incr(y0))), y1)))) -> c17(MARK(s(cons(head(tail(incr(y0))), y1)))) MARK(tail(s(cons(head(tail(pairs)), y0)))) -> c17(MARK(s(cons(head(tail(pairs)), y0)))) MARK(tail(s(cons(head(tail(odds)), y0)))) -> c17(MARK(s(cons(head(tail(odds)), y0)))) MARK(tail(s(cons(head(tail(head(y0))), y1)))) -> c17(MARK(s(cons(head(tail(head(y0))), y1)))) MARK(tail(s(cons(head(tail(tail(y0))), y1)))) -> c17(MARK(s(cons(head(tail(tail(y0))), y1)))) MARK(tail(s(cons(head(tail(cons(y0, y1))), y2)))) -> c17(MARK(s(cons(head(tail(cons(y0, y1))), y2)))) MARK(tail(s(cons(head(tail(s(y0))), y1)))) -> c17(MARK(s(cons(head(tail(s(y0))), y1)))) MARK(tail(s(cons(head(tail(y0)), y1)))) -> c17(MARK(s(cons(head(tail(y0)), y1)))) MARK(tail(s(cons(tail(incr(incr(y0))), y1)))) -> c17(MARK(s(cons(tail(incr(incr(y0))), y1)))) MARK(tail(s(cons(tail(incr(odds)), y0)))) -> c17(MARK(s(cons(tail(incr(odds)), y0)))) MARK(tail(s(cons(tail(incr(head(y0))), y1)))) -> c17(MARK(s(cons(tail(incr(head(y0))), y1)))) MARK(tail(s(cons(tail(incr(tail(y0))), y1)))) -> c17(MARK(s(cons(tail(incr(tail(y0))), y1)))) MARK(tail(s(cons(tail(incr(cons(y0, y1))), y2)))) -> c17(MARK(s(cons(tail(incr(cons(y0, y1))), y2)))) MARK(tail(s(cons(tail(incr(s(y0))), y1)))) -> c17(MARK(s(cons(tail(incr(s(y0))), y1)))) MARK(tail(s(cons(tail(incr(y0)), y1)))) -> c17(MARK(s(cons(tail(incr(y0)), y1)))) MARK(tail(s(cons(tail(head(incr(y0))), y1)))) -> c17(MARK(s(cons(tail(head(incr(y0))), y1)))) MARK(tail(s(cons(tail(head(odds)), y0)))) -> c17(MARK(s(cons(tail(head(odds)), y0)))) MARK(tail(s(cons(tail(head(head(y0))), y1)))) -> c17(MARK(s(cons(tail(head(head(y0))), y1)))) MARK(tail(s(cons(tail(head(tail(y0))), y1)))) -> c17(MARK(s(cons(tail(head(tail(y0))), y1)))) MARK(tail(s(cons(tail(head(cons(y0, y1))), y2)))) -> c17(MARK(s(cons(tail(head(cons(y0, y1))), y2)))) MARK(tail(s(cons(tail(head(s(y0))), y1)))) -> c17(MARK(s(cons(tail(head(s(y0))), y1)))) MARK(tail(s(cons(tail(head(y0)), y1)))) -> c17(MARK(s(cons(tail(head(y0)), y1)))) MARK(tail(s(cons(tail(tail(incr(y0))), y1)))) -> c17(MARK(s(cons(tail(tail(incr(y0))), y1)))) MARK(tail(s(cons(tail(tail(pairs)), y0)))) -> c17(MARK(s(cons(tail(tail(pairs)), y0)))) MARK(tail(s(cons(tail(tail(odds)), y0)))) -> c17(MARK(s(cons(tail(tail(odds)), y0)))) MARK(tail(s(cons(tail(tail(head(y0))), y1)))) -> c17(MARK(s(cons(tail(tail(head(y0))), y1)))) MARK(tail(s(cons(tail(tail(tail(y0))), y1)))) -> c17(MARK(s(cons(tail(tail(tail(y0))), y1)))) MARK(tail(s(cons(tail(tail(cons(y0, y1))), y2)))) -> c17(MARK(s(cons(tail(tail(cons(y0, y1))), y2)))) MARK(tail(s(cons(tail(tail(s(y0))), y1)))) -> c17(MARK(s(cons(tail(tail(s(y0))), y1)))) MARK(tail(s(cons(tail(tail(y0)), y1)))) -> c17(MARK(s(cons(tail(tail(y0)), y1)))) MARK(tail(s(cons(tail(pairs), y0)))) -> c17(MARK(s(cons(tail(pairs), y0)))) MARK(tail(s(cons(incr(incr(nats)), y0)))) -> c17(MARK(s(cons(incr(incr(nats)), y0)))) MARK(tail(s(cons(incr(incr(pairs)), y0)))) -> c17(MARK(s(cons(incr(incr(pairs)), y0)))) MARK(tail(s(cons(incr(odds), y0)))) -> c17(MARK(s(cons(incr(odds), y0)))) MARK(tail(s(cons(head(incr(nats)), y0)))) -> c17(MARK(s(cons(head(incr(nats)), y0)))) MARK(tail(s(cons(head(incr(pairs)), y0)))) -> c17(MARK(s(cons(head(incr(pairs)), y0)))) MARK(tail(s(cons(head(odds), y0)))) -> c17(MARK(s(cons(head(odds), y0)))) MARK(tail(s(cons(tail(incr(nats)), y0)))) -> c17(MARK(s(cons(tail(incr(nats)), y0)))) MARK(tail(s(cons(tail(incr(pairs)), y0)))) -> c17(MARK(s(cons(tail(incr(pairs)), y0)))) MARK(tail(s(cons(tail(odds), y0)))) -> c17(MARK(s(cons(tail(odds), y0)))) MARK(tail(s(cons(incr(cons(nats, y0)), y1)))) -> c17(MARK(s(cons(incr(cons(nats, y0)), y1)))) MARK(tail(s(cons(incr(cons(incr(y0), y1)), y2)))) -> c17(MARK(s(cons(incr(cons(incr(y0), y1)), y2)))) MARK(tail(s(cons(incr(cons(pairs, y0)), y1)))) -> c17(MARK(s(cons(incr(cons(pairs, y0)), y1)))) MARK(tail(s(cons(incr(cons(odds, y0)), y1)))) -> c17(MARK(s(cons(incr(cons(odds, y0)), y1)))) MARK(tail(s(cons(incr(cons(head(y0), y1)), y2)))) -> c17(MARK(s(cons(incr(cons(head(y0), y1)), y2)))) MARK(tail(s(cons(incr(cons(tail(y0), y1)), y2)))) -> c17(MARK(s(cons(incr(cons(tail(y0), y1)), y2)))) MARK(tail(s(cons(incr(cons(cons(y0, y1), y2)), y3)))) -> c17(MARK(s(cons(incr(cons(cons(y0, y1), y2)), y3)))) MARK(tail(s(cons(incr(cons(s(y0), y1)), y2)))) -> c17(MARK(s(cons(incr(cons(s(y0), y1)), y2)))) MARK(tail(s(cons(incr(cons(y0, y1)), y2)))) -> c17(MARK(s(cons(incr(cons(y0, y1)), y2)))) MARK(tail(s(cons(incr(cons(0, y0)), y1)))) -> c17(MARK(s(cons(incr(cons(0, y0)), y1)))) MARK(tail(s(cons(incr(head(nats)), y0)))) -> c17(MARK(s(cons(incr(head(nats)), y0)))) MARK(tail(s(cons(incr(head(pairs)), y0)))) -> c17(MARK(s(cons(incr(head(pairs)), y0)))) MARK(tail(s(cons(head(head(nats)), y0)))) -> c17(MARK(s(cons(head(head(nats)), y0)))) MARK(tail(s(cons(head(head(pairs)), y0)))) -> c17(MARK(s(cons(head(head(pairs)), y0)))) MARK(tail(s(cons(tail(head(nats)), y0)))) -> c17(MARK(s(cons(tail(head(nats)), y0)))) MARK(tail(s(cons(tail(head(pairs)), y0)))) -> c17(MARK(s(cons(tail(head(pairs)), y0)))) MARK(tail(s(cons(head(cons(nats, y0)), y1)))) -> c17(MARK(s(cons(head(cons(nats, y0)), y1)))) MARK(tail(s(cons(head(cons(incr(y0), y1)), y2)))) -> c17(MARK(s(cons(head(cons(incr(y0), y1)), y2)))) MARK(tail(s(cons(head(cons(pairs, y0)), y1)))) -> c17(MARK(s(cons(head(cons(pairs, y0)), y1)))) MARK(tail(s(cons(head(cons(odds, y0)), y1)))) -> c17(MARK(s(cons(head(cons(odds, y0)), y1)))) MARK(tail(s(cons(head(cons(head(y0), y1)), y2)))) -> c17(MARK(s(cons(head(cons(head(y0), y1)), y2)))) MARK(tail(s(cons(head(cons(tail(y0), y1)), y2)))) -> c17(MARK(s(cons(head(cons(tail(y0), y1)), y2)))) MARK(tail(s(cons(head(cons(cons(y0, y1), y2)), y3)))) -> c17(MARK(s(cons(head(cons(cons(y0, y1), y2)), y3)))) MARK(tail(s(cons(head(cons(s(y0), y1)), y2)))) -> c17(MARK(s(cons(head(cons(s(y0), y1)), y2)))) MARK(tail(s(cons(head(cons(y0, y1)), y2)))) -> c17(MARK(s(cons(head(cons(y0, y1)), y2)))) MARK(tail(s(cons(head(cons(0, y0)), y1)))) -> c17(MARK(s(cons(head(cons(0, y0)), y1)))) MARK(tail(s(cons(incr(tail(nats)), y0)))) -> c17(MARK(s(cons(incr(tail(nats)), y0)))) MARK(tail(s(cons(head(tail(nats)), y0)))) -> c17(MARK(s(cons(head(tail(nats)), y0)))) MARK(tail(s(cons(tail(tail(nats)), y0)))) -> c17(MARK(s(cons(tail(tail(nats)), y0)))) MARK(tail(s(incr(cons(nats, y0))))) -> c17(MARK(s(incr(cons(nats, y0))))) MARK(tail(s(incr(cons(pairs, y0))))) -> c17(MARK(s(incr(cons(pairs, y0))))) MARK(tail(s(incr(cons(odds, y0))))) -> c17(MARK(s(incr(cons(odds, y0))))) MARK(tail(s(head(cons(nats, y0))))) -> c17(MARK(s(head(cons(nats, y0))))) MARK(tail(s(head(cons(pairs, y0))))) -> c17(MARK(s(head(cons(pairs, y0))))) MARK(tail(s(head(cons(odds, y0))))) -> c17(MARK(s(head(cons(odds, y0))))) MARK(incr(incr(incr(incr(y0))))) -> c13(MARK(incr(incr(incr(y0))))) MARK(incr(incr(incr(odds)))) -> c13(MARK(incr(incr(odds)))) MARK(incr(incr(incr(head(y0))))) -> c13(MARK(incr(incr(head(y0))))) MARK(incr(incr(incr(tail(y0))))) -> c13(MARK(incr(incr(tail(y0))))) MARK(incr(incr(incr(cons(y0, y1))))) -> c13(MARK(incr(incr(cons(y0, y1))))) MARK(incr(incr(incr(s(y0))))) -> c13(MARK(incr(incr(s(y0))))) MARK(incr(incr(incr(y0)))) -> c13(MARK(incr(incr(y0)))) MARK(incr(incr(head(incr(y0))))) -> c13(MARK(incr(head(incr(y0))))) MARK(incr(incr(head(odds)))) -> c13(MARK(incr(head(odds)))) MARK(incr(incr(head(head(y0))))) -> c13(MARK(incr(head(head(y0))))) MARK(incr(incr(head(tail(y0))))) -> c13(MARK(incr(head(tail(y0))))) MARK(incr(incr(head(cons(y0, y1))))) -> c13(MARK(incr(head(cons(y0, y1))))) MARK(incr(incr(head(s(y0))))) -> c13(MARK(incr(head(s(y0))))) MARK(incr(incr(head(y0)))) -> c13(MARK(incr(head(y0)))) MARK(incr(incr(tail(incr(y0))))) -> c13(MARK(incr(tail(incr(y0))))) MARK(incr(incr(tail(pairs)))) -> c13(MARK(incr(tail(pairs)))) MARK(incr(incr(tail(odds)))) -> c13(MARK(incr(tail(odds)))) MARK(incr(incr(tail(head(y0))))) -> c13(MARK(incr(tail(head(y0))))) MARK(incr(incr(tail(tail(y0))))) -> c13(MARK(incr(tail(tail(y0))))) MARK(incr(incr(tail(cons(y0, y1))))) -> c13(MARK(incr(tail(cons(y0, y1))))) MARK(incr(incr(tail(s(y0))))) -> c13(MARK(incr(tail(s(y0))))) MARK(incr(incr(tail(y0)))) -> c13(MARK(incr(tail(y0)))) MARK(incr(incr(incr(nats)))) -> c13(MARK(incr(incr(nats)))) MARK(incr(incr(incr(pairs)))) -> c13(MARK(incr(incr(pairs)))) MARK(incr(incr(odds))) -> c13(MARK(incr(odds))) MARK(incr(incr(cons(incr(y0), y1)))) -> c13(MARK(incr(cons(incr(y0), y1)))) MARK(incr(incr(cons(head(y0), y1)))) -> c13(MARK(incr(cons(head(y0), y1)))) MARK(incr(incr(cons(tail(y0), y1)))) -> c13(MARK(incr(cons(tail(y0), y1)))) MARK(incr(incr(cons(cons(y0, y1), y2)))) -> c13(MARK(incr(cons(cons(y0, y1), y2)))) MARK(incr(incr(cons(s(y0), y1)))) -> c13(MARK(incr(cons(s(y0), y1)))) MARK(incr(incr(cons(y0, y1)))) -> c13(MARK(incr(cons(y0, y1)))) MARK(incr(incr(head(nats)))) -> c13(MARK(incr(head(nats)))) MARK(incr(incr(head(pairs)))) -> c13(MARK(incr(head(pairs)))) MARK(incr(incr(tail(nats)))) -> c13(MARK(incr(tail(nats)))) MARK(incr(incr(cons(nats, y0)))) -> c13(MARK(incr(cons(nats, y0)))) MARK(incr(incr(cons(pairs, y0)))) -> c13(MARK(incr(cons(pairs, y0)))) MARK(incr(incr(cons(odds, y0)))) -> c13(MARK(incr(cons(odds, y0)))) MARK(incr(incr(s(s(y0))))) -> c13(MARK(incr(s(s(y0))))) MARK(incr(incr(s(incr(s(y0)))))) -> c13(MARK(incr(s(incr(s(y0)))))) MARK(incr(incr(s(head(s(y0)))))) -> c13(MARK(incr(s(head(s(y0)))))) MARK(incr(incr(s(tail(cons(y0, y1)))))) -> c13(MARK(incr(s(tail(cons(y0, y1)))))) MARK(incr(incr(s(tail(s(y0)))))) -> c13(MARK(incr(s(tail(s(y0)))))) MARK(incr(incr(s(incr(incr(incr(y0))))))) -> c13(MARK(incr(s(incr(incr(incr(y0))))))) MARK(incr(incr(s(incr(incr(odds)))))) -> c13(MARK(incr(s(incr(incr(odds)))))) MARK(incr(incr(s(incr(incr(head(y0))))))) -> c13(MARK(incr(s(incr(incr(head(y0))))))) MARK(incr(incr(s(incr(incr(tail(y0))))))) -> c13(MARK(incr(s(incr(incr(tail(y0))))))) MARK(incr(incr(s(incr(incr(cons(y0, y1))))))) -> c13(MARK(incr(s(incr(incr(cons(y0, y1))))))) MARK(incr(incr(s(incr(incr(s(y0))))))) -> c13(MARK(incr(s(incr(incr(s(y0))))))) MARK(incr(incr(s(incr(incr(y0)))))) -> c13(MARK(incr(s(incr(incr(y0)))))) MARK(incr(incr(s(incr(head(incr(y0))))))) -> c13(MARK(incr(s(incr(head(incr(y0))))))) MARK(incr(incr(s(incr(head(odds)))))) -> c13(MARK(incr(s(incr(head(odds)))))) MARK(incr(incr(s(incr(head(head(y0))))))) -> c13(MARK(incr(s(incr(head(head(y0))))))) MARK(incr(incr(s(incr(head(tail(y0))))))) -> c13(MARK(incr(s(incr(head(tail(y0))))))) MARK(incr(incr(s(incr(head(cons(y0, y1))))))) -> c13(MARK(incr(s(incr(head(cons(y0, y1))))))) MARK(incr(incr(s(incr(head(s(y0))))))) -> c13(MARK(incr(s(incr(head(s(y0))))))) MARK(incr(incr(s(incr(head(y0)))))) -> c13(MARK(incr(s(incr(head(y0)))))) MARK(incr(incr(s(incr(tail(incr(y0))))))) -> c13(MARK(incr(s(incr(tail(incr(y0))))))) MARK(incr(incr(s(incr(tail(pairs)))))) -> c13(MARK(incr(s(incr(tail(pairs)))))) MARK(incr(incr(s(incr(tail(odds)))))) -> c13(MARK(incr(s(incr(tail(odds)))))) MARK(incr(incr(s(incr(tail(head(y0))))))) -> c13(MARK(incr(s(incr(tail(head(y0))))))) MARK(incr(incr(s(incr(tail(tail(y0))))))) -> c13(MARK(incr(s(incr(tail(tail(y0))))))) MARK(incr(incr(s(incr(tail(cons(y0, y1))))))) -> c13(MARK(incr(s(incr(tail(cons(y0, y1))))))) MARK(incr(incr(s(incr(tail(s(y0))))))) -> c13(MARK(incr(s(incr(tail(s(y0))))))) MARK(incr(incr(s(incr(tail(y0)))))) -> c13(MARK(incr(s(incr(tail(y0)))))) MARK(incr(incr(s(head(incr(incr(y0))))))) -> c13(MARK(incr(s(head(incr(incr(y0))))))) MARK(incr(incr(s(head(incr(odds)))))) -> c13(MARK(incr(s(head(incr(odds)))))) MARK(incr(incr(s(head(incr(head(y0))))))) -> c13(MARK(incr(s(head(incr(head(y0))))))) MARK(incr(incr(s(head(incr(tail(y0))))))) -> c13(MARK(incr(s(head(incr(tail(y0))))))) MARK(incr(incr(s(head(incr(cons(y0, y1))))))) -> c13(MARK(incr(s(head(incr(cons(y0, y1))))))) MARK(incr(incr(s(head(incr(s(y0))))))) -> c13(MARK(incr(s(head(incr(s(y0))))))) MARK(incr(incr(s(head(incr(y0)))))) -> c13(MARK(incr(s(head(incr(y0)))))) MARK(incr(incr(s(head(head(incr(y0))))))) -> c13(MARK(incr(s(head(head(incr(y0))))))) MARK(incr(incr(s(head(head(odds)))))) -> c13(MARK(incr(s(head(head(odds)))))) MARK(incr(incr(s(head(head(head(y0))))))) -> c13(MARK(incr(s(head(head(head(y0))))))) MARK(incr(incr(s(head(head(tail(y0))))))) -> c13(MARK(incr(s(head(head(tail(y0))))))) MARK(incr(incr(s(head(head(cons(y0, y1))))))) -> c13(MARK(incr(s(head(head(cons(y0, y1))))))) MARK(incr(incr(s(head(head(s(y0))))))) -> c13(MARK(incr(s(head(head(s(y0))))))) MARK(incr(incr(s(head(head(y0)))))) -> c13(MARK(incr(s(head(head(y0)))))) MARK(incr(incr(s(head(tail(incr(y0))))))) -> c13(MARK(incr(s(head(tail(incr(y0))))))) MARK(incr(incr(s(head(tail(pairs)))))) -> c13(MARK(incr(s(head(tail(pairs)))))) MARK(incr(incr(s(head(tail(odds)))))) -> c13(MARK(incr(s(head(tail(odds)))))) MARK(incr(incr(s(head(tail(head(y0))))))) -> c13(MARK(incr(s(head(tail(head(y0))))))) MARK(incr(incr(s(head(tail(tail(y0))))))) -> c13(MARK(incr(s(head(tail(tail(y0))))))) MARK(incr(incr(s(head(tail(cons(y0, y1))))))) -> c13(MARK(incr(s(head(tail(cons(y0, y1))))))) MARK(incr(incr(s(head(tail(s(y0))))))) -> c13(MARK(incr(s(head(tail(s(y0))))))) MARK(incr(incr(s(head(tail(y0)))))) -> c13(MARK(incr(s(head(tail(y0)))))) MARK(incr(incr(s(tail(incr(incr(y0))))))) -> c13(MARK(incr(s(tail(incr(incr(y0))))))) MARK(incr(incr(s(tail(incr(odds)))))) -> c13(MARK(incr(s(tail(incr(odds)))))) MARK(incr(incr(s(tail(incr(head(y0))))))) -> c13(MARK(incr(s(tail(incr(head(y0))))))) MARK(incr(incr(s(tail(incr(tail(y0))))))) -> c13(MARK(incr(s(tail(incr(tail(y0))))))) MARK(incr(incr(s(tail(incr(cons(y0, y1))))))) -> c13(MARK(incr(s(tail(incr(cons(y0, y1))))))) MARK(incr(incr(s(tail(incr(s(y0))))))) -> c13(MARK(incr(s(tail(incr(s(y0))))))) MARK(incr(incr(s(tail(incr(y0)))))) -> c13(MARK(incr(s(tail(incr(y0)))))) MARK(incr(incr(s(tail(head(incr(y0))))))) -> c13(MARK(incr(s(tail(head(incr(y0))))))) MARK(incr(incr(s(tail(head(odds)))))) -> c13(MARK(incr(s(tail(head(odds)))))) MARK(incr(incr(s(tail(head(head(y0))))))) -> c13(MARK(incr(s(tail(head(head(y0))))))) MARK(incr(incr(s(tail(head(tail(y0))))))) -> c13(MARK(incr(s(tail(head(tail(y0))))))) MARK(incr(incr(s(tail(head(cons(y0, y1))))))) -> c13(MARK(incr(s(tail(head(cons(y0, y1))))))) MARK(incr(incr(s(tail(head(s(y0))))))) -> c13(MARK(incr(s(tail(head(s(y0))))))) MARK(incr(incr(s(tail(head(y0)))))) -> c13(MARK(incr(s(tail(head(y0)))))) MARK(incr(incr(s(tail(tail(incr(y0))))))) -> c13(MARK(incr(s(tail(tail(incr(y0))))))) MARK(incr(incr(s(tail(tail(pairs)))))) -> c13(MARK(incr(s(tail(tail(pairs)))))) MARK(incr(incr(s(tail(tail(odds)))))) -> c13(MARK(incr(s(tail(tail(odds)))))) MARK(incr(incr(s(tail(tail(head(y0))))))) -> c13(MARK(incr(s(tail(tail(head(y0))))))) MARK(incr(incr(s(tail(tail(tail(y0))))))) -> c13(MARK(incr(s(tail(tail(tail(y0))))))) MARK(incr(incr(s(tail(tail(cons(y0, y1))))))) -> c13(MARK(incr(s(tail(tail(cons(y0, y1))))))) MARK(incr(incr(s(tail(tail(s(y0))))))) -> c13(MARK(incr(s(tail(tail(s(y0))))))) MARK(incr(incr(s(tail(tail(y0)))))) -> c13(MARK(incr(s(tail(tail(y0)))))) MARK(incr(incr(s(tail(pairs))))) -> c13(MARK(incr(s(tail(pairs))))) MARK(incr(incr(s(incr(incr(nats)))))) -> c13(MARK(incr(s(incr(incr(nats)))))) MARK(incr(incr(s(incr(incr(pairs)))))) -> c13(MARK(incr(s(incr(incr(pairs)))))) MARK(incr(incr(s(incr(odds))))) -> c13(MARK(incr(s(incr(odds))))) MARK(incr(incr(s(head(incr(nats)))))) -> c13(MARK(incr(s(head(incr(nats)))))) MARK(incr(incr(s(head(incr(pairs)))))) -> c13(MARK(incr(s(head(incr(pairs)))))) MARK(incr(incr(s(head(odds))))) -> c13(MARK(incr(s(head(odds))))) MARK(incr(incr(s(tail(incr(nats)))))) -> c13(MARK(incr(s(tail(incr(nats)))))) MARK(incr(incr(s(tail(incr(pairs)))))) -> c13(MARK(incr(s(tail(incr(pairs)))))) MARK(incr(incr(s(tail(odds))))) -> c13(MARK(incr(s(tail(odds))))) MARK(incr(incr(s(incr(cons(incr(y0), y1)))))) -> c13(MARK(incr(s(incr(cons(incr(y0), y1)))))) MARK(incr(incr(s(incr(cons(head(y0), y1)))))) -> c13(MARK(incr(s(incr(cons(head(y0), y1)))))) MARK(incr(incr(s(incr(cons(tail(y0), y1)))))) -> c13(MARK(incr(s(incr(cons(tail(y0), y1)))))) MARK(incr(incr(s(incr(cons(cons(y0, y1), y2)))))) -> c13(MARK(incr(s(incr(cons(cons(y0, y1), y2)))))) MARK(incr(incr(s(incr(cons(s(y0), y1)))))) -> c13(MARK(incr(s(incr(cons(s(y0), y1)))))) MARK(incr(incr(s(incr(cons(y0, y1)))))) -> c13(MARK(incr(s(incr(cons(y0, y1)))))) MARK(incr(incr(s(incr(head(nats)))))) -> c13(MARK(incr(s(incr(head(nats)))))) MARK(incr(incr(s(incr(head(pairs)))))) -> c13(MARK(incr(s(incr(head(pairs)))))) MARK(incr(incr(s(head(head(nats)))))) -> c13(MARK(incr(s(head(head(nats)))))) MARK(incr(incr(s(head(head(pairs)))))) -> c13(MARK(incr(s(head(head(pairs)))))) MARK(incr(incr(s(tail(head(nats)))))) -> c13(MARK(incr(s(tail(head(nats)))))) MARK(incr(incr(s(tail(head(pairs)))))) -> c13(MARK(incr(s(tail(head(pairs)))))) MARK(incr(incr(s(head(cons(incr(y0), y1)))))) -> c13(MARK(incr(s(head(cons(incr(y0), y1)))))) MARK(incr(incr(s(head(cons(head(y0), y1)))))) -> c13(MARK(incr(s(head(cons(head(y0), y1)))))) MARK(incr(incr(s(head(cons(tail(y0), y1)))))) -> c13(MARK(incr(s(head(cons(tail(y0), y1)))))) MARK(incr(incr(s(head(cons(cons(y0, y1), y2)))))) -> c13(MARK(incr(s(head(cons(cons(y0, y1), y2)))))) MARK(incr(incr(s(head(cons(s(y0), y1)))))) -> c13(MARK(incr(s(head(cons(s(y0), y1)))))) MARK(incr(incr(s(head(cons(y0, y1)))))) -> c13(MARK(incr(s(head(cons(y0, y1)))))) MARK(incr(incr(s(incr(tail(nats)))))) -> c13(MARK(incr(s(incr(tail(nats)))))) MARK(incr(incr(s(head(tail(nats)))))) -> c13(MARK(incr(s(head(tail(nats)))))) MARK(incr(incr(s(tail(tail(nats)))))) -> c13(MARK(incr(s(tail(tail(nats)))))) MARK(incr(incr(s(cons(cons(y0, y1), y2))))) -> c13(MARK(incr(s(cons(cons(y0, y1), y2))))) MARK(incr(incr(s(cons(s(y0), y1))))) -> c13(MARK(incr(s(cons(s(y0), y1))))) MARK(incr(incr(s(cons(incr(s(y0)), y1))))) -> c13(MARK(incr(s(cons(incr(s(y0)), y1))))) MARK(incr(incr(s(cons(head(s(y0)), y1))))) -> c13(MARK(incr(s(cons(head(s(y0)), y1))))) MARK(incr(incr(s(cons(tail(cons(y0, y1)), y2))))) -> c13(MARK(incr(s(cons(tail(cons(y0, y1)), y2))))) MARK(incr(incr(s(cons(tail(s(y0)), y1))))) -> c13(MARK(incr(s(cons(tail(s(y0)), y1))))) MARK(incr(incr(s(cons(incr(incr(incr(y0))), y1))))) -> c13(MARK(incr(s(cons(incr(incr(incr(y0))), y1))))) MARK(incr(incr(s(cons(incr(incr(odds)), y0))))) -> c13(MARK(incr(s(cons(incr(incr(odds)), y0))))) MARK(incr(incr(s(cons(incr(incr(head(y0))), y1))))) -> c13(MARK(incr(s(cons(incr(incr(head(y0))), y1))))) MARK(incr(incr(s(cons(incr(incr(tail(y0))), y1))))) -> c13(MARK(incr(s(cons(incr(incr(tail(y0))), y1))))) MARK(incr(incr(s(cons(incr(incr(cons(y0, y1))), y2))))) -> c13(MARK(incr(s(cons(incr(incr(cons(y0, y1))), y2))))) MARK(incr(incr(s(cons(incr(incr(s(y0))), y1))))) -> c13(MARK(incr(s(cons(incr(incr(s(y0))), y1))))) MARK(incr(incr(s(cons(incr(incr(y0)), y1))))) -> c13(MARK(incr(s(cons(incr(incr(y0)), y1))))) MARK(incr(incr(s(cons(incr(head(incr(y0))), y1))))) -> c13(MARK(incr(s(cons(incr(head(incr(y0))), y1))))) MARK(incr(incr(s(cons(incr(head(odds)), y0))))) -> c13(MARK(incr(s(cons(incr(head(odds)), y0))))) MARK(incr(incr(s(cons(incr(head(head(y0))), y1))))) -> c13(MARK(incr(s(cons(incr(head(head(y0))), y1))))) MARK(incr(incr(s(cons(incr(head(tail(y0))), y1))))) -> c13(MARK(incr(s(cons(incr(head(tail(y0))), y1))))) MARK(incr(incr(s(cons(incr(head(cons(y0, y1))), y2))))) -> c13(MARK(incr(s(cons(incr(head(cons(y0, y1))), y2))))) MARK(incr(incr(s(cons(incr(head(s(y0))), y1))))) -> c13(MARK(incr(s(cons(incr(head(s(y0))), y1))))) MARK(incr(incr(s(cons(incr(head(y0)), y1))))) -> c13(MARK(incr(s(cons(incr(head(y0)), y1))))) MARK(incr(incr(s(cons(incr(tail(incr(y0))), y1))))) -> c13(MARK(incr(s(cons(incr(tail(incr(y0))), y1))))) MARK(incr(incr(s(cons(incr(tail(pairs)), y0))))) -> c13(MARK(incr(s(cons(incr(tail(pairs)), y0))))) MARK(incr(incr(s(cons(incr(tail(odds)), y0))))) -> c13(MARK(incr(s(cons(incr(tail(odds)), y0))))) MARK(incr(incr(s(cons(incr(tail(head(y0))), y1))))) -> c13(MARK(incr(s(cons(incr(tail(head(y0))), y1))))) MARK(incr(incr(s(cons(incr(tail(tail(y0))), y1))))) -> c13(MARK(incr(s(cons(incr(tail(tail(y0))), y1))))) MARK(incr(incr(s(cons(incr(tail(cons(y0, y1))), y2))))) -> c13(MARK(incr(s(cons(incr(tail(cons(y0, y1))), y2))))) MARK(incr(incr(s(cons(incr(tail(s(y0))), y1))))) -> c13(MARK(incr(s(cons(incr(tail(s(y0))), y1))))) MARK(incr(incr(s(cons(incr(tail(y0)), y1))))) -> c13(MARK(incr(s(cons(incr(tail(y0)), y1))))) MARK(incr(incr(s(cons(head(incr(incr(y0))), y1))))) -> c13(MARK(incr(s(cons(head(incr(incr(y0))), y1))))) MARK(incr(incr(s(cons(head(incr(odds)), y0))))) -> c13(MARK(incr(s(cons(head(incr(odds)), y0))))) MARK(incr(incr(s(cons(head(incr(head(y0))), y1))))) -> c13(MARK(incr(s(cons(head(incr(head(y0))), y1))))) MARK(incr(incr(s(cons(head(incr(tail(y0))), y1))))) -> c13(MARK(incr(s(cons(head(incr(tail(y0))), y1))))) MARK(incr(incr(s(cons(head(incr(cons(y0, y1))), y2))))) -> c13(MARK(incr(s(cons(head(incr(cons(y0, y1))), y2))))) MARK(incr(incr(s(cons(head(incr(s(y0))), y1))))) -> c13(MARK(incr(s(cons(head(incr(s(y0))), y1))))) MARK(incr(incr(s(cons(head(incr(y0)), y1))))) -> c13(MARK(incr(s(cons(head(incr(y0)), y1))))) MARK(incr(incr(s(cons(head(head(incr(y0))), y1))))) -> c13(MARK(incr(s(cons(head(head(incr(y0))), y1))))) MARK(incr(incr(s(cons(head(head(odds)), y0))))) -> c13(MARK(incr(s(cons(head(head(odds)), y0))))) MARK(incr(incr(s(cons(head(head(head(y0))), y1))))) -> c13(MARK(incr(s(cons(head(head(head(y0))), y1))))) MARK(incr(incr(s(cons(head(head(tail(y0))), y1))))) -> c13(MARK(incr(s(cons(head(head(tail(y0))), y1))))) MARK(incr(incr(s(cons(head(head(cons(y0, y1))), y2))))) -> c13(MARK(incr(s(cons(head(head(cons(y0, y1))), y2))))) MARK(incr(incr(s(cons(head(head(s(y0))), y1))))) -> c13(MARK(incr(s(cons(head(head(s(y0))), y1))))) MARK(incr(incr(s(cons(head(head(y0)), y1))))) -> c13(MARK(incr(s(cons(head(head(y0)), y1))))) MARK(incr(incr(s(cons(head(tail(incr(y0))), y1))))) -> c13(MARK(incr(s(cons(head(tail(incr(y0))), y1))))) MARK(incr(incr(s(cons(head(tail(pairs)), y0))))) -> c13(MARK(incr(s(cons(head(tail(pairs)), y0))))) MARK(incr(incr(s(cons(head(tail(odds)), y0))))) -> c13(MARK(incr(s(cons(head(tail(odds)), y0))))) MARK(incr(incr(s(cons(head(tail(head(y0))), y1))))) -> c13(MARK(incr(s(cons(head(tail(head(y0))), y1))))) MARK(incr(incr(s(cons(head(tail(tail(y0))), y1))))) -> c13(MARK(incr(s(cons(head(tail(tail(y0))), y1))))) MARK(incr(incr(s(cons(head(tail(cons(y0, y1))), y2))))) -> c13(MARK(incr(s(cons(head(tail(cons(y0, y1))), y2))))) MARK(incr(incr(s(cons(head(tail(s(y0))), y1))))) -> c13(MARK(incr(s(cons(head(tail(s(y0))), y1))))) MARK(incr(incr(s(cons(head(tail(y0)), y1))))) -> c13(MARK(incr(s(cons(head(tail(y0)), y1))))) MARK(incr(incr(s(cons(tail(incr(incr(y0))), y1))))) -> c13(MARK(incr(s(cons(tail(incr(incr(y0))), y1))))) MARK(incr(incr(s(cons(tail(incr(odds)), y0))))) -> c13(MARK(incr(s(cons(tail(incr(odds)), y0))))) MARK(incr(incr(s(cons(tail(incr(head(y0))), y1))))) -> c13(MARK(incr(s(cons(tail(incr(head(y0))), y1))))) MARK(incr(incr(s(cons(tail(incr(tail(y0))), y1))))) -> c13(MARK(incr(s(cons(tail(incr(tail(y0))), y1))))) MARK(incr(incr(s(cons(tail(incr(cons(y0, y1))), y2))))) -> c13(MARK(incr(s(cons(tail(incr(cons(y0, y1))), y2))))) MARK(incr(incr(s(cons(tail(incr(s(y0))), y1))))) -> c13(MARK(incr(s(cons(tail(incr(s(y0))), y1))))) MARK(incr(incr(s(cons(tail(incr(y0)), y1))))) -> c13(MARK(incr(s(cons(tail(incr(y0)), y1))))) MARK(incr(incr(s(cons(tail(head(incr(y0))), y1))))) -> c13(MARK(incr(s(cons(tail(head(incr(y0))), y1))))) MARK(incr(incr(s(cons(tail(head(odds)), y0))))) -> c13(MARK(incr(s(cons(tail(head(odds)), y0))))) MARK(incr(incr(s(cons(tail(head(head(y0))), y1))))) -> c13(MARK(incr(s(cons(tail(head(head(y0))), y1))))) MARK(incr(incr(s(cons(tail(head(tail(y0))), y1))))) -> c13(MARK(incr(s(cons(tail(head(tail(y0))), y1))))) MARK(incr(incr(s(cons(tail(head(cons(y0, y1))), y2))))) -> c13(MARK(incr(s(cons(tail(head(cons(y0, y1))), y2))))) MARK(incr(incr(s(cons(tail(head(s(y0))), y1))))) -> c13(MARK(incr(s(cons(tail(head(s(y0))), y1))))) MARK(incr(incr(s(cons(tail(head(y0)), y1))))) -> c13(MARK(incr(s(cons(tail(head(y0)), y1))))) MARK(incr(incr(s(cons(tail(tail(incr(y0))), y1))))) -> c13(MARK(incr(s(cons(tail(tail(incr(y0))), y1))))) MARK(incr(incr(s(cons(tail(tail(pairs)), y0))))) -> c13(MARK(incr(s(cons(tail(tail(pairs)), y0))))) MARK(incr(incr(s(cons(tail(tail(odds)), y0))))) -> c13(MARK(incr(s(cons(tail(tail(odds)), y0))))) MARK(incr(incr(s(cons(tail(tail(head(y0))), y1))))) -> c13(MARK(incr(s(cons(tail(tail(head(y0))), y1))))) MARK(incr(incr(s(cons(tail(tail(tail(y0))), y1))))) -> c13(MARK(incr(s(cons(tail(tail(tail(y0))), y1))))) MARK(incr(incr(s(cons(tail(tail(cons(y0, y1))), y2))))) -> c13(MARK(incr(s(cons(tail(tail(cons(y0, y1))), y2))))) MARK(incr(incr(s(cons(tail(tail(s(y0))), y1))))) -> c13(MARK(incr(s(cons(tail(tail(s(y0))), y1))))) MARK(incr(incr(s(cons(tail(tail(y0)), y1))))) -> c13(MARK(incr(s(cons(tail(tail(y0)), y1))))) MARK(incr(incr(s(cons(tail(pairs), y0))))) -> c13(MARK(incr(s(cons(tail(pairs), y0))))) MARK(incr(incr(s(cons(incr(incr(nats)), y0))))) -> c13(MARK(incr(s(cons(incr(incr(nats)), y0))))) MARK(incr(incr(s(cons(incr(incr(pairs)), y0))))) -> c13(MARK(incr(s(cons(incr(incr(pairs)), y0))))) MARK(incr(incr(s(cons(incr(odds), y0))))) -> c13(MARK(incr(s(cons(incr(odds), y0))))) MARK(incr(incr(s(cons(head(incr(nats)), y0))))) -> c13(MARK(incr(s(cons(head(incr(nats)), y0))))) MARK(incr(incr(s(cons(head(incr(pairs)), y0))))) -> c13(MARK(incr(s(cons(head(incr(pairs)), y0))))) MARK(incr(incr(s(cons(head(odds), y0))))) -> c13(MARK(incr(s(cons(head(odds), y0))))) MARK(incr(incr(s(cons(tail(incr(nats)), y0))))) -> c13(MARK(incr(s(cons(tail(incr(nats)), y0))))) MARK(incr(incr(s(cons(tail(incr(pairs)), y0))))) -> c13(MARK(incr(s(cons(tail(incr(pairs)), y0))))) MARK(incr(incr(s(cons(tail(odds), y0))))) -> c13(MARK(incr(s(cons(tail(odds), y0))))) MARK(incr(incr(s(cons(incr(cons(nats, y0)), y1))))) -> c13(MARK(incr(s(cons(incr(cons(nats, y0)), y1))))) MARK(incr(incr(s(cons(incr(cons(incr(y0), y1)), y2))))) -> c13(MARK(incr(s(cons(incr(cons(incr(y0), y1)), y2))))) MARK(incr(incr(s(cons(incr(cons(pairs, y0)), y1))))) -> c13(MARK(incr(s(cons(incr(cons(pairs, y0)), y1))))) MARK(incr(incr(s(cons(incr(cons(odds, y0)), y1))))) -> c13(MARK(incr(s(cons(incr(cons(odds, y0)), y1))))) MARK(incr(incr(s(cons(incr(cons(head(y0), y1)), y2))))) -> c13(MARK(incr(s(cons(incr(cons(head(y0), y1)), y2))))) MARK(incr(incr(s(cons(incr(cons(tail(y0), y1)), y2))))) -> c13(MARK(incr(s(cons(incr(cons(tail(y0), y1)), y2))))) MARK(incr(incr(s(cons(incr(cons(cons(y0, y1), y2)), y3))))) -> c13(MARK(incr(s(cons(incr(cons(cons(y0, y1), y2)), y3))))) MARK(incr(incr(s(cons(incr(cons(s(y0), y1)), y2))))) -> c13(MARK(incr(s(cons(incr(cons(s(y0), y1)), y2))))) MARK(incr(incr(s(cons(incr(cons(y0, y1)), y2))))) -> c13(MARK(incr(s(cons(incr(cons(y0, y1)), y2))))) MARK(incr(incr(s(cons(incr(cons(0, y0)), y1))))) -> c13(MARK(incr(s(cons(incr(cons(0, y0)), y1))))) MARK(incr(incr(s(cons(incr(head(nats)), y0))))) -> c13(MARK(incr(s(cons(incr(head(nats)), y0))))) MARK(incr(incr(s(cons(incr(head(pairs)), y0))))) -> c13(MARK(incr(s(cons(incr(head(pairs)), y0))))) MARK(incr(incr(s(cons(head(head(nats)), y0))))) -> c13(MARK(incr(s(cons(head(head(nats)), y0))))) MARK(incr(incr(s(cons(head(head(pairs)), y0))))) -> c13(MARK(incr(s(cons(head(head(pairs)), y0))))) MARK(incr(incr(s(cons(tail(head(nats)), y0))))) -> c13(MARK(incr(s(cons(tail(head(nats)), y0))))) MARK(incr(incr(s(cons(tail(head(pairs)), y0))))) -> c13(MARK(incr(s(cons(tail(head(pairs)), y0))))) MARK(incr(incr(s(cons(head(cons(nats, y0)), y1))))) -> c13(MARK(incr(s(cons(head(cons(nats, y0)), y1))))) MARK(incr(incr(s(cons(head(cons(incr(y0), y1)), y2))))) -> c13(MARK(incr(s(cons(head(cons(incr(y0), y1)), y2))))) MARK(incr(incr(s(cons(head(cons(pairs, y0)), y1))))) -> c13(MARK(incr(s(cons(head(cons(pairs, y0)), y1))))) MARK(incr(incr(s(cons(head(cons(odds, y0)), y1))))) -> c13(MARK(incr(s(cons(head(cons(odds, y0)), y1))))) MARK(incr(incr(s(cons(head(cons(head(y0), y1)), y2))))) -> c13(MARK(incr(s(cons(head(cons(head(y0), y1)), y2))))) MARK(incr(incr(s(cons(head(cons(tail(y0), y1)), y2))))) -> c13(MARK(incr(s(cons(head(cons(tail(y0), y1)), y2))))) MARK(incr(incr(s(cons(head(cons(cons(y0, y1), y2)), y3))))) -> c13(MARK(incr(s(cons(head(cons(cons(y0, y1), y2)), y3))))) MARK(incr(incr(s(cons(head(cons(s(y0), y1)), y2))))) -> c13(MARK(incr(s(cons(head(cons(s(y0), y1)), y2))))) MARK(incr(incr(s(cons(head(cons(y0, y1)), y2))))) -> c13(MARK(incr(s(cons(head(cons(y0, y1)), y2))))) MARK(incr(incr(s(cons(head(cons(0, y0)), y1))))) -> c13(MARK(incr(s(cons(head(cons(0, y0)), y1))))) MARK(incr(incr(s(cons(incr(tail(nats)), y0))))) -> c13(MARK(incr(s(cons(incr(tail(nats)), y0))))) MARK(incr(incr(s(cons(head(tail(nats)), y0))))) -> c13(MARK(incr(s(cons(head(tail(nats)), y0))))) MARK(incr(incr(s(cons(tail(tail(nats)), y0))))) -> c13(MARK(incr(s(cons(tail(tail(nats)), y0))))) MARK(incr(incr(s(incr(cons(nats, y0)))))) -> c13(MARK(incr(s(incr(cons(nats, y0)))))) MARK(incr(incr(s(incr(cons(pairs, y0)))))) -> c13(MARK(incr(s(incr(cons(pairs, y0)))))) MARK(incr(incr(s(incr(cons(odds, y0)))))) -> c13(MARK(incr(s(incr(cons(odds, y0)))))) MARK(incr(incr(s(head(cons(nats, y0)))))) -> c13(MARK(incr(s(head(cons(nats, y0)))))) MARK(incr(incr(s(head(cons(pairs, y0)))))) -> c13(MARK(incr(s(head(cons(pairs, y0)))))) MARK(incr(incr(s(head(cons(odds, y0)))))) -> c13(MARK(incr(s(head(cons(odds, y0)))))) S tuples: MARK(tail(cons(z0, z1))) -> c17(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(incr(incr(incr(z0)))) -> c13(A__INCR(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(incr(incr(odds))) -> c13(A__INCR(a__incr(a__odds)), MARK(incr(odds))) MARK(incr(incr(head(z0)))) -> c13(A__INCR(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(incr(incr(tail(z0)))) -> c13(A__INCR(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(incr(incr(cons(z0, z1)))) -> c13(A__INCR(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(incr(incr(s(z0)))) -> c13(A__INCR(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(incr(head(incr(z0)))) -> c13(A__INCR(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(incr(head(odds))) -> c13(A__INCR(a__head(a__odds)), MARK(head(odds))) MARK(incr(head(head(z0)))) -> c13(A__INCR(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(incr(head(tail(z0)))) -> c13(A__INCR(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(incr(head(cons(z0, z1)))) -> c13(A__INCR(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(incr(head(s(z0)))) -> c13(A__INCR(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(incr(head(x0))) -> c13(MARK(head(x0))) MARK(incr(tail(incr(z0)))) -> c13(A__INCR(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(incr(tail(pairs))) -> c13(A__INCR(a__tail(a__pairs)), MARK(tail(pairs))) MARK(incr(tail(odds))) -> c13(A__INCR(a__tail(a__odds)), MARK(tail(odds))) MARK(incr(tail(head(z0)))) -> c13(A__INCR(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(incr(tail(tail(z0)))) -> c13(A__INCR(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(incr(tail(cons(z0, z1)))) -> c13(A__INCR(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(incr(tail(s(z0)))) -> c13(A__INCR(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(incr(tail(x0))) -> c13(MARK(tail(x0))) MARK(head(incr(incr(z0)))) -> c16(A__HEAD(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(head(incr(odds))) -> c16(A__HEAD(a__incr(a__odds)), MARK(incr(odds))) MARK(head(incr(head(z0)))) -> c16(A__HEAD(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(head(incr(tail(z0)))) -> c16(A__HEAD(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(head(incr(cons(z0, z1)))) -> c16(A__HEAD(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(head(incr(s(z0)))) -> c16(A__HEAD(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(head(incr(x0))) -> c16(MARK(incr(x0))) MARK(head(head(incr(z0)))) -> c16(A__HEAD(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(head(head(odds))) -> c16(A__HEAD(a__head(a__odds)), MARK(head(odds))) MARK(head(head(head(z0)))) -> c16(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c16(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(cons(z0, z1)))) -> c16(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(s(z0)))) -> c16(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(x0))) -> c16(MARK(head(x0))) MARK(head(tail(incr(z0)))) -> c16(A__HEAD(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(head(tail(pairs))) -> c16(A__HEAD(a__tail(a__pairs)), MARK(tail(pairs))) MARK(head(tail(odds))) -> c16(A__HEAD(a__tail(a__odds)), MARK(tail(odds))) MARK(head(tail(head(z0)))) -> c16(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c16(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(cons(z0, z1)))) -> c16(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(s(z0)))) -> c16(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(x0))) -> c16(MARK(tail(x0))) MARK(tail(incr(incr(z0)))) -> c17(A__TAIL(a__incr(a__incr(mark(z0)))), MARK(incr(incr(z0)))) MARK(tail(incr(odds))) -> c17(A__TAIL(a__incr(a__odds)), MARK(incr(odds))) MARK(tail(incr(head(z0)))) -> c17(A__TAIL(a__incr(a__head(mark(z0)))), MARK(incr(head(z0)))) MARK(tail(incr(tail(z0)))) -> c17(A__TAIL(a__incr(a__tail(mark(z0)))), MARK(incr(tail(z0)))) MARK(tail(incr(cons(z0, z1)))) -> c17(A__TAIL(a__incr(cons(mark(z0), z1))), MARK(incr(cons(z0, z1)))) MARK(tail(incr(s(z0)))) -> c17(A__TAIL(a__incr(s(mark(z0)))), MARK(incr(s(z0)))) MARK(tail(incr(x0))) -> c17(MARK(incr(x0))) MARK(tail(head(incr(z0)))) -> c17(A__TAIL(a__head(a__incr(mark(z0)))), MARK(head(incr(z0)))) MARK(tail(head(odds))) -> c17(A__TAIL(a__head(a__odds)), MARK(head(odds))) MARK(tail(head(head(z0)))) -> c17(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c17(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(cons(z0, z1)))) -> c17(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(s(z0)))) -> c17(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(x0))) -> c17(MARK(head(x0))) MARK(tail(tail(incr(z0)))) -> c17(A__TAIL(a__tail(a__incr(mark(z0)))), MARK(tail(incr(z0)))) MARK(tail(tail(pairs))) -> c17(A__TAIL(a__tail(a__pairs)), MARK(tail(pairs))) MARK(tail(tail(odds))) -> c17(A__TAIL(a__tail(a__odds)), MARK(tail(odds))) 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c6(MARK(incr(incr(nats)))) A__INCR(cons(incr(incr(incr(y0))), z1)) -> c6(MARK(incr(incr(incr(y0))))) A__INCR(cons(incr(incr(pairs)), z1)) -> c6(MARK(incr(incr(pairs)))) A__INCR(cons(incr(incr(odds)), z1)) -> c6(MARK(incr(incr(odds)))) A__INCR(cons(incr(incr(head(y0))), z1)) -> c6(MARK(incr(incr(head(y0))))) A__INCR(cons(incr(incr(tail(y0))), z1)) -> c6(MARK(incr(incr(tail(y0))))) A__INCR(cons(incr(incr(cons(y0, y1))), z1)) -> c6(MARK(incr(incr(cons(y0, y1))))) A__INCR(cons(incr(incr(s(y0))), z1)) -> c6(MARK(incr(incr(s(y0))))) A__INCR(cons(incr(incr(y0)), z1)) -> c6(MARK(incr(incr(y0)))) A__INCR(cons(incr(odds), z1)) -> c6(MARK(incr(odds))) A__INCR(cons(incr(head(nats)), z1)) -> c6(MARK(incr(head(nats)))) A__INCR(cons(incr(head(incr(y0))), z1)) -> c6(MARK(incr(head(incr(y0))))) A__INCR(cons(incr(head(pairs)), z1)) -> c6(MARK(incr(head(pairs)))) A__INCR(cons(incr(head(odds)), z1)) -> c6(MARK(incr(head(odds)))) A__INCR(cons(incr(head(head(y0))), z1)) -> c6(MARK(incr(head(head(y0))))) 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-> c18(MARK(tail(incr(cons(y0, y1))))) MARK(cons(tail(incr(s(y0))), z1)) -> c18(MARK(tail(incr(s(y0))))) MARK(cons(tail(incr(y0)), z1)) -> c18(MARK(tail(incr(y0)))) MARK(cons(tail(head(incr(y0))), z1)) -> c18(MARK(tail(head(incr(y0))))) MARK(cons(tail(head(odds)), z1)) -> c18(MARK(tail(head(odds)))) MARK(cons(tail(head(head(y0))), z1)) -> c18(MARK(tail(head(head(y0))))) MARK(cons(tail(head(tail(y0))), z1)) -> c18(MARK(tail(head(tail(y0))))) MARK(cons(tail(head(cons(y0, y1))), z1)) -> c18(MARK(tail(head(cons(y0, y1))))) MARK(cons(tail(head(s(y0))), z1)) -> c18(MARK(tail(head(s(y0))))) MARK(cons(tail(head(y0)), z1)) -> c18(MARK(tail(head(y0)))) MARK(cons(tail(tail(incr(y0))), z1)) -> c18(MARK(tail(tail(incr(y0))))) MARK(cons(tail(tail(pairs)), z1)) -> c18(MARK(tail(tail(pairs)))) MARK(cons(tail(tail(odds)), z1)) -> c18(MARK(tail(tail(odds)))) MARK(cons(tail(tail(head(y0))), z1)) -> c18(MARK(tail(tail(head(y0))))) MARK(cons(tail(tail(tail(y0))), z1)) -> c18(MARK(tail(tail(tail(y0))))) 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c18(MARK(incr(cons(pairs, y0)))) MARK(cons(incr(cons(odds, y0)), z1)) -> c18(MARK(incr(cons(odds, y0)))) MARK(cons(incr(cons(head(y0), y1)), z1)) -> c18(MARK(incr(cons(head(y0), y1)))) MARK(cons(incr(cons(tail(y0), y1)), z1)) -> c18(MARK(incr(cons(tail(y0), y1)))) MARK(cons(incr(cons(cons(y0, y1), y2)), z1)) -> c18(MARK(incr(cons(cons(y0, y1), y2)))) MARK(cons(incr(cons(s(y0), y1)), z1)) -> c18(MARK(incr(cons(s(y0), y1)))) MARK(cons(incr(cons(y0, y1)), z1)) -> c18(MARK(incr(cons(y0, y1)))) MARK(cons(incr(cons(0, y0)), z1)) -> c18(MARK(incr(cons(0, y0)))) MARK(cons(incr(head(nats)), z1)) -> c18(MARK(incr(head(nats)))) MARK(cons(incr(head(pairs)), z1)) -> c18(MARK(incr(head(pairs)))) MARK(cons(head(head(nats)), z1)) -> c18(MARK(head(head(nats)))) MARK(cons(head(head(pairs)), z1)) -> c18(MARK(head(head(pairs)))) MARK(cons(tail(head(nats)), z1)) -> c18(MARK(tail(head(nats)))) MARK(cons(tail(head(pairs)), z1)) -> c18(MARK(tail(head(pairs)))) MARK(cons(head(cons(nats, y0)), z1)) -> c18(MARK(head(cons(nats, y0)))) MARK(cons(head(cons(incr(y0), y1)), z1)) -> c18(MARK(head(cons(incr(y0), y1)))) MARK(cons(head(cons(pairs, y0)), z1)) -> c18(MARK(head(cons(pairs, y0)))) MARK(cons(head(cons(odds, y0)), z1)) -> c18(MARK(head(cons(odds, y0)))) MARK(cons(head(cons(head(y0), y1)), z1)) -> c18(MARK(head(cons(head(y0), y1)))) MARK(cons(head(cons(tail(y0), y1)), z1)) -> c18(MARK(head(cons(tail(y0), y1)))) MARK(cons(head(cons(cons(y0, y1), y2)), z1)) -> c18(MARK(head(cons(cons(y0, y1), y2)))) MARK(cons(head(cons(s(y0), y1)), z1)) -> c18(MARK(head(cons(s(y0), y1)))) MARK(cons(head(cons(y0, y1)), z1)) -> c18(MARK(head(cons(y0, y1)))) MARK(cons(head(cons(0, y0)), z1)) -> c18(MARK(head(cons(0, y0)))) MARK(cons(incr(tail(nats)), z1)) -> c18(MARK(incr(tail(nats)))) MARK(cons(head(tail(nats)), z1)) -> c18(MARK(head(tail(nats)))) MARK(cons(tail(tail(nats)), z1)) -> c18(MARK(tail(tail(nats)))) MARK(incr(cons(nats, x1))) -> c13(A__INCR(cons(a__nats, x1))) MARK(incr(cons(pairs, x1))) -> c13(A__INCR(cons(a__pairs, x1))) MARK(incr(cons(odds, x1))) -> c13(A__INCR(cons(a__odds, x1))) MARK(head(cons(nats, x1))) -> c16(A__HEAD(cons(a__nats, x1))) MARK(head(cons(pairs, x1))) -> c16(A__HEAD(cons(a__pairs, x1))) MARK(head(cons(odds, x1))) -> c16(A__HEAD(cons(a__odds, x1))) MARK(s(s(y0))) -> c20(MARK(s(y0))) MARK(s(incr(s(y0)))) -> c20(MARK(incr(s(y0)))) MARK(s(head(s(y0)))) -> c20(MARK(head(s(y0)))) MARK(s(tail(cons(y0, y1)))) -> c20(MARK(tail(cons(y0, y1)))) MARK(s(tail(s(y0)))) -> c20(MARK(tail(s(y0)))) MARK(s(incr(incr(incr(y0))))) -> c20(MARK(incr(incr(incr(y0))))) MARK(s(incr(incr(odds)))) -> c20(MARK(incr(incr(odds)))) MARK(s(incr(incr(head(y0))))) -> c20(MARK(incr(incr(head(y0))))) MARK(s(incr(incr(tail(y0))))) -> c20(MARK(incr(incr(tail(y0))))) MARK(s(incr(incr(cons(y0, y1))))) -> c20(MARK(incr(incr(cons(y0, y1))))) MARK(s(incr(incr(s(y0))))) -> c20(MARK(incr(incr(s(y0))))) MARK(s(incr(incr(y0)))) -> c20(MARK(incr(incr(y0)))) MARK(s(incr(head(incr(y0))))) -> c20(MARK(incr(head(incr(y0))))) MARK(s(incr(head(odds)))) -> c20(MARK(incr(head(odds)))) MARK(s(incr(head(head(y0))))) -> c20(MARK(incr(head(head(y0))))) MARK(s(incr(head(tail(y0))))) -> c20(MARK(incr(head(tail(y0))))) MARK(s(incr(head(cons(y0, y1))))) -> c20(MARK(incr(head(cons(y0, y1))))) MARK(s(incr(head(s(y0))))) -> c20(MARK(incr(head(s(y0))))) MARK(s(incr(head(y0)))) -> c20(MARK(incr(head(y0)))) MARK(s(incr(tail(incr(y0))))) -> c20(MARK(incr(tail(incr(y0))))) MARK(s(incr(tail(pairs)))) -> c20(MARK(incr(tail(pairs)))) MARK(s(incr(tail(odds)))) -> c20(MARK(incr(tail(odds)))) MARK(s(incr(tail(head(y0))))) -> c20(MARK(incr(tail(head(y0))))) MARK(s(incr(tail(tail(y0))))) -> c20(MARK(incr(tail(tail(y0))))) MARK(s(incr(tail(cons(y0, y1))))) -> c20(MARK(incr(tail(cons(y0, y1))))) MARK(s(incr(tail(s(y0))))) -> c20(MARK(incr(tail(s(y0))))) MARK(s(incr(tail(y0)))) -> c20(MARK(incr(tail(y0)))) MARK(s(head(incr(incr(y0))))) -> c20(MARK(head(incr(incr(y0))))) MARK(s(head(incr(odds)))) -> c20(MARK(head(incr(odds)))) MARK(s(head(incr(head(y0))))) -> c20(MARK(head(incr(head(y0))))) MARK(s(head(incr(tail(y0))))) -> c20(MARK(head(incr(tail(y0))))) MARK(s(head(incr(cons(y0, y1))))) -> c20(MARK(head(incr(cons(y0, y1))))) MARK(s(head(incr(s(y0))))) -> c20(MARK(head(incr(s(y0))))) MARK(s(head(incr(y0)))) -> c20(MARK(head(incr(y0)))) MARK(s(head(head(incr(y0))))) -> c20(MARK(head(head(incr(y0))))) MARK(s(head(head(odds)))) -> c20(MARK(head(head(odds)))) MARK(s(head(head(head(y0))))) -> c20(MARK(head(head(head(y0))))) MARK(s(head(head(tail(y0))))) -> c20(MARK(head(head(tail(y0))))) MARK(s(head(head(cons(y0, y1))))) -> c20(MARK(head(head(cons(y0, y1))))) MARK(s(head(head(s(y0))))) -> c20(MARK(head(head(s(y0))))) MARK(s(head(head(y0)))) -> c20(MARK(head(head(y0)))) MARK(s(head(tail(incr(y0))))) -> c20(MARK(head(tail(incr(y0))))) MARK(s(head(tail(pairs)))) -> c20(MARK(head(tail(pairs)))) MARK(s(head(tail(odds)))) -> c20(MARK(head(tail(odds)))) MARK(s(head(tail(head(y0))))) -> c20(MARK(head(tail(head(y0))))) MARK(s(head(tail(tail(y0))))) -> c20(MARK(head(tail(tail(y0))))) MARK(s(head(tail(cons(y0, y1))))) -> c20(MARK(head(tail(cons(y0, y1))))) MARK(s(head(tail(s(y0))))) -> c20(MARK(head(tail(s(y0))))) MARK(s(head(tail(y0)))) -> c20(MARK(head(tail(y0)))) MARK(s(tail(incr(incr(y0))))) -> c20(MARK(tail(incr(incr(y0))))) MARK(s(tail(incr(odds)))) -> c20(MARK(tail(incr(odds)))) MARK(s(tail(incr(head(y0))))) -> c20(MARK(tail(incr(head(y0))))) MARK(s(tail(incr(tail(y0))))) -> c20(MARK(tail(incr(tail(y0))))) MARK(s(tail(incr(cons(y0, y1))))) -> c20(MARK(tail(incr(cons(y0, y1))))) MARK(s(tail(incr(s(y0))))) -> c20(MARK(tail(incr(s(y0))))) MARK(s(tail(incr(y0)))) -> c20(MARK(tail(incr(y0)))) MARK(s(tail(head(incr(y0))))) -> c20(MARK(tail(head(incr(y0))))) MARK(s(tail(head(odds)))) -> c20(MARK(tail(head(odds)))) MARK(s(tail(head(head(y0))))) -> c20(MARK(tail(head(head(y0))))) MARK(s(tail(head(tail(y0))))) -> c20(MARK(tail(head(tail(y0))))) MARK(s(tail(head(cons(y0, y1))))) -> c20(MARK(tail(head(cons(y0, y1))))) MARK(s(tail(head(s(y0))))) -> c20(MARK(tail(head(s(y0))))) MARK(s(tail(head(y0)))) -> c20(MARK(tail(head(y0)))) MARK(s(tail(tail(incr(y0))))) -> c20(MARK(tail(tail(incr(y0))))) MARK(s(tail(tail(pairs)))) -> c20(MARK(tail(tail(pairs)))) MARK(s(tail(tail(odds)))) -> c20(MARK(tail(tail(odds)))) MARK(s(tail(tail(head(y0))))) -> c20(MARK(tail(tail(head(y0))))) MARK(s(tail(tail(tail(y0))))) -> c20(MARK(tail(tail(tail(y0))))) MARK(s(tail(tail(cons(y0, y1))))) -> c20(MARK(tail(tail(cons(y0, y1))))) MARK(s(tail(tail(s(y0))))) -> c20(MARK(tail(tail(s(y0))))) MARK(s(tail(tail(y0)))) -> c20(MARK(tail(tail(y0)))) MARK(s(tail(pairs))) -> c20(MARK(tail(pairs))) MARK(s(incr(incr(nats)))) -> c20(MARK(incr(incr(nats)))) MARK(s(incr(incr(pairs)))) -> c20(MARK(incr(incr(pairs)))) MARK(s(incr(odds))) -> c20(MARK(incr(odds))) MARK(s(head(incr(nats)))) -> c20(MARK(head(incr(nats)))) MARK(s(head(incr(pairs)))) -> c20(MARK(head(incr(pairs)))) MARK(s(head(odds))) -> c20(MARK(head(odds))) MARK(s(tail(incr(nats)))) -> c20(MARK(tail(incr(nats)))) MARK(s(tail(incr(pairs)))) -> c20(MARK(tail(incr(pairs)))) MARK(s(tail(odds))) -> c20(MARK(tail(odds))) MARK(s(incr(cons(incr(y0), y1)))) -> c20(MARK(incr(cons(incr(y0), y1)))) MARK(s(incr(cons(head(y0), y1)))) -> c20(MARK(incr(cons(head(y0), y1)))) MARK(s(incr(cons(tail(y0), y1)))) -> c20(MARK(incr(cons(tail(y0), y1)))) MARK(s(incr(cons(cons(y0, y1), y2)))) -> c20(MARK(incr(cons(cons(y0, y1), y2)))) MARK(s(incr(cons(s(y0), y1)))) -> c20(MARK(incr(cons(s(y0), y1)))) MARK(s(incr(cons(y0, y1)))) -> c20(MARK(incr(cons(y0, y1)))) MARK(s(incr(head(nats)))) -> c20(MARK(incr(head(nats)))) MARK(s(incr(head(pairs)))) -> c20(MARK(incr(head(pairs)))) MARK(s(head(head(nats)))) -> c20(MARK(head(head(nats)))) MARK(s(head(head(pairs)))) -> c20(MARK(head(head(pairs)))) MARK(s(tail(head(nats)))) -> c20(MARK(tail(head(nats)))) MARK(s(tail(head(pairs)))) -> c20(MARK(tail(head(pairs)))) MARK(s(head(cons(incr(y0), y1)))) -> c20(MARK(head(cons(incr(y0), y1)))) MARK(s(head(cons(head(y0), y1)))) -> c20(MARK(head(cons(head(y0), y1)))) MARK(s(head(cons(tail(y0), y1)))) -> c20(MARK(head(cons(tail(y0), y1)))) MARK(s(head(cons(cons(y0, y1), y2)))) -> c20(MARK(head(cons(cons(y0, y1), y2)))) MARK(s(head(cons(s(y0), y1)))) -> c20(MARK(head(cons(s(y0), y1)))) MARK(s(head(cons(y0, y1)))) -> c20(MARK(head(cons(y0, y1)))) MARK(s(incr(tail(nats)))) -> c20(MARK(incr(tail(nats)))) MARK(s(head(tail(nats)))) -> c20(MARK(head(tail(nats)))) MARK(s(tail(tail(nats)))) -> c20(MARK(tail(tail(nats)))) MARK(s(cons(cons(y0, y1), y2))) -> c20(MARK(cons(cons(y0, y1), y2))) MARK(s(cons(s(y0), y1))) -> c20(MARK(cons(s(y0), y1))) MARK(s(cons(incr(s(y0)), y1))) -> c20(MARK(cons(incr(s(y0)), y1))) MARK(s(cons(head(s(y0)), y1))) -> c20(MARK(cons(head(s(y0)), y1))) MARK(s(cons(tail(cons(y0, y1)), y2))) -> c20(MARK(cons(tail(cons(y0, y1)), y2))) MARK(s(cons(tail(s(y0)), y1))) -> c20(MARK(cons(tail(s(y0)), y1))) MARK(s(cons(incr(incr(incr(y0))), y1))) -> c20(MARK(cons(incr(incr(incr(y0))), y1))) MARK(s(cons(incr(incr(odds)), y0))) -> c20(MARK(cons(incr(incr(odds)), y0))) MARK(s(cons(incr(incr(head(y0))), y1))) -> c20(MARK(cons(incr(incr(head(y0))), y1))) MARK(s(cons(incr(incr(tail(y0))), y1))) -> c20(MARK(cons(incr(incr(tail(y0))), y1))) MARK(s(cons(incr(incr(cons(y0, y1))), y2))) -> c20(MARK(cons(incr(incr(cons(y0, y1))), y2))) MARK(s(cons(incr(incr(s(y0))), y1))) -> c20(MARK(cons(incr(incr(s(y0))), y1))) MARK(s(cons(incr(incr(y0)), y1))) -> c20(MARK(cons(incr(incr(y0)), y1))) MARK(s(cons(incr(head(incr(y0))), y1))) -> c20(MARK(cons(incr(head(incr(y0))), y1))) MARK(s(cons(incr(head(odds)), y0))) -> c20(MARK(cons(incr(head(odds)), y0))) MARK(s(cons(incr(head(head(y0))), y1))) -> c20(MARK(cons(incr(head(head(y0))), y1))) MARK(s(cons(incr(head(tail(y0))), y1))) -> c20(MARK(cons(incr(head(tail(y0))), y1))) MARK(s(cons(incr(head(cons(y0, y1))), y2))) -> c20(MARK(cons(incr(head(cons(y0, y1))), y2))) MARK(s(cons(incr(head(s(y0))), y1))) -> c20(MARK(cons(incr(head(s(y0))), y1))) MARK(s(cons(incr(head(y0)), y1))) -> c20(MARK(cons(incr(head(y0)), y1))) MARK(s(cons(incr(tail(incr(y0))), y1))) -> c20(MARK(cons(incr(tail(incr(y0))), y1))) MARK(s(cons(incr(tail(pairs)), y0))) -> c20(MARK(cons(incr(tail(pairs)), y0))) MARK(s(cons(incr(tail(odds)), y0))) -> c20(MARK(cons(incr(tail(odds)), y0))) MARK(s(cons(incr(tail(head(y0))), y1))) -> c20(MARK(cons(incr(tail(head(y0))), y1))) MARK(s(cons(incr(tail(tail(y0))), y1))) -> c20(MARK(cons(incr(tail(tail(y0))), y1))) MARK(s(cons(incr(tail(cons(y0, y1))), y2))) -> c20(MARK(cons(incr(tail(cons(y0, y1))), y2))) MARK(s(cons(incr(tail(s(y0))), y1))) -> c20(MARK(cons(incr(tail(s(y0))), y1))) MARK(s(cons(incr(tail(y0)), y1))) -> c20(MARK(cons(incr(tail(y0)), y1))) MARK(s(cons(head(incr(incr(y0))), y1))) -> c20(MARK(cons(head(incr(incr(y0))), y1))) MARK(s(cons(head(incr(odds)), y0))) -> c20(MARK(cons(head(incr(odds)), y0))) MARK(s(cons(head(incr(head(y0))), y1))) -> c20(MARK(cons(head(incr(head(y0))), y1))) MARK(s(cons(head(incr(tail(y0))), y1))) -> c20(MARK(cons(head(incr(tail(y0))), y1))) MARK(s(cons(head(incr(cons(y0, y1))), y2))) -> c20(MARK(cons(head(incr(cons(y0, y1))), y2))) MARK(s(cons(head(incr(s(y0))), y1))) -> c20(MARK(cons(head(incr(s(y0))), y1))) MARK(s(cons(head(incr(y0)), y1))) -> c20(MARK(cons(head(incr(y0)), y1))) MARK(s(cons(head(head(incr(y0))), y1))) -> c20(MARK(cons(head(head(incr(y0))), y1))) MARK(s(cons(head(head(odds)), y0))) -> c20(MARK(cons(head(head(odds)), y0))) MARK(s(cons(head(head(head(y0))), y1))) -> c20(MARK(cons(head(head(head(y0))), y1))) MARK(s(cons(head(head(tail(y0))), y1))) -> c20(MARK(cons(head(head(tail(y0))), y1))) MARK(s(cons(head(head(cons(y0, y1))), y2))) -> c20(MARK(cons(head(head(cons(y0, y1))), y2))) MARK(s(cons(head(head(s(y0))), y1))) -> c20(MARK(cons(head(head(s(y0))), y1))) MARK(s(cons(head(head(y0)), y1))) -> c20(MARK(cons(head(head(y0)), y1))) MARK(s(cons(head(tail(incr(y0))), y1))) -> c20(MARK(cons(head(tail(incr(y0))), y1))) MARK(s(cons(head(tail(pairs)), y0))) -> c20(MARK(cons(head(tail(pairs)), y0))) MARK(s(cons(head(tail(odds)), y0))) -> c20(MARK(cons(head(tail(odds)), y0))) MARK(s(cons(head(tail(head(y0))), y1))) -> c20(MARK(cons(head(tail(head(y0))), y1))) MARK(s(cons(head(tail(tail(y0))), y1))) -> c20(MARK(cons(head(tail(tail(y0))), y1))) MARK(s(cons(head(tail(cons(y0, y1))), y2))) -> c20(MARK(cons(head(tail(cons(y0, y1))), y2))) MARK(s(cons(head(tail(s(y0))), y1))) -> c20(MARK(cons(head(tail(s(y0))), y1))) MARK(s(cons(head(tail(y0)), y1))) -> c20(MARK(cons(head(tail(y0)), y1))) MARK(s(cons(tail(incr(incr(y0))), y1))) -> c20(MARK(cons(tail(incr(incr(y0))), y1))) MARK(s(cons(tail(incr(odds)), y0))) -> c20(MARK(cons(tail(incr(odds)), y0))) MARK(s(cons(tail(incr(head(y0))), y1))) -> c20(MARK(cons(tail(incr(head(y0))), y1))) MARK(s(cons(tail(incr(tail(y0))), y1))) -> c20(MARK(cons(tail(incr(tail(y0))), y1))) MARK(s(cons(tail(incr(cons(y0, y1))), y2))) -> c20(MARK(cons(tail(incr(cons(y0, y1))), y2))) MARK(s(cons(tail(incr(s(y0))), y1))) -> c20(MARK(cons(tail(incr(s(y0))), y1))) MARK(s(cons(tail(incr(y0)), y1))) -> c20(MARK(cons(tail(incr(y0)), y1))) MARK(s(cons(tail(head(incr(y0))), y1))) -> c20(MARK(cons(tail(head(incr(y0))), y1))) MARK(s(cons(tail(head(odds)), y0))) -> c20(MARK(cons(tail(head(odds)), y0))) MARK(s(cons(tail(head(head(y0))), y1))) -> c20(MARK(cons(tail(head(head(y0))), y1))) MARK(s(cons(tail(head(tail(y0))), y1))) -> c20(MARK(cons(tail(head(tail(y0))), y1))) MARK(s(cons(tail(head(cons(y0, y1))), y2))) -> c20(MARK(cons(tail(head(cons(y0, y1))), y2))) MARK(s(cons(tail(head(s(y0))), y1))) -> c20(MARK(cons(tail(head(s(y0))), y1))) MARK(s(cons(tail(head(y0)), y1))) -> c20(MARK(cons(tail(head(y0)), y1))) MARK(s(cons(tail(tail(incr(y0))), y1))) -> c20(MARK(cons(tail(tail(incr(y0))), y1))) MARK(s(cons(tail(tail(pairs)), y0))) -> c20(MARK(cons(tail(tail(pairs)), y0))) MARK(s(cons(tail(tail(odds)), y0))) -> c20(MARK(cons(tail(tail(odds)), y0))) MARK(s(cons(tail(tail(head(y0))), y1))) -> c20(MARK(cons(tail(tail(head(y0))), y1))) MARK(s(cons(tail(tail(tail(y0))), y1))) -> c20(MARK(cons(tail(tail(tail(y0))), y1))) MARK(s(cons(tail(tail(cons(y0, y1))), y2))) -> c20(MARK(cons(tail(tail(cons(y0, y1))), y2))) MARK(s(cons(tail(tail(s(y0))), y1))) -> c20(MARK(cons(tail(tail(s(y0))), y1))) MARK(s(cons(tail(tail(y0)), y1))) -> c20(MARK(cons(tail(tail(y0)), y1))) MARK(s(cons(tail(pairs), y0))) -> c20(MARK(cons(tail(pairs), y0))) MARK(s(cons(incr(incr(nats)), y0))) -> c20(MARK(cons(incr(incr(nats)), y0))) MARK(s(cons(incr(incr(pairs)), y0))) -> c20(MARK(cons(incr(incr(pairs)), y0))) MARK(s(cons(incr(odds), y0))) -> c20(MARK(cons(incr(odds), y0))) MARK(s(cons(head(incr(nats)), y0))) -> c20(MARK(cons(head(incr(nats)), y0))) MARK(s(cons(head(incr(pairs)), y0))) -> c20(MARK(cons(head(incr(pairs)), y0))) MARK(s(cons(head(odds), y0))) -> c20(MARK(cons(head(odds), y0))) MARK(s(cons(tail(incr(nats)), y0))) -> c20(MARK(cons(tail(incr(nats)), y0))) MARK(s(cons(tail(incr(pairs)), y0))) -> c20(MARK(cons(tail(incr(pairs)), y0))) MARK(s(cons(tail(odds), y0))) -> c20(MARK(cons(tail(odds), y0))) MARK(s(cons(incr(cons(nats, y0)), y1))) -> c20(MARK(cons(incr(cons(nats, y0)), y1))) MARK(s(cons(incr(cons(incr(y0), y1)), y2))) -> c20(MARK(cons(incr(cons(incr(y0), y1)), y2))) MARK(s(cons(incr(cons(pairs, y0)), y1))) -> c20(MARK(cons(incr(cons(pairs, y0)), y1))) MARK(s(cons(incr(cons(odds, y0)), y1))) -> c20(MARK(cons(incr(cons(odds, y0)), y1))) MARK(s(cons(incr(cons(head(y0), y1)), y2))) -> c20(MARK(cons(incr(cons(head(y0), y1)), y2))) MARK(s(cons(incr(cons(tail(y0), y1)), y2))) -> c20(MARK(cons(incr(cons(tail(y0), y1)), y2))) MARK(s(cons(incr(cons(cons(y0, y1), y2)), y3))) -> c20(MARK(cons(incr(cons(cons(y0, y1), y2)), y3))) MARK(s(cons(incr(cons(s(y0), y1)), y2))) -> c20(MARK(cons(incr(cons(s(y0), y1)), y2))) MARK(s(cons(incr(cons(y0, y1)), y2))) -> c20(MARK(cons(incr(cons(y0, y1)), y2))) MARK(s(cons(incr(cons(0, y0)), y1))) -> c20(MARK(cons(incr(cons(0, y0)), y1))) MARK(s(cons(incr(head(nats)), y0))) -> c20(MARK(cons(incr(head(nats)), y0))) MARK(s(cons(incr(head(pairs)), y0))) -> c20(MARK(cons(incr(head(pairs)), y0))) MARK(s(cons(head(head(nats)), y0))) -> c20(MARK(cons(head(head(nats)), y0))) MARK(s(cons(head(head(pairs)), y0))) -> c20(MARK(cons(head(head(pairs)), y0))) MARK(s(cons(tail(head(nats)), y0))) -> c20(MARK(cons(tail(head(nats)), y0))) MARK(s(cons(tail(head(pairs)), y0))) -> c20(MARK(cons(tail(head(pairs)), y0))) MARK(s(cons(head(cons(nats, y0)), y1))) -> c20(MARK(cons(head(cons(nats, y0)), y1))) MARK(s(cons(head(cons(incr(y0), y1)), y2))) -> c20(MARK(cons(head(cons(incr(y0), y1)), y2))) MARK(s(cons(head(cons(pairs, y0)), y1))) -> c20(MARK(cons(head(cons(pairs, y0)), y1))) MARK(s(cons(head(cons(odds, y0)), y1))) -> c20(MARK(cons(head(cons(odds, y0)), y1))) MARK(s(cons(head(cons(head(y0), y1)), y2))) -> c20(MARK(cons(head(cons(head(y0), y1)), y2))) MARK(s(cons(head(cons(tail(y0), y1)), y2))) -> c20(MARK(cons(head(cons(tail(y0), y1)), y2))) MARK(s(cons(head(cons(cons(y0, y1), y2)), y3))) -> c20(MARK(cons(head(cons(cons(y0, y1), y2)), y3))) MARK(s(cons(head(cons(s(y0), y1)), y2))) -> c20(MARK(cons(head(cons(s(y0), y1)), y2))) MARK(s(cons(head(cons(y0, y1)), y2))) -> c20(MARK(cons(head(cons(y0, y1)), y2))) MARK(s(cons(head(cons(0, y0)), y1))) -> c20(MARK(cons(head(cons(0, y0)), y1))) MARK(s(cons(incr(tail(nats)), y0))) -> c20(MARK(cons(incr(tail(nats)), y0))) MARK(s(cons(head(tail(nats)), y0))) -> c20(MARK(cons(head(tail(nats)), y0))) MARK(s(cons(tail(tail(nats)), y0))) -> c20(MARK(cons(tail(tail(nats)), y0))) MARK(s(incr(cons(nats, y0)))) -> c20(MARK(incr(cons(nats, y0)))) MARK(s(incr(cons(pairs, y0)))) -> c20(MARK(incr(cons(pairs, y0)))) MARK(s(incr(cons(odds, y0)))) -> c20(MARK(incr(cons(odds, y0)))) MARK(s(head(cons(nats, y0)))) -> c20(MARK(head(cons(nats, y0)))) MARK(s(head(cons(pairs, y0)))) -> c20(MARK(head(cons(pairs, y0)))) MARK(s(head(cons(odds, y0)))) -> c20(MARK(head(cons(odds, y0)))) MARK(incr(s(s(y0)))) -> c13(MARK(s(s(y0)))) MARK(incr(s(incr(s(y0))))) -> c13(MARK(s(incr(s(y0))))) MARK(incr(s(head(s(y0))))) -> c13(MARK(s(head(s(y0))))) MARK(incr(s(tail(cons(y0, y1))))) -> c13(MARK(s(tail(cons(y0, y1))))) MARK(incr(s(tail(s(y0))))) -> c13(MARK(s(tail(s(y0))))) MARK(incr(s(incr(incr(incr(y0)))))) -> c13(MARK(s(incr(incr(incr(y0)))))) MARK(incr(s(incr(incr(odds))))) -> c13(MARK(s(incr(incr(odds))))) MARK(incr(s(incr(incr(head(y0)))))) -> c13(MARK(s(incr(incr(head(y0)))))) MARK(incr(s(incr(incr(tail(y0)))))) -> c13(MARK(s(incr(incr(tail(y0)))))) MARK(incr(s(incr(incr(cons(y0, y1)))))) -> c13(MARK(s(incr(incr(cons(y0, y1)))))) MARK(incr(s(incr(incr(s(y0)))))) -> c13(MARK(s(incr(incr(s(y0)))))) MARK(incr(s(incr(incr(y0))))) -> c13(MARK(s(incr(incr(y0))))) MARK(incr(s(incr(head(incr(y0)))))) -> c13(MARK(s(incr(head(incr(y0)))))) MARK(incr(s(incr(head(odds))))) -> c13(MARK(s(incr(head(odds))))) MARK(incr(s(incr(head(head(y0)))))) -> c13(MARK(s(incr(head(head(y0)))))) MARK(incr(s(incr(head(tail(y0)))))) -> c13(MARK(s(incr(head(tail(y0)))))) MARK(incr(s(incr(head(cons(y0, y1)))))) -> c13(MARK(s(incr(head(cons(y0, y1)))))) MARK(incr(s(incr(head(s(y0)))))) -> c13(MARK(s(incr(head(s(y0)))))) MARK(incr(s(incr(head(y0))))) -> c13(MARK(s(incr(head(y0))))) MARK(incr(s(incr(tail(incr(y0)))))) -> c13(MARK(s(incr(tail(incr(y0)))))) MARK(incr(s(incr(tail(pairs))))) -> c13(MARK(s(incr(tail(pairs))))) MARK(incr(s(incr(tail(odds))))) -> c13(MARK(s(incr(tail(odds))))) MARK(incr(s(incr(tail(head(y0)))))) -> c13(MARK(s(incr(tail(head(y0)))))) MARK(incr(s(incr(tail(tail(y0)))))) -> c13(MARK(s(incr(tail(tail(y0)))))) MARK(incr(s(incr(tail(cons(y0, y1)))))) -> c13(MARK(s(incr(tail(cons(y0, y1)))))) MARK(incr(s(incr(tail(s(y0)))))) -> c13(MARK(s(incr(tail(s(y0)))))) MARK(incr(s(incr(tail(y0))))) -> c13(MARK(s(incr(tail(y0))))) MARK(incr(s(head(incr(incr(y0)))))) -> c13(MARK(s(head(incr(incr(y0)))))) MARK(incr(s(head(incr(odds))))) -> c13(MARK(s(head(incr(odds))))) MARK(incr(s(head(incr(head(y0)))))) -> c13(MARK(s(head(incr(head(y0)))))) MARK(incr(s(head(incr(tail(y0)))))) -> c13(MARK(s(head(incr(tail(y0)))))) MARK(incr(s(head(incr(cons(y0, y1)))))) -> c13(MARK(s(head(incr(cons(y0, y1)))))) MARK(incr(s(head(incr(s(y0)))))) -> c13(MARK(s(head(incr(s(y0)))))) MARK(incr(s(head(incr(y0))))) -> c13(MARK(s(head(incr(y0))))) MARK(incr(s(head(head(incr(y0)))))) -> c13(MARK(s(head(head(incr(y0)))))) MARK(incr(s(head(head(odds))))) -> c13(MARK(s(head(head(odds))))) MARK(incr(s(head(head(head(y0)))))) -> c13(MARK(s(head(head(head(y0)))))) MARK(incr(s(head(head(tail(y0)))))) -> c13(MARK(s(head(head(tail(y0)))))) MARK(incr(s(head(head(cons(y0, y1)))))) -> c13(MARK(s(head(head(cons(y0, y1)))))) MARK(incr(s(head(head(s(y0)))))) -> c13(MARK(s(head(head(s(y0)))))) MARK(incr(s(head(head(y0))))) -> c13(MARK(s(head(head(y0))))) MARK(incr(s(head(tail(incr(y0)))))) -> c13(MARK(s(head(tail(incr(y0)))))) MARK(incr(s(head(tail(pairs))))) -> c13(MARK(s(head(tail(pairs))))) MARK(incr(s(head(tail(odds))))) -> c13(MARK(s(head(tail(odds))))) MARK(incr(s(head(tail(head(y0)))))) -> c13(MARK(s(head(tail(head(y0)))))) MARK(incr(s(head(tail(tail(y0)))))) -> c13(MARK(s(head(tail(tail(y0)))))) MARK(incr(s(head(tail(cons(y0, y1)))))) -> c13(MARK(s(head(tail(cons(y0, y1)))))) MARK(incr(s(head(tail(s(y0)))))) -> c13(MARK(s(head(tail(s(y0)))))) MARK(incr(s(head(tail(y0))))) -> c13(MARK(s(head(tail(y0))))) MARK(incr(s(tail(incr(incr(y0)))))) -> c13(MARK(s(tail(incr(incr(y0)))))) MARK(incr(s(tail(incr(odds))))) -> c13(MARK(s(tail(incr(odds))))) MARK(incr(s(tail(incr(head(y0)))))) -> c13(MARK(s(tail(incr(head(y0)))))) MARK(incr(s(tail(incr(tail(y0)))))) -> c13(MARK(s(tail(incr(tail(y0)))))) MARK(incr(s(tail(incr(cons(y0, y1)))))) -> c13(MARK(s(tail(incr(cons(y0, y1)))))) MARK(incr(s(tail(incr(s(y0)))))) -> c13(MARK(s(tail(incr(s(y0)))))) MARK(incr(s(tail(incr(y0))))) -> c13(MARK(s(tail(incr(y0))))) MARK(incr(s(tail(head(incr(y0)))))) -> c13(MARK(s(tail(head(incr(y0)))))) MARK(incr(s(tail(head(odds))))) -> c13(MARK(s(tail(head(odds))))) MARK(incr(s(tail(head(head(y0)))))) -> c13(MARK(s(tail(head(head(y0)))))) MARK(incr(s(tail(head(tail(y0)))))) -> c13(MARK(s(tail(head(tail(y0)))))) MARK(incr(s(tail(head(cons(y0, y1)))))) -> c13(MARK(s(tail(head(cons(y0, y1)))))) MARK(incr(s(tail(head(s(y0)))))) -> c13(MARK(s(tail(head(s(y0)))))) MARK(incr(s(tail(head(y0))))) -> c13(MARK(s(tail(head(y0))))) MARK(incr(s(tail(tail(incr(y0)))))) -> c13(MARK(s(tail(tail(incr(y0)))))) MARK(incr(s(tail(tail(pairs))))) -> c13(MARK(s(tail(tail(pairs))))) MARK(incr(s(tail(tail(odds))))) -> c13(MARK(s(tail(tail(odds))))) MARK(incr(s(tail(tail(head(y0)))))) -> c13(MARK(s(tail(tail(head(y0)))))) MARK(incr(s(tail(tail(tail(y0)))))) -> c13(MARK(s(tail(tail(tail(y0)))))) MARK(incr(s(tail(tail(cons(y0, y1)))))) -> c13(MARK(s(tail(tail(cons(y0, y1)))))) MARK(incr(s(tail(tail(s(y0)))))) -> c13(MARK(s(tail(tail(s(y0)))))) MARK(incr(s(tail(tail(y0))))) -> c13(MARK(s(tail(tail(y0))))) MARK(incr(s(tail(pairs)))) -> c13(MARK(s(tail(pairs)))) MARK(incr(s(incr(incr(nats))))) -> c13(MARK(s(incr(incr(nats))))) MARK(incr(s(incr(incr(pairs))))) -> c13(MARK(s(incr(incr(pairs))))) MARK(incr(s(incr(odds)))) -> c13(MARK(s(incr(odds)))) MARK(incr(s(head(incr(nats))))) -> c13(MARK(s(head(incr(nats))))) MARK(incr(s(head(incr(pairs))))) -> c13(MARK(s(head(incr(pairs))))) MARK(incr(s(head(odds)))) -> c13(MARK(s(head(odds)))) MARK(incr(s(tail(incr(nats))))) -> c13(MARK(s(tail(incr(nats))))) MARK(incr(s(tail(incr(pairs))))) -> c13(MARK(s(tail(incr(pairs))))) MARK(incr(s(tail(odds)))) -> c13(MARK(s(tail(odds)))) MARK(incr(s(incr(cons(incr(y0), y1))))) -> c13(MARK(s(incr(cons(incr(y0), y1))))) MARK(incr(s(incr(cons(head(y0), y1))))) -> c13(MARK(s(incr(cons(head(y0), y1))))) MARK(incr(s(incr(cons(tail(y0), y1))))) -> c13(MARK(s(incr(cons(tail(y0), y1))))) MARK(incr(s(incr(cons(cons(y0, y1), y2))))) -> c13(MARK(s(incr(cons(cons(y0, y1), y2))))) MARK(incr(s(incr(cons(s(y0), y1))))) -> c13(MARK(s(incr(cons(s(y0), y1))))) MARK(incr(s(incr(cons(y0, y1))))) -> c13(MARK(s(incr(cons(y0, y1))))) MARK(incr(s(incr(head(nats))))) -> c13(MARK(s(incr(head(nats))))) MARK(incr(s(incr(head(pairs))))) -> c13(MARK(s(incr(head(pairs))))) MARK(incr(s(head(head(nats))))) -> c13(MARK(s(head(head(nats))))) MARK(incr(s(head(head(pairs))))) -> c13(MARK(s(head(head(pairs))))) MARK(incr(s(tail(head(nats))))) -> c13(MARK(s(tail(head(nats))))) MARK(incr(s(tail(head(pairs))))) -> c13(MARK(s(tail(head(pairs))))) MARK(incr(s(head(cons(incr(y0), y1))))) -> c13(MARK(s(head(cons(incr(y0), y1))))) MARK(incr(s(head(cons(head(y0), y1))))) -> c13(MARK(s(head(cons(head(y0), y1))))) MARK(incr(s(head(cons(tail(y0), y1))))) -> c13(MARK(s(head(cons(tail(y0), y1))))) MARK(incr(s(head(cons(cons(y0, y1), y2))))) -> c13(MARK(s(head(cons(cons(y0, y1), y2))))) MARK(incr(s(head(cons(s(y0), y1))))) -> c13(MARK(s(head(cons(s(y0), y1))))) MARK(incr(s(head(cons(y0, y1))))) -> c13(MARK(s(head(cons(y0, y1))))) MARK(incr(s(incr(tail(nats))))) -> c13(MARK(s(incr(tail(nats))))) MARK(incr(s(head(tail(nats))))) -> c13(MARK(s(head(tail(nats))))) MARK(incr(s(tail(tail(nats))))) -> c13(MARK(s(tail(tail(nats))))) MARK(incr(s(cons(cons(y0, y1), y2)))) -> c13(MARK(s(cons(cons(y0, y1), y2)))) MARK(incr(s(cons(s(y0), y1)))) -> c13(MARK(s(cons(s(y0), y1)))) MARK(incr(s(cons(incr(s(y0)), y1)))) -> c13(MARK(s(cons(incr(s(y0)), y1)))) MARK(incr(s(cons(head(s(y0)), y1)))) -> c13(MARK(s(cons(head(s(y0)), y1)))) MARK(incr(s(cons(tail(cons(y0, y1)), y2)))) -> c13(MARK(s(cons(tail(cons(y0, y1)), y2)))) MARK(incr(s(cons(tail(s(y0)), y1)))) -> c13(MARK(s(cons(tail(s(y0)), y1)))) MARK(incr(s(cons(incr(incr(incr(y0))), y1)))) -> c13(MARK(s(cons(incr(incr(incr(y0))), y1)))) MARK(incr(s(cons(incr(incr(odds)), y0)))) -> c13(MARK(s(cons(incr(incr(odds)), y0)))) MARK(incr(s(cons(incr(incr(head(y0))), y1)))) -> c13(MARK(s(cons(incr(incr(head(y0))), y1)))) MARK(incr(s(cons(incr(incr(tail(y0))), y1)))) -> c13(MARK(s(cons(incr(incr(tail(y0))), y1)))) MARK(incr(s(cons(incr(incr(cons(y0, y1))), y2)))) -> c13(MARK(s(cons(incr(incr(cons(y0, y1))), y2)))) MARK(incr(s(cons(incr(incr(s(y0))), y1)))) -> c13(MARK(s(cons(incr(incr(s(y0))), y1)))) MARK(incr(s(cons(incr(incr(y0)), y1)))) -> c13(MARK(s(cons(incr(incr(y0)), y1)))) MARK(incr(s(cons(incr(head(incr(y0))), y1)))) -> c13(MARK(s(cons(incr(head(incr(y0))), y1)))) MARK(incr(s(cons(incr(head(odds)), y0)))) -> c13(MARK(s(cons(incr(head(odds)), y0)))) MARK(incr(s(cons(incr(head(head(y0))), y1)))) -> c13(MARK(s(cons(incr(head(head(y0))), y1)))) MARK(incr(s(cons(incr(head(tail(y0))), y1)))) -> c13(MARK(s(cons(incr(head(tail(y0))), y1)))) MARK(incr(s(cons(incr(head(cons(y0, y1))), y2)))) -> c13(MARK(s(cons(incr(head(cons(y0, y1))), y2)))) MARK(incr(s(cons(incr(head(s(y0))), y1)))) -> c13(MARK(s(cons(incr(head(s(y0))), y1)))) MARK(incr(s(cons(incr(head(y0)), y1)))) -> c13(MARK(s(cons(incr(head(y0)), y1)))) MARK(incr(s(cons(incr(tail(incr(y0))), y1)))) -> c13(MARK(s(cons(incr(tail(incr(y0))), y1)))) MARK(incr(s(cons(incr(tail(pairs)), y0)))) -> c13(MARK(s(cons(incr(tail(pairs)), y0)))) MARK(incr(s(cons(incr(tail(odds)), y0)))) -> c13(MARK(s(cons(incr(tail(odds)), y0)))) MARK(incr(s(cons(incr(tail(head(y0))), y1)))) -> c13(MARK(s(cons(incr(tail(head(y0))), y1)))) MARK(incr(s(cons(incr(tail(tail(y0))), y1)))) -> c13(MARK(s(cons(incr(tail(tail(y0))), y1)))) MARK(incr(s(cons(incr(tail(cons(y0, y1))), y2)))) -> c13(MARK(s(cons(incr(tail(cons(y0, y1))), y2)))) MARK(incr(s(cons(incr(tail(s(y0))), y1)))) -> c13(MARK(s(cons(incr(tail(s(y0))), y1)))) MARK(incr(s(cons(incr(tail(y0)), y1)))) -> c13(MARK(s(cons(incr(tail(y0)), y1)))) MARK(incr(s(cons(head(incr(incr(y0))), y1)))) -> c13(MARK(s(cons(head(incr(incr(y0))), y1)))) MARK(incr(s(cons(head(incr(odds)), y0)))) -> c13(MARK(s(cons(head(incr(odds)), y0)))) MARK(incr(s(cons(head(incr(head(y0))), y1)))) -> c13(MARK(s(cons(head(incr(head(y0))), y1)))) MARK(incr(s(cons(head(incr(tail(y0))), y1)))) -> c13(MARK(s(cons(head(incr(tail(y0))), y1)))) MARK(incr(s(cons(head(incr(cons(y0, y1))), y2)))) -> c13(MARK(s(cons(head(incr(cons(y0, y1))), y2)))) MARK(incr(s(cons(head(incr(s(y0))), y1)))) -> c13(MARK(s(cons(head(incr(s(y0))), y1)))) MARK(incr(s(cons(head(incr(y0)), y1)))) -> c13(MARK(s(cons(head(incr(y0)), y1)))) MARK(incr(s(cons(head(head(incr(y0))), y1)))) -> c13(MARK(s(cons(head(head(incr(y0))), y1)))) MARK(incr(s(cons(head(head(odds)), y0)))) -> c13(MARK(s(cons(head(head(odds)), y0)))) MARK(incr(s(cons(head(head(head(y0))), y1)))) -> c13(MARK(s(cons(head(head(head(y0))), y1)))) MARK(incr(s(cons(head(head(tail(y0))), y1)))) -> c13(MARK(s(cons(head(head(tail(y0))), y1)))) MARK(incr(s(cons(head(head(cons(y0, y1))), y2)))) -> c13(MARK(s(cons(head(head(cons(y0, y1))), y2)))) MARK(incr(s(cons(head(head(s(y0))), y1)))) -> c13(MARK(s(cons(head(head(s(y0))), y1)))) MARK(incr(s(cons(head(head(y0)), y1)))) -> c13(MARK(s(cons(head(head(y0)), y1)))) MARK(incr(s(cons(head(tail(incr(y0))), y1)))) -> c13(MARK(s(cons(head(tail(incr(y0))), y1)))) MARK(incr(s(cons(head(tail(pairs)), y0)))) -> c13(MARK(s(cons(head(tail(pairs)), y0)))) MARK(incr(s(cons(head(tail(odds)), y0)))) -> c13(MARK(s(cons(head(tail(odds)), y0)))) MARK(incr(s(cons(head(tail(head(y0))), y1)))) -> c13(MARK(s(cons(head(tail(head(y0))), y1)))) MARK(incr(s(cons(head(tail(tail(y0))), y1)))) -> c13(MARK(s(cons(head(tail(tail(y0))), y1)))) MARK(incr(s(cons(head(tail(cons(y0, y1))), y2)))) -> c13(MARK(s(cons(head(tail(cons(y0, y1))), y2)))) MARK(incr(s(cons(head(tail(s(y0))), y1)))) -> c13(MARK(s(cons(head(tail(s(y0))), y1)))) MARK(incr(s(cons(head(tail(y0)), y1)))) -> c13(MARK(s(cons(head(tail(y0)), y1)))) MARK(incr(s(cons(tail(incr(incr(y0))), y1)))) -> c13(MARK(s(cons(tail(incr(incr(y0))), y1)))) MARK(incr(s(cons(tail(incr(odds)), y0)))) -> c13(MARK(s(cons(tail(incr(odds)), y0)))) MARK(incr(s(cons(tail(incr(head(y0))), y1)))) -> c13(MARK(s(cons(tail(incr(head(y0))), y1)))) MARK(incr(s(cons(tail(incr(tail(y0))), y1)))) -> c13(MARK(s(cons(tail(incr(tail(y0))), y1)))) MARK(incr(s(cons(tail(incr(cons(y0, y1))), y2)))) -> c13(MARK(s(cons(tail(incr(cons(y0, y1))), y2)))) MARK(incr(s(cons(tail(incr(s(y0))), y1)))) -> c13(MARK(s(cons(tail(incr(s(y0))), y1)))) MARK(incr(s(cons(tail(incr(y0)), y1)))) -> c13(MARK(s(cons(tail(incr(y0)), y1)))) MARK(incr(s(cons(tail(head(incr(y0))), y1)))) -> c13(MARK(s(cons(tail(head(incr(y0))), y1)))) MARK(incr(s(cons(tail(head(odds)), y0)))) -> c13(MARK(s(cons(tail(head(odds)), y0)))) MARK(incr(s(cons(tail(head(head(y0))), y1)))) -> c13(MARK(s(cons(tail(head(head(y0))), y1)))) MARK(incr(s(cons(tail(head(tail(y0))), y1)))) -> c13(MARK(s(cons(tail(head(tail(y0))), y1)))) MARK(incr(s(cons(tail(head(cons(y0, y1))), y2)))) -> c13(MARK(s(cons(tail(head(cons(y0, y1))), y2)))) MARK(incr(s(cons(tail(head(s(y0))), y1)))) -> c13(MARK(s(cons(tail(head(s(y0))), y1)))) MARK(incr(s(cons(tail(head(y0)), y1)))) -> c13(MARK(s(cons(tail(head(y0)), y1)))) MARK(incr(s(cons(tail(tail(incr(y0))), y1)))) -> c13(MARK(s(cons(tail(tail(incr(y0))), y1)))) MARK(incr(s(cons(tail(tail(pairs)), y0)))) -> c13(MARK(s(cons(tail(tail(pairs)), y0)))) MARK(incr(s(cons(tail(tail(odds)), y0)))) -> c13(MARK(s(cons(tail(tail(odds)), y0)))) MARK(incr(s(cons(tail(tail(head(y0))), y1)))) -> c13(MARK(s(cons(tail(tail(head(y0))), y1)))) MARK(incr(s(cons(tail(tail(tail(y0))), y1)))) -> c13(MARK(s(cons(tail(tail(tail(y0))), y1)))) MARK(incr(s(cons(tail(tail(cons(y0, y1))), y2)))) -> c13(MARK(s(cons(tail(tail(cons(y0, y1))), y2)))) MARK(incr(s(cons(tail(tail(s(y0))), y1)))) -> c13(MARK(s(cons(tail(tail(s(y0))), y1)))) MARK(incr(s(cons(tail(tail(y0)), y1)))) -> c13(MARK(s(cons(tail(tail(y0)), y1)))) MARK(incr(s(cons(tail(pairs), y0)))) -> c13(MARK(s(cons(tail(pairs), y0)))) MARK(incr(s(cons(incr(incr(nats)), y0)))) -> c13(MARK(s(cons(incr(incr(nats)), y0)))) MARK(incr(s(cons(incr(incr(pairs)), y0)))) -> c13(MARK(s(cons(incr(incr(pairs)), y0)))) MARK(incr(s(cons(incr(odds), y0)))) -> c13(MARK(s(cons(incr(odds), y0)))) MARK(incr(s(cons(head(incr(nats)), y0)))) -> c13(MARK(s(cons(head(incr(nats)), y0)))) MARK(incr(s(cons(head(incr(pairs)), y0)))) -> c13(MARK(s(cons(head(incr(pairs)), y0)))) MARK(incr(s(cons(head(odds), y0)))) -> c13(MARK(s(cons(head(odds), y0)))) MARK(incr(s(cons(tail(incr(nats)), y0)))) -> c13(MARK(s(cons(tail(incr(nats)), y0)))) MARK(incr(s(cons(tail(incr(pairs)), y0)))) -> c13(MARK(s(cons(tail(incr(pairs)), y0)))) MARK(incr(s(cons(tail(odds), y0)))) -> c13(MARK(s(cons(tail(odds), y0)))) MARK(incr(s(cons(incr(cons(nats, y0)), y1)))) -> c13(MARK(s(cons(incr(cons(nats, y0)), y1)))) MARK(incr(s(cons(incr(cons(incr(y0), y1)), y2)))) -> c13(MARK(s(cons(incr(cons(incr(y0), y1)), y2)))) MARK(incr(s(cons(incr(cons(pairs, y0)), y1)))) -> c13(MARK(s(cons(incr(cons(pairs, y0)), y1)))) MARK(incr(s(cons(incr(cons(odds, y0)), y1)))) -> c13(MARK(s(cons(incr(cons(odds, y0)), y1)))) MARK(incr(s(cons(incr(cons(head(y0), y1)), y2)))) -> c13(MARK(s(cons(incr(cons(head(y0), y1)), y2)))) MARK(incr(s(cons(incr(cons(tail(y0), y1)), y2)))) -> c13(MARK(s(cons(incr(cons(tail(y0), y1)), y2)))) MARK(incr(s(cons(incr(cons(cons(y0, y1), y2)), y3)))) -> c13(MARK(s(cons(incr(cons(cons(y0, y1), y2)), y3)))) MARK(incr(s(cons(incr(cons(s(y0), y1)), y2)))) -> c13(MARK(s(cons(incr(cons(s(y0), y1)), y2)))) MARK(incr(s(cons(incr(cons(y0, y1)), y2)))) -> c13(MARK(s(cons(incr(cons(y0, y1)), y2)))) MARK(incr(s(cons(incr(cons(0, y0)), y1)))) -> c13(MARK(s(cons(incr(cons(0, y0)), y1)))) MARK(incr(s(cons(incr(head(nats)), y0)))) -> c13(MARK(s(cons(incr(head(nats)), y0)))) MARK(incr(s(cons(incr(head(pairs)), y0)))) -> c13(MARK(s(cons(incr(head(pairs)), y0)))) MARK(incr(s(cons(head(head(nats)), y0)))) -> c13(MARK(s(cons(head(head(nats)), y0)))) MARK(incr(s(cons(head(head(pairs)), y0)))) -> c13(MARK(s(cons(head(head(pairs)), y0)))) MARK(incr(s(cons(tail(head(nats)), y0)))) -> c13(MARK(s(cons(tail(head(nats)), y0)))) MARK(incr(s(cons(tail(head(pairs)), y0)))) -> c13(MARK(s(cons(tail(head(pairs)), y0)))) MARK(incr(s(cons(head(cons(nats, y0)), y1)))) -> c13(MARK(s(cons(head(cons(nats, y0)), y1)))) MARK(incr(s(cons(head(cons(incr(y0), y1)), y2)))) -> c13(MARK(s(cons(head(cons(incr(y0), y1)), y2)))) MARK(incr(s(cons(head(cons(pairs, y0)), y1)))) -> c13(MARK(s(cons(head(cons(pairs, y0)), y1)))) MARK(incr(s(cons(head(cons(odds, y0)), y1)))) -> c13(MARK(s(cons(head(cons(odds, y0)), y1)))) MARK(incr(s(cons(head(cons(head(y0), y1)), y2)))) -> c13(MARK(s(cons(head(cons(head(y0), y1)), y2)))) MARK(incr(s(cons(head(cons(tail(y0), y1)), y2)))) -> c13(MARK(s(cons(head(cons(tail(y0), y1)), y2)))) MARK(incr(s(cons(head(cons(cons(y0, y1), y2)), y3)))) -> c13(MARK(s(cons(head(cons(cons(y0, y1), y2)), y3)))) MARK(incr(s(cons(head(cons(s(y0), y1)), y2)))) -> c13(MARK(s(cons(head(cons(s(y0), y1)), y2)))) MARK(incr(s(cons(head(cons(y0, y1)), y2)))) -> c13(MARK(s(cons(head(cons(y0, y1)), y2)))) MARK(incr(s(cons(head(cons(0, y0)), y1)))) -> c13(MARK(s(cons(head(cons(0, y0)), y1)))) MARK(incr(s(cons(incr(tail(nats)), y0)))) -> c13(MARK(s(cons(incr(tail(nats)), y0)))) MARK(incr(s(cons(head(tail(nats)), y0)))) -> c13(MARK(s(cons(head(tail(nats)), y0)))) MARK(incr(s(cons(tail(tail(nats)), y0)))) -> c13(MARK(s(cons(tail(tail(nats)), y0)))) MARK(incr(s(incr(cons(nats, y0))))) -> c13(MARK(s(incr(cons(nats, y0))))) MARK(incr(s(incr(cons(pairs, y0))))) -> c13(MARK(s(incr(cons(pairs, y0))))) MARK(incr(s(incr(cons(odds, y0))))) -> c13(MARK(s(incr(cons(odds, y0))))) MARK(incr(s(head(cons(nats, y0))))) -> c13(MARK(s(head(cons(nats, y0))))) MARK(incr(s(head(cons(pairs, y0))))) -> c13(MARK(s(head(cons(pairs, y0))))) MARK(incr(s(head(cons(odds, y0))))) -> c13(MARK(s(head(cons(odds, y0))))) MARK(head(s(s(y0)))) -> c16(MARK(s(s(y0)))) MARK(head(s(incr(s(y0))))) -> c16(MARK(s(incr(s(y0))))) MARK(head(s(head(s(y0))))) -> c16(MARK(s(head(s(y0))))) MARK(head(s(tail(cons(y0, y1))))) -> c16(MARK(s(tail(cons(y0, y1))))) MARK(head(s(tail(s(y0))))) -> c16(MARK(s(tail(s(y0))))) MARK(head(s(incr(incr(incr(y0)))))) -> c16(MARK(s(incr(incr(incr(y0)))))) MARK(head(s(incr(incr(odds))))) -> c16(MARK(s(incr(incr(odds))))) MARK(head(s(incr(incr(head(y0)))))) -> c16(MARK(s(incr(incr(head(y0)))))) MARK(head(s(incr(incr(tail(y0)))))) -> c16(MARK(s(incr(incr(tail(y0)))))) MARK(head(s(incr(incr(cons(y0, y1)))))) -> c16(MARK(s(incr(incr(cons(y0, y1)))))) MARK(head(s(incr(incr(s(y0)))))) -> c16(MARK(s(incr(incr(s(y0)))))) MARK(head(s(incr(incr(y0))))) -> c16(MARK(s(incr(incr(y0))))) MARK(head(s(incr(head(incr(y0)))))) -> c16(MARK(s(incr(head(incr(y0)))))) MARK(head(s(incr(head(odds))))) -> c16(MARK(s(incr(head(odds))))) MARK(head(s(incr(head(head(y0)))))) -> c16(MARK(s(incr(head(head(y0)))))) MARK(head(s(incr(head(tail(y0)))))) -> c16(MARK(s(incr(head(tail(y0)))))) MARK(head(s(incr(head(cons(y0, y1)))))) -> c16(MARK(s(incr(head(cons(y0, y1)))))) MARK(head(s(incr(head(s(y0)))))) -> c16(MARK(s(incr(head(s(y0)))))) MARK(head(s(incr(head(y0))))) -> c16(MARK(s(incr(head(y0))))) MARK(head(s(incr(tail(incr(y0)))))) -> c16(MARK(s(incr(tail(incr(y0)))))) MARK(head(s(incr(tail(pairs))))) -> c16(MARK(s(incr(tail(pairs))))) MARK(head(s(incr(tail(odds))))) -> c16(MARK(s(incr(tail(odds))))) MARK(head(s(incr(tail(head(y0)))))) -> c16(MARK(s(incr(tail(head(y0)))))) MARK(head(s(incr(tail(tail(y0)))))) -> c16(MARK(s(incr(tail(tail(y0)))))) MARK(head(s(incr(tail(cons(y0, y1)))))) -> c16(MARK(s(incr(tail(cons(y0, y1)))))) MARK(head(s(incr(tail(s(y0)))))) -> c16(MARK(s(incr(tail(s(y0)))))) MARK(head(s(incr(tail(y0))))) -> c16(MARK(s(incr(tail(y0))))) MARK(head(s(head(incr(incr(y0)))))) -> c16(MARK(s(head(incr(incr(y0)))))) MARK(head(s(head(incr(odds))))) -> c16(MARK(s(head(incr(odds))))) MARK(head(s(head(incr(head(y0)))))) -> c16(MARK(s(head(incr(head(y0)))))) MARK(head(s(head(incr(tail(y0)))))) -> c16(MARK(s(head(incr(tail(y0)))))) MARK(head(s(head(incr(cons(y0, y1)))))) -> c16(MARK(s(head(incr(cons(y0, y1)))))) MARK(head(s(head(incr(s(y0)))))) -> c16(MARK(s(head(incr(s(y0)))))) MARK(head(s(head(incr(y0))))) -> c16(MARK(s(head(incr(y0))))) MARK(head(s(head(head(incr(y0)))))) -> c16(MARK(s(head(head(incr(y0)))))) MARK(head(s(head(head(odds))))) -> c16(MARK(s(head(head(odds))))) MARK(head(s(head(head(head(y0)))))) -> c16(MARK(s(head(head(head(y0)))))) MARK(head(s(head(head(tail(y0)))))) -> c16(MARK(s(head(head(tail(y0)))))) MARK(head(s(head(head(cons(y0, y1)))))) -> c16(MARK(s(head(head(cons(y0, y1)))))) MARK(head(s(head(head(s(y0)))))) -> c16(MARK(s(head(head(s(y0)))))) MARK(head(s(head(head(y0))))) -> c16(MARK(s(head(head(y0))))) MARK(head(s(head(tail(incr(y0)))))) -> c16(MARK(s(head(tail(incr(y0)))))) MARK(head(s(head(tail(pairs))))) -> c16(MARK(s(head(tail(pairs))))) MARK(head(s(head(tail(odds))))) -> c16(MARK(s(head(tail(odds))))) MARK(head(s(head(tail(head(y0)))))) -> c16(MARK(s(head(tail(head(y0)))))) MARK(head(s(head(tail(tail(y0)))))) -> c16(MARK(s(head(tail(tail(y0)))))) MARK(head(s(head(tail(cons(y0, y1)))))) -> c16(MARK(s(head(tail(cons(y0, y1)))))) MARK(head(s(head(tail(s(y0)))))) -> c16(MARK(s(head(tail(s(y0)))))) MARK(head(s(head(tail(y0))))) -> c16(MARK(s(head(tail(y0))))) MARK(head(s(tail(incr(incr(y0)))))) -> c16(MARK(s(tail(incr(incr(y0)))))) MARK(head(s(tail(incr(odds))))) -> c16(MARK(s(tail(incr(odds))))) MARK(head(s(tail(incr(head(y0)))))) -> c16(MARK(s(tail(incr(head(y0)))))) MARK(head(s(tail(incr(tail(y0)))))) -> c16(MARK(s(tail(incr(tail(y0)))))) MARK(head(s(tail(incr(cons(y0, y1)))))) -> c16(MARK(s(tail(incr(cons(y0, y1)))))) MARK(head(s(tail(incr(s(y0)))))) -> c16(MARK(s(tail(incr(s(y0)))))) MARK(head(s(tail(incr(y0))))) -> c16(MARK(s(tail(incr(y0))))) MARK(head(s(tail(head(incr(y0)))))) -> c16(MARK(s(tail(head(incr(y0)))))) MARK(head(s(tail(head(odds))))) -> c16(MARK(s(tail(head(odds))))) MARK(head(s(tail(head(head(y0)))))) -> c16(MARK(s(tail(head(head(y0)))))) MARK(head(s(tail(head(tail(y0)))))) -> c16(MARK(s(tail(head(tail(y0)))))) MARK(head(s(tail(head(cons(y0, y1)))))) -> c16(MARK(s(tail(head(cons(y0, y1)))))) MARK(head(s(tail(head(s(y0)))))) -> c16(MARK(s(tail(head(s(y0)))))) MARK(head(s(tail(head(y0))))) -> c16(MARK(s(tail(head(y0))))) MARK(head(s(tail(tail(incr(y0)))))) -> c16(MARK(s(tail(tail(incr(y0)))))) MARK(head(s(tail(tail(pairs))))) -> c16(MARK(s(tail(tail(pairs))))) MARK(head(s(tail(tail(odds))))) -> c16(MARK(s(tail(tail(odds))))) MARK(head(s(tail(tail(head(y0)))))) -> c16(MARK(s(tail(tail(head(y0)))))) MARK(head(s(tail(tail(tail(y0)))))) -> c16(MARK(s(tail(tail(tail(y0)))))) MARK(head(s(tail(tail(cons(y0, y1)))))) -> c16(MARK(s(tail(tail(cons(y0, y1)))))) MARK(head(s(tail(tail(s(y0)))))) -> c16(MARK(s(tail(tail(s(y0)))))) MARK(head(s(tail(tail(y0))))) -> c16(MARK(s(tail(tail(y0))))) MARK(head(s(tail(pairs)))) -> c16(MARK(s(tail(pairs)))) MARK(head(s(incr(incr(nats))))) -> c16(MARK(s(incr(incr(nats))))) MARK(head(s(incr(incr(pairs))))) -> c16(MARK(s(incr(incr(pairs))))) MARK(head(s(incr(odds)))) -> c16(MARK(s(incr(odds)))) MARK(head(s(head(incr(nats))))) -> c16(MARK(s(head(incr(nats))))) MARK(head(s(head(incr(pairs))))) -> c16(MARK(s(head(incr(pairs))))) MARK(head(s(head(odds)))) -> c16(MARK(s(head(odds)))) MARK(head(s(tail(incr(nats))))) -> c16(MARK(s(tail(incr(nats))))) MARK(head(s(tail(incr(pairs))))) -> c16(MARK(s(tail(incr(pairs))))) MARK(head(s(tail(odds)))) -> c16(MARK(s(tail(odds)))) MARK(head(s(incr(cons(incr(y0), y1))))) -> c16(MARK(s(incr(cons(incr(y0), y1))))) MARK(head(s(incr(cons(head(y0), y1))))) -> c16(MARK(s(incr(cons(head(y0), y1))))) MARK(head(s(incr(cons(tail(y0), y1))))) -> c16(MARK(s(incr(cons(tail(y0), y1))))) MARK(head(s(incr(cons(cons(y0, y1), y2))))) -> c16(MARK(s(incr(cons(cons(y0, y1), y2))))) MARK(head(s(incr(cons(s(y0), y1))))) -> c16(MARK(s(incr(cons(s(y0), y1))))) MARK(head(s(incr(cons(y0, y1))))) -> c16(MARK(s(incr(cons(y0, y1))))) MARK(head(s(incr(head(nats))))) -> c16(MARK(s(incr(head(nats))))) MARK(head(s(incr(head(pairs))))) -> c16(MARK(s(incr(head(pairs))))) MARK(head(s(head(head(nats))))) -> c16(MARK(s(head(head(nats))))) MARK(head(s(head(head(pairs))))) -> c16(MARK(s(head(head(pairs))))) MARK(head(s(tail(head(nats))))) -> c16(MARK(s(tail(head(nats))))) MARK(head(s(tail(head(pairs))))) -> c16(MARK(s(tail(head(pairs))))) MARK(head(s(head(cons(incr(y0), y1))))) -> c16(MARK(s(head(cons(incr(y0), y1))))) MARK(head(s(head(cons(head(y0), y1))))) -> c16(MARK(s(head(cons(head(y0), y1))))) MARK(head(s(head(cons(tail(y0), y1))))) -> c16(MARK(s(head(cons(tail(y0), y1))))) MARK(head(s(head(cons(cons(y0, y1), y2))))) -> c16(MARK(s(head(cons(cons(y0, y1), y2))))) MARK(head(s(head(cons(s(y0), y1))))) -> c16(MARK(s(head(cons(s(y0), y1))))) MARK(head(s(head(cons(y0, y1))))) -> c16(MARK(s(head(cons(y0, y1))))) MARK(head(s(incr(tail(nats))))) -> c16(MARK(s(incr(tail(nats))))) MARK(head(s(head(tail(nats))))) -> c16(MARK(s(head(tail(nats))))) MARK(head(s(tail(tail(nats))))) -> c16(MARK(s(tail(tail(nats))))) MARK(head(s(cons(cons(y0, y1), y2)))) -> c16(MARK(s(cons(cons(y0, y1), y2)))) MARK(head(s(cons(s(y0), y1)))) -> c16(MARK(s(cons(s(y0), y1)))) MARK(head(s(cons(incr(s(y0)), y1)))) -> c16(MARK(s(cons(incr(s(y0)), y1)))) MARK(head(s(cons(head(s(y0)), y1)))) -> c16(MARK(s(cons(head(s(y0)), y1)))) MARK(head(s(cons(tail(cons(y0, y1)), y2)))) -> c16(MARK(s(cons(tail(cons(y0, y1)), y2)))) MARK(head(s(cons(tail(s(y0)), y1)))) -> c16(MARK(s(cons(tail(s(y0)), y1)))) MARK(head(s(cons(incr(incr(incr(y0))), y1)))) -> c16(MARK(s(cons(incr(incr(incr(y0))), y1)))) MARK(head(s(cons(incr(incr(odds)), y0)))) -> c16(MARK(s(cons(incr(incr(odds)), y0)))) MARK(head(s(cons(incr(incr(head(y0))), y1)))) -> c16(MARK(s(cons(incr(incr(head(y0))), y1)))) MARK(head(s(cons(incr(incr(tail(y0))), y1)))) -> c16(MARK(s(cons(incr(incr(tail(y0))), y1)))) MARK(head(s(cons(incr(incr(cons(y0, y1))), y2)))) -> c16(MARK(s(cons(incr(incr(cons(y0, y1))), y2)))) MARK(head(s(cons(incr(incr(s(y0))), y1)))) -> c16(MARK(s(cons(incr(incr(s(y0))), y1)))) MARK(head(s(cons(incr(incr(y0)), y1)))) -> c16(MARK(s(cons(incr(incr(y0)), y1)))) MARK(head(s(cons(incr(head(incr(y0))), y1)))) -> c16(MARK(s(cons(incr(head(incr(y0))), y1)))) MARK(head(s(cons(incr(head(odds)), y0)))) -> c16(MARK(s(cons(incr(head(odds)), y0)))) MARK(head(s(cons(incr(head(head(y0))), y1)))) -> c16(MARK(s(cons(incr(head(head(y0))), y1)))) MARK(head(s(cons(incr(head(tail(y0))), y1)))) -> c16(MARK(s(cons(incr(head(tail(y0))), y1)))) MARK(head(s(cons(incr(head(cons(y0, y1))), y2)))) -> c16(MARK(s(cons(incr(head(cons(y0, y1))), y2)))) MARK(head(s(cons(incr(head(s(y0))), y1)))) -> c16(MARK(s(cons(incr(head(s(y0))), y1)))) MARK(head(s(cons(incr(head(y0)), y1)))) -> c16(MARK(s(cons(incr(head(y0)), y1)))) MARK(head(s(cons(incr(tail(incr(y0))), y1)))) -> c16(MARK(s(cons(incr(tail(incr(y0))), y1)))) MARK(head(s(cons(incr(tail(pairs)), y0)))) -> c16(MARK(s(cons(incr(tail(pairs)), y0)))) MARK(head(s(cons(incr(tail(odds)), y0)))) -> c16(MARK(s(cons(incr(tail(odds)), y0)))) MARK(head(s(cons(incr(tail(head(y0))), y1)))) -> c16(MARK(s(cons(incr(tail(head(y0))), y1)))) MARK(head(s(cons(incr(tail(tail(y0))), y1)))) -> c16(MARK(s(cons(incr(tail(tail(y0))), y1)))) MARK(head(s(cons(incr(tail(cons(y0, y1))), y2)))) -> c16(MARK(s(cons(incr(tail(cons(y0, y1))), y2)))) MARK(head(s(cons(incr(tail(s(y0))), y1)))) -> c16(MARK(s(cons(incr(tail(s(y0))), y1)))) MARK(head(s(cons(incr(tail(y0)), y1)))) -> c16(MARK(s(cons(incr(tail(y0)), y1)))) MARK(head(s(cons(head(incr(incr(y0))), y1)))) -> c16(MARK(s(cons(head(incr(incr(y0))), y1)))) MARK(head(s(cons(head(incr(odds)), y0)))) -> c16(MARK(s(cons(head(incr(odds)), y0)))) MARK(head(s(cons(head(incr(head(y0))), y1)))) -> c16(MARK(s(cons(head(incr(head(y0))), y1)))) MARK(head(s(cons(head(incr(tail(y0))), y1)))) -> c16(MARK(s(cons(head(incr(tail(y0))), y1)))) MARK(head(s(cons(head(incr(cons(y0, y1))), y2)))) -> c16(MARK(s(cons(head(incr(cons(y0, y1))), y2)))) MARK(head(s(cons(head(incr(s(y0))), y1)))) -> c16(MARK(s(cons(head(incr(s(y0))), y1)))) MARK(head(s(cons(head(incr(y0)), y1)))) -> c16(MARK(s(cons(head(incr(y0)), y1)))) MARK(head(s(cons(head(head(incr(y0))), y1)))) -> c16(MARK(s(cons(head(head(incr(y0))), y1)))) MARK(head(s(cons(head(head(odds)), y0)))) -> c16(MARK(s(cons(head(head(odds)), y0)))) MARK(head(s(cons(head(head(head(y0))), y1)))) -> c16(MARK(s(cons(head(head(head(y0))), y1)))) MARK(head(s(cons(head(head(tail(y0))), y1)))) -> c16(MARK(s(cons(head(head(tail(y0))), y1)))) MARK(head(s(cons(head(head(cons(y0, y1))), y2)))) -> c16(MARK(s(cons(head(head(cons(y0, y1))), y2)))) MARK(head(s(cons(head(head(s(y0))), y1)))) -> c16(MARK(s(cons(head(head(s(y0))), y1)))) MARK(head(s(cons(head(head(y0)), y1)))) -> c16(MARK(s(cons(head(head(y0)), y1)))) MARK(head(s(cons(head(tail(incr(y0))), y1)))) -> c16(MARK(s(cons(head(tail(incr(y0))), y1)))) MARK(head(s(cons(head(tail(pairs)), y0)))) -> c16(MARK(s(cons(head(tail(pairs)), y0)))) MARK(head(s(cons(head(tail(odds)), y0)))) -> c16(MARK(s(cons(head(tail(odds)), y0)))) MARK(head(s(cons(head(tail(head(y0))), y1)))) -> c16(MARK(s(cons(head(tail(head(y0))), y1)))) MARK(head(s(cons(head(tail(tail(y0))), y1)))) -> c16(MARK(s(cons(head(tail(tail(y0))), y1)))) MARK(head(s(cons(head(tail(cons(y0, y1))), y2)))) -> c16(MARK(s(cons(head(tail(cons(y0, y1))), y2)))) MARK(head(s(cons(head(tail(s(y0))), y1)))) -> c16(MARK(s(cons(head(tail(s(y0))), y1)))) MARK(head(s(cons(head(tail(y0)), y1)))) -> c16(MARK(s(cons(head(tail(y0)), y1)))) MARK(head(s(cons(tail(incr(incr(y0))), y1)))) -> c16(MARK(s(cons(tail(incr(incr(y0))), y1)))) MARK(head(s(cons(tail(incr(odds)), y0)))) -> c16(MARK(s(cons(tail(incr(odds)), y0)))) MARK(head(s(cons(tail(incr(head(y0))), y1)))) -> c16(MARK(s(cons(tail(incr(head(y0))), y1)))) MARK(head(s(cons(tail(incr(tail(y0))), y1)))) -> c16(MARK(s(cons(tail(incr(tail(y0))), y1)))) MARK(head(s(cons(tail(incr(cons(y0, y1))), y2)))) -> c16(MARK(s(cons(tail(incr(cons(y0, y1))), y2)))) MARK(head(s(cons(tail(incr(s(y0))), y1)))) -> c16(MARK(s(cons(tail(incr(s(y0))), y1)))) MARK(head(s(cons(tail(incr(y0)), y1)))) -> c16(MARK(s(cons(tail(incr(y0)), y1)))) MARK(head(s(cons(tail(head(incr(y0))), y1)))) -> c16(MARK(s(cons(tail(head(incr(y0))), y1)))) MARK(head(s(cons(tail(head(odds)), y0)))) -> c16(MARK(s(cons(tail(head(odds)), y0)))) MARK(head(s(cons(tail(head(head(y0))), y1)))) -> c16(MARK(s(cons(tail(head(head(y0))), y1)))) MARK(head(s(cons(tail(head(tail(y0))), y1)))) -> c16(MARK(s(cons(tail(head(tail(y0))), y1)))) MARK(head(s(cons(tail(head(cons(y0, y1))), y2)))) -> c16(MARK(s(cons(tail(head(cons(y0, y1))), y2)))) MARK(head(s(cons(tail(head(s(y0))), y1)))) -> c16(MARK(s(cons(tail(head(s(y0))), y1)))) MARK(head(s(cons(tail(head(y0)), y1)))) -> c16(MARK(s(cons(tail(head(y0)), y1)))) MARK(head(s(cons(tail(tail(incr(y0))), y1)))) -> c16(MARK(s(cons(tail(tail(incr(y0))), y1)))) MARK(head(s(cons(tail(tail(pairs)), y0)))) -> c16(MARK(s(cons(tail(tail(pairs)), y0)))) MARK(head(s(cons(tail(tail(odds)), y0)))) -> c16(MARK(s(cons(tail(tail(odds)), y0)))) MARK(head(s(cons(tail(tail(head(y0))), y1)))) -> c16(MARK(s(cons(tail(tail(head(y0))), y1)))) MARK(head(s(cons(tail(tail(tail(y0))), y1)))) -> c16(MARK(s(cons(tail(tail(tail(y0))), y1)))) MARK(head(s(cons(tail(tail(cons(y0, y1))), y2)))) -> c16(MARK(s(cons(tail(tail(cons(y0, y1))), y2)))) MARK(head(s(cons(tail(tail(s(y0))), y1)))) -> c16(MARK(s(cons(tail(tail(s(y0))), y1)))) MARK(head(s(cons(tail(tail(y0)), y1)))) -> c16(MARK(s(cons(tail(tail(y0)), y1)))) MARK(head(s(cons(tail(pairs), y0)))) -> c16(MARK(s(cons(tail(pairs), y0)))) MARK(head(s(cons(incr(incr(nats)), y0)))) -> c16(MARK(s(cons(incr(incr(nats)), y0)))) MARK(head(s(cons(incr(incr(pairs)), y0)))) -> c16(MARK(s(cons(incr(incr(pairs)), y0)))) MARK(head(s(cons(incr(odds), y0)))) -> c16(MARK(s(cons(incr(odds), y0)))) MARK(head(s(cons(head(incr(nats)), y0)))) -> c16(MARK(s(cons(head(incr(nats)), y0)))) MARK(head(s(cons(head(incr(pairs)), y0)))) -> c16(MARK(s(cons(head(incr(pairs)), y0)))) MARK(head(s(cons(head(odds), y0)))) -> c16(MARK(s(cons(head(odds), y0)))) MARK(head(s(cons(tail(incr(nats)), y0)))) -> c16(MARK(s(cons(tail(incr(nats)), y0)))) MARK(head(s(cons(tail(incr(pairs)), y0)))) -> c16(MARK(s(cons(tail(incr(pairs)), y0)))) MARK(head(s(cons(tail(odds), y0)))) -> c16(MARK(s(cons(tail(odds), y0)))) MARK(head(s(cons(incr(cons(nats, y0)), y1)))) -> c16(MARK(s(cons(incr(cons(nats, y0)), y1)))) MARK(head(s(cons(incr(cons(incr(y0), y1)), y2)))) -> c16(MARK(s(cons(incr(cons(incr(y0), y1)), y2)))) MARK(head(s(cons(incr(cons(pairs, y0)), y1)))) -> c16(MARK(s(cons(incr(cons(pairs, y0)), y1)))) MARK(head(s(cons(incr(cons(odds, y0)), y1)))) -> c16(MARK(s(cons(incr(cons(odds, y0)), y1)))) MARK(head(s(cons(incr(cons(head(y0), y1)), y2)))) -> c16(MARK(s(cons(incr(cons(head(y0), y1)), y2)))) MARK(head(s(cons(incr(cons(tail(y0), y1)), y2)))) -> c16(MARK(s(cons(incr(cons(tail(y0), y1)), y2)))) MARK(head(s(cons(incr(cons(cons(y0, y1), y2)), y3)))) -> c16(MARK(s(cons(incr(cons(cons(y0, y1), y2)), y3)))) MARK(head(s(cons(incr(cons(s(y0), y1)), y2)))) -> c16(MARK(s(cons(incr(cons(s(y0), y1)), y2)))) MARK(head(s(cons(incr(cons(y0, y1)), y2)))) -> c16(MARK(s(cons(incr(cons(y0, y1)), y2)))) MARK(head(s(cons(incr(cons(0, y0)), y1)))) -> c16(MARK(s(cons(incr(cons(0, y0)), y1)))) MARK(head(s(cons(incr(head(nats)), y0)))) -> c16(MARK(s(cons(incr(head(nats)), y0)))) MARK(head(s(cons(incr(head(pairs)), y0)))) -> c16(MARK(s(cons(incr(head(pairs)), y0)))) MARK(head(s(cons(head(head(nats)), y0)))) -> c16(MARK(s(cons(head(head(nats)), y0)))) MARK(head(s(cons(head(head(pairs)), y0)))) -> c16(MARK(s(cons(head(head(pairs)), y0)))) MARK(head(s(cons(tail(head(nats)), y0)))) -> c16(MARK(s(cons(tail(head(nats)), y0)))) MARK(head(s(cons(tail(head(pairs)), y0)))) -> c16(MARK(s(cons(tail(head(pairs)), y0)))) MARK(head(s(cons(head(cons(nats, y0)), y1)))) -> c16(MARK(s(cons(head(cons(nats, y0)), y1)))) MARK(head(s(cons(head(cons(incr(y0), y1)), y2)))) -> c16(MARK(s(cons(head(cons(incr(y0), y1)), y2)))) MARK(head(s(cons(head(cons(pairs, y0)), y1)))) -> c16(MARK(s(cons(head(cons(pairs, y0)), y1)))) MARK(head(s(cons(head(cons(odds, y0)), y1)))) -> c16(MARK(s(cons(head(cons(odds, y0)), y1)))) MARK(head(s(cons(head(cons(head(y0), y1)), y2)))) -> c16(MARK(s(cons(head(cons(head(y0), y1)), y2)))) MARK(head(s(cons(head(cons(tail(y0), y1)), y2)))) -> c16(MARK(s(cons(head(cons(tail(y0), y1)), y2)))) MARK(head(s(cons(head(cons(cons(y0, y1), y2)), y3)))) -> c16(MARK(s(cons(head(cons(cons(y0, y1), y2)), y3)))) MARK(head(s(cons(head(cons(s(y0), y1)), y2)))) -> c16(MARK(s(cons(head(cons(s(y0), y1)), y2)))) MARK(head(s(cons(head(cons(y0, y1)), y2)))) -> c16(MARK(s(cons(head(cons(y0, y1)), y2)))) MARK(head(s(cons(head(cons(0, y0)), y1)))) -> c16(MARK(s(cons(head(cons(0, y0)), y1)))) MARK(head(s(cons(incr(tail(nats)), y0)))) -> c16(MARK(s(cons(incr(tail(nats)), y0)))) MARK(head(s(cons(head(tail(nats)), y0)))) -> c16(MARK(s(cons(head(tail(nats)), y0)))) MARK(head(s(cons(tail(tail(nats)), y0)))) -> c16(MARK(s(cons(tail(tail(nats)), y0)))) MARK(head(s(incr(cons(nats, y0))))) -> c16(MARK(s(incr(cons(nats, y0))))) MARK(head(s(incr(cons(pairs, y0))))) -> c16(MARK(s(incr(cons(pairs, y0))))) MARK(head(s(incr(cons(odds, y0))))) -> c16(MARK(s(incr(cons(odds, y0))))) MARK(head(s(head(cons(nats, y0))))) -> c16(MARK(s(head(cons(nats, y0))))) MARK(head(s(head(cons(pairs, y0))))) -> c16(MARK(s(head(cons(pairs, y0))))) MARK(head(s(head(cons(odds, y0))))) -> c16(MARK(s(head(cons(odds, y0))))) MARK(tail(s(s(y0)))) -> c17(MARK(s(s(y0)))) MARK(tail(s(incr(s(y0))))) -> c17(MARK(s(incr(s(y0))))) MARK(tail(s(head(s(y0))))) -> c17(MARK(s(head(s(y0))))) MARK(tail(s(tail(cons(y0, y1))))) -> c17(MARK(s(tail(cons(y0, y1))))) MARK(tail(s(tail(s(y0))))) -> c17(MARK(s(tail(s(y0))))) MARK(tail(s(incr(incr(incr(y0)))))) -> c17(MARK(s(incr(incr(incr(y0)))))) MARK(tail(s(incr(incr(odds))))) -> c17(MARK(s(incr(incr(odds))))) MARK(tail(s(incr(incr(head(y0)))))) -> c17(MARK(s(incr(incr(head(y0)))))) MARK(tail(s(incr(incr(tail(y0)))))) -> c17(MARK(s(incr(incr(tail(y0)))))) MARK(tail(s(incr(incr(cons(y0, y1)))))) -> c17(MARK(s(incr(incr(cons(y0, y1)))))) MARK(tail(s(incr(incr(s(y0)))))) -> c17(MARK(s(incr(incr(s(y0)))))) MARK(tail(s(incr(incr(y0))))) -> c17(MARK(s(incr(incr(y0))))) MARK(tail(s(incr(head(incr(y0)))))) -> c17(MARK(s(incr(head(incr(y0)))))) MARK(tail(s(incr(head(odds))))) -> c17(MARK(s(incr(head(odds))))) MARK(tail(s(incr(head(head(y0)))))) -> c17(MARK(s(incr(head(head(y0)))))) MARK(tail(s(incr(head(tail(y0)))))) -> c17(MARK(s(incr(head(tail(y0)))))) MARK(tail(s(incr(head(cons(y0, y1)))))) -> c17(MARK(s(incr(head(cons(y0, y1)))))) MARK(tail(s(incr(head(s(y0)))))) -> c17(MARK(s(incr(head(s(y0)))))) MARK(tail(s(incr(head(y0))))) -> c17(MARK(s(incr(head(y0))))) MARK(tail(s(incr(tail(incr(y0)))))) -> c17(MARK(s(incr(tail(incr(y0)))))) MARK(tail(s(incr(tail(pairs))))) -> c17(MARK(s(incr(tail(pairs))))) MARK(tail(s(incr(tail(odds))))) -> c17(MARK(s(incr(tail(odds))))) MARK(tail(s(incr(tail(head(y0)))))) -> c17(MARK(s(incr(tail(head(y0)))))) MARK(tail(s(incr(tail(tail(y0)))))) -> c17(MARK(s(incr(tail(tail(y0)))))) MARK(tail(s(incr(tail(cons(y0, y1)))))) -> c17(MARK(s(incr(tail(cons(y0, y1)))))) MARK(tail(s(incr(tail(s(y0)))))) -> c17(MARK(s(incr(tail(s(y0)))))) MARK(tail(s(incr(tail(y0))))) -> c17(MARK(s(incr(tail(y0))))) MARK(tail(s(head(incr(incr(y0)))))) -> c17(MARK(s(head(incr(incr(y0)))))) MARK(tail(s(head(incr(odds))))) -> c17(MARK(s(head(incr(odds))))) MARK(tail(s(head(incr(head(y0)))))) -> c17(MARK(s(head(incr(head(y0)))))) MARK(tail(s(head(incr(tail(y0)))))) -> c17(MARK(s(head(incr(tail(y0)))))) MARK(tail(s(head(incr(cons(y0, y1)))))) -> c17(MARK(s(head(incr(cons(y0, y1)))))) MARK(tail(s(head(incr(s(y0)))))) -> c17(MARK(s(head(incr(s(y0)))))) MARK(tail(s(head(incr(y0))))) -> c17(MARK(s(head(incr(y0))))) MARK(tail(s(head(head(incr(y0)))))) -> c17(MARK(s(head(head(incr(y0)))))) MARK(tail(s(head(head(odds))))) -> c17(MARK(s(head(head(odds))))) MARK(tail(s(head(head(head(y0)))))) -> c17(MARK(s(head(head(head(y0)))))) MARK(tail(s(head(head(tail(y0)))))) -> c17(MARK(s(head(head(tail(y0)))))) MARK(tail(s(head(head(cons(y0, y1)))))) -> c17(MARK(s(head(head(cons(y0, y1)))))) MARK(tail(s(head(head(s(y0)))))) -> c17(MARK(s(head(head(s(y0)))))) MARK(tail(s(head(head(y0))))) -> c17(MARK(s(head(head(y0))))) MARK(tail(s(head(tail(incr(y0)))))) -> c17(MARK(s(head(tail(incr(y0)))))) MARK(tail(s(head(tail(pairs))))) -> c17(MARK(s(head(tail(pairs))))) MARK(tail(s(head(tail(odds))))) -> c17(MARK(s(head(tail(odds))))) MARK(tail(s(head(tail(head(y0)))))) -> c17(MARK(s(head(tail(head(y0)))))) MARK(tail(s(head(tail(tail(y0)))))) -> c17(MARK(s(head(tail(tail(y0)))))) MARK(tail(s(head(tail(cons(y0, y1)))))) -> c17(MARK(s(head(tail(cons(y0, y1)))))) MARK(tail(s(head(tail(s(y0)))))) -> c17(MARK(s(head(tail(s(y0)))))) MARK(tail(s(head(tail(y0))))) -> c17(MARK(s(head(tail(y0))))) MARK(tail(s(tail(incr(incr(y0)))))) -> c17(MARK(s(tail(incr(incr(y0)))))) MARK(tail(s(tail(incr(odds))))) -> c17(MARK(s(tail(incr(odds))))) MARK(tail(s(tail(incr(head(y0)))))) -> c17(MARK(s(tail(incr(head(y0)))))) MARK(tail(s(tail(incr(tail(y0)))))) -> c17(MARK(s(tail(incr(tail(y0)))))) MARK(tail(s(tail(incr(cons(y0, y1)))))) -> c17(MARK(s(tail(incr(cons(y0, y1)))))) MARK(tail(s(tail(incr(s(y0)))))) -> c17(MARK(s(tail(incr(s(y0)))))) MARK(tail(s(tail(incr(y0))))) -> c17(MARK(s(tail(incr(y0))))) MARK(tail(s(tail(head(incr(y0)))))) -> c17(MARK(s(tail(head(incr(y0)))))) MARK(tail(s(tail(head(odds))))) -> c17(MARK(s(tail(head(odds))))) MARK(tail(s(tail(head(head(y0)))))) -> c17(MARK(s(tail(head(head(y0)))))) MARK(tail(s(tail(head(tail(y0)))))) -> c17(MARK(s(tail(head(tail(y0)))))) MARK(tail(s(tail(head(cons(y0, y1)))))) -> c17(MARK(s(tail(head(cons(y0, y1)))))) MARK(tail(s(tail(head(s(y0)))))) -> c17(MARK(s(tail(head(s(y0)))))) MARK(tail(s(tail(head(y0))))) -> c17(MARK(s(tail(head(y0))))) MARK(tail(s(tail(tail(incr(y0)))))) -> c17(MARK(s(tail(tail(incr(y0)))))) MARK(tail(s(tail(tail(pairs))))) -> c17(MARK(s(tail(tail(pairs))))) MARK(tail(s(tail(tail(odds))))) -> c17(MARK(s(tail(tail(odds))))) MARK(tail(s(tail(tail(head(y0)))))) -> c17(MARK(s(tail(tail(head(y0)))))) MARK(tail(s(tail(tail(tail(y0)))))) -> c17(MARK(s(tail(tail(tail(y0)))))) MARK(tail(s(tail(tail(cons(y0, y1)))))) -> c17(MARK(s(tail(tail(cons(y0, y1)))))) MARK(tail(s(tail(tail(s(y0)))))) -> c17(MARK(s(tail(tail(s(y0)))))) MARK(tail(s(tail(tail(y0))))) -> c17(MARK(s(tail(tail(y0))))) MARK(tail(s(tail(pairs)))) -> c17(MARK(s(tail(pairs)))) MARK(tail(s(incr(incr(nats))))) -> c17(MARK(s(incr(incr(nats))))) MARK(tail(s(incr(incr(pairs))))) -> c17(MARK(s(incr(incr(pairs))))) MARK(tail(s(incr(odds)))) -> c17(MARK(s(incr(odds)))) MARK(tail(s(head(incr(nats))))) -> c17(MARK(s(head(incr(nats))))) MARK(tail(s(head(incr(pairs))))) -> c17(MARK(s(head(incr(pairs))))) MARK(tail(s(head(odds)))) -> c17(MARK(s(head(odds)))) MARK(tail(s(tail(incr(nats))))) -> c17(MARK(s(tail(incr(nats))))) MARK(tail(s(tail(incr(pairs))))) -> c17(MARK(s(tail(incr(pairs))))) MARK(tail(s(tail(odds)))) -> c17(MARK(s(tail(odds)))) MARK(tail(s(incr(cons(incr(y0), y1))))) -> c17(MARK(s(incr(cons(incr(y0), y1))))) MARK(tail(s(incr(cons(head(y0), y1))))) -> c17(MARK(s(incr(cons(head(y0), y1))))) MARK(tail(s(incr(cons(tail(y0), y1))))) -> c17(MARK(s(incr(cons(tail(y0), y1))))) MARK(tail(s(incr(cons(cons(y0, y1), y2))))) -> c17(MARK(s(incr(cons(cons(y0, y1), y2))))) MARK(tail(s(incr(cons(s(y0), y1))))) -> c17(MARK(s(incr(cons(s(y0), y1))))) MARK(tail(s(incr(cons(y0, y1))))) -> c17(MARK(s(incr(cons(y0, y1))))) MARK(tail(s(incr(head(nats))))) -> c17(MARK(s(incr(head(nats))))) MARK(tail(s(incr(head(pairs))))) -> c17(MARK(s(incr(head(pairs))))) MARK(tail(s(head(head(nats))))) -> c17(MARK(s(head(head(nats))))) MARK(tail(s(head(head(pairs))))) -> c17(MARK(s(head(head(pairs))))) MARK(tail(s(tail(head(nats))))) -> c17(MARK(s(tail(head(nats))))) MARK(tail(s(tail(head(pairs))))) -> c17(MARK(s(tail(head(pairs))))) MARK(tail(s(head(cons(incr(y0), y1))))) -> c17(MARK(s(head(cons(incr(y0), y1))))) MARK(tail(s(head(cons(head(y0), y1))))) -> c17(MARK(s(head(cons(head(y0), y1))))) MARK(tail(s(head(cons(tail(y0), y1))))) -> c17(MARK(s(head(cons(tail(y0), y1))))) MARK(tail(s(head(cons(cons(y0, y1), y2))))) -> c17(MARK(s(head(cons(cons(y0, y1), y2))))) MARK(tail(s(head(cons(s(y0), y1))))) -> c17(MARK(s(head(cons(s(y0), y1))))) MARK(tail(s(head(cons(y0, y1))))) -> c17(MARK(s(head(cons(y0, y1))))) MARK(tail(s(incr(tail(nats))))) -> c17(MARK(s(incr(tail(nats))))) MARK(tail(s(head(tail(nats))))) -> c17(MARK(s(head(tail(nats))))) MARK(tail(s(tail(tail(nats))))) -> c17(MARK(s(tail(tail(nats))))) MARK(tail(s(cons(cons(y0, y1), y2)))) -> c17(MARK(s(cons(cons(y0, y1), y2)))) MARK(tail(s(cons(s(y0), y1)))) -> c17(MARK(s(cons(s(y0), y1)))) MARK(tail(s(cons(incr(s(y0)), y1)))) -> c17(MARK(s(cons(incr(s(y0)), y1)))) MARK(tail(s(cons(head(s(y0)), y1)))) -> c17(MARK(s(cons(head(s(y0)), y1)))) MARK(tail(s(cons(tail(cons(y0, y1)), y2)))) -> c17(MARK(s(cons(tail(cons(y0, y1)), y2)))) MARK(tail(s(cons(tail(s(y0)), y1)))) -> c17(MARK(s(cons(tail(s(y0)), y1)))) MARK(tail(s(cons(incr(incr(incr(y0))), y1)))) -> c17(MARK(s(cons(incr(incr(incr(y0))), y1)))) MARK(tail(s(cons(incr(incr(odds)), y0)))) -> c17(MARK(s(cons(incr(incr(odds)), y0)))) MARK(tail(s(cons(incr(incr(head(y0))), y1)))) -> c17(MARK(s(cons(incr(incr(head(y0))), y1)))) MARK(tail(s(cons(incr(incr(tail(y0))), y1)))) -> c17(MARK(s(cons(incr(incr(tail(y0))), y1)))) MARK(tail(s(cons(incr(incr(cons(y0, y1))), y2)))) -> c17(MARK(s(cons(incr(incr(cons(y0, y1))), y2)))) MARK(tail(s(cons(incr(incr(s(y0))), y1)))) -> c17(MARK(s(cons(incr(incr(s(y0))), y1)))) MARK(tail(s(cons(incr(incr(y0)), y1)))) -> c17(MARK(s(cons(incr(incr(y0)), y1)))) MARK(tail(s(cons(incr(head(incr(y0))), y1)))) -> c17(MARK(s(cons(incr(head(incr(y0))), y1)))) MARK(tail(s(cons(incr(head(odds)), y0)))) -> c17(MARK(s(cons(incr(head(odds)), y0)))) MARK(tail(s(cons(incr(head(head(y0))), y1)))) -> c17(MARK(s(cons(incr(head(head(y0))), y1)))) MARK(tail(s(cons(incr(head(tail(y0))), y1)))) -> c17(MARK(s(cons(incr(head(tail(y0))), y1)))) MARK(tail(s(cons(incr(head(cons(y0, y1))), y2)))) -> c17(MARK(s(cons(incr(head(cons(y0, y1))), y2)))) MARK(tail(s(cons(incr(head(s(y0))), y1)))) -> c17(MARK(s(cons(incr(head(s(y0))), y1)))) MARK(tail(s(cons(incr(head(y0)), y1)))) -> c17(MARK(s(cons(incr(head(y0)), y1)))) MARK(tail(s(cons(incr(tail(incr(y0))), y1)))) -> c17(MARK(s(cons(incr(tail(incr(y0))), y1)))) MARK(tail(s(cons(incr(tail(pairs)), y0)))) -> c17(MARK(s(cons(incr(tail(pairs)), y0)))) MARK(tail(s(cons(incr(tail(odds)), y0)))) -> c17(MARK(s(cons(incr(tail(odds)), y0)))) MARK(tail(s(cons(incr(tail(head(y0))), y1)))) -> c17(MARK(s(cons(incr(tail(head(y0))), y1)))) MARK(tail(s(cons(incr(tail(tail(y0))), y1)))) -> c17(MARK(s(cons(incr(tail(tail(y0))), y1)))) MARK(tail(s(cons(incr(tail(cons(y0, y1))), y2)))) -> c17(MARK(s(cons(incr(tail(cons(y0, y1))), y2)))) MARK(tail(s(cons(incr(tail(s(y0))), y1)))) -> c17(MARK(s(cons(incr(tail(s(y0))), y1)))) MARK(tail(s(cons(incr(tail(y0)), y1)))) -> c17(MARK(s(cons(incr(tail(y0)), y1)))) MARK(tail(s(cons(head(incr(incr(y0))), y1)))) -> c17(MARK(s(cons(head(incr(incr(y0))), y1)))) MARK(tail(s(cons(head(incr(odds)), y0)))) -> c17(MARK(s(cons(head(incr(odds)), y0)))) MARK(tail(s(cons(head(incr(head(y0))), y1)))) -> c17(MARK(s(cons(head(incr(head(y0))), y1)))) MARK(tail(s(cons(head(incr(tail(y0))), y1)))) -> c17(MARK(s(cons(head(incr(tail(y0))), y1)))) MARK(tail(s(cons(head(incr(cons(y0, y1))), y2)))) -> c17(MARK(s(cons(head(incr(cons(y0, y1))), y2)))) MARK(tail(s(cons(head(incr(s(y0))), y1)))) -> c17(MARK(s(cons(head(incr(s(y0))), y1)))) MARK(tail(s(cons(head(incr(y0)), y1)))) -> c17(MARK(s(cons(head(incr(y0)), y1)))) MARK(tail(s(cons(head(head(incr(y0))), y1)))) -> c17(MARK(s(cons(head(head(incr(y0))), y1)))) MARK(tail(s(cons(head(head(odds)), y0)))) -> c17(MARK(s(cons(head(head(odds)), y0)))) MARK(tail(s(cons(head(head(head(y0))), y1)))) -> c17(MARK(s(cons(head(head(head(y0))), y1)))) MARK(tail(s(cons(head(head(tail(y0))), y1)))) -> c17(MARK(s(cons(head(head(tail(y0))), y1)))) MARK(tail(s(cons(head(head(cons(y0, y1))), y2)))) -> c17(MARK(s(cons(head(head(cons(y0, y1))), y2)))) MARK(tail(s(cons(head(head(s(y0))), y1)))) -> c17(MARK(s(cons(head(head(s(y0))), y1)))) MARK(tail(s(cons(head(head(y0)), y1)))) -> c17(MARK(s(cons(head(head(y0)), y1)))) MARK(tail(s(cons(head(tail(incr(y0))), y1)))) -> c17(MARK(s(cons(head(tail(incr(y0))), y1)))) MARK(tail(s(cons(head(tail(pairs)), y0)))) -> c17(MARK(s(cons(head(tail(pairs)), y0)))) MARK(tail(s(cons(head(tail(odds)), y0)))) -> c17(MARK(s(cons(head(tail(odds)), y0)))) MARK(tail(s(cons(head(tail(head(y0))), y1)))) -> c17(MARK(s(cons(head(tail(head(y0))), y1)))) MARK(tail(s(cons(head(tail(tail(y0))), y1)))) -> c17(MARK(s(cons(head(tail(tail(y0))), y1)))) MARK(tail(s(cons(head(tail(cons(y0, y1))), y2)))) -> c17(MARK(s(cons(head(tail(cons(y0, y1))), y2)))) MARK(tail(s(cons(head(tail(s(y0))), y1)))) -> c17(MARK(s(cons(head(tail(s(y0))), y1)))) MARK(tail(s(cons(head(tail(y0)), y1)))) -> c17(MARK(s(cons(head(tail(y0)), y1)))) MARK(tail(s(cons(tail(incr(incr(y0))), y1)))) -> c17(MARK(s(cons(tail(incr(incr(y0))), y1)))) MARK(tail(s(cons(tail(incr(odds)), y0)))) -> c17(MARK(s(cons(tail(incr(odds)), y0)))) MARK(tail(s(cons(tail(incr(head(y0))), y1)))) -> c17(MARK(s(cons(tail(incr(head(y0))), y1)))) MARK(tail(s(cons(tail(incr(tail(y0))), y1)))) -> c17(MARK(s(cons(tail(incr(tail(y0))), y1)))) MARK(tail(s(cons(tail(incr(cons(y0, y1))), y2)))) -> c17(MARK(s(cons(tail(incr(cons(y0, y1))), y2)))) MARK(tail(s(cons(tail(incr(s(y0))), y1)))) -> c17(MARK(s(cons(tail(incr(s(y0))), y1)))) MARK(tail(s(cons(tail(incr(y0)), y1)))) -> c17(MARK(s(cons(tail(incr(y0)), y1)))) MARK(tail(s(cons(tail(head(incr(y0))), y1)))) -> c17(MARK(s(cons(tail(head(incr(y0))), y1)))) MARK(tail(s(cons(tail(head(odds)), y0)))) -> c17(MARK(s(cons(tail(head(odds)), y0)))) MARK(tail(s(cons(tail(head(head(y0))), y1)))) -> c17(MARK(s(cons(tail(head(head(y0))), y1)))) MARK(tail(s(cons(tail(head(tail(y0))), y1)))) -> c17(MARK(s(cons(tail(head(tail(y0))), y1)))) MARK(tail(s(cons(tail(head(cons(y0, y1))), y2)))) -> c17(MARK(s(cons(tail(head(cons(y0, y1))), y2)))) MARK(tail(s(cons(tail(head(s(y0))), y1)))) -> c17(MARK(s(cons(tail(head(s(y0))), y1)))) MARK(tail(s(cons(tail(head(y0)), y1)))) -> c17(MARK(s(cons(tail(head(y0)), y1)))) MARK(tail(s(cons(tail(tail(incr(y0))), y1)))) -> c17(MARK(s(cons(tail(tail(incr(y0))), y1)))) MARK(tail(s(cons(tail(tail(pairs)), y0)))) -> c17(MARK(s(cons(tail(tail(pairs)), y0)))) MARK(tail(s(cons(tail(tail(odds)), y0)))) -> c17(MARK(s(cons(tail(tail(odds)), y0)))) MARK(tail(s(cons(tail(tail(head(y0))), y1)))) -> c17(MARK(s(cons(tail(tail(head(y0))), y1)))) MARK(tail(s(cons(tail(tail(tail(y0))), y1)))) -> c17(MARK(s(cons(tail(tail(tail(y0))), y1)))) MARK(tail(s(cons(tail(tail(cons(y0, y1))), y2)))) -> c17(MARK(s(cons(tail(tail(cons(y0, y1))), y2)))) MARK(tail(s(cons(tail(tail(s(y0))), y1)))) -> c17(MARK(s(cons(tail(tail(s(y0))), y1)))) MARK(tail(s(cons(tail(tail(y0)), y1)))) -> c17(MARK(s(cons(tail(tail(y0)), y1)))) MARK(tail(s(cons(tail(pairs), y0)))) -> c17(MARK(s(cons(tail(pairs), y0)))) MARK(tail(s(cons(incr(incr(nats)), y0)))) -> c17(MARK(s(cons(incr(incr(nats)), y0)))) MARK(tail(s(cons(incr(incr(pairs)), y0)))) -> c17(MARK(s(cons(incr(incr(pairs)), y0)))) MARK(tail(s(cons(incr(odds), y0)))) -> c17(MARK(s(cons(incr(odds), y0)))) MARK(tail(s(cons(head(incr(nats)), y0)))) -> c17(MARK(s(cons(head(incr(nats)), y0)))) MARK(tail(s(cons(head(incr(pairs)), y0)))) -> c17(MARK(s(cons(head(incr(pairs)), y0)))) MARK(tail(s(cons(head(odds), y0)))) -> c17(MARK(s(cons(head(odds), y0)))) MARK(tail(s(cons(tail(incr(nats)), y0)))) -> c17(MARK(s(cons(tail(incr(nats)), y0)))) MARK(tail(s(cons(tail(incr(pairs)), y0)))) -> c17(MARK(s(cons(tail(incr(pairs)), y0)))) MARK(tail(s(cons(tail(odds), y0)))) -> c17(MARK(s(cons(tail(odds), y0)))) MARK(tail(s(cons(incr(cons(nats, y0)), y1)))) -> c17(MARK(s(cons(incr(cons(nats, y0)), y1)))) MARK(tail(s(cons(incr(cons(incr(y0), y1)), y2)))) -> c17(MARK(s(cons(incr(cons(incr(y0), y1)), y2)))) MARK(tail(s(cons(incr(cons(pairs, y0)), y1)))) -> c17(MARK(s(cons(incr(cons(pairs, y0)), y1)))) MARK(tail(s(cons(incr(cons(odds, y0)), y1)))) -> c17(MARK(s(cons(incr(cons(odds, y0)), y1)))) MARK(tail(s(cons(incr(cons(head(y0), y1)), y2)))) -> c17(MARK(s(cons(incr(cons(head(y0), y1)), y2)))) MARK(tail(s(cons(incr(cons(tail(y0), y1)), y2)))) -> c17(MARK(s(cons(incr(cons(tail(y0), y1)), y2)))) MARK(tail(s(cons(incr(cons(cons(y0, y1), y2)), y3)))) -> c17(MARK(s(cons(incr(cons(cons(y0, y1), y2)), y3)))) MARK(tail(s(cons(incr(cons(s(y0), y1)), y2)))) -> c17(MARK(s(cons(incr(cons(s(y0), y1)), y2)))) MARK(tail(s(cons(incr(cons(y0, y1)), y2)))) -> c17(MARK(s(cons(incr(cons(y0, y1)), y2)))) MARK(tail(s(cons(incr(cons(0, y0)), y1)))) -> c17(MARK(s(cons(incr(cons(0, y0)), y1)))) MARK(tail(s(cons(incr(head(nats)), y0)))) -> c17(MARK(s(cons(incr(head(nats)), y0)))) MARK(tail(s(cons(incr(head(pairs)), y0)))) -> c17(MARK(s(cons(incr(head(pairs)), y0)))) MARK(tail(s(cons(head(head(nats)), y0)))) -> c17(MARK(s(cons(head(head(nats)), y0)))) MARK(tail(s(cons(head(head(pairs)), y0)))) -> c17(MARK(s(cons(head(head(pairs)), y0)))) MARK(tail(s(cons(tail(head(nats)), y0)))) -> c17(MARK(s(cons(tail(head(nats)), y0)))) MARK(tail(s(cons(tail(head(pairs)), y0)))) -> c17(MARK(s(cons(tail(head(pairs)), y0)))) MARK(tail(s(cons(head(cons(nats, y0)), y1)))) -> c17(MARK(s(cons(head(cons(nats, y0)), y1)))) MARK(tail(s(cons(head(cons(incr(y0), y1)), y2)))) -> c17(MARK(s(cons(head(cons(incr(y0), y1)), y2)))) MARK(tail(s(cons(head(cons(pairs, y0)), y1)))) -> c17(MARK(s(cons(head(cons(pairs, y0)), y1)))) MARK(tail(s(cons(head(cons(odds, y0)), y1)))) -> c17(MARK(s(cons(head(cons(odds, y0)), y1)))) MARK(tail(s(cons(head(cons(head(y0), y1)), y2)))) -> c17(MARK(s(cons(head(cons(head(y0), y1)), y2)))) MARK(tail(s(cons(head(cons(tail(y0), y1)), y2)))) -> c17(MARK(s(cons(head(cons(tail(y0), y1)), y2)))) MARK(tail(s(cons(head(cons(cons(y0, y1), y2)), y3)))) -> c17(MARK(s(cons(head(cons(cons(y0, y1), y2)), y3)))) MARK(tail(s(cons(head(cons(s(y0), y1)), y2)))) -> c17(MARK(s(cons(head(cons(s(y0), y1)), y2)))) MARK(tail(s(cons(head(cons(y0, y1)), y2)))) -> c17(MARK(s(cons(head(cons(y0, y1)), y2)))) MARK(tail(s(cons(head(cons(0, y0)), y1)))) -> c17(MARK(s(cons(head(cons(0, y0)), y1)))) MARK(tail(s(cons(incr(tail(nats)), y0)))) -> c17(MARK(s(cons(incr(tail(nats)), y0)))) MARK(tail(s(cons(head(tail(nats)), y0)))) -> c17(MARK(s(cons(head(tail(nats)), y0)))) MARK(tail(s(cons(tail(tail(nats)), y0)))) -> c17(MARK(s(cons(tail(tail(nats)), y0)))) MARK(tail(s(incr(cons(nats, y0))))) -> c17(MARK(s(incr(cons(nats, y0))))) MARK(tail(s(incr(cons(pairs, y0))))) -> c17(MARK(s(incr(cons(pairs, y0))))) MARK(tail(s(incr(cons(odds, y0))))) -> c17(MARK(s(incr(cons(odds, y0))))) MARK(tail(s(head(cons(nats, y0))))) -> c17(MARK(s(head(cons(nats, y0))))) MARK(tail(s(head(cons(pairs, y0))))) -> c17(MARK(s(head(cons(pairs, y0))))) MARK(tail(s(head(cons(odds, y0))))) -> c17(MARK(s(head(cons(odds, y0))))) MARK(incr(incr(incr(incr(y0))))) -> c13(MARK(incr(incr(incr(y0))))) MARK(incr(incr(incr(odds)))) -> c13(MARK(incr(incr(odds)))) MARK(incr(incr(incr(head(y0))))) -> c13(MARK(incr(incr(head(y0))))) MARK(incr(incr(incr(tail(y0))))) -> c13(MARK(incr(incr(tail(y0))))) MARK(incr(incr(incr(cons(y0, y1))))) -> c13(MARK(incr(incr(cons(y0, y1))))) MARK(incr(incr(incr(s(y0))))) -> c13(MARK(incr(incr(s(y0))))) MARK(incr(incr(incr(y0)))) -> c13(MARK(incr(incr(y0)))) MARK(incr(incr(head(incr(y0))))) -> c13(MARK(incr(head(incr(y0))))) MARK(incr(incr(head(odds)))) -> c13(MARK(incr(head(odds)))) MARK(incr(incr(head(head(y0))))) -> c13(MARK(incr(head(head(y0))))) MARK(incr(incr(head(tail(y0))))) -> c13(MARK(incr(head(tail(y0))))) MARK(incr(incr(head(cons(y0, y1))))) -> c13(MARK(incr(head(cons(y0, y1))))) MARK(incr(incr(head(s(y0))))) -> c13(MARK(incr(head(s(y0))))) MARK(incr(incr(head(y0)))) -> c13(MARK(incr(head(y0)))) MARK(incr(incr(tail(incr(y0))))) -> c13(MARK(incr(tail(incr(y0))))) MARK(incr(incr(tail(pairs)))) -> c13(MARK(incr(tail(pairs)))) MARK(incr(incr(tail(odds)))) -> c13(MARK(incr(tail(odds)))) MARK(incr(incr(tail(head(y0))))) -> c13(MARK(incr(tail(head(y0))))) MARK(incr(incr(tail(tail(y0))))) -> c13(MARK(incr(tail(tail(y0))))) MARK(incr(incr(tail(cons(y0, y1))))) -> c13(MARK(incr(tail(cons(y0, y1))))) MARK(incr(incr(tail(s(y0))))) -> c13(MARK(incr(tail(s(y0))))) MARK(incr(incr(tail(y0)))) -> c13(MARK(incr(tail(y0)))) MARK(incr(incr(incr(nats)))) -> c13(MARK(incr(incr(nats)))) MARK(incr(incr(incr(pairs)))) -> c13(MARK(incr(incr(pairs)))) MARK(incr(incr(odds))) -> c13(MARK(incr(odds))) MARK(incr(incr(cons(incr(y0), y1)))) -> c13(MARK(incr(cons(incr(y0), y1)))) MARK(incr(incr(cons(head(y0), y1)))) -> c13(MARK(incr(cons(head(y0), y1)))) MARK(incr(incr(cons(tail(y0), y1)))) -> c13(MARK(incr(cons(tail(y0), y1)))) MARK(incr(incr(cons(cons(y0, y1), y2)))) -> c13(MARK(incr(cons(cons(y0, y1), y2)))) MARK(incr(incr(cons(s(y0), y1)))) -> c13(MARK(incr(cons(s(y0), y1)))) MARK(incr(incr(cons(y0, y1)))) -> c13(MARK(incr(cons(y0, y1)))) MARK(incr(incr(head(nats)))) -> c13(MARK(incr(head(nats)))) MARK(incr(incr(head(pairs)))) -> c13(MARK(incr(head(pairs)))) MARK(incr(incr(tail(nats)))) -> c13(MARK(incr(tail(nats)))) MARK(incr(incr(cons(nats, y0)))) -> c13(MARK(incr(cons(nats, y0)))) MARK(incr(incr(cons(pairs, y0)))) -> c13(MARK(incr(cons(pairs, y0)))) MARK(incr(incr(cons(odds, y0)))) -> c13(MARK(incr(cons(odds, y0)))) MARK(incr(incr(s(s(y0))))) -> c13(MARK(incr(s(s(y0))))) MARK(incr(incr(s(incr(s(y0)))))) -> c13(MARK(incr(s(incr(s(y0)))))) MARK(incr(incr(s(head(s(y0)))))) -> c13(MARK(incr(s(head(s(y0)))))) MARK(incr(incr(s(tail(cons(y0, y1)))))) -> c13(MARK(incr(s(tail(cons(y0, y1)))))) MARK(incr(incr(s(tail(s(y0)))))) -> c13(MARK(incr(s(tail(s(y0)))))) MARK(incr(incr(s(incr(incr(incr(y0))))))) -> c13(MARK(incr(s(incr(incr(incr(y0))))))) MARK(incr(incr(s(incr(incr(odds)))))) -> c13(MARK(incr(s(incr(incr(odds)))))) MARK(incr(incr(s(incr(incr(head(y0))))))) -> c13(MARK(incr(s(incr(incr(head(y0))))))) MARK(incr(incr(s(incr(incr(tail(y0))))))) -> c13(MARK(incr(s(incr(incr(tail(y0))))))) MARK(incr(incr(s(incr(incr(cons(y0, y1))))))) -> c13(MARK(incr(s(incr(incr(cons(y0, y1))))))) MARK(incr(incr(s(incr(incr(s(y0))))))) -> c13(MARK(incr(s(incr(incr(s(y0))))))) MARK(incr(incr(s(incr(incr(y0)))))) -> c13(MARK(incr(s(incr(incr(y0)))))) MARK(incr(incr(s(incr(head(incr(y0))))))) -> c13(MARK(incr(s(incr(head(incr(y0))))))) MARK(incr(incr(s(incr(head(odds)))))) -> c13(MARK(incr(s(incr(head(odds)))))) MARK(incr(incr(s(incr(head(head(y0))))))) -> c13(MARK(incr(s(incr(head(head(y0))))))) MARK(incr(incr(s(incr(head(tail(y0))))))) -> c13(MARK(incr(s(incr(head(tail(y0))))))) MARK(incr(incr(s(incr(head(cons(y0, y1))))))) -> c13(MARK(incr(s(incr(head(cons(y0, y1))))))) MARK(incr(incr(s(incr(head(s(y0))))))) -> c13(MARK(incr(s(incr(head(s(y0))))))) MARK(incr(incr(s(incr(head(y0)))))) -> c13(MARK(incr(s(incr(head(y0)))))) MARK(incr(incr(s(incr(tail(incr(y0))))))) -> c13(MARK(incr(s(incr(tail(incr(y0))))))) MARK(incr(incr(s(incr(tail(pairs)))))) -> c13(MARK(incr(s(incr(tail(pairs)))))) MARK(incr(incr(s(incr(tail(odds)))))) -> c13(MARK(incr(s(incr(tail(odds)))))) MARK(incr(incr(s(incr(tail(head(y0))))))) -> c13(MARK(incr(s(incr(tail(head(y0))))))) MARK(incr(incr(s(incr(tail(tail(y0))))))) -> c13(MARK(incr(s(incr(tail(tail(y0))))))) MARK(incr(incr(s(incr(tail(cons(y0, y1))))))) -> c13(MARK(incr(s(incr(tail(cons(y0, y1))))))) MARK(incr(incr(s(incr(tail(s(y0))))))) -> c13(MARK(incr(s(incr(tail(s(y0))))))) MARK(incr(incr(s(incr(tail(y0)))))) -> c13(MARK(incr(s(incr(tail(y0)))))) MARK(incr(incr(s(head(incr(incr(y0))))))) -> c13(MARK(incr(s(head(incr(incr(y0))))))) MARK(incr(incr(s(head(incr(odds)))))) -> c13(MARK(incr(s(head(incr(odds)))))) MARK(incr(incr(s(head(incr(head(y0))))))) -> c13(MARK(incr(s(head(incr(head(y0))))))) MARK(incr(incr(s(head(incr(tail(y0))))))) -> c13(MARK(incr(s(head(incr(tail(y0))))))) MARK(incr(incr(s(head(incr(cons(y0, y1))))))) -> c13(MARK(incr(s(head(incr(cons(y0, y1))))))) MARK(incr(incr(s(head(incr(s(y0))))))) -> c13(MARK(incr(s(head(incr(s(y0))))))) MARK(incr(incr(s(head(incr(y0)))))) -> c13(MARK(incr(s(head(incr(y0)))))) MARK(incr(incr(s(head(head(incr(y0))))))) -> c13(MARK(incr(s(head(head(incr(y0))))))) MARK(incr(incr(s(head(head(odds)))))) -> c13(MARK(incr(s(head(head(odds)))))) MARK(incr(incr(s(head(head(head(y0))))))) -> c13(MARK(incr(s(head(head(head(y0))))))) MARK(incr(incr(s(head(head(tail(y0))))))) -> c13(MARK(incr(s(head(head(tail(y0))))))) MARK(incr(incr(s(head(head(cons(y0, y1))))))) -> c13(MARK(incr(s(head(head(cons(y0, y1))))))) MARK(incr(incr(s(head(head(s(y0))))))) -> c13(MARK(incr(s(head(head(s(y0))))))) MARK(incr(incr(s(head(head(y0)))))) -> c13(MARK(incr(s(head(head(y0)))))) MARK(incr(incr(s(head(tail(incr(y0))))))) -> c13(MARK(incr(s(head(tail(incr(y0))))))) MARK(incr(incr(s(head(tail(pairs)))))) -> c13(MARK(incr(s(head(tail(pairs)))))) MARK(incr(incr(s(head(tail(odds)))))) -> c13(MARK(incr(s(head(tail(odds)))))) MARK(incr(incr(s(head(tail(head(y0))))))) -> c13(MARK(incr(s(head(tail(head(y0))))))) MARK(incr(incr(s(head(tail(tail(y0))))))) -> c13(MARK(incr(s(head(tail(tail(y0))))))) MARK(incr(incr(s(head(tail(cons(y0, y1))))))) -> c13(MARK(incr(s(head(tail(cons(y0, y1))))))) MARK(incr(incr(s(head(tail(s(y0))))))) -> c13(MARK(incr(s(head(tail(s(y0))))))) MARK(incr(incr(s(head(tail(y0)))))) -> c13(MARK(incr(s(head(tail(y0)))))) MARK(incr(incr(s(tail(incr(incr(y0))))))) -> c13(MARK(incr(s(tail(incr(incr(y0))))))) MARK(incr(incr(s(tail(incr(odds)))))) -> c13(MARK(incr(s(tail(incr(odds)))))) MARK(incr(incr(s(tail(incr(head(y0))))))) -> c13(MARK(incr(s(tail(incr(head(y0))))))) MARK(incr(incr(s(tail(incr(tail(y0))))))) -> c13(MARK(incr(s(tail(incr(tail(y0))))))) MARK(incr(incr(s(tail(incr(cons(y0, y1))))))) -> c13(MARK(incr(s(tail(incr(cons(y0, y1))))))) MARK(incr(incr(s(tail(incr(s(y0))))))) -> c13(MARK(incr(s(tail(incr(s(y0))))))) MARK(incr(incr(s(tail(incr(y0)))))) -> c13(MARK(incr(s(tail(incr(y0)))))) MARK(incr(incr(s(tail(head(incr(y0))))))) -> c13(MARK(incr(s(tail(head(incr(y0))))))) MARK(incr(incr(s(tail(head(odds)))))) -> c13(MARK(incr(s(tail(head(odds)))))) MARK(incr(incr(s(tail(head(head(y0))))))) -> c13(MARK(incr(s(tail(head(head(y0))))))) MARK(incr(incr(s(tail(head(tail(y0))))))) -> c13(MARK(incr(s(tail(head(tail(y0))))))) MARK(incr(incr(s(tail(head(cons(y0, y1))))))) -> c13(MARK(incr(s(tail(head(cons(y0, y1))))))) MARK(incr(incr(s(tail(head(s(y0))))))) -> c13(MARK(incr(s(tail(head(s(y0))))))) MARK(incr(incr(s(tail(head(y0)))))) -> c13(MARK(incr(s(tail(head(y0)))))) MARK(incr(incr(s(tail(tail(incr(y0))))))) -> c13(MARK(incr(s(tail(tail(incr(y0))))))) MARK(incr(incr(s(tail(tail(pairs)))))) -> c13(MARK(incr(s(tail(tail(pairs)))))) MARK(incr(incr(s(tail(tail(odds)))))) -> c13(MARK(incr(s(tail(tail(odds)))))) MARK(incr(incr(s(tail(tail(head(y0))))))) -> c13(MARK(incr(s(tail(tail(head(y0))))))) MARK(incr(incr(s(tail(tail(tail(y0))))))) -> c13(MARK(incr(s(tail(tail(tail(y0))))))) MARK(incr(incr(s(tail(tail(cons(y0, y1))))))) -> c13(MARK(incr(s(tail(tail(cons(y0, y1))))))) MARK(incr(incr(s(tail(tail(s(y0))))))) -> c13(MARK(incr(s(tail(tail(s(y0))))))) MARK(incr(incr(s(tail(tail(y0)))))) -> c13(MARK(incr(s(tail(tail(y0)))))) MARK(incr(incr(s(tail(pairs))))) -> c13(MARK(incr(s(tail(pairs))))) MARK(incr(incr(s(incr(incr(nats)))))) -> c13(MARK(incr(s(incr(incr(nats)))))) MARK(incr(incr(s(incr(incr(pairs)))))) -> c13(MARK(incr(s(incr(incr(pairs)))))) MARK(incr(incr(s(incr(odds))))) -> c13(MARK(incr(s(incr(odds))))) MARK(incr(incr(s(head(incr(nats)))))) -> c13(MARK(incr(s(head(incr(nats)))))) MARK(incr(incr(s(head(incr(pairs)))))) -> c13(MARK(incr(s(head(incr(pairs)))))) MARK(incr(incr(s(head(odds))))) -> c13(MARK(incr(s(head(odds))))) MARK(incr(incr(s(tail(incr(nats)))))) -> c13(MARK(incr(s(tail(incr(nats)))))) MARK(incr(incr(s(tail(incr(pairs)))))) -> c13(MARK(incr(s(tail(incr(pairs)))))) MARK(incr(incr(s(tail(odds))))) -> c13(MARK(incr(s(tail(odds))))) MARK(incr(incr(s(incr(cons(incr(y0), y1)))))) -> c13(MARK(incr(s(incr(cons(incr(y0), y1)))))) MARK(incr(incr(s(incr(cons(head(y0), y1)))))) -> c13(MARK(incr(s(incr(cons(head(y0), y1)))))) MARK(incr(incr(s(incr(cons(tail(y0), y1)))))) -> c13(MARK(incr(s(incr(cons(tail(y0), y1)))))) MARK(incr(incr(s(incr(cons(cons(y0, y1), y2)))))) -> c13(MARK(incr(s(incr(cons(cons(y0, y1), y2)))))) MARK(incr(incr(s(incr(cons(s(y0), y1)))))) -> c13(MARK(incr(s(incr(cons(s(y0), y1)))))) MARK(incr(incr(s(incr(cons(y0, y1)))))) -> c13(MARK(incr(s(incr(cons(y0, y1)))))) MARK(incr(incr(s(incr(head(nats)))))) -> c13(MARK(incr(s(incr(head(nats)))))) MARK(incr(incr(s(incr(head(pairs)))))) -> c13(MARK(incr(s(incr(head(pairs)))))) MARK(incr(incr(s(head(head(nats)))))) -> c13(MARK(incr(s(head(head(nats)))))) MARK(incr(incr(s(head(head(pairs)))))) -> c13(MARK(incr(s(head(head(pairs)))))) MARK(incr(incr(s(tail(head(nats)))))) -> c13(MARK(incr(s(tail(head(nats)))))) MARK(incr(incr(s(tail(head(pairs)))))) -> c13(MARK(incr(s(tail(head(pairs)))))) MARK(incr(incr(s(head(cons(incr(y0), y1)))))) -> c13(MARK(incr(s(head(cons(incr(y0), y1)))))) MARK(incr(incr(s(head(cons(head(y0), y1)))))) -> c13(MARK(incr(s(head(cons(head(y0), y1)))))) MARK(incr(incr(s(head(cons(tail(y0), y1)))))) -> c13(MARK(incr(s(head(cons(tail(y0), y1)))))) MARK(incr(incr(s(head(cons(cons(y0, y1), y2)))))) -> c13(MARK(incr(s(head(cons(cons(y0, y1), y2)))))) MARK(incr(incr(s(head(cons(s(y0), y1)))))) -> c13(MARK(incr(s(head(cons(s(y0), y1)))))) MARK(incr(incr(s(head(cons(y0, y1)))))) -> c13(MARK(incr(s(head(cons(y0, y1)))))) MARK(incr(incr(s(incr(tail(nats)))))) -> c13(MARK(incr(s(incr(tail(nats)))))) MARK(incr(incr(s(head(tail(nats)))))) -> c13(MARK(incr(s(head(tail(nats)))))) MARK(incr(incr(s(tail(tail(nats)))))) -> c13(MARK(incr(s(tail(tail(nats)))))) MARK(incr(incr(s(cons(cons(y0, y1), y2))))) -> c13(MARK(incr(s(cons(cons(y0, y1), y2))))) MARK(incr(incr(s(cons(s(y0), y1))))) -> c13(MARK(incr(s(cons(s(y0), y1))))) MARK(incr(incr(s(cons(incr(s(y0)), y1))))) -> c13(MARK(incr(s(cons(incr(s(y0)), y1))))) MARK(incr(incr(s(cons(head(s(y0)), y1))))) -> c13(MARK(incr(s(cons(head(s(y0)), y1))))) MARK(incr(incr(s(cons(tail(cons(y0, y1)), y2))))) -> c13(MARK(incr(s(cons(tail(cons(y0, y1)), y2))))) MARK(incr(incr(s(cons(tail(s(y0)), y1))))) -> c13(MARK(incr(s(cons(tail(s(y0)), y1))))) MARK(incr(incr(s(cons(incr(incr(incr(y0))), y1))))) -> c13(MARK(incr(s(cons(incr(incr(incr(y0))), y1))))) MARK(incr(incr(s(cons(incr(incr(odds)), y0))))) -> c13(MARK(incr(s(cons(incr(incr(odds)), y0))))) MARK(incr(incr(s(cons(incr(incr(head(y0))), y1))))) -> c13(MARK(incr(s(cons(incr(incr(head(y0))), y1))))) MARK(incr(incr(s(cons(incr(incr(tail(y0))), y1))))) -> c13(MARK(incr(s(cons(incr(incr(tail(y0))), y1))))) MARK(incr(incr(s(cons(incr(incr(cons(y0, y1))), y2))))) -> c13(MARK(incr(s(cons(incr(incr(cons(y0, y1))), y2))))) MARK(incr(incr(s(cons(incr(incr(s(y0))), y1))))) -> c13(MARK(incr(s(cons(incr(incr(s(y0))), y1))))) MARK(incr(incr(s(cons(incr(incr(y0)), y1))))) -> c13(MARK(incr(s(cons(incr(incr(y0)), y1))))) MARK(incr(incr(s(cons(incr(head(incr(y0))), y1))))) -> c13(MARK(incr(s(cons(incr(head(incr(y0))), y1))))) MARK(incr(incr(s(cons(incr(head(odds)), y0))))) -> c13(MARK(incr(s(cons(incr(head(odds)), y0))))) MARK(incr(incr(s(cons(incr(head(head(y0))), y1))))) -> c13(MARK(incr(s(cons(incr(head(head(y0))), y1))))) MARK(incr(incr(s(cons(incr(head(tail(y0))), y1))))) -> c13(MARK(incr(s(cons(incr(head(tail(y0))), y1))))) MARK(incr(incr(s(cons(incr(head(cons(y0, y1))), y2))))) -> c13(MARK(incr(s(cons(incr(head(cons(y0, y1))), y2))))) MARK(incr(incr(s(cons(incr(head(s(y0))), y1))))) -> c13(MARK(incr(s(cons(incr(head(s(y0))), y1))))) MARK(incr(incr(s(cons(incr(head(y0)), y1))))) -> c13(MARK(incr(s(cons(incr(head(y0)), y1))))) MARK(incr(incr(s(cons(incr(tail(incr(y0))), y1))))) -> c13(MARK(incr(s(cons(incr(tail(incr(y0))), y1))))) MARK(incr(incr(s(cons(incr(tail(pairs)), y0))))) -> c13(MARK(incr(s(cons(incr(tail(pairs)), y0))))) MARK(incr(incr(s(cons(incr(tail(odds)), y0))))) -> c13(MARK(incr(s(cons(incr(tail(odds)), y0))))) MARK(incr(incr(s(cons(incr(tail(head(y0))), y1))))) -> c13(MARK(incr(s(cons(incr(tail(head(y0))), y1))))) MARK(incr(incr(s(cons(incr(tail(tail(y0))), y1))))) -> c13(MARK(incr(s(cons(incr(tail(tail(y0))), y1))))) MARK(incr(incr(s(cons(incr(tail(cons(y0, y1))), y2))))) -> c13(MARK(incr(s(cons(incr(tail(cons(y0, y1))), y2))))) MARK(incr(incr(s(cons(incr(tail(s(y0))), y1))))) -> c13(MARK(incr(s(cons(incr(tail(s(y0))), y1))))) MARK(incr(incr(s(cons(incr(tail(y0)), y1))))) -> c13(MARK(incr(s(cons(incr(tail(y0)), y1))))) MARK(incr(incr(s(cons(head(incr(incr(y0))), y1))))) -> c13(MARK(incr(s(cons(head(incr(incr(y0))), y1))))) MARK(incr(incr(s(cons(head(incr(odds)), y0))))) -> c13(MARK(incr(s(cons(head(incr(odds)), y0))))) MARK(incr(incr(s(cons(head(incr(head(y0))), y1))))) -> c13(MARK(incr(s(cons(head(incr(head(y0))), y1))))) MARK(incr(incr(s(cons(head(incr(tail(y0))), y1))))) -> c13(MARK(incr(s(cons(head(incr(tail(y0))), y1))))) MARK(incr(incr(s(cons(head(incr(cons(y0, y1))), y2))))) -> c13(MARK(incr(s(cons(head(incr(cons(y0, y1))), y2))))) MARK(incr(incr(s(cons(head(incr(s(y0))), y1))))) -> c13(MARK(incr(s(cons(head(incr(s(y0))), y1))))) MARK(incr(incr(s(cons(head(incr(y0)), y1))))) -> c13(MARK(incr(s(cons(head(incr(y0)), y1))))) MARK(incr(incr(s(cons(head(head(incr(y0))), y1))))) -> c13(MARK(incr(s(cons(head(head(incr(y0))), y1))))) MARK(incr(incr(s(cons(head(head(odds)), y0))))) -> c13(MARK(incr(s(cons(head(head(odds)), y0))))) MARK(incr(incr(s(cons(head(head(head(y0))), y1))))) -> c13(MARK(incr(s(cons(head(head(head(y0))), y1))))) MARK(incr(incr(s(cons(head(head(tail(y0))), y1))))) -> c13(MARK(incr(s(cons(head(head(tail(y0))), y1))))) MARK(incr(incr(s(cons(head(head(cons(y0, y1))), y2))))) -> c13(MARK(incr(s(cons(head(head(cons(y0, y1))), y2))))) MARK(incr(incr(s(cons(head(head(s(y0))), y1))))) -> c13(MARK(incr(s(cons(head(head(s(y0))), y1))))) MARK(incr(incr(s(cons(head(head(y0)), y1))))) -> c13(MARK(incr(s(cons(head(head(y0)), y1))))) MARK(incr(incr(s(cons(head(tail(incr(y0))), y1))))) -> c13(MARK(incr(s(cons(head(tail(incr(y0))), y1))))) MARK(incr(incr(s(cons(head(tail(pairs)), y0))))) -> c13(MARK(incr(s(cons(head(tail(pairs)), y0))))) MARK(incr(incr(s(cons(head(tail(odds)), y0))))) -> c13(MARK(incr(s(cons(head(tail(odds)), y0))))) MARK(incr(incr(s(cons(head(tail(head(y0))), y1))))) -> c13(MARK(incr(s(cons(head(tail(head(y0))), y1))))) MARK(incr(incr(s(cons(head(tail(tail(y0))), y1))))) -> c13(MARK(incr(s(cons(head(tail(tail(y0))), y1))))) MARK(incr(incr(s(cons(head(tail(cons(y0, y1))), y2))))) -> c13(MARK(incr(s(cons(head(tail(cons(y0, y1))), y2))))) MARK(incr(incr(s(cons(head(tail(s(y0))), y1))))) -> c13(MARK(incr(s(cons(head(tail(s(y0))), y1))))) MARK(incr(incr(s(cons(head(tail(y0)), y1))))) -> c13(MARK(incr(s(cons(head(tail(y0)), y1))))) MARK(incr(incr(s(cons(tail(incr(incr(y0))), y1))))) -> c13(MARK(incr(s(cons(tail(incr(incr(y0))), y1))))) MARK(incr(incr(s(cons(tail(incr(odds)), y0))))) -> c13(MARK(incr(s(cons(tail(incr(odds)), y0))))) MARK(incr(incr(s(cons(tail(incr(head(y0))), y1))))) -> c13(MARK(incr(s(cons(tail(incr(head(y0))), y1))))) MARK(incr(incr(s(cons(tail(incr(tail(y0))), y1))))) -> c13(MARK(incr(s(cons(tail(incr(tail(y0))), y1))))) MARK(incr(incr(s(cons(tail(incr(cons(y0, y1))), y2))))) -> c13(MARK(incr(s(cons(tail(incr(cons(y0, y1))), y2))))) MARK(incr(incr(s(cons(tail(incr(s(y0))), y1))))) -> c13(MARK(incr(s(cons(tail(incr(s(y0))), y1))))) MARK(incr(incr(s(cons(tail(incr(y0)), y1))))) -> c13(MARK(incr(s(cons(tail(incr(y0)), y1))))) MARK(incr(incr(s(cons(tail(head(incr(y0))), y1))))) -> c13(MARK(incr(s(cons(tail(head(incr(y0))), y1))))) MARK(incr(incr(s(cons(tail(head(odds)), y0))))) -> c13(MARK(incr(s(cons(tail(head(odds)), y0))))) MARK(incr(incr(s(cons(tail(head(head(y0))), y1))))) -> c13(MARK(incr(s(cons(tail(head(head(y0))), y1))))) MARK(incr(incr(s(cons(tail(head(tail(y0))), y1))))) -> c13(MARK(incr(s(cons(tail(head(tail(y0))), y1))))) MARK(incr(incr(s(cons(tail(head(cons(y0, y1))), y2))))) -> c13(MARK(incr(s(cons(tail(head(cons(y0, y1))), y2))))) MARK(incr(incr(s(cons(tail(head(s(y0))), y1))))) -> c13(MARK(incr(s(cons(tail(head(s(y0))), y1))))) MARK(incr(incr(s(cons(tail(head(y0)), y1))))) -> c13(MARK(incr(s(cons(tail(head(y0)), y1))))) MARK(incr(incr(s(cons(tail(tail(incr(y0))), y1))))) -> c13(MARK(incr(s(cons(tail(tail(incr(y0))), y1))))) MARK(incr(incr(s(cons(tail(tail(pairs)), y0))))) -> c13(MARK(incr(s(cons(tail(tail(pairs)), y0))))) MARK(incr(incr(s(cons(tail(tail(odds)), y0))))) -> c13(MARK(incr(s(cons(tail(tail(odds)), y0))))) MARK(incr(incr(s(cons(tail(tail(head(y0))), y1))))) -> c13(MARK(incr(s(cons(tail(tail(head(y0))), y1))))) MARK(incr(incr(s(cons(tail(tail(tail(y0))), y1))))) -> c13(MARK(incr(s(cons(tail(tail(tail(y0))), y1))))) MARK(incr(incr(s(cons(tail(tail(cons(y0, y1))), y2))))) -> c13(MARK(incr(s(cons(tail(tail(cons(y0, y1))), y2))))) MARK(incr(incr(s(cons(tail(tail(s(y0))), y1))))) -> c13(MARK(incr(s(cons(tail(tail(s(y0))), y1))))) MARK(incr(incr(s(cons(tail(tail(y0)), y1))))) -> c13(MARK(incr(s(cons(tail(tail(y0)), y1))))) MARK(incr(incr(s(cons(tail(pairs), y0))))) -> c13(MARK(incr(s(cons(tail(pairs), y0))))) MARK(incr(incr(s(cons(incr(incr(nats)), y0))))) -> c13(MARK(incr(s(cons(incr(incr(nats)), y0))))) MARK(incr(incr(s(cons(incr(incr(pairs)), y0))))) -> c13(MARK(incr(s(cons(incr(incr(pairs)), y0))))) MARK(incr(incr(s(cons(incr(odds), y0))))) -> c13(MARK(incr(s(cons(incr(odds), y0))))) MARK(incr(incr(s(cons(head(incr(nats)), y0))))) -> c13(MARK(incr(s(cons(head(incr(nats)), y0))))) MARK(incr(incr(s(cons(head(incr(pairs)), y0))))) -> c13(MARK(incr(s(cons(head(incr(pairs)), y0))))) MARK(incr(incr(s(cons(head(odds), y0))))) -> c13(MARK(incr(s(cons(head(odds), y0))))) MARK(incr(incr(s(cons(tail(incr(nats)), y0))))) -> c13(MARK(incr(s(cons(tail(incr(nats)), y0))))) MARK(incr(incr(s(cons(tail(incr(pairs)), y0))))) -> c13(MARK(incr(s(cons(tail(incr(pairs)), y0))))) MARK(incr(incr(s(cons(tail(odds), y0))))) -> c13(MARK(incr(s(cons(tail(odds), y0))))) MARK(incr(incr(s(cons(incr(cons(nats, y0)), y1))))) -> c13(MARK(incr(s(cons(incr(cons(nats, y0)), y1))))) MARK(incr(incr(s(cons(incr(cons(incr(y0), y1)), y2))))) -> c13(MARK(incr(s(cons(incr(cons(incr(y0), y1)), y2))))) MARK(incr(incr(s(cons(incr(cons(pairs, y0)), y1))))) -> c13(MARK(incr(s(cons(incr(cons(pairs, y0)), y1))))) MARK(incr(incr(s(cons(incr(cons(odds, y0)), y1))))) -> c13(MARK(incr(s(cons(incr(cons(odds, y0)), y1))))) MARK(incr(incr(s(cons(incr(cons(head(y0), y1)), y2))))) -> c13(MARK(incr(s(cons(incr(cons(head(y0), y1)), y2))))) MARK(incr(incr(s(cons(incr(cons(tail(y0), y1)), y2))))) -> c13(MARK(incr(s(cons(incr(cons(tail(y0), y1)), y2))))) MARK(incr(incr(s(cons(incr(cons(cons(y0, y1), y2)), y3))))) -> c13(MARK(incr(s(cons(incr(cons(cons(y0, y1), y2)), y3))))) MARK(incr(incr(s(cons(incr(cons(s(y0), y1)), y2))))) -> c13(MARK(incr(s(cons(incr(cons(s(y0), y1)), y2))))) MARK(incr(incr(s(cons(incr(cons(y0, y1)), y2))))) -> c13(MARK(incr(s(cons(incr(cons(y0, y1)), y2))))) MARK(incr(incr(s(cons(incr(cons(0, y0)), y1))))) -> c13(MARK(incr(s(cons(incr(cons(0, y0)), y1))))) MARK(incr(incr(s(cons(incr(head(nats)), y0))))) -> c13(MARK(incr(s(cons(incr(head(nats)), y0))))) MARK(incr(incr(s(cons(incr(head(pairs)), y0))))) -> c13(MARK(incr(s(cons(incr(head(pairs)), y0))))) MARK(incr(incr(s(cons(head(head(nats)), y0))))) -> c13(MARK(incr(s(cons(head(head(nats)), y0))))) MARK(incr(incr(s(cons(head(head(pairs)), y0))))) -> c13(MARK(incr(s(cons(head(head(pairs)), y0))))) MARK(incr(incr(s(cons(tail(head(nats)), y0))))) -> c13(MARK(incr(s(cons(tail(head(nats)), y0))))) MARK(incr(incr(s(cons(tail(head(pairs)), y0))))) -> c13(MARK(incr(s(cons(tail(head(pairs)), y0))))) MARK(incr(incr(s(cons(head(cons(nats, y0)), y1))))) -> c13(MARK(incr(s(cons(head(cons(nats, y0)), y1))))) MARK(incr(incr(s(cons(head(cons(incr(y0), y1)), y2))))) -> c13(MARK(incr(s(cons(head(cons(incr(y0), y1)), y2))))) MARK(incr(incr(s(cons(head(cons(pairs, y0)), y1))))) -> c13(MARK(incr(s(cons(head(cons(pairs, y0)), y1))))) MARK(incr(incr(s(cons(head(cons(odds, y0)), y1))))) -> c13(MARK(incr(s(cons(head(cons(odds, y0)), y1))))) MARK(incr(incr(s(cons(head(cons(head(y0), y1)), y2))))) -> c13(MARK(incr(s(cons(head(cons(head(y0), y1)), y2))))) MARK(incr(incr(s(cons(head(cons(tail(y0), y1)), y2))))) -> c13(MARK(incr(s(cons(head(cons(tail(y0), y1)), y2))))) MARK(incr(incr(s(cons(head(cons(cons(y0, y1), y2)), y3))))) -> c13(MARK(incr(s(cons(head(cons(cons(y0, y1), y2)), y3))))) MARK(incr(incr(s(cons(head(cons(s(y0), y1)), y2))))) -> c13(MARK(incr(s(cons(head(cons(s(y0), y1)), y2))))) MARK(incr(incr(s(cons(head(cons(y0, y1)), y2))))) -> c13(MARK(incr(s(cons(head(cons(y0, y1)), y2))))) MARK(incr(incr(s(cons(head(cons(0, y0)), y1))))) -> c13(MARK(incr(s(cons(head(cons(0, y0)), y1))))) MARK(incr(incr(s(cons(incr(tail(nats)), y0))))) -> c13(MARK(incr(s(cons(incr(tail(nats)), y0))))) MARK(incr(incr(s(cons(head(tail(nats)), y0))))) -> c13(MARK(incr(s(cons(head(tail(nats)), y0))))) MARK(incr(incr(s(cons(tail(tail(nats)), y0))))) -> c13(MARK(incr(s(cons(tail(tail(nats)), y0))))) MARK(incr(incr(s(incr(cons(nats, y0)))))) -> c13(MARK(incr(s(incr(cons(nats, y0)))))) MARK(incr(incr(s(incr(cons(pairs, y0)))))) -> c13(MARK(incr(s(incr(cons(pairs, y0)))))) MARK(incr(incr(s(incr(cons(odds, y0)))))) -> c13(MARK(incr(s(incr(cons(odds, y0)))))) MARK(incr(incr(s(head(cons(nats, y0)))))) -> c13(MARK(incr(s(head(cons(nats, y0)))))) MARK(incr(incr(s(head(cons(pairs, y0)))))) -> c13(MARK(incr(s(head(cons(pairs, y0)))))) MARK(incr(incr(s(head(cons(odds, y0)))))) -> c13(MARK(incr(s(head(cons(odds, y0)))))) K tuples:none Defined Rule Symbols: a__nats, a__pairs, a__odds, a__incr_1, a__head_1, a__tail_1, mark_1 Defined Pair Symbols: MARK_1, A__INCR_1, A__HEAD_1, A__TAIL_1 Compound Symbols: c17_2, c13_2, c13_1, c16_2, c16_1, c17_1, c6_1, c8_1, c10_1, c18_1, c20_1