KILLED proof of input_NRgacXClRt.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).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), 138 ms] (12) CpxTypedWeightedCompleteTrs (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (14) CpxRNTS (15) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (16) CpxRNTS (17) CompletionProof [UPPER BOUND(ID), 0 ms] (18) CpxTypedWeightedCompleteTrs (19) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (20) CpxRNTS (21) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (22) CdtProblem (23) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (24) CdtProblem (25) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (26) CdtProblem (27) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (28) CdtProblem (29) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 2 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) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (44) CdtProblem (45) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (46) CdtProblem (47) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (48) CdtProblem (49) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (50) CdtProblem (51) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (52) CdtProblem (53) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 61 ms] (54) CdtProblem (55) CdtLeafRemovalProof [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) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 4 ms] (62) CdtProblem (63) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (64) CdtProblem (65) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 82 ms] (66) CdtProblem (67) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (68) CdtProblem (69) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (70) CdtProblem (71) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 30 ms] (72) CdtProblem (73) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (74) CdtProblem (75) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 18 ms] (76) CdtProblem (77) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 109 ms] (78) CdtProblem (79) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (80) CdtProblem (81) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (82) CdtProblem (83) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 66 ms] (84) CdtProblem (85) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (86) CdtProblem (87) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (88) CdtProblem (89) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 112 ms] (90) CdtProblem (91) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (92) CdtProblem (93) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 29 ms] (94) CdtProblem (95) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 230 ms] (96) CdtProblem (97) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (98) CdtProblem (99) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 9 ms] (100) CdtProblem (101) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 110 ms] (102) CdtProblem (103) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 27 ms] (104) CdtProblem (105) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (106) CdtProblem (107) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 187 ms] (108) CdtProblem (109) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (110) CdtProblem (111) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (112) CdtProblem (113) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 143 ms] (114) CdtProblem (115) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (116) CdtProblem (117) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (118) CdtProblem (119) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 206 ms] (120) CdtProblem (121) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (122) CdtProblem (123) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (124) CdtProblem (125) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 325 ms] (126) CdtProblem (127) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 38 ms] (128) CdtProblem (129) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 99 ms] (130) CdtProblem (131) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (132) CdtProblem (133) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 139 ms] (134) CdtProblem (135) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (136) CdtProblem (137) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 104 ms] (138) CdtProblem (139) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 1 ms] (140) CdtProblem (141) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 164 ms] (142) CdtProblem (143) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 44 ms] (144) CdtProblem (145) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 432 ms] (146) CdtProblem (147) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 62 ms] (148) CdtProblem (149) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (150) CdtProblem (151) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 373 ms] (152) CdtProblem (153) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 26 ms] (154) CdtProblem (155) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 102 ms] (156) CdtProblem (157) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 301 ms] (158) CdtProblem (159) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 82 ms] (160) CdtProblem (161) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 87 ms] (162) CdtProblem (163) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 282 ms] (164) CdtProblem (165) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 70 ms] (166) CdtProblem (167) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 36 ms] (168) CdtProblem (169) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 361 ms] (170) CdtProblem (171) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 58 ms] (172) CdtProblem (173) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 93 ms] (174) CdtProblem (175) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 238 ms] (176) CdtProblem (177) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 27 ms] (178) CdtProblem (179) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 77 ms] (180) CdtProblem (181) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 423 ms] (182) CdtProblem (183) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 140 ms] (184) CdtProblem (185) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 105 ms] (186) CdtProblem (187) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 684 ms] (188) CdtProblem (189) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 93 ms] (190) CdtProblem (191) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 258 ms] (192) CdtProblem (193) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 100 ms] (194) CdtProblem (195) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 260 ms] (196) CdtProblem (197) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 114 ms] (198) CdtProblem (199) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 309 ms] (200) CdtProblem (201) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 107 ms] (202) CdtProblem (203) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 326 ms] (204) CdtProblem (205) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 90 ms] (206) CdtProblem (207) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 776 ms] (208) CdtProblem (209) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 187 ms] (210) CdtProblem (211) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 143 ms] (212) CdtProblem (213) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 564 ms] (214) CdtProblem (215) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 131 ms] (216) CdtProblem (217) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 115 ms] (218) CdtProblem (219) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 689 ms] (220) CdtProblem (221) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 235 ms] (222) CdtProblem (223) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 168 ms] (224) CdtProblem (225) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 459 ms] (226) CdtProblem (227) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 157 ms] (228) CdtProblem (229) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 109 ms] (230) 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__filter(cons(X, Y), 0, M) -> cons(0, filter(Y, M, M)) a__filter(cons(X, Y), s(N), M) -> cons(mark(X), filter(Y, N, M)) a__sieve(cons(0, Y)) -> cons(0, sieve(Y)) a__sieve(cons(s(N), Y)) -> cons(s(mark(N)), sieve(filter(Y, N, N))) a__nats(N) -> cons(mark(N), nats(s(N))) a__zprimes -> a__sieve(a__nats(s(s(0)))) mark(filter(X1, X2, X3)) -> a__filter(mark(X1), mark(X2), mark(X3)) mark(sieve(X)) -> a__sieve(mark(X)) mark(nats(X)) -> a__nats(mark(X)) mark(zprimes) -> a__zprimes mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0) -> 0 mark(s(X)) -> s(mark(X)) a__filter(X1, X2, X3) -> filter(X1, X2, X3) a__sieve(X) -> sieve(X) a__nats(X) -> nats(X) a__zprimes -> zprimes 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__filter(cons(X, Y), 0', M) -> cons(0', filter(Y, M, M)) a__filter(cons(X, Y), s(N), M) -> cons(mark(X), filter(Y, N, M)) a__sieve(cons(0', Y)) -> cons(0', sieve(Y)) a__sieve(cons(s(N), Y)) -> cons(s(mark(N)), sieve(filter(Y, N, N))) a__nats(N) -> cons(mark(N), nats(s(N))) a__zprimes -> a__sieve(a__nats(s(s(0')))) mark(filter(X1, X2, X3)) -> a__filter(mark(X1), mark(X2), mark(X3)) mark(sieve(X)) -> a__sieve(mark(X)) mark(nats(X)) -> a__nats(mark(X)) mark(zprimes) -> a__zprimes mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0') -> 0' mark(s(X)) -> s(mark(X)) a__filter(X1, X2, X3) -> filter(X1, X2, X3) a__sieve(X) -> sieve(X) a__nats(X) -> nats(X) a__zprimes -> zprimes 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__filter(cons(X, Y), 0, M) -> cons(0, filter(Y, M, M)) a__filter(cons(X, Y), s(N), M) -> cons(mark(X), filter(Y, N, M)) a__sieve(cons(0, Y)) -> cons(0, sieve(Y)) a__sieve(cons(s(N), Y)) -> cons(s(mark(N)), sieve(filter(Y, N, N))) a__nats(N) -> cons(mark(N), nats(s(N))) a__zprimes -> a__sieve(a__nats(s(s(0)))) mark(filter(X1, X2, X3)) -> a__filter(mark(X1), mark(X2), mark(X3)) mark(sieve(X)) -> a__sieve(mark(X)) mark(nats(X)) -> a__nats(mark(X)) mark(zprimes) -> a__zprimes mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(0) -> 0 mark(s(X)) -> s(mark(X)) a__filter(X1, X2, X3) -> filter(X1, X2, X3) a__sieve(X) -> sieve(X) a__nats(X) -> nats(X) a__zprimes -> zprimes 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__filter(cons(X, Y), 0, M) -> cons(0, filter(Y, M, M)) [1] a__filter(cons(X, Y), s(N), M) -> cons(mark(X), filter(Y, N, M)) [1] a__sieve(cons(0, Y)) -> cons(0, sieve(Y)) [1] a__sieve(cons(s(N), Y)) -> cons(s(mark(N)), sieve(filter(Y, N, N))) [1] a__nats(N) -> cons(mark(N), nats(s(N))) [1] a__zprimes -> a__sieve(a__nats(s(s(0)))) [1] mark(filter(X1, X2, X3)) -> a__filter(mark(X1), mark(X2), mark(X3)) [1] mark(sieve(X)) -> a__sieve(mark(X)) [1] mark(nats(X)) -> a__nats(mark(X)) [1] mark(zprimes) -> a__zprimes [1] mark(cons(X1, X2)) -> cons(mark(X1), X2) [1] mark(0) -> 0 [1] mark(s(X)) -> s(mark(X)) [1] a__filter(X1, X2, X3) -> filter(X1, X2, X3) [1] a__sieve(X) -> sieve(X) [1] a__nats(X) -> nats(X) [1] a__zprimes -> zprimes [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__filter(cons(X, Y), 0, M) -> cons(0, filter(Y, M, M)) [1] a__filter(cons(X, Y), s(N), M) -> cons(mark(X), filter(Y, N, M)) [1] a__sieve(cons(0, Y)) -> cons(0, sieve(Y)) [1] a__sieve(cons(s(N), Y)) -> cons(s(mark(N)), sieve(filter(Y, N, N))) [1] a__nats(N) -> cons(mark(N), nats(s(N))) [1] a__zprimes -> a__sieve(a__nats(s(s(0)))) [1] mark(filter(X1, X2, X3)) -> a__filter(mark(X1), mark(X2), mark(X3)) [1] mark(sieve(X)) -> a__sieve(mark(X)) [1] mark(nats(X)) -> a__nats(mark(X)) [1] mark(zprimes) -> a__zprimes [1] mark(cons(X1, X2)) -> cons(mark(X1), X2) [1] mark(0) -> 0 [1] mark(s(X)) -> s(mark(X)) [1] a__filter(X1, X2, X3) -> filter(X1, X2, X3) [1] a__sieve(X) -> sieve(X) [1] a__nats(X) -> nats(X) [1] a__zprimes -> zprimes [1] The TRS has the following type information: a__filter :: cons:0:filter:s:sieve:nats:zprimes -> cons:0:filter:s:sieve:nats:zprimes -> cons:0:filter:s:sieve:nats:zprimes -> cons:0:filter:s:sieve:nats:zprimes cons :: cons:0:filter:s:sieve:nats:zprimes -> cons:0:filter:s:sieve:nats:zprimes -> cons:0:filter:s:sieve:nats:zprimes 0 :: cons:0:filter:s:sieve:nats:zprimes filter :: cons:0:filter:s:sieve:nats:zprimes -> cons:0:filter:s:sieve:nats:zprimes -> cons:0:filter:s:sieve:nats:zprimes -> cons:0:filter:s:sieve:nats:zprimes s :: cons:0:filter:s:sieve:nats:zprimes -> cons:0:filter:s:sieve:nats:zprimes mark :: cons:0:filter:s:sieve:nats:zprimes -> cons:0:filter:s:sieve:nats:zprimes a__sieve :: cons:0:filter:s:sieve:nats:zprimes -> cons:0:filter:s:sieve:nats:zprimes sieve :: cons:0:filter:s:sieve:nats:zprimes -> cons:0:filter:s:sieve:nats:zprimes a__nats :: cons:0:filter:s:sieve:nats:zprimes -> cons:0:filter:s:sieve:nats:zprimes nats :: cons:0:filter:s:sieve:nats:zprimes -> cons:0:filter:s:sieve:nats:zprimes a__zprimes :: cons:0:filter:s:sieve:nats:zprimes zprimes :: cons:0:filter:s:sieve:nats:zprimes 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__nats_1 a__sieve_1 a__zprimes a__filter_3 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__filter(cons(X, Y), 0, M) -> cons(0, filter(Y, M, M)) [1] a__filter(cons(X, Y), s(N), M) -> cons(mark(X), filter(Y, N, M)) [1] a__sieve(cons(0, Y)) -> cons(0, sieve(Y)) [1] a__sieve(cons(s(N), Y)) -> cons(s(mark(N)), sieve(filter(Y, N, N))) [1] a__nats(N) -> cons(mark(N), nats(s(N))) [1] a__zprimes -> a__sieve(a__nats(s(s(0)))) [1] mark(filter(X1, X2, X3)) -> a__filter(mark(X1), mark(X2), mark(X3)) [1] mark(sieve(X)) -> a__sieve(mark(X)) [1] mark(nats(X)) -> a__nats(mark(X)) [1] mark(zprimes) -> a__zprimes [1] mark(cons(X1, X2)) -> cons(mark(X1), X2) [1] mark(0) -> 0 [1] mark(s(X)) -> s(mark(X)) [1] a__filter(X1, X2, X3) -> filter(X1, X2, X3) [1] a__sieve(X) -> sieve(X) [1] a__nats(X) -> nats(X) [1] a__zprimes -> zprimes [1] The TRS has the following type information: a__filter :: cons:0:filter:s:sieve:nats:zprimes -> cons:0:filter:s:sieve:nats:zprimes -> cons:0:filter:s:sieve:nats:zprimes -> cons:0:filter:s:sieve:nats:zprimes cons :: cons:0:filter:s:sieve:nats:zprimes -> cons:0:filter:s:sieve:nats:zprimes -> cons:0:filter:s:sieve:nats:zprimes 0 :: cons:0:filter:s:sieve:nats:zprimes filter :: cons:0:filter:s:sieve:nats:zprimes -> cons:0:filter:s:sieve:nats:zprimes -> cons:0:filter:s:sieve:nats:zprimes -> cons:0:filter:s:sieve:nats:zprimes s :: cons:0:filter:s:sieve:nats:zprimes -> cons:0:filter:s:sieve:nats:zprimes mark :: cons:0:filter:s:sieve:nats:zprimes -> cons:0:filter:s:sieve:nats:zprimes a__sieve :: cons:0:filter:s:sieve:nats:zprimes -> cons:0:filter:s:sieve:nats:zprimes sieve :: cons:0:filter:s:sieve:nats:zprimes -> cons:0:filter:s:sieve:nats:zprimes a__nats :: cons:0:filter:s:sieve:nats:zprimes -> cons:0:filter:s:sieve:nats:zprimes nats :: cons:0:filter:s:sieve:nats:zprimes -> cons:0:filter:s:sieve:nats:zprimes a__zprimes :: cons:0:filter:s:sieve:nats:zprimes zprimes :: cons:0:filter:s:sieve:nats:zprimes 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__filter(cons(X, Y), 0, M) -> cons(0, filter(Y, M, M)) [1] a__filter(cons(X, Y), s(N), M) -> cons(mark(X), filter(Y, N, M)) [1] a__sieve(cons(0, Y)) -> cons(0, sieve(Y)) [1] a__sieve(cons(s(N), Y)) -> cons(s(mark(N)), sieve(filter(Y, N, N))) [1] a__nats(N) -> cons(mark(N), nats(s(N))) [1] a__zprimes -> a__sieve(cons(mark(s(s(0))), nats(s(s(s(0)))))) [2] a__zprimes -> a__sieve(nats(s(s(0)))) [2] mark(filter(X1, X2, X3)) -> a__filter(mark(X1), mark(X2), mark(X3)) [1] mark(sieve(filter(X1139, X2125, X388))) -> a__sieve(a__filter(mark(X1139), mark(X2125), mark(X388))) [2] mark(sieve(sieve(X389))) -> a__sieve(a__sieve(mark(X389))) [2] mark(sieve(nats(X390))) -> a__sieve(a__nats(mark(X390))) [2] mark(sieve(zprimes)) -> a__sieve(a__zprimes) [2] mark(sieve(cons(X1140, X2126))) -> a__sieve(cons(mark(X1140), X2126)) [2] mark(sieve(0)) -> a__sieve(0) [2] mark(sieve(s(X391))) -> a__sieve(s(mark(X391))) [2] mark(nats(filter(X1141, X2127, X392))) -> a__nats(a__filter(mark(X1141), mark(X2127), mark(X392))) [2] mark(nats(sieve(X393))) -> a__nats(a__sieve(mark(X393))) [2] mark(nats(nats(X394))) -> a__nats(a__nats(mark(X394))) [2] mark(nats(zprimes)) -> a__nats(a__zprimes) [2] mark(nats(cons(X1142, X2128))) -> a__nats(cons(mark(X1142), X2128)) [2] mark(nats(0)) -> a__nats(0) [2] mark(nats(s(X395))) -> a__nats(s(mark(X395))) [2] mark(zprimes) -> a__zprimes [1] mark(cons(X1, X2)) -> cons(mark(X1), X2) [1] mark(0) -> 0 [1] mark(s(X)) -> s(mark(X)) [1] a__filter(X1, X2, X3) -> filter(X1, X2, X3) [1] a__sieve(X) -> sieve(X) [1] a__nats(X) -> nats(X) [1] a__zprimes -> zprimes [1] The TRS has the following type information: a__filter :: cons:0:filter:s:sieve:nats:zprimes -> cons:0:filter:s:sieve:nats:zprimes -> cons:0:filter:s:sieve:nats:zprimes -> cons:0:filter:s:sieve:nats:zprimes cons :: cons:0:filter:s:sieve:nats:zprimes -> cons:0:filter:s:sieve:nats:zprimes -> cons:0:filter:s:sieve:nats:zprimes 0 :: cons:0:filter:s:sieve:nats:zprimes filter :: cons:0:filter:s:sieve:nats:zprimes -> cons:0:filter:s:sieve:nats:zprimes -> cons:0:filter:s:sieve:nats:zprimes -> cons:0:filter:s:sieve:nats:zprimes s :: cons:0:filter:s:sieve:nats:zprimes -> cons:0:filter:s:sieve:nats:zprimes mark :: cons:0:filter:s:sieve:nats:zprimes -> cons:0:filter:s:sieve:nats:zprimes a__sieve :: cons:0:filter:s:sieve:nats:zprimes -> cons:0:filter:s:sieve:nats:zprimes sieve :: cons:0:filter:s:sieve:nats:zprimes -> cons:0:filter:s:sieve:nats:zprimes a__nats :: cons:0:filter:s:sieve:nats:zprimes -> cons:0:filter:s:sieve:nats:zprimes nats :: cons:0:filter:s:sieve:nats:zprimes -> cons:0:filter:s:sieve:nats:zprimes a__zprimes :: cons:0:filter:s:sieve:nats:zprimes zprimes :: cons:0:filter:s:sieve:nats:zprimes 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 zprimes => 1 ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: a__filter(z, z', z'') -{ 1 }-> 1 + mark(X) + (1 + Y + N + M) :|: z' = 1 + N, Y >= 0, z = 1 + X + Y, X >= 0, M >= 0, z'' = M, N >= 0 a__filter(z, z', z'') -{ 1 }-> 1 + 0 + (1 + Y + M + M) :|: Y >= 0, z = 1 + X + Y, X >= 0, M >= 0, z' = 0, z'' = M a__filter(z, z', z'') -{ 1 }-> 1 + X1 + X2 + X3 :|: X1 >= 0, X3 >= 0, X2 >= 0, z = X1, z' = X2, z'' = X3 a__nats(z) -{ 1 }-> 1 + X :|: X >= 0, z = X a__nats(z) -{ 1 }-> 1 + mark(N) + (1 + (1 + N)) :|: z = N, N >= 0 a__sieve(z) -{ 1 }-> 1 + X :|: X >= 0, z = X a__sieve(z) -{ 1 }-> 1 + 0 + (1 + Y) :|: Y >= 0, z = 1 + 0 + Y a__sieve(z) -{ 1 }-> 1 + (1 + mark(N)) + (1 + (1 + Y + N + N)) :|: Y >= 0, z = 1 + (1 + N) + Y, N >= 0 a__zprimes -{ 2 }-> a__sieve(1 + (1 + (1 + 0))) :|: a__zprimes -{ 2 }-> a__sieve(1 + mark(1 + (1 + 0)) + (1 + (1 + (1 + (1 + 0))))) :|: a__zprimes -{ 1 }-> 1 :|: mark(z) -{ 1 }-> a__zprimes :|: z = 1 mark(z) -{ 2 }-> a__sieve(a__zprimes) :|: z = 1 + 1 mark(z) -{ 2 }-> a__sieve(a__sieve(mark(X389))) :|: X389 >= 0, z = 1 + (1 + X389) mark(z) -{ 2 }-> a__sieve(a__nats(mark(X390))) :|: X390 >= 0, z = 1 + (1 + X390) mark(z) -{ 2 }-> a__sieve(a__filter(mark(X1139), mark(X2125), mark(X388))) :|: X388 >= 0, z = 1 + (1 + X1139 + X2125 + X388), X1139 >= 0, X2125 >= 0 mark(z) -{ 2 }-> a__sieve(0) :|: z = 1 + 0 mark(z) -{ 2 }-> a__sieve(1 + mark(X391)) :|: X391 >= 0, z = 1 + (1 + X391) mark(z) -{ 2 }-> a__sieve(1 + mark(X1140) + X2126) :|: X1140 >= 0, z = 1 + (1 + X1140 + X2126), X2126 >= 0 mark(z) -{ 2 }-> a__nats(a__zprimes) :|: z = 1 + 1 mark(z) -{ 2 }-> a__nats(a__sieve(mark(X393))) :|: z = 1 + (1 + X393), X393 >= 0 mark(z) -{ 2 }-> a__nats(a__nats(mark(X394))) :|: z = 1 + (1 + X394), X394 >= 0 mark(z) -{ 2 }-> a__nats(a__filter(mark(X1141), mark(X2127), mark(X392))) :|: X2127 >= 0, z = 1 + (1 + X1141 + X2127 + X392), X392 >= 0, X1141 >= 0 mark(z) -{ 2 }-> a__nats(0) :|: z = 1 + 0 mark(z) -{ 2 }-> a__nats(1 + mark(X395)) :|: X395 >= 0, z = 1 + (1 + X395) mark(z) -{ 2 }-> a__nats(1 + mark(X1142) + X2128) :|: z = 1 + (1 + X1142 + X2128), X2128 >= 0, X1142 >= 0 mark(z) -{ 1 }-> a__filter(mark(X1), mark(X2), mark(X3)) :|: X1 >= 0, X3 >= 0, z = 1 + X1 + X2 + X3, 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) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (16) Obligation: Complexity RNTS consisting of the following rules: a__filter(z, z', z'') -{ 1 }-> 1 + mark(X) + (1 + Y + (z' - 1) + z'') :|: Y >= 0, z = 1 + X + Y, X >= 0, z'' >= 0, z' - 1 >= 0 a__filter(z, z', z'') -{ 1 }-> 1 + 0 + (1 + Y + z'' + z'') :|: Y >= 0, z = 1 + X + Y, X >= 0, z'' >= 0, z' = 0 a__filter(z, z', z'') -{ 1 }-> 1 + z + z' + z'' :|: z >= 0, z'' >= 0, z' >= 0 a__nats(z) -{ 1 }-> 1 + z :|: z >= 0 a__nats(z) -{ 1 }-> 1 + mark(z) + (1 + (1 + z)) :|: z >= 0 a__sieve(z) -{ 1 }-> 1 + z :|: z >= 0 a__sieve(z) -{ 1 }-> 1 + 0 + (1 + (z - 1)) :|: z - 1 >= 0 a__sieve(z) -{ 1 }-> 1 + (1 + mark(N)) + (1 + (1 + Y + N + N)) :|: Y >= 0, z = 1 + (1 + N) + Y, N >= 0 a__zprimes -{ 2 }-> a__sieve(1 + (1 + (1 + 0))) :|: a__zprimes -{ 2 }-> a__sieve(1 + mark(1 + (1 + 0)) + (1 + (1 + (1 + (1 + 0))))) :|: a__zprimes -{ 1 }-> 1 :|: mark(z) -{ 1 }-> a__zprimes :|: z = 1 mark(z) -{ 2 }-> a__sieve(a__zprimes) :|: z = 1 + 1 mark(z) -{ 2 }-> a__sieve(a__sieve(mark(z - 2))) :|: z - 2 >= 0 mark(z) -{ 2 }-> a__sieve(a__nats(mark(z - 2))) :|: z - 2 >= 0 mark(z) -{ 2 }-> a__sieve(a__filter(mark(X1139), mark(X2125), mark(X388))) :|: X388 >= 0, z = 1 + (1 + X1139 + X2125 + X388), X1139 >= 0, X2125 >= 0 mark(z) -{ 2 }-> a__sieve(0) :|: z = 1 + 0 mark(z) -{ 2 }-> a__sieve(1 + mark(z - 2)) :|: z - 2 >= 0 mark(z) -{ 2 }-> a__sieve(1 + mark(X1140) + X2126) :|: X1140 >= 0, z = 1 + (1 + X1140 + X2126), X2126 >= 0 mark(z) -{ 2 }-> a__nats(a__zprimes) :|: z = 1 + 1 mark(z) -{ 2 }-> a__nats(a__sieve(mark(z - 2))) :|: z - 2 >= 0 mark(z) -{ 2 }-> a__nats(a__nats(mark(z - 2))) :|: z - 2 >= 0 mark(z) -{ 2 }-> a__nats(a__filter(mark(X1141), mark(X2127), mark(X392))) :|: X2127 >= 0, z = 1 + (1 + X1141 + X2127 + X392), X392 >= 0, X1141 >= 0 mark(z) -{ 2 }-> a__nats(0) :|: z = 1 + 0 mark(z) -{ 2 }-> a__nats(1 + mark(z - 2)) :|: z - 2 >= 0 mark(z) -{ 2 }-> a__nats(1 + mark(X1142) + X2128) :|: z = 1 + (1 + X1142 + X2128), X2128 >= 0, X1142 >= 0 mark(z) -{ 1 }-> a__filter(mark(X1), mark(X2), mark(X3)) :|: X1 >= 0, X3 >= 0, z = 1 + X1 + X2 + X3, X2 >= 0 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 ---------------------------------------- (17) 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 ---------------------------------------- (18) 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__filter(cons(X, Y), 0, M) -> cons(0, filter(Y, M, M)) [1] a__filter(cons(X, Y), s(N), M) -> cons(mark(X), filter(Y, N, M)) [1] a__sieve(cons(0, Y)) -> cons(0, sieve(Y)) [1] a__sieve(cons(s(N), Y)) -> cons(s(mark(N)), sieve(filter(Y, N, N))) [1] a__nats(N) -> cons(mark(N), nats(s(N))) [1] a__zprimes -> a__sieve(a__nats(s(s(0)))) [1] mark(filter(X1, X2, X3)) -> a__filter(mark(X1), mark(X2), mark(X3)) [1] mark(sieve(X)) -> a__sieve(mark(X)) [1] mark(nats(X)) -> a__nats(mark(X)) [1] mark(zprimes) -> a__zprimes [1] mark(cons(X1, X2)) -> cons(mark(X1), X2) [1] mark(0) -> 0 [1] mark(s(X)) -> s(mark(X)) [1] a__filter(X1, X2, X3) -> filter(X1, X2, X3) [1] a__sieve(X) -> sieve(X) [1] a__nats(X) -> nats(X) [1] a__zprimes -> zprimes [1] The TRS has the following type information: a__filter :: cons:0:filter:s:sieve:nats:zprimes -> cons:0:filter:s:sieve:nats:zprimes -> cons:0:filter:s:sieve:nats:zprimes -> cons:0:filter:s:sieve:nats:zprimes cons :: cons:0:filter:s:sieve:nats:zprimes -> cons:0:filter:s:sieve:nats:zprimes -> cons:0:filter:s:sieve:nats:zprimes 0 :: cons:0:filter:s:sieve:nats:zprimes filter :: cons:0:filter:s:sieve:nats:zprimes -> cons:0:filter:s:sieve:nats:zprimes -> cons:0:filter:s:sieve:nats:zprimes -> cons:0:filter:s:sieve:nats:zprimes s :: cons:0:filter:s:sieve:nats:zprimes -> cons:0:filter:s:sieve:nats:zprimes mark :: cons:0:filter:s:sieve:nats:zprimes -> cons:0:filter:s:sieve:nats:zprimes a__sieve :: cons:0:filter:s:sieve:nats:zprimes -> cons:0:filter:s:sieve:nats:zprimes sieve :: cons:0:filter:s:sieve:nats:zprimes -> cons:0:filter:s:sieve:nats:zprimes a__nats :: cons:0:filter:s:sieve:nats:zprimes -> cons:0:filter:s:sieve:nats:zprimes nats :: cons:0:filter:s:sieve:nats:zprimes -> cons:0:filter:s:sieve:nats:zprimes a__zprimes :: cons:0:filter:s:sieve:nats:zprimes zprimes :: cons:0:filter:s:sieve:nats:zprimes Rewrite Strategy: INNERMOST ---------------------------------------- (19) 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 zprimes => 1 ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: a__filter(z, z', z'') -{ 1 }-> 1 + mark(X) + (1 + Y + N + M) :|: z' = 1 + N, Y >= 0, z = 1 + X + Y, X >= 0, M >= 0, z'' = M, N >= 0 a__filter(z, z', z'') -{ 1 }-> 1 + 0 + (1 + Y + M + M) :|: Y >= 0, z = 1 + X + Y, X >= 0, M >= 0, z' = 0, z'' = M a__filter(z, z', z'') -{ 1 }-> 1 + X1 + X2 + X3 :|: X1 >= 0, X3 >= 0, X2 >= 0, z = X1, z' = X2, z'' = X3 a__nats(z) -{ 1 }-> 1 + X :|: X >= 0, z = X a__nats(z) -{ 1 }-> 1 + mark(N) + (1 + (1 + N)) :|: z = N, N >= 0 a__sieve(z) -{ 1 }-> 1 + X :|: X >= 0, z = X a__sieve(z) -{ 1 }-> 1 + 0 + (1 + Y) :|: Y >= 0, z = 1 + 0 + Y a__sieve(z) -{ 1 }-> 1 + (1 + mark(N)) + (1 + (1 + Y + N + N)) :|: Y >= 0, z = 1 + (1 + N) + Y, N >= 0 a__zprimes -{ 1 }-> a__sieve(a__nats(1 + (1 + 0))) :|: a__zprimes -{ 1 }-> 1 :|: mark(z) -{ 1 }-> a__zprimes :|: z = 1 mark(z) -{ 1 }-> a__sieve(mark(X)) :|: z = 1 + X, X >= 0 mark(z) -{ 1 }-> a__nats(mark(X)) :|: z = 1 + X, X >= 0 mark(z) -{ 1 }-> a__filter(mark(X1), mark(X2), mark(X3)) :|: X1 >= 0, X3 >= 0, z = 1 + X1 + X2 + X3, 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 Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (21) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (22) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), 0, z2) -> c A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__FILTER(z0, z1, z2) -> c2 A__SIEVE(cons(0, z0)) -> c3 A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__SIEVE(z0) -> c5 A__NATS(z0) -> c6(MARK(z0)) A__NATS(z0) -> c7 A__ZPRIMES -> c8(A__SIEVE(a__nats(s(s(0)))), A__NATS(s(s(0)))) A__ZPRIMES -> c9 MARK(filter(z0, z1, z2)) -> c10(A__FILTER(mark(z0), mark(z1), mark(z2)), MARK(z0)) MARK(filter(z0, z1, z2)) -> c11(A__FILTER(mark(z0), mark(z1), mark(z2)), MARK(z1)) MARK(filter(z0, z1, z2)) -> c12(A__FILTER(mark(z0), mark(z1), mark(z2)), MARK(z2)) MARK(sieve(z0)) -> c13(A__SIEVE(mark(z0)), MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(0) -> c17 MARK(s(z0)) -> c18(MARK(z0)) S tuples: A__FILTER(cons(z0, z1), 0, z2) -> c A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__FILTER(z0, z1, z2) -> c2 A__SIEVE(cons(0, z0)) -> c3 A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__SIEVE(z0) -> c5 A__NATS(z0) -> c6(MARK(z0)) A__NATS(z0) -> c7 A__ZPRIMES -> c8(A__SIEVE(a__nats(s(s(0)))), A__NATS(s(s(0)))) A__ZPRIMES -> c9 MARK(filter(z0, z1, z2)) -> c10(A__FILTER(mark(z0), mark(z1), mark(z2)), MARK(z0)) MARK(filter(z0, z1, z2)) -> c11(A__FILTER(mark(z0), mark(z1), mark(z2)), MARK(z1)) MARK(filter(z0, z1, z2)) -> c12(A__FILTER(mark(z0), mark(z1), mark(z2)), MARK(z2)) MARK(sieve(z0)) -> c13(A__SIEVE(mark(z0)), MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(0) -> c17 MARK(s(z0)) -> c18(MARK(z0)) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, A__ZPRIMES, MARK_1 Compound Symbols: c, c1_1, c2, c3, c4_1, c5, c6_1, c7, c8_2, c9, c10_2, c11_2, c12_2, c13_2, c14_2, c15_1, c16_1, c17, c18_1 ---------------------------------------- (23) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 7 trailing nodes: A__FILTER(z0, z1, z2) -> c2 A__NATS(z0) -> c7 A__ZPRIMES -> c9 A__SIEVE(z0) -> c5 A__FILTER(cons(z0, z1), 0, z2) -> c A__SIEVE(cons(0, z0)) -> c3 MARK(0) -> c17 ---------------------------------------- (24) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) A__ZPRIMES -> c8(A__SIEVE(a__nats(s(s(0)))), A__NATS(s(s(0)))) MARK(filter(z0, z1, z2)) -> c10(A__FILTER(mark(z0), mark(z1), mark(z2)), MARK(z0)) MARK(filter(z0, z1, z2)) -> c11(A__FILTER(mark(z0), mark(z1), mark(z2)), MARK(z1)) MARK(filter(z0, z1, z2)) -> c12(A__FILTER(mark(z0), mark(z1), mark(z2)), MARK(z2)) MARK(sieve(z0)) -> c13(A__SIEVE(mark(z0)), MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) S tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) A__ZPRIMES -> c8(A__SIEVE(a__nats(s(s(0)))), A__NATS(s(s(0)))) MARK(filter(z0, z1, z2)) -> c10(A__FILTER(mark(z0), mark(z1), mark(z2)), MARK(z0)) MARK(filter(z0, z1, z2)) -> c11(A__FILTER(mark(z0), mark(z1), mark(z2)), MARK(z1)) MARK(filter(z0, z1, z2)) -> c12(A__FILTER(mark(z0), mark(z1), mark(z2)), MARK(z2)) MARK(sieve(z0)) -> c13(A__SIEVE(mark(z0)), MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, A__ZPRIMES, MARK_1 Compound Symbols: c1_1, c4_1, c6_1, c8_2, c10_2, c11_2, c12_2, c13_2, c14_2, c15_1, c16_1, c18_1 ---------------------------------------- (25) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace A__ZPRIMES -> c8(A__SIEVE(a__nats(s(s(0)))), A__NATS(s(s(0)))) by A__ZPRIMES -> c8(A__SIEVE(cons(mark(s(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) A__ZPRIMES -> c8(A__SIEVE(nats(s(s(0)))), A__NATS(s(s(0)))) ---------------------------------------- (26) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(filter(z0, z1, z2)) -> c10(A__FILTER(mark(z0), mark(z1), mark(z2)), MARK(z0)) MARK(filter(z0, z1, z2)) -> c11(A__FILTER(mark(z0), mark(z1), mark(z2)), MARK(z1)) MARK(filter(z0, z1, z2)) -> c12(A__FILTER(mark(z0), mark(z1), mark(z2)), MARK(z2)) MARK(sieve(z0)) -> c13(A__SIEVE(mark(z0)), MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__SIEVE(cons(mark(s(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) A__ZPRIMES -> c8(A__SIEVE(nats(s(s(0)))), A__NATS(s(s(0)))) S tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(filter(z0, z1, z2)) -> c10(A__FILTER(mark(z0), mark(z1), mark(z2)), MARK(z0)) MARK(filter(z0, z1, z2)) -> c11(A__FILTER(mark(z0), mark(z1), mark(z2)), MARK(z1)) MARK(filter(z0, z1, z2)) -> c12(A__FILTER(mark(z0), mark(z1), mark(z2)), MARK(z2)) MARK(sieve(z0)) -> c13(A__SIEVE(mark(z0)), MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__SIEVE(cons(mark(s(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) A__ZPRIMES -> c8(A__SIEVE(nats(s(s(0)))), A__NATS(s(s(0)))) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c10_2, c11_2, c12_2, c13_2, c14_2, c15_1, c16_1, c18_1, c8_2 ---------------------------------------- (27) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (28) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(filter(z0, z1, z2)) -> c10(A__FILTER(mark(z0), mark(z1), mark(z2)), MARK(z0)) MARK(filter(z0, z1, z2)) -> c11(A__FILTER(mark(z0), mark(z1), mark(z2)), MARK(z1)) MARK(filter(z0, z1, z2)) -> c12(A__FILTER(mark(z0), mark(z1), mark(z2)), MARK(z2)) MARK(sieve(z0)) -> c13(A__SIEVE(mark(z0)), MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__SIEVE(cons(mark(s(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) A__ZPRIMES -> c8(A__NATS(s(s(0)))) S tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(filter(z0, z1, z2)) -> c10(A__FILTER(mark(z0), mark(z1), mark(z2)), MARK(z0)) MARK(filter(z0, z1, z2)) -> c11(A__FILTER(mark(z0), mark(z1), mark(z2)), MARK(z1)) MARK(filter(z0, z1, z2)) -> c12(A__FILTER(mark(z0), mark(z1), mark(z2)), MARK(z2)) MARK(sieve(z0)) -> c13(A__SIEVE(mark(z0)), MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__SIEVE(cons(mark(s(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) A__ZPRIMES -> c8(A__NATS(s(s(0)))) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c10_2, c11_2, c12_2, c13_2, c14_2, c15_1, c16_1, c18_1, c8_2, c8_1 ---------------------------------------- (29) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(filter(z0, z1, z2)) -> c10(A__FILTER(mark(z0), mark(z1), mark(z2)), MARK(z0)) by MARK(filter(x0, x1, filter(z0, z1, z2))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x0)) MARK(filter(x0, x1, sieve(z0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, x1, nats(z0))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x0)) MARK(filter(x0, x1, zprimes)) -> c10(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x0)) MARK(filter(x0, x1, cons(z0, z1))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x0)) MARK(filter(x0, x1, 0)) -> c10(A__FILTER(mark(x0), mark(x1), 0), MARK(x0)) MARK(filter(x0, x1, s(z0))) -> c10(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(z0), x2)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, nats(z0), x2)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, zprimes, x2)) -> c10(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x0)) MARK(filter(x0, cons(z0, z1), x2)) -> c10(A__FILTER(mark(x0), cons(mark(z0), z1), mark(x2)), MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(A__FILTER(mark(x0), 0, mark(x2)), MARK(x0)) MARK(filter(x0, s(z0), x2)) -> c10(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, x2)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, x2)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, x1, x2)) -> c10(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, x2)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c10(A__FILTER(0, mark(x1), mark(x2)), MARK(0)) MARK(filter(s(z0), x1, x2)) -> c10(A__FILTER(s(mark(z0)), mark(x1), mark(x2)), MARK(s(z0))) ---------------------------------------- (30) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(filter(z0, z1, z2)) -> c11(A__FILTER(mark(z0), mark(z1), mark(z2)), MARK(z1)) MARK(filter(z0, z1, z2)) -> c12(A__FILTER(mark(z0), mark(z1), mark(z2)), MARK(z2)) MARK(sieve(z0)) -> c13(A__SIEVE(mark(z0)), MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__SIEVE(cons(mark(s(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x0)) MARK(filter(x0, x1, sieve(z0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, x1, nats(z0))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x0)) MARK(filter(x0, x1, zprimes)) -> c10(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x0)) MARK(filter(x0, x1, cons(z0, z1))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x0)) MARK(filter(x0, x1, 0)) -> c10(A__FILTER(mark(x0), mark(x1), 0), MARK(x0)) MARK(filter(x0, x1, s(z0))) -> c10(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(z0), x2)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, nats(z0), x2)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, zprimes, x2)) -> c10(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x0)) MARK(filter(x0, cons(z0, z1), x2)) -> c10(A__FILTER(mark(x0), cons(mark(z0), z1), mark(x2)), MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(A__FILTER(mark(x0), 0, mark(x2)), MARK(x0)) MARK(filter(x0, s(z0), x2)) -> c10(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, x2)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, x2)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, x1, x2)) -> c10(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, x2)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c10(A__FILTER(0, mark(x1), mark(x2)), MARK(0)) MARK(filter(s(z0), x1, x2)) -> c10(A__FILTER(s(mark(z0)), mark(x1), mark(x2)), MARK(s(z0))) S tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(filter(z0, z1, z2)) -> c11(A__FILTER(mark(z0), mark(z1), mark(z2)), MARK(z1)) MARK(filter(z0, z1, z2)) -> c12(A__FILTER(mark(z0), mark(z1), mark(z2)), MARK(z2)) MARK(sieve(z0)) -> c13(A__SIEVE(mark(z0)), MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__SIEVE(cons(mark(s(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x0)) MARK(filter(x0, x1, sieve(z0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, x1, nats(z0))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x0)) MARK(filter(x0, x1, zprimes)) -> c10(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x0)) MARK(filter(x0, x1, cons(z0, z1))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x0)) MARK(filter(x0, x1, 0)) -> c10(A__FILTER(mark(x0), mark(x1), 0), MARK(x0)) MARK(filter(x0, x1, s(z0))) -> c10(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(z0), x2)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, nats(z0), x2)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, zprimes, x2)) -> c10(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x0)) MARK(filter(x0, cons(z0, z1), x2)) -> c10(A__FILTER(mark(x0), cons(mark(z0), z1), mark(x2)), MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(A__FILTER(mark(x0), 0, mark(x2)), MARK(x0)) MARK(filter(x0, s(z0), x2)) -> c10(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, x2)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, x2)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, x1, x2)) -> c10(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, x2)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c10(A__FILTER(0, mark(x1), mark(x2)), MARK(0)) MARK(filter(s(z0), x1, x2)) -> c10(A__FILTER(s(mark(z0)), mark(x1), mark(x2)), MARK(s(z0))) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c11_2, c12_2, c13_2, c14_2, c15_1, c16_1, c18_1, c8_2, c8_1, c10_2 ---------------------------------------- (31) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(filter(0, x1, x2)) -> c10(A__FILTER(0, mark(x1), mark(x2)), MARK(0)) ---------------------------------------- (32) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(filter(z0, z1, z2)) -> c11(A__FILTER(mark(z0), mark(z1), mark(z2)), MARK(z1)) MARK(filter(z0, z1, z2)) -> c12(A__FILTER(mark(z0), mark(z1), mark(z2)), MARK(z2)) MARK(sieve(z0)) -> c13(A__SIEVE(mark(z0)), MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__SIEVE(cons(mark(s(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x0)) MARK(filter(x0, x1, sieve(z0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, x1, nats(z0))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x0)) MARK(filter(x0, x1, zprimes)) -> c10(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x0)) MARK(filter(x0, x1, cons(z0, z1))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x0)) MARK(filter(x0, x1, 0)) -> c10(A__FILTER(mark(x0), mark(x1), 0), MARK(x0)) MARK(filter(x0, x1, s(z0))) -> c10(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(z0), x2)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, nats(z0), x2)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, zprimes, x2)) -> c10(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x0)) MARK(filter(x0, cons(z0, z1), x2)) -> c10(A__FILTER(mark(x0), cons(mark(z0), z1), mark(x2)), MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(A__FILTER(mark(x0), 0, mark(x2)), MARK(x0)) MARK(filter(x0, s(z0), x2)) -> c10(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, x2)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, x2)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, x1, x2)) -> c10(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, x2)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(s(z0), x1, x2)) -> c10(A__FILTER(s(mark(z0)), mark(x1), mark(x2)), MARK(s(z0))) S tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(filter(z0, z1, z2)) -> c11(A__FILTER(mark(z0), mark(z1), mark(z2)), MARK(z1)) MARK(filter(z0, z1, z2)) -> c12(A__FILTER(mark(z0), mark(z1), mark(z2)), MARK(z2)) MARK(sieve(z0)) -> c13(A__SIEVE(mark(z0)), MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__SIEVE(cons(mark(s(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x0)) MARK(filter(x0, x1, sieve(z0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, x1, nats(z0))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x0)) MARK(filter(x0, x1, zprimes)) -> c10(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x0)) MARK(filter(x0, x1, cons(z0, z1))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x0)) MARK(filter(x0, x1, 0)) -> c10(A__FILTER(mark(x0), mark(x1), 0), MARK(x0)) MARK(filter(x0, x1, s(z0))) -> c10(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(z0), x2)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, nats(z0), x2)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, zprimes, x2)) -> c10(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x0)) MARK(filter(x0, cons(z0, z1), x2)) -> c10(A__FILTER(mark(x0), cons(mark(z0), z1), mark(x2)), MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(A__FILTER(mark(x0), 0, mark(x2)), MARK(x0)) MARK(filter(x0, s(z0), x2)) -> c10(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, x2)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, x2)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, x1, x2)) -> c10(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, x2)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(s(z0), x1, x2)) -> c10(A__FILTER(s(mark(z0)), mark(x1), mark(x2)), MARK(s(z0))) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c11_2, c12_2, c13_2, c14_2, c15_1, c16_1, c18_1, c8_2, c8_1, c10_2 ---------------------------------------- (33) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (34) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(filter(z0, z1, z2)) -> c11(A__FILTER(mark(z0), mark(z1), mark(z2)), MARK(z1)) MARK(filter(z0, z1, z2)) -> c12(A__FILTER(mark(z0), mark(z1), mark(z2)), MARK(z2)) MARK(sieve(z0)) -> c13(A__SIEVE(mark(z0)), MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__SIEVE(cons(mark(s(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x0)) MARK(filter(x0, x1, sieve(z0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, x1, nats(z0))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x0)) MARK(filter(x0, x1, zprimes)) -> c10(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x0)) MARK(filter(x0, x1, cons(z0, z1))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x0)) MARK(filter(x0, x1, 0)) -> c10(A__FILTER(mark(x0), mark(x1), 0), MARK(x0)) MARK(filter(x0, x1, s(z0))) -> c10(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(z0), x2)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, nats(z0), x2)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, zprimes, x2)) -> c10(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x0)) MARK(filter(x0, s(z0), x2)) -> c10(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, x2)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, x2)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, x1, x2)) -> c10(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, x2)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) S tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(filter(z0, z1, z2)) -> c11(A__FILTER(mark(z0), mark(z1), mark(z2)), MARK(z1)) MARK(filter(z0, z1, z2)) -> c12(A__FILTER(mark(z0), mark(z1), mark(z2)), MARK(z2)) MARK(sieve(z0)) -> c13(A__SIEVE(mark(z0)), MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__SIEVE(cons(mark(s(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x0)) MARK(filter(x0, x1, sieve(z0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, x1, nats(z0))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x0)) MARK(filter(x0, x1, zprimes)) -> c10(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x0)) MARK(filter(x0, x1, cons(z0, z1))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x0)) MARK(filter(x0, x1, 0)) -> c10(A__FILTER(mark(x0), mark(x1), 0), MARK(x0)) MARK(filter(x0, x1, s(z0))) -> c10(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(z0), x2)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, nats(z0), x2)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, zprimes, x2)) -> c10(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x0)) MARK(filter(x0, s(z0), x2)) -> c10(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, x2)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, x2)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, x1, x2)) -> c10(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, x2)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c11_2, c12_2, c13_2, c14_2, c15_1, c16_1, c18_1, c8_2, c8_1, c10_2, c10_1 ---------------------------------------- (35) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(filter(z0, z1, z2)) -> c11(A__FILTER(mark(z0), mark(z1), mark(z2)), MARK(z1)) by MARK(filter(x0, x1, filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x1)) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(A__FILTER(mark(x0), cons(mark(z0), z1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, 0, x2)) -> c11(A__FILTER(mark(x0), 0, mark(x2)), MARK(0)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(0, x1, x2)) -> c11(A__FILTER(0, mark(x1), mark(x2)), MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(A__FILTER(s(mark(z0)), mark(x1), mark(x2)), MARK(x1)) ---------------------------------------- (36) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(filter(z0, z1, z2)) -> c12(A__FILTER(mark(z0), mark(z1), mark(z2)), MARK(z2)) MARK(sieve(z0)) -> c13(A__SIEVE(mark(z0)), MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__SIEVE(cons(mark(s(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x0)) MARK(filter(x0, x1, sieve(z0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, x1, nats(z0))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x0)) MARK(filter(x0, x1, zprimes)) -> c10(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x0)) MARK(filter(x0, x1, cons(z0, z1))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x0)) MARK(filter(x0, x1, 0)) -> c10(A__FILTER(mark(x0), mark(x1), 0), MARK(x0)) MARK(filter(x0, x1, s(z0))) -> c10(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(z0), x2)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, nats(z0), x2)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, zprimes, x2)) -> c10(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x0)) MARK(filter(x0, s(z0), x2)) -> c10(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, x2)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, x2)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, x1, x2)) -> c10(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, x2)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x1)) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(A__FILTER(mark(x0), cons(mark(z0), z1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, 0, x2)) -> c11(A__FILTER(mark(x0), 0, mark(x2)), MARK(0)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(0, x1, x2)) -> c11(A__FILTER(0, mark(x1), mark(x2)), MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(A__FILTER(s(mark(z0)), mark(x1), mark(x2)), MARK(x1)) S tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(filter(z0, z1, z2)) -> c12(A__FILTER(mark(z0), mark(z1), mark(z2)), MARK(z2)) MARK(sieve(z0)) -> c13(A__SIEVE(mark(z0)), MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__SIEVE(cons(mark(s(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x0)) MARK(filter(x0, x1, sieve(z0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, x1, nats(z0))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x0)) MARK(filter(x0, x1, zprimes)) -> c10(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x0)) MARK(filter(x0, x1, cons(z0, z1))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x0)) MARK(filter(x0, x1, 0)) -> c10(A__FILTER(mark(x0), mark(x1), 0), MARK(x0)) MARK(filter(x0, x1, s(z0))) -> c10(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(z0), x2)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, nats(z0), x2)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, zprimes, x2)) -> c10(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x0)) MARK(filter(x0, s(z0), x2)) -> c10(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, x2)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, x2)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, x1, x2)) -> c10(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, x2)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x1)) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(A__FILTER(mark(x0), cons(mark(z0), z1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, 0, x2)) -> c11(A__FILTER(mark(x0), 0, mark(x2)), MARK(0)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(0, x1, x2)) -> c11(A__FILTER(0, mark(x1), mark(x2)), MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(A__FILTER(s(mark(z0)), mark(x1), mark(x2)), MARK(x1)) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c12_2, c13_2, c14_2, c15_1, c16_1, c18_1, c8_2, c8_1, c10_2, c10_1, c11_2 ---------------------------------------- (37) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(filter(x0, 0, x2)) -> c11(A__FILTER(mark(x0), 0, mark(x2)), MARK(0)) ---------------------------------------- (38) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(filter(z0, z1, z2)) -> c12(A__FILTER(mark(z0), mark(z1), mark(z2)), MARK(z2)) MARK(sieve(z0)) -> c13(A__SIEVE(mark(z0)), MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__SIEVE(cons(mark(s(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x0)) MARK(filter(x0, x1, sieve(z0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, x1, nats(z0))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x0)) MARK(filter(x0, x1, zprimes)) -> c10(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x0)) MARK(filter(x0, x1, cons(z0, z1))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x0)) MARK(filter(x0, x1, 0)) -> c10(A__FILTER(mark(x0), mark(x1), 0), MARK(x0)) MARK(filter(x0, x1, s(z0))) -> c10(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(z0), x2)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, nats(z0), x2)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, zprimes, x2)) -> c10(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x0)) MARK(filter(x0, s(z0), x2)) -> c10(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, x2)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, x2)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, x1, x2)) -> c10(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, x2)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x1)) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(A__FILTER(mark(x0), cons(mark(z0), z1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(0, x1, x2)) -> c11(A__FILTER(0, mark(x1), mark(x2)), MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(A__FILTER(s(mark(z0)), mark(x1), mark(x2)), MARK(x1)) S tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(filter(z0, z1, z2)) -> c12(A__FILTER(mark(z0), mark(z1), mark(z2)), MARK(z2)) MARK(sieve(z0)) -> c13(A__SIEVE(mark(z0)), MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__SIEVE(cons(mark(s(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x0)) MARK(filter(x0, x1, sieve(z0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, x1, nats(z0))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x0)) MARK(filter(x0, x1, zprimes)) -> c10(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x0)) MARK(filter(x0, x1, cons(z0, z1))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x0)) MARK(filter(x0, x1, 0)) -> c10(A__FILTER(mark(x0), mark(x1), 0), MARK(x0)) MARK(filter(x0, x1, s(z0))) -> c10(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(z0), x2)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, nats(z0), x2)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, zprimes, x2)) -> c10(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x0)) MARK(filter(x0, s(z0), x2)) -> c10(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, x2)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, x2)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, x1, x2)) -> c10(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, x2)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x1)) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(A__FILTER(mark(x0), cons(mark(z0), z1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(0, x1, x2)) -> c11(A__FILTER(0, mark(x1), mark(x2)), MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(A__FILTER(s(mark(z0)), mark(x1), mark(x2)), MARK(x1)) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c12_2, c13_2, c14_2, c15_1, c16_1, c18_1, c8_2, c8_1, c10_2, c10_1, c11_2 ---------------------------------------- (39) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (40) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(filter(z0, z1, z2)) -> c12(A__FILTER(mark(z0), mark(z1), mark(z2)), MARK(z2)) MARK(sieve(z0)) -> c13(A__SIEVE(mark(z0)), MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__SIEVE(cons(mark(s(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x0)) MARK(filter(x0, x1, sieve(z0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, x1, nats(z0))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x0)) MARK(filter(x0, x1, zprimes)) -> c10(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x0)) MARK(filter(x0, x1, cons(z0, z1))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x0)) MARK(filter(x0, x1, 0)) -> c10(A__FILTER(mark(x0), mark(x1), 0), MARK(x0)) MARK(filter(x0, x1, s(z0))) -> c10(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(z0), x2)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, nats(z0), x2)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, zprimes, x2)) -> c10(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x0)) MARK(filter(x0, s(z0), x2)) -> c10(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, x2)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, x2)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, x1, x2)) -> c10(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, x2)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x1)) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) S tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(filter(z0, z1, z2)) -> c12(A__FILTER(mark(z0), mark(z1), mark(z2)), MARK(z2)) MARK(sieve(z0)) -> c13(A__SIEVE(mark(z0)), MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__SIEVE(cons(mark(s(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x0)) MARK(filter(x0, x1, sieve(z0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, x1, nats(z0))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x0)) MARK(filter(x0, x1, zprimes)) -> c10(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x0)) MARK(filter(x0, x1, cons(z0, z1))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x0)) MARK(filter(x0, x1, 0)) -> c10(A__FILTER(mark(x0), mark(x1), 0), MARK(x0)) MARK(filter(x0, x1, s(z0))) -> c10(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(z0), x2)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, nats(z0), x2)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, zprimes, x2)) -> c10(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x0)) MARK(filter(x0, s(z0), x2)) -> c10(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, x2)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, x2)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, x1, x2)) -> c10(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, x2)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x1)) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c12_2, c13_2, c14_2, c15_1, c16_1, c18_1, c8_2, c8_1, c10_2, c10_1, c11_2, c11_1 ---------------------------------------- (41) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(filter(z0, z1, z2)) -> c12(A__FILTER(mark(z0), mark(z1), mark(z2)), MARK(z2)) by MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0), MARK(0)) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(A__FILTER(mark(x0), cons(mark(z0), z1), mark(x2)), MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(A__FILTER(mark(x0), 0, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(0, x1, x2)) -> c12(A__FILTER(0, mark(x1), mark(x2)), MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(A__FILTER(s(mark(z0)), mark(x1), mark(x2)), MARK(x2)) ---------------------------------------- (42) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(sieve(z0)) -> c13(A__SIEVE(mark(z0)), MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__SIEVE(cons(mark(s(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x0)) MARK(filter(x0, x1, sieve(z0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, x1, nats(z0))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x0)) MARK(filter(x0, x1, zprimes)) -> c10(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x0)) MARK(filter(x0, x1, cons(z0, z1))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x0)) MARK(filter(x0, x1, 0)) -> c10(A__FILTER(mark(x0), mark(x1), 0), MARK(x0)) MARK(filter(x0, x1, s(z0))) -> c10(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(z0), x2)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, nats(z0), x2)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, zprimes, x2)) -> c10(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x0)) MARK(filter(x0, s(z0), x2)) -> c10(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, x2)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, x2)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, x1, x2)) -> c10(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, x2)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x1)) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0), MARK(0)) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(A__FILTER(mark(x0), cons(mark(z0), z1), mark(x2)), MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(A__FILTER(mark(x0), 0, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(0, x1, x2)) -> c12(A__FILTER(0, mark(x1), mark(x2)), MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(A__FILTER(s(mark(z0)), mark(x1), mark(x2)), MARK(x2)) S tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(sieve(z0)) -> c13(A__SIEVE(mark(z0)), MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__SIEVE(cons(mark(s(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x0)) MARK(filter(x0, x1, sieve(z0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, x1, nats(z0))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x0)) MARK(filter(x0, x1, zprimes)) -> c10(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x0)) MARK(filter(x0, x1, cons(z0, z1))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x0)) MARK(filter(x0, x1, 0)) -> c10(A__FILTER(mark(x0), mark(x1), 0), MARK(x0)) MARK(filter(x0, x1, s(z0))) -> c10(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(z0), x2)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, nats(z0), x2)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, zprimes, x2)) -> c10(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x0)) MARK(filter(x0, s(z0), x2)) -> c10(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, x2)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, x2)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, x1, x2)) -> c10(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, x2)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x1)) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0), MARK(0)) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(A__FILTER(mark(x0), cons(mark(z0), z1), mark(x2)), MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(A__FILTER(mark(x0), 0, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(0, x1, x2)) -> c12(A__FILTER(0, mark(x1), mark(x2)), MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(A__FILTER(s(mark(z0)), mark(x1), mark(x2)), MARK(x2)) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c13_2, c14_2, c15_1, c16_1, c18_1, c8_2, c8_1, c10_2, c10_1, c11_2, c11_1, c12_2 ---------------------------------------- (43) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 5 trailing tuple parts ---------------------------------------- (44) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(sieve(z0)) -> c13(A__SIEVE(mark(z0)), MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__SIEVE(cons(mark(s(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x0)) MARK(filter(x0, x1, sieve(z0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, x1, nats(z0))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x0)) MARK(filter(x0, x1, zprimes)) -> c10(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x0)) MARK(filter(x0, x1, cons(z0, z1))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x0)) MARK(filter(x0, x1, 0)) -> c10(A__FILTER(mark(x0), mark(x1), 0), MARK(x0)) MARK(filter(x0, x1, s(z0))) -> c10(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(z0), x2)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, nats(z0), x2)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, zprimes, x2)) -> c10(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x0)) MARK(filter(x0, s(z0), x2)) -> c10(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, x2)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, x2)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, x1, x2)) -> c10(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, x2)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x1)) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) S tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(sieve(z0)) -> c13(A__SIEVE(mark(z0)), MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__SIEVE(cons(mark(s(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x0)) MARK(filter(x0, x1, sieve(z0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, x1, nats(z0))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x0)) MARK(filter(x0, x1, zprimes)) -> c10(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x0)) MARK(filter(x0, x1, cons(z0, z1))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x0)) MARK(filter(x0, x1, 0)) -> c10(A__FILTER(mark(x0), mark(x1), 0), MARK(x0)) MARK(filter(x0, x1, s(z0))) -> c10(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(z0), x2)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, nats(z0), x2)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, zprimes, x2)) -> c10(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x0)) MARK(filter(x0, s(z0), x2)) -> c10(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, x2)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, x2)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, x1, x2)) -> c10(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, x2)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x1)) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c13_2, c14_2, c15_1, c16_1, c18_1, c8_2, c8_1, c10_2, c10_1, c11_2, c11_1, c12_2, c12_1 ---------------------------------------- (45) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(sieve(z0)) -> c13(A__SIEVE(mark(z0)), MARK(z0)) by MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(0)) -> c13(A__SIEVE(0), MARK(0)) MARK(sieve(s(z0))) -> c13(A__SIEVE(s(mark(z0))), MARK(s(z0))) ---------------------------------------- (46) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__SIEVE(cons(mark(s(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x0)) MARK(filter(x0, x1, sieve(z0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, x1, nats(z0))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x0)) MARK(filter(x0, x1, zprimes)) -> c10(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x0)) MARK(filter(x0, x1, cons(z0, z1))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x0)) MARK(filter(x0, x1, 0)) -> c10(A__FILTER(mark(x0), mark(x1), 0), MARK(x0)) MARK(filter(x0, x1, s(z0))) -> c10(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(z0), x2)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, nats(z0), x2)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, zprimes, x2)) -> c10(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x0)) MARK(filter(x0, s(z0), x2)) -> c10(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, x2)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, x2)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, x1, x2)) -> c10(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, x2)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x1)) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(0)) -> c13(A__SIEVE(0), MARK(0)) MARK(sieve(s(z0))) -> c13(A__SIEVE(s(mark(z0))), MARK(s(z0))) S tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__SIEVE(cons(mark(s(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x0)) MARK(filter(x0, x1, sieve(z0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, x1, nats(z0))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x0)) MARK(filter(x0, x1, zprimes)) -> c10(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x0)) MARK(filter(x0, x1, cons(z0, z1))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x0)) MARK(filter(x0, x1, 0)) -> c10(A__FILTER(mark(x0), mark(x1), 0), MARK(x0)) MARK(filter(x0, x1, s(z0))) -> c10(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(z0), x2)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, nats(z0), x2)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, zprimes, x2)) -> c10(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x0)) MARK(filter(x0, s(z0), x2)) -> c10(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, x2)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, x2)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, x1, x2)) -> c10(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, x2)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x1)) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(0)) -> c13(A__SIEVE(0), MARK(0)) MARK(sieve(s(z0))) -> c13(A__SIEVE(s(mark(z0))), MARK(s(z0))) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c14_2, c15_1, c16_1, c18_1, c8_2, c8_1, c10_2, c10_1, c11_2, c11_1, c12_2, c12_1, c13_2 ---------------------------------------- (47) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(sieve(0)) -> c13(A__SIEVE(0), MARK(0)) ---------------------------------------- (48) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__SIEVE(cons(mark(s(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x0)) MARK(filter(x0, x1, sieve(z0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, x1, nats(z0))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x0)) MARK(filter(x0, x1, zprimes)) -> c10(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x0)) MARK(filter(x0, x1, cons(z0, z1))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x0)) MARK(filter(x0, x1, 0)) -> c10(A__FILTER(mark(x0), mark(x1), 0), MARK(x0)) MARK(filter(x0, x1, s(z0))) -> c10(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(z0), x2)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, nats(z0), x2)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, zprimes, x2)) -> c10(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x0)) MARK(filter(x0, s(z0), x2)) -> c10(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, x2)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, x2)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, x1, x2)) -> c10(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, x2)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x1)) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(A__SIEVE(s(mark(z0))), MARK(s(z0))) S tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__SIEVE(cons(mark(s(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x0)) MARK(filter(x0, x1, sieve(z0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, x1, nats(z0))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x0)) MARK(filter(x0, x1, zprimes)) -> c10(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x0)) MARK(filter(x0, x1, cons(z0, z1))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x0)) MARK(filter(x0, x1, 0)) -> c10(A__FILTER(mark(x0), mark(x1), 0), MARK(x0)) MARK(filter(x0, x1, s(z0))) -> c10(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(z0), x2)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, nats(z0), x2)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, zprimes, x2)) -> c10(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x0)) MARK(filter(x0, s(z0), x2)) -> c10(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, x2)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, x2)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, x1, x2)) -> c10(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, x2)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x1)) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(A__SIEVE(s(mark(z0))), MARK(s(z0))) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c14_2, c15_1, c16_1, c18_1, c8_2, c8_1, c10_2, c10_1, c11_2, c11_1, c12_2, c12_1, c13_2 ---------------------------------------- (49) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (50) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__SIEVE(cons(mark(s(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x0)) MARK(filter(x0, x1, sieve(z0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, x1, nats(z0))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x0)) MARK(filter(x0, x1, zprimes)) -> c10(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x0)) MARK(filter(x0, x1, cons(z0, z1))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x0)) MARK(filter(x0, x1, 0)) -> c10(A__FILTER(mark(x0), mark(x1), 0), MARK(x0)) MARK(filter(x0, x1, s(z0))) -> c10(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(z0), x2)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, nats(z0), x2)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, zprimes, x2)) -> c10(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x0)) MARK(filter(x0, s(z0), x2)) -> c10(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, x2)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, x2)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, x1, x2)) -> c10(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, x2)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x1)) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) S tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__SIEVE(cons(mark(s(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x0)) MARK(filter(x0, x1, sieve(z0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, x1, nats(z0))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x0)) MARK(filter(x0, x1, zprimes)) -> c10(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x0)) MARK(filter(x0, x1, cons(z0, z1))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x0)) MARK(filter(x0, x1, 0)) -> c10(A__FILTER(mark(x0), mark(x1), 0), MARK(x0)) MARK(filter(x0, x1, s(z0))) -> c10(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(z0), x2)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, nats(z0), x2)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, zprimes, x2)) -> c10(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x0)) MARK(filter(x0, s(z0), x2)) -> c10(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, x2)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, x2)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, x1, x2)) -> c10(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, x2)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x1)) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c14_2, c15_1, c16_1, c18_1, c8_2, c8_1, c10_2, c10_1, c11_2, c11_1, c12_2, c12_1, c13_2, c13_1 ---------------------------------------- (51) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace A__ZPRIMES -> c8(A__SIEVE(cons(mark(s(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) by A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) A__ZPRIMES -> c8(A__NATS(s(s(0)))) ---------------------------------------- (52) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x0)) MARK(filter(x0, x1, sieve(z0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, x1, nats(z0))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x0)) MARK(filter(x0, x1, zprimes)) -> c10(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x0)) MARK(filter(x0, x1, cons(z0, z1))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x0)) MARK(filter(x0, x1, 0)) -> c10(A__FILTER(mark(x0), mark(x1), 0), MARK(x0)) MARK(filter(x0, x1, s(z0))) -> c10(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(z0), x2)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, nats(z0), x2)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, zprimes, x2)) -> c10(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x0)) MARK(filter(x0, s(z0), x2)) -> c10(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, x2)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, x2)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, x1, x2)) -> c10(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, x2)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x1)) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) S tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x0)) MARK(filter(x0, x1, sieve(z0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, x1, nats(z0))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x0)) MARK(filter(x0, x1, zprimes)) -> c10(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x0)) MARK(filter(x0, x1, cons(z0, z1))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x0)) MARK(filter(x0, x1, 0)) -> c10(A__FILTER(mark(x0), mark(x1), 0), MARK(x0)) MARK(filter(x0, x1, s(z0))) -> c10(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(z0), x2)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, nats(z0), x2)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, zprimes, x2)) -> c10(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x0)) MARK(filter(x0, s(z0), x2)) -> c10(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, x2)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, x2)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, x1, x2)) -> c10(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, x2)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x1)) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c14_2, c15_1, c16_1, c18_1, c8_1, c10_2, c10_1, c11_2, c11_1, c12_2, c12_1, c13_2, c13_1, c8_2 ---------------------------------------- (53) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(filter(x0, x1, filter(z0, z1, z2))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x0)) by MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), cons(mark(z0), z1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, 0, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), 0, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(cons(z0, z1))) MARK(filter(0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(0, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(0)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(s(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(s(z0))) ---------------------------------------- (54) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, x1, sieve(z0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, x1, nats(z0))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x0)) MARK(filter(x0, x1, zprimes)) -> c10(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x0)) MARK(filter(x0, x1, cons(z0, z1))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x0)) MARK(filter(x0, x1, 0)) -> c10(A__FILTER(mark(x0), mark(x1), 0), MARK(x0)) MARK(filter(x0, x1, s(z0))) -> c10(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(z0), x2)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, nats(z0), x2)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, zprimes, x2)) -> c10(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x0)) MARK(filter(x0, s(z0), x2)) -> c10(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, x2)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, x2)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, x1, x2)) -> c10(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, x2)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x1)) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), cons(mark(z0), z1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, 0, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), 0, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(cons(z0, z1))) MARK(filter(0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(0, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(0)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(s(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(s(z0))) S tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, x1, sieve(z0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, x1, nats(z0))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x0)) MARK(filter(x0, x1, zprimes)) -> c10(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x0)) MARK(filter(x0, x1, cons(z0, z1))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x0)) MARK(filter(x0, x1, 0)) -> c10(A__FILTER(mark(x0), mark(x1), 0), MARK(x0)) MARK(filter(x0, x1, s(z0))) -> c10(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(z0), x2)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, nats(z0), x2)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, zprimes, x2)) -> c10(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x0)) MARK(filter(x0, s(z0), x2)) -> c10(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, x2)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, x2)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, x1, x2)) -> c10(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, x2)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x1)) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), cons(mark(z0), z1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, 0, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), 0, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(cons(z0, z1))) MARK(filter(0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(0, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(0)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(s(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(s(z0))) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c14_2, c15_1, c16_1, c18_1, c8_1, c10_2, c10_1, c11_2, c11_1, c12_2, c12_1, c13_2, c13_1, c8_2 ---------------------------------------- (55) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(filter(0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(0, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(0)) ---------------------------------------- (56) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, x1, sieve(z0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, x1, nats(z0))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x0)) MARK(filter(x0, x1, zprimes)) -> c10(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x0)) MARK(filter(x0, x1, cons(z0, z1))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x0)) MARK(filter(x0, x1, 0)) -> c10(A__FILTER(mark(x0), mark(x1), 0), MARK(x0)) MARK(filter(x0, x1, s(z0))) -> c10(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(z0), x2)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, nats(z0), x2)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, zprimes, x2)) -> c10(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x0)) MARK(filter(x0, s(z0), x2)) -> c10(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, x2)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, x2)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, x1, x2)) -> c10(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, x2)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x1)) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), cons(mark(z0), z1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, 0, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), 0, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(cons(z0, z1))) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(s(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(s(z0))) S tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, x1, sieve(z0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, x1, nats(z0))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x0)) MARK(filter(x0, x1, zprimes)) -> c10(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x0)) MARK(filter(x0, x1, cons(z0, z1))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x0)) MARK(filter(x0, x1, 0)) -> c10(A__FILTER(mark(x0), mark(x1), 0), MARK(x0)) MARK(filter(x0, x1, s(z0))) -> c10(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(z0), x2)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, nats(z0), x2)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, zprimes, x2)) -> c10(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x0)) MARK(filter(x0, s(z0), x2)) -> c10(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, x2)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, x2)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, x1, x2)) -> c10(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, x2)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x1)) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), cons(mark(z0), z1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, 0, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), 0, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(cons(z0, z1))) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(s(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(s(z0))) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c14_2, c15_1, c16_1, c18_1, c8_1, c10_2, c10_1, c11_2, c11_1, c12_2, c12_1, c13_2, c13_1, c8_2 ---------------------------------------- (57) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (58) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, x1, sieve(z0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, x1, nats(z0))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x0)) MARK(filter(x0, x1, zprimes)) -> c10(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x0)) MARK(filter(x0, x1, cons(z0, z1))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x0)) MARK(filter(x0, x1, 0)) -> c10(A__FILTER(mark(x0), mark(x1), 0), MARK(x0)) MARK(filter(x0, x1, s(z0))) -> c10(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(z0), x2)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, nats(z0), x2)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, zprimes, x2)) -> c10(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x0)) MARK(filter(x0, s(z0), x2)) -> c10(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, x2)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, x2)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, x1, x2)) -> c10(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, x2)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x1)) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(x0, 0, filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c10(MARK(s(z0))) S tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, x1, sieve(z0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, x1, nats(z0))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x0)) MARK(filter(x0, x1, zprimes)) -> c10(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x0)) MARK(filter(x0, x1, cons(z0, z1))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x0)) MARK(filter(x0, x1, 0)) -> c10(A__FILTER(mark(x0), mark(x1), 0), MARK(x0)) MARK(filter(x0, x1, s(z0))) -> c10(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(z0), x2)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, nats(z0), x2)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, zprimes, x2)) -> c10(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x0)) MARK(filter(x0, s(z0), x2)) -> c10(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, x2)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, x2)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, x1, x2)) -> c10(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, x2)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x1)) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(x0, 0, filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c10(MARK(s(z0))) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c14_2, c15_1, c16_1, c18_1, c8_1, c10_2, c10_1, c11_2, c11_1, c12_2, c12_1, c13_2, c13_1, c8_2 ---------------------------------------- (59) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(filter(x0, x1, sieve(z0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x0)) by MARK(filter(x0, x1, sieve(x2))) -> c10(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x0)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, sieve(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, sieve(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, sieve(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(0)), MARK(x0)) MARK(filter(x0, x1, sieve(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), sieve(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, sieve(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, cons(z0, z1), sieve(x2))) -> c10(A__FILTER(mark(x0), cons(mark(z0), z1), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, 0, sieve(x2))) -> c10(A__FILTER(mark(x0), 0, a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, sieve(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__sieve(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, sieve(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, sieve(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, sieve(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), a__sieve(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, sieve(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__sieve(mark(x2))), MARK(cons(z0, z1))) MARK(filter(0, x1, sieve(x2))) -> c10(A__FILTER(0, mark(x1), a__sieve(mark(x2))), MARK(0)) MARK(filter(s(z0), x1, sieve(x2))) -> c10(A__FILTER(s(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(s(z0))) ---------------------------------------- (60) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, x1, nats(z0))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x0)) MARK(filter(x0, x1, zprimes)) -> c10(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x0)) MARK(filter(x0, x1, cons(z0, z1))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x0)) MARK(filter(x0, x1, 0)) -> c10(A__FILTER(mark(x0), mark(x1), 0), MARK(x0)) MARK(filter(x0, x1, s(z0))) -> c10(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(z0), x2)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, nats(z0), x2)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, zprimes, x2)) -> c10(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x0)) MARK(filter(x0, s(z0), x2)) -> c10(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, x2)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, x2)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, x1, x2)) -> c10(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, x2)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x1)) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(x0, 0, filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, sieve(x2))) -> c10(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x0)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, sieve(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, sieve(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, sieve(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(0)), MARK(x0)) MARK(filter(x0, x1, sieve(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), sieve(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, sieve(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, cons(z0, z1), sieve(x2))) -> c10(A__FILTER(mark(x0), cons(mark(z0), z1), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, 0, sieve(x2))) -> c10(A__FILTER(mark(x0), 0, a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, sieve(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__sieve(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, sieve(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, sieve(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, sieve(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), a__sieve(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, sieve(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__sieve(mark(x2))), MARK(cons(z0, z1))) MARK(filter(0, x1, sieve(x2))) -> c10(A__FILTER(0, mark(x1), a__sieve(mark(x2))), MARK(0)) MARK(filter(s(z0), x1, sieve(x2))) -> c10(A__FILTER(s(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(s(z0))) S tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, x1, nats(z0))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x0)) MARK(filter(x0, x1, zprimes)) -> c10(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x0)) MARK(filter(x0, x1, cons(z0, z1))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x0)) MARK(filter(x0, x1, 0)) -> c10(A__FILTER(mark(x0), mark(x1), 0), MARK(x0)) MARK(filter(x0, x1, s(z0))) -> c10(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(z0), x2)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, nats(z0), x2)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, zprimes, x2)) -> c10(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x0)) MARK(filter(x0, s(z0), x2)) -> c10(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, x2)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, x2)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, x1, x2)) -> c10(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, x2)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x1)) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(x0, 0, filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, sieve(x2))) -> c10(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x0)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, sieve(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, sieve(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, sieve(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(0)), MARK(x0)) MARK(filter(x0, x1, sieve(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), sieve(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, sieve(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, cons(z0, z1), sieve(x2))) -> c10(A__FILTER(mark(x0), cons(mark(z0), z1), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, 0, sieve(x2))) -> c10(A__FILTER(mark(x0), 0, a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, sieve(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__sieve(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, sieve(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, sieve(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, sieve(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), a__sieve(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, sieve(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__sieve(mark(x2))), MARK(cons(z0, z1))) MARK(filter(0, x1, sieve(x2))) -> c10(A__FILTER(0, mark(x1), a__sieve(mark(x2))), MARK(0)) MARK(filter(s(z0), x1, sieve(x2))) -> c10(A__FILTER(s(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(s(z0))) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c14_2, c15_1, c16_1, c18_1, c8_1, c10_2, c10_1, c11_2, c11_1, c12_2, c12_1, c13_2, c13_1, c8_2 ---------------------------------------- (61) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(filter(0, x1, sieve(x2))) -> c10(A__FILTER(0, mark(x1), a__sieve(mark(x2))), MARK(0)) ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, x1, nats(z0))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x0)) MARK(filter(x0, x1, zprimes)) -> c10(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x0)) MARK(filter(x0, x1, cons(z0, z1))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x0)) MARK(filter(x0, x1, 0)) -> c10(A__FILTER(mark(x0), mark(x1), 0), MARK(x0)) MARK(filter(x0, x1, s(z0))) -> c10(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(z0), x2)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, nats(z0), x2)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, zprimes, x2)) -> c10(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x0)) MARK(filter(x0, s(z0), x2)) -> c10(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, x2)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, x2)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, x1, x2)) -> c10(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, x2)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x1)) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, 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c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> 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MARK(x0)) MARK(filter(x0, cons(z0, z1), sieve(x2))) -> c10(A__FILTER(mark(x0), cons(mark(z0), z1), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, 0, sieve(x2))) -> c10(A__FILTER(mark(x0), 0, a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, sieve(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__sieve(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, sieve(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, sieve(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, sieve(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), a__sieve(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, sieve(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__sieve(mark(x2))), MARK(cons(z0, z1))) MARK(filter(s(z0), x1, sieve(x2))) -> c10(A__FILTER(s(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(s(z0))) S tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, x1, nats(z0))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x0)) MARK(filter(x0, x1, zprimes)) -> c10(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x0)) MARK(filter(x0, x1, cons(z0, z1))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x0)) MARK(filter(x0, x1, 0)) -> c10(A__FILTER(mark(x0), mark(x1), 0), MARK(x0)) MARK(filter(x0, x1, s(z0))) -> c10(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(z0), x2)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, nats(z0), x2)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, zprimes, x2)) -> c10(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x0)) MARK(filter(x0, s(z0), x2)) -> c10(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, x2)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, x2)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, x1, x2)) -> c10(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, x2)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x1)) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(x0, 0, filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, sieve(x2))) -> c10(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x0)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, sieve(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, sieve(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, sieve(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(0)), MARK(x0)) MARK(filter(x0, x1, sieve(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), sieve(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, sieve(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, cons(z0, z1), sieve(x2))) -> c10(A__FILTER(mark(x0), cons(mark(z0), z1), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, 0, sieve(x2))) -> c10(A__FILTER(mark(x0), 0, a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, sieve(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__sieve(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, sieve(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, sieve(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, sieve(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), a__sieve(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, sieve(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__sieve(mark(x2))), MARK(cons(z0, z1))) MARK(filter(s(z0), x1, sieve(x2))) -> c10(A__FILTER(s(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(s(z0))) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c14_2, c15_1, c16_1, c18_1, c8_1, c10_2, c10_1, c11_2, c11_1, c12_2, c12_1, c13_2, c13_1, c8_2 ---------------------------------------- (63) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (64) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, x1, nats(z0))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x0)) MARK(filter(x0, x1, zprimes)) -> c10(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x0)) MARK(filter(x0, x1, cons(z0, z1))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x0)) MARK(filter(x0, x1, 0)) -> c10(A__FILTER(mark(x0), mark(x1), 0), MARK(x0)) MARK(filter(x0, x1, s(z0))) -> c10(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(z0), x2)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, nats(z0), x2)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, zprimes, x2)) -> c10(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x0)) MARK(filter(x0, s(z0), x2)) -> c10(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, x2)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, x2)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, x1, x2)) -> c10(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, x2)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x1)) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(x0, 0, filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, sieve(x2))) -> c10(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x0)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, sieve(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, sieve(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, sieve(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(0)), MARK(x0)) MARK(filter(x0, x1, sieve(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), sieve(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, sieve(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, sieve(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__sieve(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, sieve(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, sieve(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, sieve(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), a__sieve(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, sieve(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__sieve(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), sieve(x2))) -> c10(MARK(x0)) MARK(filter(x0, 0, sieve(x2))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, sieve(x2))) -> c10(MARK(s(z0))) S tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, x1, nats(z0))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x0)) MARK(filter(x0, x1, zprimes)) -> c10(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x0)) MARK(filter(x0, x1, cons(z0, z1))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x0)) MARK(filter(x0, x1, 0)) -> c10(A__FILTER(mark(x0), mark(x1), 0), MARK(x0)) MARK(filter(x0, x1, s(z0))) -> c10(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(z0), x2)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, nats(z0), x2)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, zprimes, x2)) -> c10(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x0)) MARK(filter(x0, s(z0), x2)) -> c10(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, x2)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, x2)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, x1, x2)) -> c10(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, x2)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x1)) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(x0, 0, filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, sieve(x2))) -> c10(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x0)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, sieve(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, sieve(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, sieve(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(0)), MARK(x0)) MARK(filter(x0, x1, sieve(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), sieve(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, sieve(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, sieve(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__sieve(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, sieve(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, sieve(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, sieve(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), a__sieve(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, sieve(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__sieve(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), sieve(x2))) -> c10(MARK(x0)) MARK(filter(x0, 0, sieve(x2))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, sieve(x2))) -> c10(MARK(s(z0))) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c14_2, c15_1, c16_1, c18_1, c8_1, c10_2, c10_1, c11_2, c11_1, c12_2, c12_1, c13_2, c13_1, c8_2 ---------------------------------------- (65) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(filter(x0, x1, nats(z0))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x0)) by MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(mark(x2)), nats(s(mark(x2))))), MARK(x0)) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), nats(mark(x2))), MARK(x0)) MARK(filter(x0, x1, nats(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, nats(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, nats(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, nats(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, nats(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, nats(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(0)), MARK(x0)) MARK(filter(x0, x1, nats(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), nats(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), nats(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), nats(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, nats(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, cons(z0, z1), nats(x2))) -> c10(A__FILTER(mark(x0), cons(mark(z0), z1), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, 0, nats(x2))) -> c10(A__FILTER(mark(x0), 0, a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), nats(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, nats(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__nats(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, nats(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, nats(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, nats(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), a__nats(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, nats(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__nats(mark(x2))), MARK(cons(z0, z1))) MARK(filter(0, x1, nats(x2))) -> c10(A__FILTER(0, mark(x1), a__nats(mark(x2))), MARK(0)) MARK(filter(s(z0), x1, nats(x2))) -> c10(A__FILTER(s(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(s(z0))) ---------------------------------------- (66) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, x1, zprimes)) -> c10(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x0)) MARK(filter(x0, x1, cons(z0, z1))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x0)) MARK(filter(x0, x1, 0)) -> c10(A__FILTER(mark(x0), mark(x1), 0), MARK(x0)) MARK(filter(x0, x1, s(z0))) -> c10(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(z0), x2)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, nats(z0), x2)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, zprimes, x2)) -> c10(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x0)) MARK(filter(x0, s(z0), x2)) -> c10(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, x2)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, x2)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, x1, x2)) -> c10(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, x2)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x1)) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(x0, 0, filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, sieve(x2))) -> c10(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x0)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, sieve(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, sieve(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, sieve(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(0)), MARK(x0)) MARK(filter(x0, x1, sieve(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), sieve(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, sieve(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, sieve(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__sieve(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, sieve(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, sieve(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, sieve(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), a__sieve(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, sieve(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__sieve(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), sieve(x2))) -> c10(MARK(x0)) MARK(filter(x0, 0, sieve(x2))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, sieve(x2))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(mark(x2)), nats(s(mark(x2))))), MARK(x0)) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), nats(mark(x2))), MARK(x0)) MARK(filter(x0, x1, nats(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, nats(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, nats(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, nats(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, nats(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, nats(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(0)), MARK(x0)) MARK(filter(x0, x1, nats(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), nats(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), nats(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), nats(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, nats(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, cons(z0, z1), nats(x2))) -> c10(A__FILTER(mark(x0), cons(mark(z0), z1), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, 0, nats(x2))) -> c10(A__FILTER(mark(x0), 0, a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), nats(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, nats(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__nats(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, nats(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, nats(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, nats(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), a__nats(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, nats(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__nats(mark(x2))), MARK(cons(z0, z1))) MARK(filter(0, x1, nats(x2))) -> c10(A__FILTER(0, mark(x1), a__nats(mark(x2))), MARK(0)) MARK(filter(s(z0), x1, nats(x2))) -> c10(A__FILTER(s(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(s(z0))) S tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, x1, zprimes)) -> c10(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x0)) MARK(filter(x0, x1, cons(z0, z1))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x0)) MARK(filter(x0, x1, 0)) -> c10(A__FILTER(mark(x0), mark(x1), 0), MARK(x0)) MARK(filter(x0, x1, s(z0))) -> c10(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(z0), x2)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, nats(z0), x2)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, zprimes, x2)) -> c10(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x0)) MARK(filter(x0, s(z0), x2)) -> c10(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, x2)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, x2)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, x1, x2)) -> c10(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, x2)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x1)) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(x0, 0, filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, sieve(x2))) -> c10(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x0)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, sieve(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, sieve(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, sieve(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(0)), MARK(x0)) MARK(filter(x0, x1, sieve(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), sieve(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, sieve(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, sieve(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__sieve(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, sieve(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, sieve(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, sieve(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), a__sieve(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, sieve(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__sieve(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), sieve(x2))) -> c10(MARK(x0)) MARK(filter(x0, 0, sieve(x2))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, sieve(x2))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(mark(x2)), nats(s(mark(x2))))), MARK(x0)) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), nats(mark(x2))), MARK(x0)) MARK(filter(x0, x1, nats(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, nats(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, nats(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, nats(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, nats(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, nats(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(0)), MARK(x0)) MARK(filter(x0, x1, nats(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), nats(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), nats(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), nats(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, nats(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, cons(z0, z1), nats(x2))) -> c10(A__FILTER(mark(x0), cons(mark(z0), z1), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, 0, nats(x2))) -> c10(A__FILTER(mark(x0), 0, a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), nats(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, nats(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__nats(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, nats(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, nats(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, nats(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), a__nats(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, nats(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__nats(mark(x2))), MARK(cons(z0, z1))) MARK(filter(0, x1, nats(x2))) -> c10(A__FILTER(0, mark(x1), a__nats(mark(x2))), MARK(0)) MARK(filter(s(z0), x1, nats(x2))) -> c10(A__FILTER(s(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(s(z0))) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c14_2, c15_1, c16_1, c18_1, c8_1, c10_2, c10_1, c11_2, c11_1, c12_2, c12_1, c13_2, c13_1, c8_2 ---------------------------------------- (67) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(filter(0, x1, nats(x2))) -> c10(A__FILTER(0, mark(x1), a__nats(mark(x2))), MARK(0)) ---------------------------------------- (68) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, x1, zprimes)) -> c10(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x0)) MARK(filter(x0, x1, cons(z0, z1))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x0)) MARK(filter(x0, x1, 0)) -> c10(A__FILTER(mark(x0), mark(x1), 0), MARK(x0)) MARK(filter(x0, x1, s(z0))) -> c10(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(z0), x2)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, nats(z0), x2)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, zprimes, x2)) -> c10(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x0)) MARK(filter(x0, s(z0), x2)) -> c10(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, x2)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, x2)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, x1, x2)) -> c10(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, x2)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x1)) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) 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c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), 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a__sieve(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, sieve(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, sieve(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, sieve(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), a__sieve(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, sieve(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__sieve(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), sieve(x2))) -> c10(MARK(x0)) MARK(filter(x0, 0, sieve(x2))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, sieve(x2))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(mark(x2)), nats(s(mark(x2))))), MARK(x0)) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), nats(mark(x2))), MARK(x0)) MARK(filter(x0, x1, nats(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), 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MARK(zprimes)) MARK(filter(cons(z0, z1), x1, nats(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__nats(mark(x2))), MARK(cons(z0, z1))) MARK(filter(s(z0), x1, nats(x2))) -> c10(A__FILTER(s(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(s(z0))) S tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, x1, zprimes)) -> c10(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x0)) MARK(filter(x0, x1, cons(z0, z1))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x0)) MARK(filter(x0, x1, 0)) -> c10(A__FILTER(mark(x0), mark(x1), 0), MARK(x0)) MARK(filter(x0, x1, s(z0))) -> c10(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c10(A__FILTER(mark(x0), 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sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(x0, 0, filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, sieve(x2))) -> c10(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x0)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, sieve(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, sieve(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, sieve(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(0)), MARK(x0)) MARK(filter(x0, x1, sieve(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), sieve(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, sieve(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, sieve(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__sieve(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, sieve(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, sieve(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, sieve(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), a__sieve(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, sieve(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__sieve(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), sieve(x2))) -> c10(MARK(x0)) MARK(filter(x0, 0, sieve(x2))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, sieve(x2))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(mark(x2)), nats(s(mark(x2))))), MARK(x0)) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), nats(mark(x2))), MARK(x0)) MARK(filter(x0, x1, nats(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, nats(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, nats(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, nats(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, nats(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, nats(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(0)), MARK(x0)) MARK(filter(x0, x1, nats(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), nats(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), nats(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), nats(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, nats(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, cons(z0, z1), nats(x2))) -> c10(A__FILTER(mark(x0), cons(mark(z0), z1), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, 0, nats(x2))) -> c10(A__FILTER(mark(x0), 0, a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), nats(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, nats(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__nats(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, nats(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, nats(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, nats(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), a__nats(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, nats(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__nats(mark(x2))), MARK(cons(z0, z1))) MARK(filter(s(z0), x1, nats(x2))) -> c10(A__FILTER(s(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(s(z0))) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c14_2, c15_1, c16_1, c18_1, c8_1, c10_2, c10_1, c11_2, c11_1, c12_2, c12_1, c13_2, c13_1, c8_2 ---------------------------------------- (69) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (70) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, x1, zprimes)) -> c10(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x0)) MARK(filter(x0, x1, cons(z0, z1))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x0)) MARK(filter(x0, x1, 0)) -> c10(A__FILTER(mark(x0), mark(x1), 0), MARK(x0)) MARK(filter(x0, x1, s(z0))) -> c10(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(z0), x2)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, nats(z0), x2)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, zprimes, x2)) -> c10(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x0)) MARK(filter(x0, s(z0), x2)) -> c10(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, x2)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, x2)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, x1, x2)) -> c10(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, x2)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x1)) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(x0, 0, filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, sieve(x2))) -> c10(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x0)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, sieve(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, sieve(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, sieve(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(0)), MARK(x0)) MARK(filter(x0, x1, sieve(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), sieve(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, sieve(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, sieve(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__sieve(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, sieve(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, sieve(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, sieve(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), a__sieve(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, sieve(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__sieve(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), sieve(x2))) -> c10(MARK(x0)) MARK(filter(x0, 0, sieve(x2))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, sieve(x2))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(mark(x2)), nats(s(mark(x2))))), MARK(x0)) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), nats(mark(x2))), MARK(x0)) MARK(filter(x0, x1, nats(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, nats(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, nats(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, nats(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, nats(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, nats(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(0)), MARK(x0)) MARK(filter(x0, x1, nats(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), nats(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), nats(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), nats(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, nats(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), nats(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, nats(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__nats(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, nats(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, nats(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, nats(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), a__nats(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, nats(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__nats(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), nats(x2))) -> c10(MARK(x0)) MARK(filter(x0, 0, nats(x2))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, nats(x2))) -> c10(MARK(s(z0))) S tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, x1, zprimes)) -> c10(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x0)) MARK(filter(x0, x1, cons(z0, z1))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x0)) MARK(filter(x0, x1, 0)) -> c10(A__FILTER(mark(x0), mark(x1), 0), MARK(x0)) MARK(filter(x0, x1, s(z0))) -> c10(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(z0), x2)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, nats(z0), x2)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, zprimes, x2)) -> c10(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x0)) MARK(filter(x0, s(z0), x2)) -> c10(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, x2)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, x2)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, x1, x2)) -> c10(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, x2)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x1)) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(x0, 0, filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, sieve(x2))) -> c10(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x0)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, sieve(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, sieve(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, sieve(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(0)), MARK(x0)) MARK(filter(x0, x1, sieve(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), sieve(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, sieve(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, sieve(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__sieve(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, sieve(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, sieve(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, sieve(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), a__sieve(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, sieve(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__sieve(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), sieve(x2))) -> c10(MARK(x0)) MARK(filter(x0, 0, sieve(x2))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, sieve(x2))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(mark(x2)), nats(s(mark(x2))))), MARK(x0)) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), nats(mark(x2))), MARK(x0)) MARK(filter(x0, x1, nats(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, nats(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, nats(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, nats(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, nats(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, nats(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(0)), MARK(x0)) MARK(filter(x0, x1, nats(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), nats(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), nats(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), nats(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, nats(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), nats(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, nats(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__nats(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, nats(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, nats(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, nats(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), a__nats(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, nats(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__nats(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), nats(x2))) -> c10(MARK(x0)) MARK(filter(x0, 0, nats(x2))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, nats(x2))) -> c10(MARK(s(z0))) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c14_2, c15_1, c16_1, c18_1, c8_1, c10_2, c10_1, c11_2, c11_1, c12_2, c12_1, c13_2, c13_1, c8_2 ---------------------------------------- (71) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(filter(x0, x1, zprimes)) -> c10(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x0)) by MARK(filter(x0, x1, zprimes)) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(s(s(0))))), MARK(x0)) MARK(filter(x0, x1, zprimes)) -> c10(A__FILTER(mark(x0), mark(x1), zprimes), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), zprimes)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__zprimes), MARK(x0)) MARK(filter(x0, sieve(z0), zprimes)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__zprimes), MARK(x0)) MARK(filter(x0, nats(z0), zprimes)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__zprimes), MARK(x0)) MARK(filter(x0, zprimes, zprimes)) -> c10(A__FILTER(mark(x0), a__zprimes, a__zprimes), MARK(x0)) MARK(filter(x0, cons(z0, z1), zprimes)) -> c10(A__FILTER(mark(x0), cons(mark(z0), z1), a__zprimes), MARK(x0)) MARK(filter(x0, 0, zprimes)) -> c10(A__FILTER(mark(x0), 0, a__zprimes), MARK(x0)) MARK(filter(x0, s(z0), zprimes)) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__zprimes), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, zprimes)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__zprimes), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, zprimes)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__zprimes), MARK(sieve(z0))) MARK(filter(nats(z0), x1, zprimes)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__zprimes), MARK(nats(z0))) MARK(filter(zprimes, x1, zprimes)) -> c10(A__FILTER(a__zprimes, mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, zprimes)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__zprimes), MARK(cons(z0, z1))) MARK(filter(0, x1, zprimes)) -> c10(A__FILTER(0, mark(x1), a__zprimes), MARK(0)) MARK(filter(s(z0), x1, zprimes)) -> c10(A__FILTER(s(mark(z0)), mark(x1), a__zprimes), MARK(s(z0))) ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, x1, cons(z0, z1))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x0)) MARK(filter(x0, x1, 0)) -> c10(A__FILTER(mark(x0), mark(x1), 0), MARK(x0)) MARK(filter(x0, x1, s(z0))) -> c10(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(z0), x2)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, nats(z0), x2)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, zprimes, x2)) -> c10(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x0)) MARK(filter(x0, s(z0), x2)) -> c10(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, x2)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, x2)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, x1, x2)) -> c10(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, x2)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x1)) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(x0, 0, filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, sieve(x2))) -> c10(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x0)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, sieve(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, sieve(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, sieve(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(0)), MARK(x0)) MARK(filter(x0, x1, sieve(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), sieve(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, sieve(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, sieve(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__sieve(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, sieve(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, sieve(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, sieve(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), a__sieve(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, sieve(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__sieve(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), sieve(x2))) -> c10(MARK(x0)) MARK(filter(x0, 0, sieve(x2))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, sieve(x2))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(mark(x2)), nats(s(mark(x2))))), MARK(x0)) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), nats(mark(x2))), MARK(x0)) MARK(filter(x0, x1, nats(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, nats(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, nats(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, nats(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, nats(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, nats(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(0)), MARK(x0)) MARK(filter(x0, x1, nats(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), nats(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), nats(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), nats(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, nats(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), nats(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, nats(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__nats(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, nats(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, nats(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, nats(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), a__nats(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, nats(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__nats(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), nats(x2))) -> c10(MARK(x0)) MARK(filter(x0, 0, nats(x2))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, nats(x2))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, zprimes)) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(s(s(0))))), MARK(x0)) MARK(filter(x0, x1, zprimes)) -> c10(A__FILTER(mark(x0), mark(x1), zprimes), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), zprimes)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__zprimes), MARK(x0)) MARK(filter(x0, sieve(z0), zprimes)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__zprimes), MARK(x0)) MARK(filter(x0, nats(z0), zprimes)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__zprimes), MARK(x0)) MARK(filter(x0, zprimes, zprimes)) -> c10(A__FILTER(mark(x0), a__zprimes, a__zprimes), MARK(x0)) MARK(filter(x0, cons(z0, z1), zprimes)) -> c10(A__FILTER(mark(x0), cons(mark(z0), z1), a__zprimes), MARK(x0)) MARK(filter(x0, 0, zprimes)) -> c10(A__FILTER(mark(x0), 0, a__zprimes), MARK(x0)) MARK(filter(x0, s(z0), zprimes)) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__zprimes), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, zprimes)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__zprimes), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, zprimes)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__zprimes), MARK(sieve(z0))) MARK(filter(nats(z0), x1, zprimes)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__zprimes), MARK(nats(z0))) MARK(filter(zprimes, x1, zprimes)) -> c10(A__FILTER(a__zprimes, mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, zprimes)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__zprimes), MARK(cons(z0, z1))) MARK(filter(0, x1, zprimes)) -> c10(A__FILTER(0, mark(x1), a__zprimes), MARK(0)) MARK(filter(s(z0), x1, zprimes)) -> c10(A__FILTER(s(mark(z0)), mark(x1), a__zprimes), MARK(s(z0))) S tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, x1, cons(z0, z1))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x0)) MARK(filter(x0, x1, 0)) -> c10(A__FILTER(mark(x0), mark(x1), 0), MARK(x0)) MARK(filter(x0, x1, s(z0))) -> c10(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(z0), x2)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, nats(z0), x2)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, zprimes, x2)) -> c10(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x0)) MARK(filter(x0, s(z0), x2)) -> c10(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, x2)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, x2)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, x1, x2)) -> c10(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, x2)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x1)) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(x0, 0, filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, sieve(x2))) -> c10(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x0)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, sieve(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, sieve(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, sieve(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(0)), MARK(x0)) MARK(filter(x0, x1, sieve(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), sieve(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, sieve(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, sieve(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__sieve(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, sieve(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, sieve(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, sieve(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), a__sieve(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, sieve(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__sieve(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), sieve(x2))) -> c10(MARK(x0)) MARK(filter(x0, 0, sieve(x2))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, sieve(x2))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(mark(x2)), nats(s(mark(x2))))), MARK(x0)) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), nats(mark(x2))), MARK(x0)) MARK(filter(x0, x1, nats(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, nats(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, nats(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, nats(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, nats(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, nats(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(0)), MARK(x0)) MARK(filter(x0, x1, nats(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), nats(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), nats(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), nats(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, nats(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), nats(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, nats(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__nats(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, nats(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, nats(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, nats(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), a__nats(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, nats(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__nats(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), nats(x2))) -> c10(MARK(x0)) MARK(filter(x0, 0, nats(x2))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, nats(x2))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, zprimes)) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(s(s(0))))), MARK(x0)) MARK(filter(x0, x1, zprimes)) -> c10(A__FILTER(mark(x0), mark(x1), zprimes), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), zprimes)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__zprimes), MARK(x0)) MARK(filter(x0, sieve(z0), zprimes)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__zprimes), MARK(x0)) MARK(filter(x0, nats(z0), zprimes)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__zprimes), MARK(x0)) MARK(filter(x0, zprimes, zprimes)) -> c10(A__FILTER(mark(x0), a__zprimes, a__zprimes), MARK(x0)) MARK(filter(x0, cons(z0, z1), zprimes)) -> c10(A__FILTER(mark(x0), cons(mark(z0), z1), a__zprimes), MARK(x0)) MARK(filter(x0, 0, zprimes)) -> c10(A__FILTER(mark(x0), 0, a__zprimes), MARK(x0)) MARK(filter(x0, s(z0), zprimes)) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__zprimes), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, zprimes)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__zprimes), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, zprimes)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__zprimes), MARK(sieve(z0))) MARK(filter(nats(z0), x1, zprimes)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__zprimes), MARK(nats(z0))) MARK(filter(zprimes, x1, zprimes)) -> c10(A__FILTER(a__zprimes, mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, zprimes)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__zprimes), MARK(cons(z0, z1))) MARK(filter(0, x1, zprimes)) -> c10(A__FILTER(0, mark(x1), a__zprimes), MARK(0)) MARK(filter(s(z0), x1, zprimes)) -> c10(A__FILTER(s(mark(z0)), mark(x1), a__zprimes), MARK(s(z0))) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c14_2, c15_1, c16_1, c18_1, c8_1, c10_2, c10_1, c11_2, c11_1, c12_2, c12_1, c13_2, c13_1, c8_2 ---------------------------------------- (73) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(filter(0, x1, zprimes)) -> c10(A__FILTER(0, mark(x1), a__zprimes), MARK(0)) ---------------------------------------- (74) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, x1, cons(z0, z1))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x0)) MARK(filter(x0, x1, 0)) -> c10(A__FILTER(mark(x0), mark(x1), 0), MARK(x0)) MARK(filter(x0, x1, s(z0))) -> c10(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(z0), x2)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, nats(z0), x2)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, zprimes, x2)) -> c10(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x0)) MARK(filter(x0, s(z0), x2)) -> c10(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, x2)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, x2)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, x1, x2)) -> c10(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, x2)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x1)) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(x0, 0, filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, sieve(x2))) -> c10(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x0)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, sieve(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, sieve(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, sieve(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(0)), MARK(x0)) MARK(filter(x0, x1, sieve(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), sieve(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, sieve(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, sieve(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__sieve(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, sieve(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, sieve(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, sieve(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), a__sieve(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, sieve(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__sieve(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), sieve(x2))) -> c10(MARK(x0)) MARK(filter(x0, 0, sieve(x2))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, sieve(x2))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(mark(x2)), nats(s(mark(x2))))), MARK(x0)) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), nats(mark(x2))), MARK(x0)) MARK(filter(x0, x1, nats(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, nats(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, nats(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, nats(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, nats(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, nats(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(0)), MARK(x0)) MARK(filter(x0, x1, nats(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), nats(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), nats(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), nats(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, nats(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), nats(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, nats(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__nats(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, nats(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, nats(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, nats(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), a__nats(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, nats(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__nats(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), nats(x2))) -> c10(MARK(x0)) MARK(filter(x0, 0, nats(x2))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, nats(x2))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, zprimes)) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(s(s(0))))), MARK(x0)) MARK(filter(x0, x1, zprimes)) -> c10(A__FILTER(mark(x0), mark(x1), zprimes), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), zprimes)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__zprimes), MARK(x0)) MARK(filter(x0, sieve(z0), zprimes)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__zprimes), MARK(x0)) MARK(filter(x0, nats(z0), zprimes)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__zprimes), MARK(x0)) MARK(filter(x0, zprimes, zprimes)) -> c10(A__FILTER(mark(x0), a__zprimes, a__zprimes), MARK(x0)) MARK(filter(x0, cons(z0, z1), zprimes)) -> c10(A__FILTER(mark(x0), cons(mark(z0), z1), a__zprimes), MARK(x0)) MARK(filter(x0, 0, zprimes)) -> c10(A__FILTER(mark(x0), 0, a__zprimes), MARK(x0)) MARK(filter(x0, s(z0), zprimes)) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__zprimes), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, zprimes)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__zprimes), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, zprimes)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__zprimes), MARK(sieve(z0))) MARK(filter(nats(z0), x1, zprimes)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__zprimes), MARK(nats(z0))) MARK(filter(zprimes, x1, zprimes)) -> c10(A__FILTER(a__zprimes, mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, zprimes)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__zprimes), MARK(cons(z0, z1))) MARK(filter(s(z0), x1, zprimes)) -> c10(A__FILTER(s(mark(z0)), mark(x1), a__zprimes), MARK(s(z0))) S tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, x1, cons(z0, z1))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x0)) MARK(filter(x0, x1, 0)) -> c10(A__FILTER(mark(x0), mark(x1), 0), MARK(x0)) MARK(filter(x0, x1, s(z0))) -> c10(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(z0), x2)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, nats(z0), x2)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, zprimes, x2)) -> c10(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x0)) MARK(filter(x0, s(z0), x2)) -> c10(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, x2)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, x2)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, x1, x2)) -> c10(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, x2)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x1)) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(x0, 0, filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, sieve(x2))) -> c10(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x0)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, sieve(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, sieve(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, sieve(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(0)), MARK(x0)) MARK(filter(x0, x1, sieve(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), sieve(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, sieve(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, sieve(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__sieve(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, sieve(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, sieve(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, sieve(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), a__sieve(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, sieve(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__sieve(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), sieve(x2))) -> c10(MARK(x0)) MARK(filter(x0, 0, sieve(x2))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, sieve(x2))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(mark(x2)), nats(s(mark(x2))))), MARK(x0)) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), nats(mark(x2))), MARK(x0)) MARK(filter(x0, x1, nats(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, nats(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, nats(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, nats(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, nats(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, nats(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(0)), MARK(x0)) MARK(filter(x0, x1, nats(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), nats(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), nats(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), nats(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, nats(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), nats(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, nats(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__nats(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, nats(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, nats(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, nats(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), a__nats(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, nats(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__nats(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), nats(x2))) -> c10(MARK(x0)) MARK(filter(x0, 0, nats(x2))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, nats(x2))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, zprimes)) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(s(s(0))))), MARK(x0)) MARK(filter(x0, x1, zprimes)) -> c10(A__FILTER(mark(x0), mark(x1), zprimes), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), zprimes)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__zprimes), MARK(x0)) MARK(filter(x0, sieve(z0), zprimes)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__zprimes), MARK(x0)) MARK(filter(x0, nats(z0), zprimes)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__zprimes), MARK(x0)) MARK(filter(x0, zprimes, zprimes)) -> c10(A__FILTER(mark(x0), a__zprimes, a__zprimes), MARK(x0)) MARK(filter(x0, cons(z0, z1), zprimes)) -> c10(A__FILTER(mark(x0), cons(mark(z0), z1), a__zprimes), MARK(x0)) MARK(filter(x0, 0, zprimes)) -> c10(A__FILTER(mark(x0), 0, a__zprimes), MARK(x0)) MARK(filter(x0, s(z0), zprimes)) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__zprimes), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, zprimes)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__zprimes), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, zprimes)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__zprimes), MARK(sieve(z0))) MARK(filter(nats(z0), x1, zprimes)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__zprimes), MARK(nats(z0))) MARK(filter(zprimes, x1, zprimes)) -> c10(A__FILTER(a__zprimes, mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, zprimes)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__zprimes), MARK(cons(z0, z1))) MARK(filter(s(z0), x1, zprimes)) -> c10(A__FILTER(s(mark(z0)), mark(x1), a__zprimes), MARK(s(z0))) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c14_2, c15_1, c16_1, c18_1, c8_1, c10_2, c10_1, c11_2, c11_1, c12_2, c12_1, c13_2, c13_1, c8_2 ---------------------------------------- (75) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (76) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, x1, cons(z0, z1))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x0)) MARK(filter(x0, x1, 0)) -> c10(A__FILTER(mark(x0), mark(x1), 0), MARK(x0)) MARK(filter(x0, x1, s(z0))) -> c10(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(z0), x2)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, nats(z0), x2)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, zprimes, x2)) -> c10(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x0)) MARK(filter(x0, s(z0), x2)) -> c10(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, x2)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, x2)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, x1, x2)) -> c10(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, x2)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x1)) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) 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c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(x0, 0, filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, sieve(x2))) -> c10(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x0)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, sieve(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, sieve(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, sieve(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(0)), MARK(x0)) MARK(filter(x0, x1, sieve(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), sieve(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, sieve(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, sieve(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__sieve(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, sieve(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, sieve(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, sieve(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), a__sieve(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, sieve(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__sieve(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), sieve(x2))) -> c10(MARK(x0)) MARK(filter(x0, 0, sieve(x2))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, sieve(x2))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(mark(x2)), nats(s(mark(x2))))), MARK(x0)) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), nats(mark(x2))), MARK(x0)) MARK(filter(x0, x1, nats(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, nats(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, nats(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, nats(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, nats(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, nats(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(0)), MARK(x0)) MARK(filter(x0, x1, nats(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), nats(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), nats(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), nats(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, nats(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), nats(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, nats(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__nats(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, nats(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, nats(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, nats(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), a__nats(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, nats(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__nats(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), nats(x2))) -> c10(MARK(x0)) MARK(filter(x0, 0, nats(x2))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, nats(x2))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, zprimes)) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(s(s(0))))), MARK(x0)) MARK(filter(x0, x1, zprimes)) -> c10(A__FILTER(mark(x0), mark(x1), zprimes), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), zprimes)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__zprimes), MARK(x0)) MARK(filter(x0, sieve(z0), zprimes)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__zprimes), MARK(x0)) MARK(filter(x0, nats(z0), zprimes)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__zprimes), MARK(x0)) MARK(filter(x0, zprimes, zprimes)) -> c10(A__FILTER(mark(x0), a__zprimes, a__zprimes), MARK(x0)) MARK(filter(x0, s(z0), zprimes)) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__zprimes), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, zprimes)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__zprimes), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, zprimes)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__zprimes), MARK(sieve(z0))) MARK(filter(nats(z0), x1, zprimes)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__zprimes), MARK(nats(z0))) MARK(filter(zprimes, x1, zprimes)) -> c10(A__FILTER(a__zprimes, mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, zprimes)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__zprimes), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), zprimes)) -> c10(MARK(x0)) MARK(filter(x0, 0, zprimes)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, zprimes)) -> c10(MARK(s(z0))) S tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, x1, cons(z0, z1))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x0)) MARK(filter(x0, x1, 0)) -> c10(A__FILTER(mark(x0), mark(x1), 0), MARK(x0)) MARK(filter(x0, x1, s(z0))) -> 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a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), nats(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), nats(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, nats(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), nats(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, nats(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__nats(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, nats(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, nats(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, nats(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), a__nats(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, nats(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__nats(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), nats(x2))) -> c10(MARK(x0)) MARK(filter(x0, 0, nats(x2))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, nats(x2))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, zprimes)) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(s(s(0))))), MARK(x0)) MARK(filter(x0, x1, zprimes)) -> c10(A__FILTER(mark(x0), mark(x1), zprimes), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), zprimes)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__zprimes), MARK(x0)) MARK(filter(x0, sieve(z0), zprimes)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__zprimes), MARK(x0)) MARK(filter(x0, nats(z0), zprimes)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__zprimes), MARK(x0)) MARK(filter(x0, zprimes, zprimes)) -> c10(A__FILTER(mark(x0), a__zprimes, a__zprimes), MARK(x0)) MARK(filter(x0, s(z0), zprimes)) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__zprimes), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, zprimes)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__zprimes), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, zprimes)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__zprimes), MARK(sieve(z0))) MARK(filter(nats(z0), x1, zprimes)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__zprimes), MARK(nats(z0))) MARK(filter(zprimes, x1, zprimes)) -> c10(A__FILTER(a__zprimes, mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, zprimes)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__zprimes), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), zprimes)) -> c10(MARK(x0)) MARK(filter(x0, 0, zprimes)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, zprimes)) -> c10(MARK(s(z0))) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c14_2, c15_1, c16_1, c18_1, c8_1, c10_2, c10_1, c11_2, c11_1, c12_2, c12_1, c13_2, c13_1, c8_2 ---------------------------------------- (77) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(filter(x0, x1, cons(z0, z1))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x0)) by MARK(filter(x0, x1, cons(filter(z0, z1, z2), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(a__filter(mark(z0), mark(z1), mark(z2)), x3)), MARK(x0)) MARK(filter(x0, x1, cons(sieve(z0), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(a__sieve(mark(z0)), x3)), MARK(x0)) MARK(filter(x0, x1, cons(nats(z0), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(a__nats(mark(z0)), x3)), MARK(x0)) MARK(filter(x0, x1, cons(zprimes, x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(a__zprimes, x3)), MARK(x0)) MARK(filter(x0, x1, cons(cons(z0, z1), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(cons(mark(z0), z1), x3)), MARK(x0)) MARK(filter(x0, x1, cons(0, x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(0, x3)), MARK(x0)) MARK(filter(x0, x1, cons(s(z0), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(s(mark(z0)), x3)), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), cons(x2, x3))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), cons(mark(x2), x3)), MARK(x0)) MARK(filter(x0, sieve(z0), cons(x2, x3))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), cons(mark(x2), x3)), MARK(x0)) MARK(filter(x0, nats(z0), cons(x2, x3))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), cons(mark(x2), x3)), MARK(x0)) MARK(filter(x0, zprimes, cons(x2, x3))) -> c10(A__FILTER(mark(x0), a__zprimes, cons(mark(x2), x3)), MARK(x0)) MARK(filter(x0, cons(z0, z1), cons(x2, x3))) -> c10(A__FILTER(mark(x0), cons(mark(z0), z1), cons(mark(x2), x3)), MARK(x0)) MARK(filter(x0, 0, cons(x2, x3))) -> c10(A__FILTER(mark(x0), 0, cons(mark(x2), x3)), MARK(x0)) MARK(filter(x0, s(z0), cons(x2, x3))) -> c10(A__FILTER(mark(x0), s(mark(z0)), cons(mark(x2), x3)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, cons(x2, x3))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), cons(mark(x2), x3)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, cons(x2, x3))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), cons(mark(x2), x3)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, cons(x2, x3))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), cons(mark(x2), x3)), MARK(nats(z0))) MARK(filter(zprimes, x1, cons(x2, x3))) -> c10(A__FILTER(a__zprimes, mark(x1), cons(mark(x2), x3)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, cons(x2, x3))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), cons(mark(x2), x3)), MARK(cons(z0, z1))) MARK(filter(0, x1, cons(x2, x3))) -> c10(A__FILTER(0, mark(x1), cons(mark(x2), x3)), MARK(0)) MARK(filter(s(z0), x1, cons(x2, x3))) -> c10(A__FILTER(s(mark(z0)), mark(x1), cons(mark(x2), x3)), MARK(s(z0))) ---------------------------------------- (78) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, x1, 0)) -> c10(A__FILTER(mark(x0), mark(x1), 0), MARK(x0)) MARK(filter(x0, x1, s(z0))) -> c10(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(z0), x2)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, nats(z0), x2)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, zprimes, x2)) -> c10(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x0)) MARK(filter(x0, s(z0), x2)) -> c10(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, x2)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, x2)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, x1, x2)) -> c10(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, x2)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x1)) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, filter(x2, 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sieve(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, sieve(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, sieve(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(0)), MARK(x0)) MARK(filter(x0, x1, sieve(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), sieve(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, sieve(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, sieve(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__sieve(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, sieve(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, sieve(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, sieve(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), a__sieve(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, sieve(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__sieve(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), sieve(x2))) -> c10(MARK(x0)) MARK(filter(x0, 0, sieve(x2))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, sieve(x2))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(mark(x2)), nats(s(mark(x2))))), MARK(x0)) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), nats(mark(x2))), MARK(x0)) MARK(filter(x0, x1, nats(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, nats(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, nats(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, nats(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, nats(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, nats(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(0)), MARK(x0)) MARK(filter(x0, x1, nats(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), nats(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), nats(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), nats(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, nats(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), nats(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, nats(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__nats(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, nats(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, nats(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, nats(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), a__nats(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, nats(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__nats(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), nats(x2))) -> c10(MARK(x0)) MARK(filter(x0, 0, nats(x2))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, nats(x2))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, zprimes)) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(s(s(0))))), MARK(x0)) MARK(filter(x0, x1, zprimes)) -> c10(A__FILTER(mark(x0), mark(x1), zprimes), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), zprimes)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__zprimes), MARK(x0)) MARK(filter(x0, sieve(z0), zprimes)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__zprimes), MARK(x0)) MARK(filter(x0, nats(z0), zprimes)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__zprimes), MARK(x0)) MARK(filter(x0, zprimes, zprimes)) -> c10(A__FILTER(mark(x0), a__zprimes, a__zprimes), MARK(x0)) MARK(filter(x0, s(z0), zprimes)) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__zprimes), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, zprimes)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__zprimes), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, zprimes)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__zprimes), MARK(sieve(z0))) MARK(filter(nats(z0), x1, zprimes)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__zprimes), MARK(nats(z0))) MARK(filter(zprimes, x1, zprimes)) -> c10(A__FILTER(a__zprimes, mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, zprimes)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__zprimes), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), zprimes)) -> c10(MARK(x0)) MARK(filter(x0, 0, zprimes)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, zprimes)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, cons(filter(z0, z1, z2), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(a__filter(mark(z0), mark(z1), mark(z2)), x3)), MARK(x0)) MARK(filter(x0, x1, cons(sieve(z0), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(a__sieve(mark(z0)), x3)), MARK(x0)) MARK(filter(x0, x1, cons(nats(z0), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(a__nats(mark(z0)), x3)), MARK(x0)) MARK(filter(x0, x1, cons(zprimes, x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(a__zprimes, x3)), MARK(x0)) MARK(filter(x0, x1, cons(cons(z0, z1), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(cons(mark(z0), z1), x3)), MARK(x0)) MARK(filter(x0, x1, cons(0, x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(0, x3)), MARK(x0)) MARK(filter(x0, x1, cons(s(z0), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(s(mark(z0)), x3)), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), cons(x2, x3))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), cons(mark(x2), x3)), MARK(x0)) MARK(filter(x0, sieve(z0), cons(x2, x3))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), cons(mark(x2), x3)), MARK(x0)) MARK(filter(x0, nats(z0), cons(x2, x3))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), cons(mark(x2), x3)), MARK(x0)) MARK(filter(x0, zprimes, cons(x2, x3))) -> c10(A__FILTER(mark(x0), a__zprimes, cons(mark(x2), x3)), MARK(x0)) MARK(filter(x0, cons(z0, z1), cons(x2, x3))) -> c10(A__FILTER(mark(x0), cons(mark(z0), z1), cons(mark(x2), x3)), MARK(x0)) MARK(filter(x0, 0, cons(x2, x3))) -> c10(A__FILTER(mark(x0), 0, cons(mark(x2), x3)), MARK(x0)) MARK(filter(x0, s(z0), cons(x2, x3))) -> c10(A__FILTER(mark(x0), s(mark(z0)), cons(mark(x2), x3)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, cons(x2, x3))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), cons(mark(x2), x3)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, cons(x2, x3))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), cons(mark(x2), x3)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, cons(x2, x3))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), cons(mark(x2), x3)), MARK(nats(z0))) MARK(filter(zprimes, x1, cons(x2, x3))) -> c10(A__FILTER(a__zprimes, mark(x1), cons(mark(x2), x3)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, cons(x2, x3))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), cons(mark(x2), x3)), MARK(cons(z0, z1))) MARK(filter(0, x1, cons(x2, x3))) -> c10(A__FILTER(0, mark(x1), cons(mark(x2), x3)), MARK(0)) MARK(filter(s(z0), x1, cons(x2, x3))) -> c10(A__FILTER(s(mark(z0)), mark(x1), cons(mark(x2), x3)), MARK(s(z0))) S tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, x1, 0)) -> c10(A__FILTER(mark(x0), mark(x1), 0), MARK(x0)) MARK(filter(x0, x1, s(z0))) -> c10(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(z0), x2)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, nats(z0), x2)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, zprimes, x2)) -> c10(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x0)) MARK(filter(x0, s(z0), x2)) -> c10(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, x2)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, x2)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, x1, x2)) -> c10(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, x2)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x1)) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), 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z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(x0, 0, filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, sieve(x2))) -> c10(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x0)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, sieve(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, sieve(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, sieve(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(0)), MARK(x0)) MARK(filter(x0, x1, sieve(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), sieve(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, sieve(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, sieve(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__sieve(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, sieve(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, sieve(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, sieve(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), a__sieve(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, sieve(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__sieve(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), sieve(x2))) -> c10(MARK(x0)) MARK(filter(x0, 0, sieve(x2))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, sieve(x2))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(mark(x2)), nats(s(mark(x2))))), MARK(x0)) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), nats(mark(x2))), MARK(x0)) MARK(filter(x0, x1, nats(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, nats(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, nats(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, nats(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, nats(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, nats(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(0)), MARK(x0)) MARK(filter(x0, x1, nats(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), nats(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), nats(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), nats(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, nats(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), nats(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, nats(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__nats(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, nats(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, nats(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, nats(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), a__nats(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, nats(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__nats(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), nats(x2))) -> c10(MARK(x0)) MARK(filter(x0, 0, nats(x2))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, nats(x2))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, zprimes)) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(s(s(0))))), MARK(x0)) MARK(filter(x0, x1, zprimes)) -> c10(A__FILTER(mark(x0), mark(x1), zprimes), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), zprimes)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__zprimes), MARK(x0)) MARK(filter(x0, sieve(z0), zprimes)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__zprimes), MARK(x0)) MARK(filter(x0, nats(z0), zprimes)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__zprimes), MARK(x0)) MARK(filter(x0, zprimes, zprimes)) -> c10(A__FILTER(mark(x0), a__zprimes, a__zprimes), MARK(x0)) MARK(filter(x0, s(z0), zprimes)) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__zprimes), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, zprimes)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__zprimes), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, zprimes)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__zprimes), MARK(sieve(z0))) MARK(filter(nats(z0), x1, zprimes)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__zprimes), MARK(nats(z0))) MARK(filter(zprimes, x1, zprimes)) -> c10(A__FILTER(a__zprimes, mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, zprimes)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__zprimes), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), zprimes)) -> c10(MARK(x0)) MARK(filter(x0, 0, zprimes)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, zprimes)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, cons(filter(z0, z1, z2), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(a__filter(mark(z0), mark(z1), mark(z2)), x3)), MARK(x0)) MARK(filter(x0, x1, cons(sieve(z0), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(a__sieve(mark(z0)), x3)), MARK(x0)) MARK(filter(x0, x1, cons(nats(z0), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(a__nats(mark(z0)), x3)), MARK(x0)) MARK(filter(x0, x1, cons(zprimes, x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(a__zprimes, x3)), MARK(x0)) MARK(filter(x0, x1, cons(cons(z0, z1), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(cons(mark(z0), z1), x3)), MARK(x0)) MARK(filter(x0, x1, cons(0, x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(0, x3)), MARK(x0)) MARK(filter(x0, x1, cons(s(z0), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(s(mark(z0)), x3)), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), cons(x2, x3))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), cons(mark(x2), x3)), MARK(x0)) MARK(filter(x0, sieve(z0), cons(x2, x3))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), cons(mark(x2), x3)), MARK(x0)) MARK(filter(x0, nats(z0), cons(x2, x3))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), cons(mark(x2), x3)), MARK(x0)) MARK(filter(x0, zprimes, cons(x2, x3))) -> c10(A__FILTER(mark(x0), a__zprimes, cons(mark(x2), x3)), MARK(x0)) MARK(filter(x0, cons(z0, z1), cons(x2, x3))) -> c10(A__FILTER(mark(x0), cons(mark(z0), z1), cons(mark(x2), x3)), MARK(x0)) MARK(filter(x0, 0, cons(x2, x3))) -> c10(A__FILTER(mark(x0), 0, cons(mark(x2), x3)), MARK(x0)) MARK(filter(x0, s(z0), cons(x2, x3))) -> c10(A__FILTER(mark(x0), s(mark(z0)), cons(mark(x2), x3)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, cons(x2, x3))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), cons(mark(x2), x3)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, cons(x2, x3))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), cons(mark(x2), x3)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, cons(x2, x3))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), cons(mark(x2), x3)), MARK(nats(z0))) MARK(filter(zprimes, x1, cons(x2, x3))) -> c10(A__FILTER(a__zprimes, mark(x1), cons(mark(x2), x3)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, cons(x2, x3))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), cons(mark(x2), x3)), MARK(cons(z0, z1))) MARK(filter(0, x1, cons(x2, x3))) -> c10(A__FILTER(0, mark(x1), cons(mark(x2), x3)), MARK(0)) MARK(filter(s(z0), x1, cons(x2, x3))) -> c10(A__FILTER(s(mark(z0)), mark(x1), cons(mark(x2), x3)), MARK(s(z0))) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c14_2, c15_1, c16_1, c18_1, c8_1, c10_2, c10_1, c11_2, c11_1, c12_2, c12_1, c13_2, c13_1, c8_2 ---------------------------------------- (79) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(filter(0, x1, cons(x2, x3))) -> c10(A__FILTER(0, mark(x1), cons(mark(x2), x3)), MARK(0)) ---------------------------------------- (80) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, x1, 0)) -> c10(A__FILTER(mark(x0), mark(x1), 0), MARK(x0)) MARK(filter(x0, x1, s(z0))) -> c10(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(z0), x2)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, nats(z0), x2)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, zprimes, x2)) -> c10(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x0)) MARK(filter(x0, s(z0), x2)) -> c10(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, x2)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, x2)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, x1, x2)) -> c10(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, x2)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x1)) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(x0, 0, filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, sieve(x2))) -> c10(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x0)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, sieve(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, 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MARK(filter(filter(z0, z1, z2), x1, sieve(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__sieve(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, sieve(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, sieve(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, sieve(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), a__sieve(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, sieve(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__sieve(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), sieve(x2))) -> c10(MARK(x0)) MARK(filter(x0, 0, sieve(x2))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, sieve(x2))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(mark(x2)), nats(s(mark(x2))))), MARK(x0)) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), nats(mark(x2))), MARK(x0)) MARK(filter(x0, x1, nats(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, nats(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, nats(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, nats(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, nats(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, nats(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(0)), MARK(x0)) MARK(filter(x0, x1, nats(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), nats(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), nats(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), nats(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, nats(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), nats(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, nats(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__nats(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, nats(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, nats(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, nats(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), a__nats(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, nats(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__nats(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), nats(x2))) -> c10(MARK(x0)) MARK(filter(x0, 0, nats(x2))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, nats(x2))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, zprimes)) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(s(s(0))))), MARK(x0)) MARK(filter(x0, x1, zprimes)) -> c10(A__FILTER(mark(x0), mark(x1), zprimes), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), zprimes)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__zprimes), MARK(x0)) MARK(filter(x0, sieve(z0), zprimes)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__zprimes), MARK(x0)) MARK(filter(x0, nats(z0), zprimes)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__zprimes), MARK(x0)) MARK(filter(x0, zprimes, zprimes)) -> c10(A__FILTER(mark(x0), a__zprimes, a__zprimes), MARK(x0)) MARK(filter(x0, s(z0), zprimes)) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__zprimes), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, zprimes)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__zprimes), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, zprimes)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__zprimes), MARK(sieve(z0))) MARK(filter(nats(z0), x1, zprimes)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__zprimes), MARK(nats(z0))) MARK(filter(zprimes, x1, zprimes)) -> c10(A__FILTER(a__zprimes, mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, zprimes)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__zprimes), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), zprimes)) -> c10(MARK(x0)) MARK(filter(x0, 0, zprimes)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, zprimes)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, cons(filter(z0, z1, z2), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(a__filter(mark(z0), mark(z1), mark(z2)), x3)), MARK(x0)) MARK(filter(x0, x1, cons(sieve(z0), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(a__sieve(mark(z0)), x3)), MARK(x0)) MARK(filter(x0, x1, cons(nats(z0), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(a__nats(mark(z0)), x3)), MARK(x0)) MARK(filter(x0, x1, cons(zprimes, x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(a__zprimes, x3)), MARK(x0)) MARK(filter(x0, x1, cons(cons(z0, z1), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(cons(mark(z0), z1), x3)), MARK(x0)) MARK(filter(x0, x1, cons(0, x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(0, x3)), MARK(x0)) MARK(filter(x0, x1, cons(s(z0), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(s(mark(z0)), x3)), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), cons(x2, x3))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), cons(mark(x2), x3)), MARK(x0)) MARK(filter(x0, sieve(z0), cons(x2, x3))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), cons(mark(x2), x3)), MARK(x0)) MARK(filter(x0, nats(z0), cons(x2, x3))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), cons(mark(x2), x3)), MARK(x0)) MARK(filter(x0, zprimes, cons(x2, x3))) -> c10(A__FILTER(mark(x0), a__zprimes, cons(mark(x2), x3)), MARK(x0)) MARK(filter(x0, cons(z0, z1), cons(x2, x3))) -> c10(A__FILTER(mark(x0), cons(mark(z0), z1), cons(mark(x2), x3)), MARK(x0)) MARK(filter(x0, 0, cons(x2, x3))) -> c10(A__FILTER(mark(x0), 0, cons(mark(x2), x3)), MARK(x0)) MARK(filter(x0, s(z0), cons(x2, x3))) -> c10(A__FILTER(mark(x0), s(mark(z0)), cons(mark(x2), x3)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, cons(x2, x3))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), cons(mark(x2), x3)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, cons(x2, x3))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), cons(mark(x2), x3)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, cons(x2, x3))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), cons(mark(x2), x3)), MARK(nats(z0))) MARK(filter(zprimes, x1, cons(x2, x3))) -> c10(A__FILTER(a__zprimes, mark(x1), cons(mark(x2), x3)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, cons(x2, x3))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), cons(mark(x2), x3)), MARK(cons(z0, z1))) MARK(filter(s(z0), x1, cons(x2, x3))) -> c10(A__FILTER(s(mark(z0)), mark(x1), cons(mark(x2), x3)), MARK(s(z0))) S tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, x1, 0)) -> c10(A__FILTER(mark(x0), mark(x1), 0), MARK(x0)) MARK(filter(x0, x1, s(z0))) -> c10(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(z0), x2)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, nats(z0), x2)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, zprimes, x2)) -> c10(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x0)) MARK(filter(x0, s(z0), x2)) -> c10(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, x2)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, x2)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, x1, x2)) -> c10(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, x2)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x1)) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) 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zprimes)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__zprimes), MARK(nats(z0))) MARK(filter(zprimes, x1, zprimes)) -> c10(A__FILTER(a__zprimes, mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, zprimes)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__zprimes), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), zprimes)) -> c10(MARK(x0)) MARK(filter(x0, 0, zprimes)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, zprimes)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, cons(filter(z0, z1, z2), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(a__filter(mark(z0), mark(z1), mark(z2)), x3)), MARK(x0)) MARK(filter(x0, x1, cons(sieve(z0), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(a__sieve(mark(z0)), x3)), MARK(x0)) MARK(filter(x0, x1, cons(nats(z0), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(a__nats(mark(z0)), x3)), MARK(x0)) MARK(filter(x0, x1, cons(zprimes, x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(a__zprimes, x3)), MARK(x0)) MARK(filter(x0, x1, cons(cons(z0, z1), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(cons(mark(z0), z1), x3)), MARK(x0)) MARK(filter(x0, x1, cons(0, x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(0, x3)), MARK(x0)) MARK(filter(x0, x1, cons(s(z0), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(s(mark(z0)), x3)), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), cons(x2, x3))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), cons(mark(x2), x3)), MARK(x0)) MARK(filter(x0, sieve(z0), cons(x2, x3))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), cons(mark(x2), x3)), MARK(x0)) MARK(filter(x0, nats(z0), cons(x2, x3))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), cons(mark(x2), x3)), MARK(x0)) MARK(filter(x0, zprimes, cons(x2, x3))) -> c10(A__FILTER(mark(x0), a__zprimes, cons(mark(x2), x3)), MARK(x0)) MARK(filter(x0, cons(z0, z1), cons(x2, x3))) -> c10(A__FILTER(mark(x0), cons(mark(z0), z1), cons(mark(x2), x3)), MARK(x0)) MARK(filter(x0, 0, cons(x2, x3))) -> c10(A__FILTER(mark(x0), 0, cons(mark(x2), x3)), MARK(x0)) MARK(filter(x0, s(z0), cons(x2, x3))) -> c10(A__FILTER(mark(x0), s(mark(z0)), cons(mark(x2), x3)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, cons(x2, x3))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), cons(mark(x2), x3)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, cons(x2, x3))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), cons(mark(x2), x3)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, cons(x2, x3))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), cons(mark(x2), x3)), MARK(nats(z0))) MARK(filter(zprimes, x1, cons(x2, x3))) -> c10(A__FILTER(a__zprimes, mark(x1), cons(mark(x2), x3)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, cons(x2, x3))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), cons(mark(x2), x3)), MARK(cons(z0, z1))) MARK(filter(s(z0), x1, cons(x2, x3))) -> c10(A__FILTER(s(mark(z0)), mark(x1), cons(mark(x2), x3)), MARK(s(z0))) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c14_2, c15_1, c16_1, c18_1, c8_1, c10_2, c10_1, c11_2, c11_1, c12_2, c12_1, c13_2, c13_1, c8_2 ---------------------------------------- (81) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (82) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, x1, 0)) -> c10(A__FILTER(mark(x0), mark(x1), 0), MARK(x0)) MARK(filter(x0, x1, s(z0))) -> c10(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(z0), x2)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, nats(z0), x2)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, zprimes, x2)) -> c10(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x0)) MARK(filter(x0, s(z0), x2)) -> c10(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, x2)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, x2)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, x1, x2)) -> c10(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, x2)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x1)) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(x0, 0, filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, sieve(x2))) -> c10(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x0)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, sieve(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, sieve(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, sieve(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(0)), MARK(x0)) MARK(filter(x0, x1, sieve(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), sieve(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, sieve(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, sieve(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__sieve(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, sieve(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, sieve(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, sieve(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), a__sieve(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, sieve(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__sieve(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), sieve(x2))) -> c10(MARK(x0)) MARK(filter(x0, 0, sieve(x2))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, sieve(x2))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(mark(x2)), nats(s(mark(x2))))), MARK(x0)) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), nats(mark(x2))), MARK(x0)) MARK(filter(x0, x1, nats(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, nats(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, nats(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, nats(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, nats(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, nats(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(0)), MARK(x0)) MARK(filter(x0, x1, nats(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), nats(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), nats(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), nats(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, nats(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), nats(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, nats(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__nats(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, nats(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, nats(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, nats(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), a__nats(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, nats(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__nats(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), nats(x2))) -> c10(MARK(x0)) MARK(filter(x0, 0, nats(x2))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, nats(x2))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, zprimes)) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(s(s(0))))), MARK(x0)) MARK(filter(x0, x1, zprimes)) -> c10(A__FILTER(mark(x0), mark(x1), zprimes), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), zprimes)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__zprimes), MARK(x0)) MARK(filter(x0, sieve(z0), zprimes)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__zprimes), MARK(x0)) MARK(filter(x0, nats(z0), zprimes)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__zprimes), MARK(x0)) MARK(filter(x0, zprimes, zprimes)) -> c10(A__FILTER(mark(x0), a__zprimes, a__zprimes), MARK(x0)) MARK(filter(x0, s(z0), zprimes)) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__zprimes), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, zprimes)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__zprimes), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, zprimes)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__zprimes), MARK(sieve(z0))) MARK(filter(nats(z0), x1, zprimes)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__zprimes), MARK(nats(z0))) MARK(filter(zprimes, x1, zprimes)) -> c10(A__FILTER(a__zprimes, mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, zprimes)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__zprimes), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), zprimes)) -> c10(MARK(x0)) MARK(filter(x0, 0, zprimes)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, zprimes)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, cons(filter(z0, z1, z2), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(a__filter(mark(z0), mark(z1), mark(z2)), x3)), MARK(x0)) MARK(filter(x0, x1, cons(sieve(z0), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(a__sieve(mark(z0)), x3)), MARK(x0)) MARK(filter(x0, x1, cons(nats(z0), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(a__nats(mark(z0)), x3)), MARK(x0)) MARK(filter(x0, x1, cons(zprimes, x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(a__zprimes, x3)), MARK(x0)) MARK(filter(x0, x1, cons(cons(z0, z1), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(cons(mark(z0), z1), x3)), MARK(x0)) MARK(filter(x0, x1, cons(0, x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(0, x3)), MARK(x0)) MARK(filter(x0, x1, cons(s(z0), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(s(mark(z0)), x3)), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), cons(x2, x3))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), cons(mark(x2), x3)), MARK(x0)) MARK(filter(x0, sieve(z0), cons(x2, x3))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), cons(mark(x2), x3)), MARK(x0)) MARK(filter(x0, nats(z0), cons(x2, x3))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), cons(mark(x2), x3)), MARK(x0)) MARK(filter(x0, zprimes, cons(x2, x3))) -> c10(A__FILTER(mark(x0), a__zprimes, cons(mark(x2), x3)), MARK(x0)) MARK(filter(x0, s(z0), cons(x2, x3))) -> c10(A__FILTER(mark(x0), s(mark(z0)), cons(mark(x2), x3)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, cons(x2, x3))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), cons(mark(x2), x3)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, cons(x2, x3))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), cons(mark(x2), x3)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, cons(x2, x3))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), cons(mark(x2), x3)), MARK(nats(z0))) MARK(filter(zprimes, x1, cons(x2, x3))) -> c10(A__FILTER(a__zprimes, mark(x1), cons(mark(x2), x3)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, cons(x2, x3))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), cons(mark(x2), x3)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), cons(x2, x3))) -> c10(MARK(x0)) MARK(filter(x0, 0, cons(x2, x3))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, cons(x2, x3))) -> c10(MARK(s(z0))) S tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, x1, 0)) -> c10(A__FILTER(mark(x0), mark(x1), 0), MARK(x0)) MARK(filter(x0, x1, s(z0))) -> c10(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(z0), x2)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, nats(z0), x2)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, zprimes, x2)) -> c10(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x0)) MARK(filter(x0, s(z0), x2)) -> c10(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, x2)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, x2)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, x1, x2)) -> c10(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, x2)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x1)) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(x0, 0, filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, sieve(x2))) -> c10(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x0)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, sieve(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, sieve(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, sieve(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(0)), MARK(x0)) MARK(filter(x0, x1, sieve(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), sieve(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, sieve(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, sieve(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__sieve(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, sieve(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, sieve(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, sieve(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), a__sieve(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, sieve(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__sieve(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), sieve(x2))) -> c10(MARK(x0)) MARK(filter(x0, 0, sieve(x2))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, sieve(x2))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(mark(x2)), nats(s(mark(x2))))), MARK(x0)) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), nats(mark(x2))), MARK(x0)) MARK(filter(x0, x1, nats(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, nats(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, nats(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, nats(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, nats(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, nats(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(0)), MARK(x0)) MARK(filter(x0, x1, nats(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), nats(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), nats(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), nats(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, nats(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), nats(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, nats(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__nats(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, nats(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, nats(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, nats(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), a__nats(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, nats(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__nats(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), nats(x2))) -> c10(MARK(x0)) MARK(filter(x0, 0, nats(x2))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, nats(x2))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, zprimes)) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(s(s(0))))), MARK(x0)) MARK(filter(x0, x1, zprimes)) -> c10(A__FILTER(mark(x0), mark(x1), zprimes), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), zprimes)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__zprimes), MARK(x0)) MARK(filter(x0, sieve(z0), zprimes)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__zprimes), MARK(x0)) MARK(filter(x0, nats(z0), zprimes)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__zprimes), MARK(x0)) MARK(filter(x0, zprimes, zprimes)) -> c10(A__FILTER(mark(x0), a__zprimes, a__zprimes), MARK(x0)) MARK(filter(x0, s(z0), zprimes)) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__zprimes), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, zprimes)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__zprimes), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, zprimes)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__zprimes), MARK(sieve(z0))) MARK(filter(nats(z0), x1, zprimes)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__zprimes), MARK(nats(z0))) MARK(filter(zprimes, x1, zprimes)) -> c10(A__FILTER(a__zprimes, mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, zprimes)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__zprimes), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), zprimes)) -> c10(MARK(x0)) MARK(filter(x0, 0, zprimes)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, zprimes)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, cons(filter(z0, z1, z2), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(a__filter(mark(z0), mark(z1), mark(z2)), x3)), MARK(x0)) MARK(filter(x0, x1, cons(sieve(z0), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(a__sieve(mark(z0)), x3)), MARK(x0)) MARK(filter(x0, x1, cons(nats(z0), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(a__nats(mark(z0)), x3)), MARK(x0)) MARK(filter(x0, x1, cons(zprimes, x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(a__zprimes, x3)), MARK(x0)) MARK(filter(x0, x1, cons(cons(z0, z1), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(cons(mark(z0), z1), x3)), MARK(x0)) MARK(filter(x0, x1, cons(0, x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(0, x3)), MARK(x0)) MARK(filter(x0, x1, cons(s(z0), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(s(mark(z0)), x3)), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), cons(x2, x3))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), cons(mark(x2), x3)), MARK(x0)) MARK(filter(x0, sieve(z0), cons(x2, x3))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), cons(mark(x2), x3)), MARK(x0)) MARK(filter(x0, nats(z0), cons(x2, x3))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), cons(mark(x2), x3)), MARK(x0)) MARK(filter(x0, zprimes, cons(x2, x3))) -> c10(A__FILTER(mark(x0), a__zprimes, cons(mark(x2), x3)), MARK(x0)) MARK(filter(x0, s(z0), cons(x2, x3))) -> c10(A__FILTER(mark(x0), s(mark(z0)), cons(mark(x2), x3)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, cons(x2, x3))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), cons(mark(x2), x3)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, cons(x2, x3))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), cons(mark(x2), x3)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, cons(x2, x3))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), cons(mark(x2), x3)), MARK(nats(z0))) MARK(filter(zprimes, x1, cons(x2, x3))) -> c10(A__FILTER(a__zprimes, mark(x1), cons(mark(x2), x3)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, cons(x2, x3))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), cons(mark(x2), x3)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), cons(x2, x3))) -> c10(MARK(x0)) MARK(filter(x0, 0, cons(x2, x3))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, cons(x2, x3))) -> c10(MARK(s(z0))) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c14_2, c15_1, c16_1, c18_1, c8_1, c10_2, c10_1, c11_2, c11_1, c12_2, c12_1, c13_2, c13_1, c8_2 ---------------------------------------- (83) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(filter(x0, x1, 0)) -> c10(A__FILTER(mark(x0), mark(x1), 0), MARK(x0)) by MARK(filter(x0, filter(z0, z1, z2), 0)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), 0), MARK(x0)) MARK(filter(x0, sieve(z0), 0)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), 0), MARK(x0)) MARK(filter(x0, nats(z0), 0)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), 0), MARK(x0)) MARK(filter(x0, zprimes, 0)) -> c10(A__FILTER(mark(x0), a__zprimes, 0), MARK(x0)) MARK(filter(x0, cons(z0, z1), 0)) -> c10(A__FILTER(mark(x0), cons(mark(z0), z1), 0), MARK(x0)) MARK(filter(x0, 0, 0)) -> c10(A__FILTER(mark(x0), 0, 0), MARK(x0)) MARK(filter(x0, s(z0), 0)) -> c10(A__FILTER(mark(x0), s(mark(z0)), 0), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, 0)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), 0), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, 0)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), 0), MARK(sieve(z0))) MARK(filter(nats(z0), x1, 0)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), 0), MARK(nats(z0))) MARK(filter(zprimes, x1, 0)) -> c10(A__FILTER(a__zprimes, mark(x1), 0), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, 0)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), 0), MARK(cons(z0, z1))) MARK(filter(0, x1, 0)) -> c10(A__FILTER(0, mark(x1), 0), MARK(0)) MARK(filter(s(z0), x1, 0)) -> c10(A__FILTER(s(mark(z0)), mark(x1), 0), MARK(s(z0))) ---------------------------------------- (84) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, x1, s(z0))) -> c10(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(z0), x2)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, nats(z0), x2)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, zprimes, x2)) -> c10(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x0)) MARK(filter(x0, s(z0), x2)) -> c10(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, x2)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, x2)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, x1, x2)) -> c10(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, x2)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x1)) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(x0, 0, filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, sieve(x2))) -> c10(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x0)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, sieve(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, sieve(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, sieve(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(0)), MARK(x0)) MARK(filter(x0, x1, sieve(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), sieve(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), 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MARK(x0)) MARK(filter(x0, x1, cons(sieve(z0), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(a__sieve(mark(z0)), x3)), MARK(x0)) MARK(filter(x0, x1, cons(nats(z0), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(a__nats(mark(z0)), x3)), MARK(x0)) MARK(filter(x0, x1, cons(zprimes, x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(a__zprimes, x3)), MARK(x0)) MARK(filter(x0, x1, cons(cons(z0, z1), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(cons(mark(z0), z1), x3)), MARK(x0)) MARK(filter(x0, x1, cons(0, x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(0, x3)), MARK(x0)) MARK(filter(x0, x1, cons(s(z0), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(s(mark(z0)), x3)), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), cons(x2, x3))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), cons(mark(x2), x3)), MARK(x0)) MARK(filter(x0, sieve(z0), cons(x2, x3))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), cons(mark(x2), x3)), MARK(x0)) MARK(filter(x0, nats(z0), cons(x2, x3))) -> 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x3))) -> c10(MARK(x0)) MARK(filter(x0, 0, cons(x2, x3))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, cons(x2, x3))) -> c10(MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), 0)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), 0), MARK(x0)) MARK(filter(x0, sieve(z0), 0)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), 0), MARK(x0)) MARK(filter(x0, nats(z0), 0)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), 0), MARK(x0)) MARK(filter(x0, zprimes, 0)) -> c10(A__FILTER(mark(x0), a__zprimes, 0), MARK(x0)) MARK(filter(x0, cons(z0, z1), 0)) -> c10(A__FILTER(mark(x0), cons(mark(z0), z1), 0), MARK(x0)) MARK(filter(x0, 0, 0)) -> c10(A__FILTER(mark(x0), 0, 0), MARK(x0)) MARK(filter(x0, s(z0), 0)) -> c10(A__FILTER(mark(x0), s(mark(z0)), 0), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, 0)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), 0), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, 0)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), 0), MARK(sieve(z0))) MARK(filter(nats(z0), x1, 0)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), 0), MARK(nats(z0))) MARK(filter(zprimes, x1, 0)) -> c10(A__FILTER(a__zprimes, mark(x1), 0), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, 0)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), 0), MARK(cons(z0, z1))) MARK(filter(0, x1, 0)) -> c10(A__FILTER(0, mark(x1), 0), MARK(0)) MARK(filter(s(z0), x1, 0)) -> c10(A__FILTER(s(mark(z0)), mark(x1), 0), MARK(s(z0))) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c14_2, c15_1, c16_1, c18_1, c8_1, c10_2, c10_1, c11_2, c11_1, c12_2, c12_1, c13_2, c13_1, c8_2 ---------------------------------------- (85) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(filter(0, x1, 0)) -> c10(A__FILTER(0, mark(x1), 0), MARK(0)) ---------------------------------------- (86) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, x1, s(z0))) -> c10(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(z0), x2)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, nats(z0), x2)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, zprimes, x2)) -> c10(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x0)) MARK(filter(x0, s(z0), x2)) -> c10(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, x2)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, x2)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, x1, x2)) -> c10(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, x2)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x1)) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(x0, 0, filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, sieve(x2))) -> c10(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x0)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, sieve(sieve(z0)))) -> 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MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, zprimes)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__zprimes), MARK(sieve(z0))) MARK(filter(nats(z0), x1, zprimes)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__zprimes), MARK(nats(z0))) MARK(filter(zprimes, x1, zprimes)) -> c10(A__FILTER(a__zprimes, mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, zprimes)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__zprimes), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), zprimes)) -> c10(MARK(x0)) MARK(filter(x0, 0, zprimes)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, zprimes)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, cons(filter(z0, z1, z2), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(a__filter(mark(z0), mark(z1), mark(z2)), x3)), MARK(x0)) MARK(filter(x0, x1, cons(sieve(z0), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(a__sieve(mark(z0)), x3)), MARK(x0)) MARK(filter(x0, x1, cons(nats(z0), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(a__nats(mark(z0)), x3)), MARK(x0)) MARK(filter(x0, x1, cons(zprimes, x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(a__zprimes, x3)), MARK(x0)) MARK(filter(x0, x1, cons(cons(z0, z1), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(cons(mark(z0), z1), x3)), MARK(x0)) MARK(filter(x0, x1, cons(0, x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(0, x3)), MARK(x0)) MARK(filter(x0, x1, cons(s(z0), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(s(mark(z0)), x3)), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), cons(x2, x3))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), cons(mark(x2), x3)), MARK(x0)) MARK(filter(x0, sieve(z0), cons(x2, x3))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), cons(mark(x2), x3)), MARK(x0)) MARK(filter(x0, nats(z0), cons(x2, x3))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), cons(mark(x2), x3)), MARK(x0)) MARK(filter(x0, zprimes, cons(x2, x3))) -> c10(A__FILTER(mark(x0), a__zprimes, cons(mark(x2), x3)), MARK(x0)) MARK(filter(x0, s(z0), cons(x2, x3))) -> c10(A__FILTER(mark(x0), s(mark(z0)), cons(mark(x2), x3)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, cons(x2, x3))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), cons(mark(x2), x3)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, cons(x2, x3))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), cons(mark(x2), x3)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, cons(x2, x3))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), cons(mark(x2), x3)), MARK(nats(z0))) MARK(filter(zprimes, x1, cons(x2, x3))) -> c10(A__FILTER(a__zprimes, mark(x1), cons(mark(x2), x3)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, cons(x2, x3))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), cons(mark(x2), x3)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), cons(x2, x3))) -> c10(MARK(x0)) MARK(filter(x0, 0, cons(x2, x3))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, cons(x2, x3))) -> c10(MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), 0)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), 0), MARK(x0)) MARK(filter(x0, sieve(z0), 0)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), 0), MARK(x0)) MARK(filter(x0, nats(z0), 0)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), 0), MARK(x0)) MARK(filter(x0, zprimes, 0)) -> c10(A__FILTER(mark(x0), a__zprimes, 0), MARK(x0)) MARK(filter(x0, cons(z0, z1), 0)) -> c10(A__FILTER(mark(x0), cons(mark(z0), z1), 0), MARK(x0)) MARK(filter(x0, 0, 0)) -> c10(A__FILTER(mark(x0), 0, 0), MARK(x0)) MARK(filter(x0, s(z0), 0)) -> c10(A__FILTER(mark(x0), s(mark(z0)), 0), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, 0)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), 0), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, 0)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), 0), MARK(sieve(z0))) MARK(filter(nats(z0), x1, 0)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), 0), MARK(nats(z0))) MARK(filter(zprimes, x1, 0)) -> c10(A__FILTER(a__zprimes, mark(x1), 0), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, 0)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), 0), MARK(cons(z0, z1))) MARK(filter(s(z0), x1, 0)) -> c10(A__FILTER(s(mark(z0)), mark(x1), 0), MARK(s(z0))) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c14_2, c15_1, c16_1, c18_1, c8_1, c10_2, c10_1, c11_2, c11_1, c12_2, c12_1, c13_2, c13_1, c8_2 ---------------------------------------- (87) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (88) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, x1, s(z0))) -> c10(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(z0), x2)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, nats(z0), x2)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, zprimes, x2)) -> c10(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x0)) MARK(filter(x0, s(z0), x2)) -> c10(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, x2)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, x2)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, x1, x2)) -> c10(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, x2)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x1)) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(x0, 0, filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, sieve(x2))) -> c10(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x0)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, sieve(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, sieve(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, sieve(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(0)), MARK(x0)) MARK(filter(x0, x1, sieve(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), sieve(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, sieve(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, sieve(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__sieve(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, sieve(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, sieve(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, sieve(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), a__sieve(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, sieve(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__sieve(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), sieve(x2))) -> c10(MARK(x0)) MARK(filter(x0, 0, sieve(x2))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, sieve(x2))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(mark(x2)), nats(s(mark(x2))))), MARK(x0)) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), nats(mark(x2))), MARK(x0)) MARK(filter(x0, x1, nats(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, nats(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, nats(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, nats(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, nats(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, nats(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(0)), MARK(x0)) MARK(filter(x0, x1, nats(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), nats(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), nats(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), nats(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, nats(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), nats(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, nats(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__nats(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, nats(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, nats(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, nats(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), a__nats(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, nats(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__nats(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), nats(x2))) -> c10(MARK(x0)) MARK(filter(x0, 0, nats(x2))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, nats(x2))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, zprimes)) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(s(s(0))))), MARK(x0)) MARK(filter(x0, x1, zprimes)) -> c10(A__FILTER(mark(x0), mark(x1), zprimes), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), zprimes)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__zprimes), MARK(x0)) MARK(filter(x0, sieve(z0), zprimes)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__zprimes), MARK(x0)) MARK(filter(x0, nats(z0), zprimes)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__zprimes), MARK(x0)) MARK(filter(x0, zprimes, zprimes)) -> c10(A__FILTER(mark(x0), a__zprimes, a__zprimes), MARK(x0)) MARK(filter(x0, s(z0), zprimes)) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__zprimes), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, zprimes)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__zprimes), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, zprimes)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__zprimes), MARK(sieve(z0))) MARK(filter(nats(z0), x1, zprimes)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__zprimes), MARK(nats(z0))) MARK(filter(zprimes, x1, zprimes)) -> c10(A__FILTER(a__zprimes, mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, zprimes)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__zprimes), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), zprimes)) -> c10(MARK(x0)) MARK(filter(x0, 0, zprimes)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, zprimes)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, cons(filter(z0, z1, z2), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(a__filter(mark(z0), mark(z1), mark(z2)), x3)), MARK(x0)) MARK(filter(x0, x1, cons(sieve(z0), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(a__sieve(mark(z0)), x3)), MARK(x0)) MARK(filter(x0, x1, cons(nats(z0), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(a__nats(mark(z0)), x3)), MARK(x0)) MARK(filter(x0, x1, cons(zprimes, x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(a__zprimes, x3)), MARK(x0)) MARK(filter(x0, x1, cons(cons(z0, z1), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(cons(mark(z0), z1), x3)), MARK(x0)) MARK(filter(x0, x1, cons(0, x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(0, x3)), MARK(x0)) MARK(filter(x0, x1, cons(s(z0), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(s(mark(z0)), x3)), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), cons(x2, x3))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), cons(mark(x2), x3)), MARK(x0)) MARK(filter(x0, sieve(z0), cons(x2, x3))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), cons(mark(x2), x3)), MARK(x0)) MARK(filter(x0, nats(z0), cons(x2, x3))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), cons(mark(x2), x3)), MARK(x0)) MARK(filter(x0, zprimes, cons(x2, x3))) -> c10(A__FILTER(mark(x0), a__zprimes, cons(mark(x2), x3)), MARK(x0)) MARK(filter(x0, s(z0), cons(x2, x3))) -> c10(A__FILTER(mark(x0), s(mark(z0)), cons(mark(x2), x3)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, cons(x2, x3))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), cons(mark(x2), x3)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, cons(x2, x3))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), cons(mark(x2), x3)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, cons(x2, x3))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), cons(mark(x2), x3)), MARK(nats(z0))) MARK(filter(zprimes, x1, cons(x2, x3))) -> c10(A__FILTER(a__zprimes, mark(x1), cons(mark(x2), x3)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, cons(x2, x3))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), cons(mark(x2), x3)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), cons(x2, x3))) -> c10(MARK(x0)) MARK(filter(x0, 0, cons(x2, x3))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, cons(x2, x3))) -> c10(MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), 0)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), 0), MARK(x0)) MARK(filter(x0, sieve(z0), 0)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), 0), MARK(x0)) MARK(filter(x0, nats(z0), 0)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), 0), MARK(x0)) MARK(filter(x0, zprimes, 0)) -> c10(A__FILTER(mark(x0), a__zprimes, 0), MARK(x0)) MARK(filter(x0, s(z0), 0)) -> c10(A__FILTER(mark(x0), s(mark(z0)), 0), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, 0)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), 0), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, 0)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), 0), MARK(sieve(z0))) MARK(filter(nats(z0), x1, 0)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), 0), MARK(nats(z0))) MARK(filter(zprimes, x1, 0)) -> c10(A__FILTER(a__zprimes, mark(x1), 0), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, 0)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), 0), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), 0)) -> c10(MARK(x0)) MARK(filter(x0, 0, 0)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, 0)) -> c10(MARK(s(z0))) S tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, x1, s(z0))) -> c10(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(z0), x2)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, nats(z0), x2)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, zprimes, x2)) -> c10(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x0)) MARK(filter(x0, s(z0), x2)) -> c10(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, x2)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, x2)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, x1, x2)) -> c10(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, x2)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x1)) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) 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c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__zprimes), MARK(x0)) MARK(filter(x0, sieve(z0), zprimes)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__zprimes), MARK(x0)) MARK(filter(x0, nats(z0), zprimes)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__zprimes), MARK(x0)) MARK(filter(x0, zprimes, zprimes)) -> c10(A__FILTER(mark(x0), a__zprimes, a__zprimes), MARK(x0)) MARK(filter(x0, s(z0), zprimes)) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__zprimes), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, zprimes)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__zprimes), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, zprimes)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__zprimes), MARK(sieve(z0))) MARK(filter(nats(z0), x1, zprimes)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__zprimes), MARK(nats(z0))) MARK(filter(zprimes, x1, zprimes)) -> c10(A__FILTER(a__zprimes, mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, zprimes)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__zprimes), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), zprimes)) -> c10(MARK(x0)) MARK(filter(x0, 0, zprimes)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, zprimes)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, cons(filter(z0, z1, z2), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(a__filter(mark(z0), mark(z1), mark(z2)), x3)), MARK(x0)) MARK(filter(x0, x1, cons(sieve(z0), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(a__sieve(mark(z0)), x3)), MARK(x0)) MARK(filter(x0, x1, cons(nats(z0), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(a__nats(mark(z0)), x3)), MARK(x0)) MARK(filter(x0, x1, cons(zprimes, x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(a__zprimes, x3)), MARK(x0)) MARK(filter(x0, x1, cons(cons(z0, z1), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(cons(mark(z0), z1), x3)), MARK(x0)) MARK(filter(x0, x1, cons(0, x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(0, x3)), MARK(x0)) MARK(filter(x0, x1, cons(s(z0), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(s(mark(z0)), x3)), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), cons(x2, x3))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), cons(mark(x2), x3)), MARK(x0)) MARK(filter(x0, sieve(z0), cons(x2, x3))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), cons(mark(x2), x3)), MARK(x0)) MARK(filter(x0, nats(z0), cons(x2, x3))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), cons(mark(x2), x3)), MARK(x0)) MARK(filter(x0, zprimes, cons(x2, x3))) -> c10(A__FILTER(mark(x0), a__zprimes, cons(mark(x2), x3)), MARK(x0)) MARK(filter(x0, s(z0), cons(x2, x3))) -> c10(A__FILTER(mark(x0), s(mark(z0)), cons(mark(x2), x3)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, cons(x2, x3))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), cons(mark(x2), x3)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, cons(x2, x3))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), cons(mark(x2), x3)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, cons(x2, x3))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), cons(mark(x2), x3)), MARK(nats(z0))) MARK(filter(zprimes, x1, cons(x2, x3))) -> c10(A__FILTER(a__zprimes, mark(x1), cons(mark(x2), x3)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, cons(x2, x3))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), cons(mark(x2), x3)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), cons(x2, x3))) -> c10(MARK(x0)) MARK(filter(x0, 0, cons(x2, x3))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, cons(x2, x3))) -> c10(MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), 0)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), 0), MARK(x0)) MARK(filter(x0, sieve(z0), 0)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), 0), MARK(x0)) MARK(filter(x0, nats(z0), 0)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), 0), MARK(x0)) MARK(filter(x0, zprimes, 0)) -> c10(A__FILTER(mark(x0), a__zprimes, 0), MARK(x0)) MARK(filter(x0, s(z0), 0)) -> c10(A__FILTER(mark(x0), s(mark(z0)), 0), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, 0)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), 0), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, 0)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), 0), MARK(sieve(z0))) MARK(filter(nats(z0), x1, 0)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), 0), MARK(nats(z0))) MARK(filter(zprimes, x1, 0)) -> c10(A__FILTER(a__zprimes, mark(x1), 0), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, 0)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), 0), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), 0)) -> c10(MARK(x0)) MARK(filter(x0, 0, 0)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, 0)) -> c10(MARK(s(z0))) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c14_2, c15_1, c16_1, c18_1, c8_1, c10_2, c10_1, c11_2, c11_1, c12_2, c12_1, c13_2, c13_1, c8_2 ---------------------------------------- (89) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(filter(x0, x1, s(z0))) -> c10(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x0)) by MARK(filter(x0, x1, s(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), s(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, s(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), s(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, s(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), s(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, s(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), s(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, s(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), s(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, s(0))) -> c10(A__FILTER(mark(x0), mark(x1), s(0)), MARK(x0)) MARK(filter(x0, x1, s(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), s(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), s(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), s(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), s(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), s(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), s(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), s(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, s(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, s(mark(x2))), MARK(x0)) MARK(filter(x0, cons(z0, z1), s(x2))) -> c10(A__FILTER(mark(x0), cons(mark(z0), z1), s(mark(x2))), MARK(x0)) MARK(filter(x0, 0, s(x2))) -> c10(A__FILTER(mark(x0), 0, s(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), s(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), s(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, s(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), s(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, s(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), s(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, s(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), s(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, s(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), s(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, s(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), s(mark(x2))), MARK(cons(z0, z1))) MARK(filter(0, x1, s(x2))) -> c10(A__FILTER(0, mark(x1), s(mark(x2))), MARK(0)) MARK(filter(s(z0), x1, s(x2))) -> c10(A__FILTER(s(mark(z0)), mark(x1), s(mark(x2))), MARK(s(z0))) ---------------------------------------- (90) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(z0), x2)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, nats(z0), x2)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, zprimes, x2)) -> c10(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x0)) MARK(filter(x0, s(z0), x2)) -> c10(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, x2)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, x2)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, x1, x2)) -> c10(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, x2)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x1)) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), 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MARK(filter(x0, 0, s(x2))) -> c10(A__FILTER(mark(x0), 0, s(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), s(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), s(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, s(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), s(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, s(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), s(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, s(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), s(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, s(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), s(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, s(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), s(mark(x2))), MARK(cons(z0, z1))) MARK(filter(0, x1, s(x2))) -> c10(A__FILTER(0, mark(x1), s(mark(x2))), MARK(0)) MARK(filter(s(z0), x1, s(x2))) -> c10(A__FILTER(s(mark(z0)), mark(x1), s(mark(x2))), MARK(s(z0))) S tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(z0), x2)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, nats(z0), x2)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, zprimes, x2)) -> c10(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x0)) MARK(filter(x0, s(z0), x2)) -> c10(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, x2)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, x2)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, x1, x2)) -> c10(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, x2)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x1)) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(x0, 0, filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, sieve(x2))) -> c10(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x0)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, sieve(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, sieve(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, sieve(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(0)), MARK(x0)) MARK(filter(x0, x1, sieve(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), sieve(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, sieve(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, sieve(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__sieve(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, sieve(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, sieve(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, sieve(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), a__sieve(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, sieve(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__sieve(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), sieve(x2))) -> c10(MARK(x0)) MARK(filter(x0, 0, sieve(x2))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, sieve(x2))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(mark(x2)), nats(s(mark(x2))))), MARK(x0)) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), nats(mark(x2))), MARK(x0)) MARK(filter(x0, x1, nats(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, nats(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, nats(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, nats(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, nats(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, nats(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(0)), MARK(x0)) MARK(filter(x0, x1, nats(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), nats(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), nats(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), nats(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, nats(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), nats(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, nats(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__nats(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, nats(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, nats(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, nats(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), a__nats(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, nats(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__nats(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), nats(x2))) -> c10(MARK(x0)) MARK(filter(x0, 0, nats(x2))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, nats(x2))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, zprimes)) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(s(s(0))))), MARK(x0)) MARK(filter(x0, x1, zprimes)) -> c10(A__FILTER(mark(x0), mark(x1), zprimes), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), zprimes)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__zprimes), MARK(x0)) MARK(filter(x0, sieve(z0), zprimes)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__zprimes), MARK(x0)) MARK(filter(x0, nats(z0), zprimes)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__zprimes), MARK(x0)) MARK(filter(x0, zprimes, zprimes)) -> c10(A__FILTER(mark(x0), a__zprimes, a__zprimes), MARK(x0)) MARK(filter(x0, s(z0), zprimes)) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__zprimes), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, zprimes)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__zprimes), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, zprimes)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__zprimes), MARK(sieve(z0))) MARK(filter(nats(z0), x1, zprimes)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__zprimes), MARK(nats(z0))) MARK(filter(zprimes, x1, zprimes)) -> c10(A__FILTER(a__zprimes, mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, zprimes)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__zprimes), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), zprimes)) -> c10(MARK(x0)) MARK(filter(x0, 0, zprimes)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, zprimes)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, cons(filter(z0, z1, z2), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(a__filter(mark(z0), mark(z1), mark(z2)), x3)), MARK(x0)) MARK(filter(x0, x1, cons(sieve(z0), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(a__sieve(mark(z0)), x3)), MARK(x0)) MARK(filter(x0, x1, cons(nats(z0), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(a__nats(mark(z0)), x3)), MARK(x0)) MARK(filter(x0, x1, cons(zprimes, x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(a__zprimes, x3)), MARK(x0)) MARK(filter(x0, x1, cons(cons(z0, z1), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(cons(mark(z0), z1), x3)), MARK(x0)) MARK(filter(x0, x1, cons(0, x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(0, x3)), MARK(x0)) MARK(filter(x0, x1, cons(s(z0), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(s(mark(z0)), x3)), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), cons(x2, x3))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), cons(mark(x2), x3)), MARK(x0)) MARK(filter(x0, sieve(z0), cons(x2, x3))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), cons(mark(x2), x3)), MARK(x0)) MARK(filter(x0, nats(z0), cons(x2, x3))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), cons(mark(x2), x3)), MARK(x0)) MARK(filter(x0, zprimes, cons(x2, x3))) -> c10(A__FILTER(mark(x0), a__zprimes, cons(mark(x2), x3)), MARK(x0)) MARK(filter(x0, s(z0), cons(x2, x3))) -> c10(A__FILTER(mark(x0), s(mark(z0)), cons(mark(x2), x3)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, cons(x2, x3))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), cons(mark(x2), x3)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, cons(x2, x3))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), cons(mark(x2), x3)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, cons(x2, x3))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), cons(mark(x2), x3)), MARK(nats(z0))) MARK(filter(zprimes, x1, cons(x2, x3))) -> c10(A__FILTER(a__zprimes, mark(x1), cons(mark(x2), x3)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, cons(x2, x3))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), cons(mark(x2), x3)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), cons(x2, x3))) -> c10(MARK(x0)) MARK(filter(x0, 0, cons(x2, x3))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, cons(x2, x3))) -> c10(MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), 0)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), 0), MARK(x0)) MARK(filter(x0, sieve(z0), 0)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), 0), MARK(x0)) MARK(filter(x0, nats(z0), 0)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), 0), MARK(x0)) MARK(filter(x0, zprimes, 0)) -> c10(A__FILTER(mark(x0), a__zprimes, 0), MARK(x0)) MARK(filter(x0, s(z0), 0)) -> c10(A__FILTER(mark(x0), s(mark(z0)), 0), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, 0)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), 0), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, 0)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), 0), MARK(sieve(z0))) MARK(filter(nats(z0), x1, 0)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), 0), MARK(nats(z0))) MARK(filter(zprimes, x1, 0)) -> c10(A__FILTER(a__zprimes, mark(x1), 0), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, 0)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), 0), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), 0)) -> c10(MARK(x0)) MARK(filter(x0, 0, 0)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, 0)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, s(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), s(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, s(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), s(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, s(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), s(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, s(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), s(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, s(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), s(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, s(0))) -> c10(A__FILTER(mark(x0), mark(x1), s(0)), MARK(x0)) MARK(filter(x0, x1, s(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), s(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), s(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), s(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), s(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), s(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), s(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), s(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, s(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, s(mark(x2))), MARK(x0)) MARK(filter(x0, cons(z0, z1), s(x2))) -> c10(A__FILTER(mark(x0), cons(mark(z0), z1), s(mark(x2))), MARK(x0)) MARK(filter(x0, 0, s(x2))) -> c10(A__FILTER(mark(x0), 0, s(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), s(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), s(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, s(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), s(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, s(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), s(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, s(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), s(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, s(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), s(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, s(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), s(mark(x2))), MARK(cons(z0, z1))) MARK(filter(0, x1, s(x2))) -> c10(A__FILTER(0, mark(x1), s(mark(x2))), MARK(0)) MARK(filter(s(z0), x1, s(x2))) -> c10(A__FILTER(s(mark(z0)), mark(x1), s(mark(x2))), MARK(s(z0))) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c14_2, c15_1, c16_1, c18_1, c8_1, c10_2, c10_1, c11_2, c11_1, c12_2, c12_1, c13_2, c13_1, c8_2 ---------------------------------------- (91) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(filter(0, x1, s(x2))) -> c10(A__FILTER(0, mark(x1), s(mark(x2))), MARK(0)) ---------------------------------------- (92) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(z0), x2)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, nats(z0), x2)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, zprimes, x2)) -> c10(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x0)) MARK(filter(x0, s(z0), x2)) -> c10(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, x2)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, x2)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, x1, x2)) -> c10(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, x2)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x1)) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), 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MARK(filter(x0, 0, zprimes)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, zprimes)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, cons(filter(z0, z1, z2), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(a__filter(mark(z0), mark(z1), mark(z2)), x3)), MARK(x0)) MARK(filter(x0, x1, cons(sieve(z0), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(a__sieve(mark(z0)), x3)), MARK(x0)) MARK(filter(x0, x1, cons(nats(z0), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(a__nats(mark(z0)), x3)), MARK(x0)) MARK(filter(x0, x1, cons(zprimes, x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(a__zprimes, x3)), MARK(x0)) MARK(filter(x0, x1, cons(cons(z0, z1), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(cons(mark(z0), z1), x3)), MARK(x0)) MARK(filter(x0, x1, cons(0, x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(0, x3)), MARK(x0)) MARK(filter(x0, x1, cons(s(z0), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(s(mark(z0)), x3)), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), cons(x2, x3))) -> 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z1)))) -> c10(A__FILTER(mark(x0), mark(x1), s(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, s(0))) -> c10(A__FILTER(mark(x0), mark(x1), s(0)), MARK(x0)) MARK(filter(x0, x1, s(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), s(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), s(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), s(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), s(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), s(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), s(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), s(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, s(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, s(mark(x2))), MARK(x0)) MARK(filter(x0, cons(z0, z1), s(x2))) -> c10(A__FILTER(mark(x0), cons(mark(z0), z1), s(mark(x2))), MARK(x0)) MARK(filter(x0, 0, s(x2))) -> c10(A__FILTER(mark(x0), 0, s(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), s(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), s(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, s(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), s(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, s(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), s(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, s(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), s(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, s(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), s(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, s(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), s(mark(x2))), MARK(cons(z0, z1))) MARK(filter(s(z0), x1, s(x2))) -> c10(A__FILTER(s(mark(z0)), mark(x1), s(mark(x2))), MARK(s(z0))) S tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(z0), x2)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, nats(z0), x2)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, zprimes, x2)) -> c10(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x0)) MARK(filter(x0, s(z0), x2)) -> c10(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, x2)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, x2)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, x1, x2)) -> c10(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, x2)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x1)) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> 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MARK(filter(zprimes, x1, s(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), s(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, s(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), s(mark(x2))), MARK(cons(z0, z1))) MARK(filter(s(z0), x1, s(x2))) -> c10(A__FILTER(s(mark(z0)), mark(x1), s(mark(x2))), MARK(s(z0))) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c14_2, c15_1, c16_1, c18_1, c8_1, c10_2, c10_1, c11_2, c11_1, c12_2, c12_1, c13_2, c13_1, c8_2 ---------------------------------------- (93) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (94) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(z0), x2)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, nats(z0), x2)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, zprimes, x2)) -> c10(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x0)) MARK(filter(x0, s(z0), x2)) -> c10(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, x2)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, x2)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, x1, x2)) -> c10(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, x2)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x1)) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(x0, 0, filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, sieve(x2))) -> c10(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x0)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, sieve(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, sieve(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, sieve(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(0)), MARK(x0)) MARK(filter(x0, x1, sieve(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), sieve(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, sieve(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, sieve(x2))) -> c10(A__FILTER(a__filter(mark(z0), 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sieve(z0), zprimes)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__zprimes), MARK(x0)) MARK(filter(x0, nats(z0), zprimes)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__zprimes), MARK(x0)) MARK(filter(x0, zprimes, zprimes)) -> c10(A__FILTER(mark(x0), a__zprimes, a__zprimes), MARK(x0)) MARK(filter(x0, s(z0), zprimes)) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__zprimes), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, zprimes)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__zprimes), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, zprimes)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__zprimes), MARK(sieve(z0))) MARK(filter(nats(z0), x1, zprimes)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__zprimes), MARK(nats(z0))) MARK(filter(zprimes, x1, zprimes)) -> c10(A__FILTER(a__zprimes, mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, zprimes)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__zprimes), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), zprimes)) -> c10(MARK(x0)) MARK(filter(x0, 0, zprimes)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, zprimes)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, cons(filter(z0, z1, z2), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(a__filter(mark(z0), mark(z1), mark(z2)), x3)), MARK(x0)) MARK(filter(x0, x1, cons(sieve(z0), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(a__sieve(mark(z0)), x3)), MARK(x0)) MARK(filter(x0, x1, cons(nats(z0), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(a__nats(mark(z0)), x3)), MARK(x0)) MARK(filter(x0, x1, cons(zprimes, x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(a__zprimes, x3)), MARK(x0)) MARK(filter(x0, x1, cons(cons(z0, z1), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(cons(mark(z0), z1), x3)), MARK(x0)) MARK(filter(x0, x1, cons(0, x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(0, x3)), MARK(x0)) MARK(filter(x0, x1, cons(s(z0), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(s(mark(z0)), x3)), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), cons(x2, x3))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), cons(mark(x2), x3)), MARK(x0)) MARK(filter(x0, sieve(z0), cons(x2, x3))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), cons(mark(x2), x3)), MARK(x0)) MARK(filter(x0, nats(z0), cons(x2, x3))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), cons(mark(x2), x3)), MARK(x0)) MARK(filter(x0, zprimes, cons(x2, x3))) -> c10(A__FILTER(mark(x0), a__zprimes, cons(mark(x2), x3)), MARK(x0)) MARK(filter(x0, s(z0), cons(x2, x3))) -> c10(A__FILTER(mark(x0), s(mark(z0)), cons(mark(x2), x3)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, cons(x2, x3))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), cons(mark(x2), x3)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, cons(x2, x3))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), cons(mark(x2), x3)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, cons(x2, x3))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), cons(mark(x2), x3)), MARK(nats(z0))) MARK(filter(zprimes, x1, cons(x2, x3))) -> c10(A__FILTER(a__zprimes, mark(x1), cons(mark(x2), x3)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, cons(x2, x3))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), cons(mark(x2), x3)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), cons(x2, x3))) -> c10(MARK(x0)) MARK(filter(x0, 0, cons(x2, x3))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, cons(x2, x3))) -> c10(MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), 0)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), 0), MARK(x0)) MARK(filter(x0, sieve(z0), 0)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), 0), MARK(x0)) MARK(filter(x0, nats(z0), 0)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), 0), MARK(x0)) MARK(filter(x0, zprimes, 0)) -> c10(A__FILTER(mark(x0), a__zprimes, 0), MARK(x0)) MARK(filter(x0, s(z0), 0)) -> c10(A__FILTER(mark(x0), s(mark(z0)), 0), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, 0)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), 0), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, 0)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), 0), MARK(sieve(z0))) MARK(filter(nats(z0), x1, 0)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), 0), MARK(nats(z0))) MARK(filter(zprimes, x1, 0)) -> c10(A__FILTER(a__zprimes, mark(x1), 0), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, 0)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), 0), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), 0)) -> c10(MARK(x0)) MARK(filter(x0, 0, 0)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, 0)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, s(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), s(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, s(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), s(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, s(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), s(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, s(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), s(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, s(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), s(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, s(0))) -> c10(A__FILTER(mark(x0), mark(x1), s(0)), MARK(x0)) MARK(filter(x0, x1, s(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), s(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), s(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), s(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), s(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), s(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), s(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), s(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, s(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, s(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), s(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), s(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, s(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), s(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, s(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), s(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, s(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), s(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, s(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), s(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, s(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), s(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), s(x2))) -> c10(MARK(x0)) MARK(filter(x0, 0, s(x2))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, s(x2))) -> c10(MARK(s(z0))) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c14_2, c15_1, c16_1, c18_1, c8_1, c10_2, c10_1, c11_2, c11_1, c12_2, c12_1, c13_2, c13_1, c8_2 ---------------------------------------- (95) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(filter(x0, filter(z0, z1, z2), x2)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x0)) by MARK(filter(x0, filter(x1, x2, x3), filter(z0, z1, z2))) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), mark(x3)), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x0)) MARK(filter(x0, filter(x1, x2, x3), sieve(z0))) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), mark(x3)), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, filter(x1, x2, x3), nats(z0))) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), mark(x3)), a__nats(mark(z0))), MARK(x0)) MARK(filter(x0, filter(x1, x2, x3), zprimes)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), mark(x3)), a__zprimes), MARK(x0)) MARK(filter(x0, filter(x1, x2, x3), cons(z0, z1))) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), mark(x3)), cons(mark(z0), z1)), MARK(x0)) MARK(filter(x0, filter(x1, x2, x3), 0)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), mark(x3)), 0), MARK(x0)) MARK(filter(x0, filter(x1, x2, x3), s(z0))) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), mark(x3)), s(mark(z0))), MARK(x0)) MARK(filter(x0, filter(x1, x2, x3), x4)) -> c10(A__FILTER(mark(x0), filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, filter(z0, z1, z2)), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__filter(mark(z0), mark(z1), mark(z2))), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, sieve(z0)), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__sieve(mark(z0))), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, nats(z0)), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__nats(mark(z0))), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, zprimes), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__zprimes), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, cons(z0, z1)), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), cons(mark(z0), z1)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, 0), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), 0), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, s(z0)), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), s(mark(z0))), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, filter(z0, z1, z2), x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), a__filter(mark(z0), mark(z1), mark(z2)), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, sieve(z0), x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), a__sieve(mark(z0)), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, nats(z0), x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), a__nats(mark(z0)), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, zprimes, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), a__zprimes, mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, cons(z0, z1), x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), cons(mark(z0), z1), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, 0, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), 0, mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, s(z0), x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), s(mark(z0)), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(filter(z0, z1, z2), x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(sieve(z0), x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(a__sieve(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(nats(z0), x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(a__nats(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(zprimes, x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(a__zprimes, mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(cons(z0, z1), x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(cons(mark(z0), z1), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(0, x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(0, mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(s(z0), x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(s(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(filter(z0, z1, z2), filter(x1, x2, x3), x4)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), filter(x1, x2, x3), x4)) -> c10(A__FILTER(a__sieve(mark(z0)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(sieve(z0))) MARK(filter(nats(z0), filter(x1, x2, x3), x4)) -> c10(A__FILTER(a__nats(mark(z0)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(nats(z0))) MARK(filter(zprimes, filter(x1, x2, x3), x4)) -> c10(A__FILTER(a__zprimes, a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(zprimes)) MARK(filter(cons(z0, z1), filter(x1, x2, x3), x4)) -> c10(A__FILTER(cons(mark(z0), z1), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(cons(z0, z1))) MARK(filter(0, filter(x1, x2, x3), x4)) -> c10(A__FILTER(0, a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(0)) MARK(filter(s(z0), filter(x1, x2, x3), x4)) -> c10(A__FILTER(s(mark(z0)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(s(z0))) ---------------------------------------- (96) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, sieve(z0), x2)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, nats(z0), x2)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, zprimes, x2)) -> c10(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x0)) MARK(filter(x0, s(z0), x2)) -> c10(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c10(A__FILTER(a__filter(mark(z0), 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c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) 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a__zprimes), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), zprimes)) -> c10(MARK(x0)) MARK(filter(x0, 0, zprimes)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, zprimes)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, cons(filter(z0, z1, z2), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(a__filter(mark(z0), mark(z1), mark(z2)), x3)), MARK(x0)) MARK(filter(x0, x1, cons(sieve(z0), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(a__sieve(mark(z0)), x3)), MARK(x0)) MARK(filter(x0, x1, cons(nats(z0), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(a__nats(mark(z0)), x3)), MARK(x0)) MARK(filter(x0, x1, cons(zprimes, x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(a__zprimes, x3)), MARK(x0)) MARK(filter(x0, x1, cons(cons(z0, z1), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(cons(mark(z0), z1), x3)), MARK(x0)) MARK(filter(x0, x1, cons(0, x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(0, x3)), MARK(x0)) MARK(filter(x0, x1, cons(s(z0), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(s(mark(z0)), x3)), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), cons(x2, x3))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), cons(mark(x2), x3)), MARK(x0)) MARK(filter(x0, sieve(z0), cons(x2, x3))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), cons(mark(x2), x3)), MARK(x0)) MARK(filter(x0, nats(z0), cons(x2, x3))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), cons(mark(x2), x3)), MARK(x0)) MARK(filter(x0, zprimes, cons(x2, x3))) -> c10(A__FILTER(mark(x0), a__zprimes, cons(mark(x2), x3)), MARK(x0)) MARK(filter(x0, s(z0), cons(x2, x3))) -> c10(A__FILTER(mark(x0), s(mark(z0)), cons(mark(x2), x3)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, cons(x2, x3))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), cons(mark(x2), x3)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, cons(x2, x3))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), cons(mark(x2), x3)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, cons(x2, x3))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), cons(mark(x2), x3)), MARK(nats(z0))) MARK(filter(zprimes, x1, cons(x2, x3))) -> c10(A__FILTER(a__zprimes, mark(x1), cons(mark(x2), x3)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, cons(x2, x3))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), cons(mark(x2), x3)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), cons(x2, x3))) -> c10(MARK(x0)) MARK(filter(x0, 0, cons(x2, x3))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, cons(x2, x3))) -> c10(MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), 0)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), 0), MARK(x0)) MARK(filter(x0, sieve(z0), 0)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), 0), MARK(x0)) MARK(filter(x0, nats(z0), 0)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), 0), MARK(x0)) MARK(filter(x0, zprimes, 0)) -> c10(A__FILTER(mark(x0), a__zprimes, 0), MARK(x0)) MARK(filter(x0, s(z0), 0)) -> c10(A__FILTER(mark(x0), s(mark(z0)), 0), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, 0)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), 0), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, 0)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), 0), MARK(sieve(z0))) MARK(filter(nats(z0), x1, 0)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), 0), MARK(nats(z0))) MARK(filter(zprimes, x1, 0)) -> c10(A__FILTER(a__zprimes, mark(x1), 0), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, 0)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), 0), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), 0)) -> c10(MARK(x0)) MARK(filter(x0, 0, 0)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, 0)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, s(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), s(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, s(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), s(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, s(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), s(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, s(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), s(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, s(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), s(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, s(0))) -> c10(A__FILTER(mark(x0), mark(x1), s(0)), MARK(x0)) MARK(filter(x0, x1, s(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), s(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), s(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), s(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), s(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), s(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), s(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), s(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, s(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, s(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), s(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), s(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, s(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), s(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, s(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), s(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, s(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), s(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, s(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), s(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, s(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), s(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), s(x2))) -> c10(MARK(x0)) MARK(filter(x0, 0, s(x2))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, s(x2))) -> c10(MARK(s(z0))) MARK(filter(x0, filter(x1, x2, x3), x4)) -> c10(A__FILTER(mark(x0), filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, filter(z0, z1, z2)), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__filter(mark(z0), mark(z1), mark(z2))), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, sieve(z0)), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__sieve(mark(z0))), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, nats(z0)), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__nats(mark(z0))), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, zprimes), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__zprimes), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, cons(z0, z1)), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), cons(mark(z0), z1)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, 0), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), 0), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, s(z0)), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), s(mark(z0))), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, filter(z0, z1, z2), x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), a__filter(mark(z0), mark(z1), mark(z2)), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, sieve(z0), x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), a__sieve(mark(z0)), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, nats(z0), x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), a__nats(mark(z0)), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, zprimes, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), a__zprimes, mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, cons(z0, z1), x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), cons(mark(z0), z1), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, 0, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), 0, mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, s(z0), x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), s(mark(z0)), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(filter(z0, z1, z2), x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(sieve(z0), x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(a__sieve(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(nats(z0), x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(a__nats(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(zprimes, x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(a__zprimes, mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(cons(z0, z1), x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(cons(mark(z0), z1), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(0, x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(0, mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(s(z0), x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(s(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(filter(z0, z1, z2), filter(x1, x2, x3), x4)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), filter(x1, x2, x3), x4)) -> c10(A__FILTER(a__sieve(mark(z0)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(sieve(z0))) MARK(filter(nats(z0), filter(x1, x2, x3), x4)) -> c10(A__FILTER(a__nats(mark(z0)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(nats(z0))) MARK(filter(zprimes, filter(x1, x2, x3), x4)) -> c10(A__FILTER(a__zprimes, a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(zprimes)) MARK(filter(cons(z0, z1), filter(x1, x2, x3), x4)) -> c10(A__FILTER(cons(mark(z0), z1), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(cons(z0, z1))) MARK(filter(0, filter(x1, x2, x3), x4)) -> c10(A__FILTER(0, a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(0)) MARK(filter(s(z0), filter(x1, x2, x3), x4)) -> c10(A__FILTER(s(mark(z0)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(s(z0))) S tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, sieve(z0), x2)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, nats(z0), x2)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, zprimes, x2)) -> c10(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x0)) MARK(filter(x0, s(z0), x2)) -> c10(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, x2)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, x2)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, x1, x2)) -> c10(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, x2)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x1)) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) 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a__nats(mark(z0)), cons(mark(x2), x3)), MARK(x0)) MARK(filter(x0, zprimes, cons(x2, x3))) -> c10(A__FILTER(mark(x0), a__zprimes, cons(mark(x2), x3)), MARK(x0)) MARK(filter(x0, s(z0), cons(x2, x3))) -> c10(A__FILTER(mark(x0), s(mark(z0)), cons(mark(x2), x3)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, cons(x2, x3))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), cons(mark(x2), x3)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, cons(x2, x3))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), cons(mark(x2), x3)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, cons(x2, x3))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), cons(mark(x2), x3)), MARK(nats(z0))) MARK(filter(zprimes, x1, cons(x2, x3))) -> c10(A__FILTER(a__zprimes, mark(x1), cons(mark(x2), x3)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, cons(x2, x3))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), cons(mark(x2), x3)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), cons(x2, x3))) -> c10(MARK(x0)) MARK(filter(x0, 0, cons(x2, x3))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, cons(x2, x3))) -> c10(MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), 0)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), 0), MARK(x0)) MARK(filter(x0, sieve(z0), 0)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), 0), MARK(x0)) MARK(filter(x0, nats(z0), 0)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), 0), MARK(x0)) MARK(filter(x0, zprimes, 0)) -> c10(A__FILTER(mark(x0), a__zprimes, 0), MARK(x0)) MARK(filter(x0, s(z0), 0)) -> c10(A__FILTER(mark(x0), s(mark(z0)), 0), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, 0)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), 0), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, 0)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), 0), MARK(sieve(z0))) MARK(filter(nats(z0), x1, 0)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), 0), MARK(nats(z0))) MARK(filter(zprimes, x1, 0)) -> c10(A__FILTER(a__zprimes, mark(x1), 0), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, 0)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), 0), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), 0)) -> c10(MARK(x0)) MARK(filter(x0, 0, 0)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, 0)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, s(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), s(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, s(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), s(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, s(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), s(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, s(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), s(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, s(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), s(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, s(0))) -> c10(A__FILTER(mark(x0), mark(x1), s(0)), MARK(x0)) MARK(filter(x0, x1, s(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), s(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), s(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), s(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), s(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), s(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), s(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), s(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, s(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, s(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), s(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), s(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, s(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), s(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, s(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), s(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, s(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), s(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, s(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), s(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, s(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), s(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), s(x2))) -> c10(MARK(x0)) MARK(filter(x0, 0, s(x2))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, s(x2))) -> c10(MARK(s(z0))) MARK(filter(x0, filter(x1, x2, x3), x4)) -> c10(A__FILTER(mark(x0), filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, filter(z0, z1, z2)), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__filter(mark(z0), mark(z1), mark(z2))), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, sieve(z0)), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__sieve(mark(z0))), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, nats(z0)), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__nats(mark(z0))), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, zprimes), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__zprimes), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, cons(z0, z1)), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), cons(mark(z0), z1)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, 0), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), 0), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, s(z0)), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), s(mark(z0))), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, filter(z0, z1, z2), x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), a__filter(mark(z0), mark(z1), mark(z2)), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, sieve(z0), x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), a__sieve(mark(z0)), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, nats(z0), x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), a__nats(mark(z0)), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, zprimes, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), a__zprimes, mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, cons(z0, z1), x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), cons(mark(z0), z1), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, 0, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), 0, mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, s(z0), x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), s(mark(z0)), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(filter(z0, z1, z2), x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(sieve(z0), x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(a__sieve(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(nats(z0), x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(a__nats(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(zprimes, x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(a__zprimes, mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(cons(z0, z1), x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(cons(mark(z0), z1), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(0, x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(0, mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(s(z0), x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(s(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(filter(z0, z1, z2), filter(x1, x2, x3), x4)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), filter(x1, x2, x3), x4)) -> c10(A__FILTER(a__sieve(mark(z0)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(sieve(z0))) MARK(filter(nats(z0), filter(x1, x2, x3), x4)) -> c10(A__FILTER(a__nats(mark(z0)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(nats(z0))) MARK(filter(zprimes, filter(x1, x2, x3), x4)) -> c10(A__FILTER(a__zprimes, a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(zprimes)) MARK(filter(cons(z0, z1), filter(x1, x2, x3), x4)) -> c10(A__FILTER(cons(mark(z0), z1), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(cons(z0, z1))) MARK(filter(0, filter(x1, x2, x3), x4)) -> c10(A__FILTER(0, a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(0)) MARK(filter(s(z0), filter(x1, x2, x3), x4)) -> c10(A__FILTER(s(mark(z0)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(s(z0))) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c14_2, c15_1, c16_1, c18_1, c8_1, c10_2, c10_1, c11_2, c11_1, c12_2, c12_1, c13_2, c13_1, c8_2 ---------------------------------------- (97) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(filter(0, filter(x1, x2, x3), x4)) -> c10(A__FILTER(0, a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(0)) ---------------------------------------- (98) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, sieve(z0), x2)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, nats(z0), x2)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, zprimes, x2)) -> c10(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x0)) MARK(filter(x0, s(z0), x2)) -> c10(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, x2)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, x2)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, x1, x2)) -> c10(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, x2)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x1)) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), 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c10(A__FILTER(mark(x0), s(mark(z0)), 0), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, 0)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), 0), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, 0)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), 0), MARK(sieve(z0))) MARK(filter(nats(z0), x1, 0)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), 0), MARK(nats(z0))) MARK(filter(zprimes, x1, 0)) -> c10(A__FILTER(a__zprimes, mark(x1), 0), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, 0)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), 0), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), 0)) -> c10(MARK(x0)) MARK(filter(x0, 0, 0)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, 0)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, s(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), s(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, s(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), s(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, s(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), s(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, s(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), s(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, s(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), s(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, s(0))) -> c10(A__FILTER(mark(x0), mark(x1), s(0)), MARK(x0)) MARK(filter(x0, x1, s(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), s(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), s(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), s(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), s(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), s(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), s(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), s(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, s(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, s(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), s(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), s(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, s(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), s(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, s(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), s(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, s(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), s(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, s(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), s(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, s(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), s(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), s(x2))) -> c10(MARK(x0)) MARK(filter(x0, 0, s(x2))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, s(x2))) -> c10(MARK(s(z0))) MARK(filter(x0, filter(x1, x2, x3), x4)) -> c10(A__FILTER(mark(x0), filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, filter(z0, z1, z2)), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__filter(mark(z0), mark(z1), mark(z2))), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, sieve(z0)), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__sieve(mark(z0))), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, nats(z0)), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__nats(mark(z0))), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, zprimes), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__zprimes), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, cons(z0, z1)), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), cons(mark(z0), z1)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, 0), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), 0), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, s(z0)), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), s(mark(z0))), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, filter(z0, z1, z2), x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), a__filter(mark(z0), mark(z1), mark(z2)), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, sieve(z0), x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), a__sieve(mark(z0)), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, nats(z0), x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), a__nats(mark(z0)), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, zprimes, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), a__zprimes, mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, cons(z0, z1), x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), cons(mark(z0), z1), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, 0, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), 0, mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, s(z0), x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), s(mark(z0)), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(filter(z0, z1, z2), x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(sieve(z0), x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(a__sieve(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(nats(z0), x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(a__nats(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(zprimes, x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(a__zprimes, mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(cons(z0, z1), x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(cons(mark(z0), z1), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(0, x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(0, mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(s(z0), x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(s(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(filter(z0, z1, z2), filter(x1, x2, x3), x4)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), filter(x1, x2, x3), x4)) -> c10(A__FILTER(a__sieve(mark(z0)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(sieve(z0))) MARK(filter(nats(z0), filter(x1, x2, x3), x4)) -> c10(A__FILTER(a__nats(mark(z0)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(nats(z0))) MARK(filter(zprimes, filter(x1, x2, x3), x4)) -> c10(A__FILTER(a__zprimes, a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(zprimes)) MARK(filter(cons(z0, z1), filter(x1, x2, x3), x4)) -> c10(A__FILTER(cons(mark(z0), z1), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(cons(z0, z1))) MARK(filter(s(z0), filter(x1, x2, x3), x4)) -> c10(A__FILTER(s(mark(z0)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(s(z0))) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c14_2, c15_1, c16_1, c18_1, c8_1, c10_2, c10_1, c11_2, c11_1, c12_2, c12_1, c13_2, c13_1, c8_2 ---------------------------------------- (99) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (100) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, sieve(z0), x2)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, nats(z0), x2)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, zprimes, x2)) -> c10(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x0)) MARK(filter(x0, s(z0), x2)) -> c10(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, x2)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, x2)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, x1, x2)) -> c10(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, x2)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x1)) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), 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MARK(filter(sieve(z0), x1, 0)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), 0), MARK(sieve(z0))) MARK(filter(nats(z0), x1, 0)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), 0), MARK(nats(z0))) MARK(filter(zprimes, x1, 0)) -> c10(A__FILTER(a__zprimes, mark(x1), 0), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, 0)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), 0), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), 0)) -> c10(MARK(x0)) MARK(filter(x0, 0, 0)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, 0)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, s(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), s(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, s(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), s(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, s(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), s(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, s(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), s(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, s(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), s(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, s(0))) -> c10(A__FILTER(mark(x0), mark(x1), s(0)), MARK(x0)) MARK(filter(x0, x1, s(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), s(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), s(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), s(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), s(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), s(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), s(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), s(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, s(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, s(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), s(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), s(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, s(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), s(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, s(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), s(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, s(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), s(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, s(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), s(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, s(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), s(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), s(x2))) -> c10(MARK(x0)) MARK(filter(x0, 0, s(x2))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, s(x2))) -> c10(MARK(s(z0))) MARK(filter(x0, filter(x1, x2, filter(z0, z1, z2)), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__filter(mark(z0), mark(z1), mark(z2))), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, sieve(z0)), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__sieve(mark(z0))), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, nats(z0)), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), 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a__filter(mark(x1), a__nats(mark(z0)), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, zprimes, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), a__zprimes, mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, cons(z0, z1), x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), cons(mark(z0), z1), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, 0, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), 0, mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, s(z0), x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), s(mark(z0)), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(filter(z0, z1, z2), x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(sieve(z0), x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(a__sieve(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(nats(z0), x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(a__nats(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(zprimes, x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(a__zprimes, mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(cons(z0, z1), x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(cons(mark(z0), z1), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(0, x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(0, mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(s(z0), x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(s(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(filter(z0, z1, z2), filter(x1, x2, x3), x4)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), filter(x1, x2, x3), x4)) -> c10(A__FILTER(a__sieve(mark(z0)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(sieve(z0))) MARK(filter(nats(z0), filter(x1, x2, x3), x4)) -> c10(A__FILTER(a__nats(mark(z0)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(nats(z0))) MARK(filter(zprimes, filter(x1, x2, x3), x4)) -> c10(A__FILTER(a__zprimes, a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(zprimes)) MARK(filter(cons(z0, z1), filter(x1, x2, x3), x4)) -> c10(A__FILTER(cons(mark(z0), z1), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(cons(z0, z1))) MARK(filter(x0, filter(x1, x2, x3), x4)) -> c10(MARK(x0)) MARK(filter(s(z0), filter(x1, x2, x3), x4)) -> c10(MARK(s(z0))) S tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, sieve(z0), x2)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, nats(z0), x2)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, zprimes, x2)) -> c10(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x0)) MARK(filter(x0, s(z0), x2)) -> c10(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, x2)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, x2)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, x1, x2)) -> c10(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, x2)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x1)) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), 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c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), 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MARK(filter(x0, nats(z0), 0)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), 0), MARK(x0)) MARK(filter(x0, zprimes, 0)) -> c10(A__FILTER(mark(x0), a__zprimes, 0), MARK(x0)) MARK(filter(x0, s(z0), 0)) -> c10(A__FILTER(mark(x0), s(mark(z0)), 0), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, 0)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), 0), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, 0)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), 0), MARK(sieve(z0))) MARK(filter(nats(z0), x1, 0)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), 0), MARK(nats(z0))) MARK(filter(zprimes, x1, 0)) -> c10(A__FILTER(a__zprimes, mark(x1), 0), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, 0)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), 0), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), 0)) -> c10(MARK(x0)) MARK(filter(x0, 0, 0)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, 0)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, s(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), s(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, s(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), s(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, s(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), s(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, s(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), s(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, s(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), s(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, s(0))) -> c10(A__FILTER(mark(x0), mark(x1), s(0)), MARK(x0)) MARK(filter(x0, x1, s(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), s(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), s(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), s(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), s(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), s(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), s(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), s(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, s(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, s(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), s(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), s(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, s(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), s(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, s(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), s(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, s(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), s(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, s(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), s(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, s(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), s(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), s(x2))) -> c10(MARK(x0)) MARK(filter(x0, 0, s(x2))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, s(x2))) -> c10(MARK(s(z0))) MARK(filter(x0, filter(x1, x2, filter(z0, z1, z2)), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__filter(mark(z0), mark(z1), mark(z2))), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, sieve(z0)), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__sieve(mark(z0))), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, nats(z0)), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__nats(mark(z0))), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, zprimes), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__zprimes), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, cons(z0, z1)), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), cons(mark(z0), z1)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, 0), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), 0), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, s(z0)), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), s(mark(z0))), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, filter(z0, z1, z2), x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), a__filter(mark(z0), mark(z1), mark(z2)), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, sieve(z0), x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), a__sieve(mark(z0)), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, nats(z0), x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), a__nats(mark(z0)), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, zprimes, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), a__zprimes, mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, cons(z0, z1), x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), cons(mark(z0), z1), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, 0, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), 0, mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, s(z0), x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), s(mark(z0)), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(filter(z0, z1, z2), x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(sieve(z0), x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(a__sieve(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(nats(z0), x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(a__nats(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(zprimes, x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(a__zprimes, mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(cons(z0, z1), x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(cons(mark(z0), z1), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(0, x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(0, mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(s(z0), x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(s(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(filter(z0, z1, z2), filter(x1, x2, x3), x4)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), filter(x1, x2, x3), x4)) -> c10(A__FILTER(a__sieve(mark(z0)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(sieve(z0))) MARK(filter(nats(z0), filter(x1, x2, x3), x4)) -> c10(A__FILTER(a__nats(mark(z0)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(nats(z0))) MARK(filter(zprimes, filter(x1, x2, x3), x4)) -> c10(A__FILTER(a__zprimes, a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(zprimes)) MARK(filter(cons(z0, z1), filter(x1, x2, x3), x4)) -> c10(A__FILTER(cons(mark(z0), z1), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(cons(z0, z1))) MARK(filter(x0, filter(x1, x2, x3), x4)) -> c10(MARK(x0)) MARK(filter(s(z0), filter(x1, x2, x3), x4)) -> c10(MARK(s(z0))) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c14_2, c15_1, c16_1, c18_1, c8_1, c10_2, c10_1, c11_2, c11_1, c12_2, c12_1, c13_2, c13_1, c8_2 ---------------------------------------- (101) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(filter(x0, sieve(z0), x2)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x0)) by MARK(filter(x0, sieve(x1), filter(z0, z1, z2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(x1)), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x0)) MARK(filter(x0, sieve(x1), sieve(z0))) -> c10(A__FILTER(mark(x0), a__sieve(mark(x1)), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, sieve(x1), nats(z0))) -> c10(A__FILTER(mark(x0), a__sieve(mark(x1)), a__nats(mark(z0))), MARK(x0)) MARK(filter(x0, sieve(x1), zprimes)) -> c10(A__FILTER(mark(x0), a__sieve(mark(x1)), a__zprimes), MARK(x0)) MARK(filter(x0, sieve(x1), cons(z0, z1))) -> c10(A__FILTER(mark(x0), a__sieve(mark(x1)), cons(mark(z0), z1)), MARK(x0)) MARK(filter(x0, sieve(x1), 0)) -> c10(A__FILTER(mark(x0), a__sieve(mark(x1)), 0), MARK(x0)) MARK(filter(x0, sieve(x1), s(z0))) -> c10(A__FILTER(mark(x0), a__sieve(mark(x1)), s(mark(z0))), MARK(x0)) MARK(filter(x0, sieve(x1), x2)) -> c10(A__FILTER(mark(x0), sieve(mark(x1)), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(filter(z0, z1, z2)), x2)) -> c10(A__FILTER(mark(x0), a__sieve(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(sieve(z0)), x2)) -> c10(A__FILTER(mark(x0), a__sieve(a__sieve(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(nats(z0)), x2)) -> c10(A__FILTER(mark(x0), a__sieve(a__nats(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(zprimes), x2)) -> c10(A__FILTER(mark(x0), a__sieve(a__zprimes), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(cons(z0, z1)), x2)) -> c10(A__FILTER(mark(x0), a__sieve(cons(mark(z0), z1)), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(0), x2)) -> c10(A__FILTER(mark(x0), a__sieve(0), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(s(z0)), x2)) -> c10(A__FILTER(mark(x0), a__sieve(s(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), sieve(x1), x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x1)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), sieve(x1), x2)) -> c10(A__FILTER(a__sieve(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), sieve(x1), x2)) -> c10(A__FILTER(a__nats(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, sieve(x1), x2)) -> c10(A__FILTER(a__zprimes, a__sieve(mark(x1)), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), sieve(x1), x2)) -> c10(A__FILTER(cons(mark(z0), z1), a__sieve(mark(x1)), mark(x2)), MARK(cons(z0, z1))) MARK(filter(0, sieve(x1), x2)) -> c10(A__FILTER(0, a__sieve(mark(x1)), mark(x2)), MARK(0)) MARK(filter(s(z0), sieve(x1), x2)) -> c10(A__FILTER(s(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(s(z0))) ---------------------------------------- (102) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, nats(z0), x2)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, zprimes, x2)) -> c10(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x0)) MARK(filter(x0, s(z0), x2)) -> c10(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, x2)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, x2)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, x1, x2)) -> c10(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, x2)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x1)) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) 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MARK(filter(x0, filter(z0, z1, z2), 0)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), 0), MARK(x0)) MARK(filter(x0, sieve(z0), 0)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), 0), MARK(x0)) MARK(filter(x0, nats(z0), 0)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), 0), MARK(x0)) MARK(filter(x0, zprimes, 0)) -> c10(A__FILTER(mark(x0), a__zprimes, 0), MARK(x0)) MARK(filter(x0, s(z0), 0)) -> c10(A__FILTER(mark(x0), s(mark(z0)), 0), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, 0)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), 0), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, 0)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), 0), MARK(sieve(z0))) MARK(filter(nats(z0), x1, 0)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), 0), MARK(nats(z0))) MARK(filter(zprimes, x1, 0)) -> c10(A__FILTER(a__zprimes, mark(x1), 0), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, 0)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), 0), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), 0)) -> c10(MARK(x0)) MARK(filter(x0, 0, 0)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, 0)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, s(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), s(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, s(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), s(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, s(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), s(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, s(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), s(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, s(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), s(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, s(0))) -> c10(A__FILTER(mark(x0), mark(x1), s(0)), MARK(x0)) MARK(filter(x0, x1, s(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), s(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), s(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), s(mark(x2))), MARK(x0)) 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a__sieve(s(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), sieve(x1), x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x1)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), sieve(x1), x2)) -> c10(A__FILTER(a__sieve(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), sieve(x1), x2)) -> c10(A__FILTER(a__nats(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, sieve(x1), x2)) -> c10(A__FILTER(a__zprimes, a__sieve(mark(x1)), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), sieve(x1), x2)) -> c10(A__FILTER(cons(mark(z0), z1), a__sieve(mark(x1)), mark(x2)), MARK(cons(z0, z1))) MARK(filter(0, sieve(x1), x2)) -> c10(A__FILTER(0, a__sieve(mark(x1)), mark(x2)), MARK(0)) MARK(filter(s(z0), sieve(x1), x2)) -> c10(A__FILTER(s(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(s(z0))) S tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, nats(z0), x2)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, zprimes, x2)) -> c10(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x0)) MARK(filter(x0, s(z0), x2)) -> c10(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, x2)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, x2)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, x1, x2)) -> c10(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, x2)) -> 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x1, cons(x2, x3))) -> c10(MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), 0)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), 0), MARK(x0)) MARK(filter(x0, sieve(z0), 0)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), 0), MARK(x0)) MARK(filter(x0, nats(z0), 0)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), 0), MARK(x0)) MARK(filter(x0, zprimes, 0)) -> c10(A__FILTER(mark(x0), a__zprimes, 0), MARK(x0)) MARK(filter(x0, s(z0), 0)) -> c10(A__FILTER(mark(x0), s(mark(z0)), 0), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, 0)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), 0), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, 0)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), 0), MARK(sieve(z0))) MARK(filter(nats(z0), x1, 0)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), 0), MARK(nats(z0))) MARK(filter(zprimes, x1, 0)) -> c10(A__FILTER(a__zprimes, mark(x1), 0), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, 0)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), 0), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), 0)) -> c10(MARK(x0)) MARK(filter(x0, 0, 0)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, 0)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, s(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), s(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, s(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), s(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, s(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), s(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, s(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), s(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, s(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), s(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, s(0))) -> c10(A__FILTER(mark(x0), mark(x1), s(0)), MARK(x0)) MARK(filter(x0, x1, s(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), s(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), s(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), s(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), s(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), s(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), s(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), s(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, s(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, s(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), s(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), s(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, s(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), s(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, s(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), s(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, s(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), s(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, s(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), s(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, s(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), s(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), s(x2))) -> c10(MARK(x0)) MARK(filter(x0, 0, s(x2))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, s(x2))) -> c10(MARK(s(z0))) MARK(filter(x0, filter(x1, x2, filter(z0, z1, z2)), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__filter(mark(z0), mark(z1), mark(z2))), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, sieve(z0)), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__sieve(mark(z0))), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, nats(z0)), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__nats(mark(z0))), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, zprimes), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__zprimes), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, cons(z0, z1)), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), cons(mark(z0), z1)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, 0), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), 0), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, s(z0)), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), s(mark(z0))), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, filter(z0, z1, z2), x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), a__filter(mark(z0), mark(z1), mark(z2)), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, sieve(z0), x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), a__sieve(mark(z0)), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, nats(z0), x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), a__nats(mark(z0)), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, zprimes, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), a__zprimes, mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, cons(z0, z1), x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), cons(mark(z0), z1), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, 0, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), 0, mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, s(z0), x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), s(mark(z0)), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(filter(z0, z1, z2), x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(sieve(z0), x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(a__sieve(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(nats(z0), x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(a__nats(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(zprimes, x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(a__zprimes, mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(cons(z0, z1), x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(cons(mark(z0), z1), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(0, x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(0, mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(s(z0), x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(s(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(filter(z0, z1, z2), filter(x1, x2, x3), x4)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), filter(x1, x2, x3), x4)) -> c10(A__FILTER(a__sieve(mark(z0)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(sieve(z0))) MARK(filter(nats(z0), filter(x1, x2, x3), x4)) -> c10(A__FILTER(a__nats(mark(z0)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(nats(z0))) MARK(filter(zprimes, filter(x1, x2, x3), x4)) -> c10(A__FILTER(a__zprimes, a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(zprimes)) MARK(filter(cons(z0, z1), filter(x1, x2, x3), x4)) -> c10(A__FILTER(cons(mark(z0), z1), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(cons(z0, z1))) MARK(filter(x0, filter(x1, x2, x3), x4)) -> c10(MARK(x0)) MARK(filter(s(z0), filter(x1, x2, x3), x4)) -> c10(MARK(s(z0))) MARK(filter(x0, sieve(x1), x2)) -> c10(A__FILTER(mark(x0), sieve(mark(x1)), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(filter(z0, z1, z2)), x2)) -> c10(A__FILTER(mark(x0), a__sieve(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(sieve(z0)), x2)) -> c10(A__FILTER(mark(x0), a__sieve(a__sieve(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(nats(z0)), x2)) -> c10(A__FILTER(mark(x0), a__sieve(a__nats(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(zprimes), x2)) -> c10(A__FILTER(mark(x0), a__sieve(a__zprimes), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(cons(z0, z1)), x2)) -> c10(A__FILTER(mark(x0), a__sieve(cons(mark(z0), z1)), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(0), x2)) -> c10(A__FILTER(mark(x0), a__sieve(0), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(s(z0)), x2)) -> c10(A__FILTER(mark(x0), a__sieve(s(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), sieve(x1), x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x1)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), sieve(x1), x2)) -> c10(A__FILTER(a__sieve(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), sieve(x1), x2)) -> c10(A__FILTER(a__nats(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, sieve(x1), x2)) -> c10(A__FILTER(a__zprimes, a__sieve(mark(x1)), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), sieve(x1), x2)) -> c10(A__FILTER(cons(mark(z0), z1), a__sieve(mark(x1)), mark(x2)), MARK(cons(z0, z1))) MARK(filter(0, sieve(x1), x2)) -> c10(A__FILTER(0, a__sieve(mark(x1)), mark(x2)), MARK(0)) MARK(filter(s(z0), sieve(x1), x2)) -> c10(A__FILTER(s(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(s(z0))) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c14_2, c15_1, c16_1, c18_1, c8_1, c10_2, c10_1, c11_2, c11_1, c12_2, c12_1, c13_2, c13_1, c8_2 ---------------------------------------- (103) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(filter(0, sieve(x1), x2)) -> c10(A__FILTER(0, a__sieve(mark(x1)), mark(x2)), MARK(0)) ---------------------------------------- (104) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, nats(z0), x2)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, zprimes, x2)) -> c10(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x0)) MARK(filter(x0, s(z0), x2)) -> c10(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, x2)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, x2)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, x1, x2)) -> c10(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, x2)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x1)) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, 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c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), cons(mark(x2), x3)), MARK(x0)) MARK(filter(x0, sieve(z0), cons(x2, x3))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), cons(mark(x2), x3)), MARK(x0)) MARK(filter(x0, nats(z0), cons(x2, x3))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), cons(mark(x2), x3)), MARK(x0)) MARK(filter(x0, zprimes, cons(x2, x3))) -> c10(A__FILTER(mark(x0), a__zprimes, cons(mark(x2), x3)), MARK(x0)) MARK(filter(x0, s(z0), cons(x2, x3))) -> c10(A__FILTER(mark(x0), s(mark(z0)), cons(mark(x2), x3)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, cons(x2, x3))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), cons(mark(x2), x3)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, cons(x2, x3))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), cons(mark(x2), x3)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, cons(x2, x3))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), cons(mark(x2), x3)), MARK(nats(z0))) MARK(filter(zprimes, x1, cons(x2, x3))) -> c10(A__FILTER(a__zprimes, mark(x1), cons(mark(x2), x3)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, cons(x2, x3))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), cons(mark(x2), x3)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), cons(x2, x3))) -> c10(MARK(x0)) MARK(filter(x0, 0, cons(x2, x3))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, cons(x2, x3))) -> c10(MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), 0)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), 0), MARK(x0)) MARK(filter(x0, sieve(z0), 0)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), 0), MARK(x0)) MARK(filter(x0, nats(z0), 0)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), 0), MARK(x0)) MARK(filter(x0, zprimes, 0)) -> c10(A__FILTER(mark(x0), a__zprimes, 0), MARK(x0)) MARK(filter(x0, s(z0), 0)) -> c10(A__FILTER(mark(x0), s(mark(z0)), 0), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, 0)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), 0), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, 0)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), 0), MARK(sieve(z0))) MARK(filter(nats(z0), x1, 0)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), 0), MARK(nats(z0))) MARK(filter(zprimes, x1, 0)) -> c10(A__FILTER(a__zprimes, mark(x1), 0), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, 0)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), 0), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), 0)) -> c10(MARK(x0)) MARK(filter(x0, 0, 0)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, 0)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, s(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), s(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, s(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), s(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, s(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), s(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, s(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), s(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, s(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), s(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, s(0))) -> c10(A__FILTER(mark(x0), mark(x1), s(0)), MARK(x0)) MARK(filter(x0, x1, s(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), s(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), s(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), s(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), s(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), s(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), s(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), s(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, s(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, s(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), s(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), s(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, s(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), s(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, s(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), s(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, s(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), s(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, s(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), s(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, s(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), s(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), s(x2))) -> c10(MARK(x0)) MARK(filter(x0, 0, s(x2))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, s(x2))) -> c10(MARK(s(z0))) MARK(filter(x0, filter(x1, x2, filter(z0, z1, z2)), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__filter(mark(z0), mark(z1), mark(z2))), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, sieve(z0)), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__sieve(mark(z0))), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, nats(z0)), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__nats(mark(z0))), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, zprimes), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__zprimes), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, cons(z0, z1)), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), cons(mark(z0), z1)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, 0), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), 0), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, s(z0)), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), s(mark(z0))), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, filter(z0, z1, z2), x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), a__filter(mark(z0), mark(z1), mark(z2)), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, sieve(z0), x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), a__sieve(mark(z0)), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, nats(z0), x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), a__nats(mark(z0)), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, zprimes, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), a__zprimes, mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, cons(z0, z1), x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), cons(mark(z0), z1), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, 0, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), 0, mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, s(z0), x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), s(mark(z0)), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(filter(z0, z1, z2), x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(sieve(z0), x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(a__sieve(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(nats(z0), x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(a__nats(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(zprimes, x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(a__zprimes, mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(cons(z0, z1), x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(cons(mark(z0), z1), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(0, x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(0, mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(s(z0), x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(s(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(filter(z0, z1, z2), filter(x1, x2, x3), x4)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), filter(x1, x2, x3), x4)) -> c10(A__FILTER(a__sieve(mark(z0)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(sieve(z0))) MARK(filter(nats(z0), filter(x1, x2, x3), x4)) -> c10(A__FILTER(a__nats(mark(z0)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(nats(z0))) MARK(filter(zprimes, filter(x1, x2, x3), x4)) -> c10(A__FILTER(a__zprimes, a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(zprimes)) MARK(filter(cons(z0, z1), filter(x1, x2, x3), x4)) -> c10(A__FILTER(cons(mark(z0), z1), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(cons(z0, z1))) MARK(filter(x0, filter(x1, x2, x3), x4)) -> c10(MARK(x0)) MARK(filter(s(z0), filter(x1, x2, x3), x4)) -> c10(MARK(s(z0))) MARK(filter(x0, sieve(x1), x2)) -> c10(A__FILTER(mark(x0), sieve(mark(x1)), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(filter(z0, z1, z2)), x2)) -> c10(A__FILTER(mark(x0), a__sieve(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(sieve(z0)), x2)) -> c10(A__FILTER(mark(x0), a__sieve(a__sieve(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(nats(z0)), x2)) -> c10(A__FILTER(mark(x0), a__sieve(a__nats(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(zprimes), x2)) -> c10(A__FILTER(mark(x0), a__sieve(a__zprimes), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(cons(z0, z1)), x2)) -> c10(A__FILTER(mark(x0), a__sieve(cons(mark(z0), z1)), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(0), x2)) -> c10(A__FILTER(mark(x0), a__sieve(0), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(s(z0)), x2)) -> c10(A__FILTER(mark(x0), a__sieve(s(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), sieve(x1), x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x1)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), sieve(x1), x2)) -> c10(A__FILTER(a__sieve(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), sieve(x1), x2)) -> c10(A__FILTER(a__nats(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, sieve(x1), x2)) -> c10(A__FILTER(a__zprimes, a__sieve(mark(x1)), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), sieve(x1), x2)) -> c10(A__FILTER(cons(mark(z0), z1), a__sieve(mark(x1)), mark(x2)), MARK(cons(z0, z1))) MARK(filter(s(z0), sieve(x1), x2)) -> c10(A__FILTER(s(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(s(z0))) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c14_2, c15_1, c16_1, c18_1, c8_1, c10_2, c10_1, c11_2, c11_1, c12_2, c12_1, c13_2, c13_1, c8_2 ---------------------------------------- (105) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (106) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, nats(z0), x2)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(x0, zprimes, x2)) -> c10(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x0)) MARK(filter(x0, 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z1, z2))) MARK(filter(sieve(z0), x1, nats(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, nats(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, nats(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), a__nats(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, nats(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__nats(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), nats(x2))) -> c10(MARK(x0)) MARK(filter(x0, 0, nats(x2))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, nats(x2))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, zprimes)) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(s(s(0))))), MARK(x0)) MARK(filter(x0, x1, zprimes)) -> c10(A__FILTER(mark(x0), mark(x1), zprimes), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), zprimes)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__zprimes), MARK(x0)) MARK(filter(x0, 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MARK(filter(x0, cons(z0, z1), zprimes)) -> c10(MARK(x0)) MARK(filter(x0, 0, zprimes)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, zprimes)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, cons(filter(z0, z1, z2), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(a__filter(mark(z0), mark(z1), mark(z2)), x3)), MARK(x0)) MARK(filter(x0, x1, cons(sieve(z0), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(a__sieve(mark(z0)), x3)), MARK(x0)) MARK(filter(x0, x1, cons(nats(z0), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(a__nats(mark(z0)), x3)), MARK(x0)) MARK(filter(x0, x1, cons(zprimes, x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(a__zprimes, x3)), MARK(x0)) MARK(filter(x0, x1, cons(cons(z0, z1), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(cons(mark(z0), z1), x3)), MARK(x0)) MARK(filter(x0, x1, cons(0, x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(0, x3)), MARK(x0)) MARK(filter(x0, x1, cons(s(z0), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(s(mark(z0)), x3)), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), cons(x2, x3))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), cons(mark(x2), x3)), MARK(x0)) MARK(filter(x0, sieve(z0), cons(x2, x3))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), cons(mark(x2), x3)), MARK(x0)) MARK(filter(x0, nats(z0), cons(x2, x3))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), cons(mark(x2), x3)), MARK(x0)) MARK(filter(x0, zprimes, cons(x2, x3))) -> c10(A__FILTER(mark(x0), a__zprimes, cons(mark(x2), x3)), MARK(x0)) MARK(filter(x0, s(z0), cons(x2, x3))) -> c10(A__FILTER(mark(x0), s(mark(z0)), cons(mark(x2), x3)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, cons(x2, x3))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), cons(mark(x2), x3)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, cons(x2, x3))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), cons(mark(x2), x3)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, cons(x2, x3))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), cons(mark(x2), x3)), MARK(nats(z0))) MARK(filter(zprimes, x1, cons(x2, x3))) -> c10(A__FILTER(a__zprimes, mark(x1), cons(mark(x2), x3)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, cons(x2, x3))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), cons(mark(x2), x3)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), cons(x2, x3))) -> c10(MARK(x0)) MARK(filter(x0, 0, cons(x2, x3))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, cons(x2, x3))) -> c10(MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), 0)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), 0), MARK(x0)) MARK(filter(x0, sieve(z0), 0)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), 0), MARK(x0)) MARK(filter(x0, nats(z0), 0)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), 0), MARK(x0)) MARK(filter(x0, zprimes, 0)) -> c10(A__FILTER(mark(x0), a__zprimes, 0), MARK(x0)) MARK(filter(x0, s(z0), 0)) -> c10(A__FILTER(mark(x0), s(mark(z0)), 0), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, 0)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), 0), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, 0)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), 0), MARK(sieve(z0))) MARK(filter(nats(z0), x1, 0)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), 0), MARK(nats(z0))) MARK(filter(zprimes, x1, 0)) -> c10(A__FILTER(a__zprimes, mark(x1), 0), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, 0)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), 0), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), 0)) -> c10(MARK(x0)) MARK(filter(x0, 0, 0)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, 0)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, s(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), s(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, s(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), s(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, s(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), s(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, s(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), s(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, s(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), s(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, s(0))) -> c10(A__FILTER(mark(x0), mark(x1), s(0)), MARK(x0)) MARK(filter(x0, x1, s(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), s(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), s(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), s(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), s(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), s(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), s(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), s(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, s(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, s(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), s(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), s(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, s(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), s(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, s(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), s(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, s(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), s(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, s(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), s(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, s(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), s(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), s(x2))) -> c10(MARK(x0)) MARK(filter(x0, 0, s(x2))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, s(x2))) -> c10(MARK(s(z0))) MARK(filter(x0, filter(x1, x2, filter(z0, z1, z2)), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__filter(mark(z0), mark(z1), mark(z2))), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, sieve(z0)), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__sieve(mark(z0))), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, nats(z0)), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__nats(mark(z0))), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, zprimes), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__zprimes), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, cons(z0, z1)), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), cons(mark(z0), z1)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, 0), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), 0), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, s(z0)), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), s(mark(z0))), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, filter(z0, z1, z2), x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), a__filter(mark(z0), mark(z1), mark(z2)), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, sieve(z0), x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), a__sieve(mark(z0)), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, nats(z0), x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), a__nats(mark(z0)), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, zprimes, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), a__zprimes, mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, cons(z0, z1), x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), cons(mark(z0), z1), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, 0, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), 0, mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, s(z0), x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), s(mark(z0)), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(filter(z0, z1, z2), x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(sieve(z0), x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(a__sieve(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(nats(z0), x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(a__nats(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(zprimes, x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(a__zprimes, mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(cons(z0, z1), x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(cons(mark(z0), z1), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(0, x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(0, mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(s(z0), x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(s(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(filter(z0, z1, z2), filter(x1, x2, x3), x4)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), filter(x1, x2, x3), x4)) -> c10(A__FILTER(a__sieve(mark(z0)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(sieve(z0))) MARK(filter(nats(z0), filter(x1, x2, x3), x4)) -> c10(A__FILTER(a__nats(mark(z0)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(nats(z0))) MARK(filter(zprimes, filter(x1, x2, x3), x4)) -> c10(A__FILTER(a__zprimes, a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(zprimes)) MARK(filter(cons(z0, z1), filter(x1, x2, x3), x4)) -> c10(A__FILTER(cons(mark(z0), z1), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(cons(z0, z1))) MARK(filter(x0, filter(x1, x2, x3), x4)) -> c10(MARK(x0)) MARK(filter(s(z0), filter(x1, x2, x3), x4)) -> c10(MARK(s(z0))) MARK(filter(x0, sieve(filter(z0, z1, z2)), x2)) -> c10(A__FILTER(mark(x0), a__sieve(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(sieve(z0)), x2)) -> c10(A__FILTER(mark(x0), a__sieve(a__sieve(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(nats(z0)), x2)) -> c10(A__FILTER(mark(x0), a__sieve(a__nats(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(zprimes), x2)) -> c10(A__FILTER(mark(x0), a__sieve(a__zprimes), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(cons(z0, z1)), x2)) -> c10(A__FILTER(mark(x0), a__sieve(cons(mark(z0), z1)), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(0), x2)) -> c10(A__FILTER(mark(x0), a__sieve(0), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(s(z0)), x2)) -> c10(A__FILTER(mark(x0), a__sieve(s(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), sieve(x1), x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x1)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), sieve(x1), x2)) -> c10(A__FILTER(a__sieve(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), sieve(x1), x2)) -> c10(A__FILTER(a__nats(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, sieve(x1), x2)) -> c10(A__FILTER(a__zprimes, a__sieve(mark(x1)), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), sieve(x1), x2)) -> c10(A__FILTER(cons(mark(z0), z1), a__sieve(mark(x1)), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, sieve(x1), x2)) -> c10(MARK(x0)) MARK(filter(s(z0), sieve(x1), x2)) -> c10(MARK(s(z0))) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c14_2, c15_1, c16_1, c18_1, c8_1, c10_2, c10_1, c11_2, c11_1, c12_2, c12_1, c13_2, c13_1, c8_2 ---------------------------------------- (107) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(filter(x0, nats(z0), x2)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x0)) by MARK(filter(x0, nats(x1), filter(z0, z1, z2))) -> c10(A__FILTER(mark(x0), a__nats(mark(x1)), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x0)) MARK(filter(x0, nats(x1), sieve(z0))) -> c10(A__FILTER(mark(x0), a__nats(mark(x1)), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, nats(x1), nats(z0))) -> c10(A__FILTER(mark(x0), a__nats(mark(x1)), a__nats(mark(z0))), MARK(x0)) MARK(filter(x0, nats(x1), zprimes)) -> c10(A__FILTER(mark(x0), a__nats(mark(x1)), a__zprimes), MARK(x0)) MARK(filter(x0, nats(x1), cons(z0, z1))) -> c10(A__FILTER(mark(x0), a__nats(mark(x1)), cons(mark(z0), z1)), MARK(x0)) MARK(filter(x0, nats(x1), 0)) -> c10(A__FILTER(mark(x0), a__nats(mark(x1)), 0), MARK(x0)) MARK(filter(x0, nats(x1), s(z0))) -> c10(A__FILTER(mark(x0), a__nats(mark(x1)), s(mark(z0))), MARK(x0)) MARK(filter(x0, nats(x1), x2)) -> c10(A__FILTER(mark(x0), cons(mark(mark(x1)), nats(s(mark(x1)))), mark(x2)), MARK(x0)) MARK(filter(x0, nats(x1), x2)) -> c10(A__FILTER(mark(x0), nats(mark(x1)), mark(x2)), MARK(x0)) MARK(filter(x0, nats(filter(z0, z1, z2)), x2)) -> c10(A__FILTER(mark(x0), a__nats(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(x0)) MARK(filter(x0, nats(sieve(z0)), x2)) -> c10(A__FILTER(mark(x0), a__nats(a__sieve(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, nats(nats(z0)), x2)) -> c10(A__FILTER(mark(x0), a__nats(a__nats(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, nats(zprimes), x2)) -> c10(A__FILTER(mark(x0), a__nats(a__zprimes), mark(x2)), MARK(x0)) MARK(filter(x0, nats(cons(z0, z1)), x2)) -> c10(A__FILTER(mark(x0), a__nats(cons(mark(z0), z1)), mark(x2)), MARK(x0)) MARK(filter(x0, nats(0), x2)) -> c10(A__FILTER(mark(x0), a__nats(0), mark(x2)), MARK(x0)) MARK(filter(x0, nats(s(z0)), x2)) -> c10(A__FILTER(mark(x0), a__nats(s(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), nats(x1), x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x1)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), nats(x1), x2)) -> c10(A__FILTER(a__sieve(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), nats(x1), x2)) -> c10(A__FILTER(a__nats(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, nats(x1), x2)) -> c10(A__FILTER(a__zprimes, a__nats(mark(x1)), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), nats(x1), x2)) -> c10(A__FILTER(cons(mark(z0), z1), a__nats(mark(x1)), mark(x2)), MARK(cons(z0, z1))) MARK(filter(0, nats(x1), x2)) -> c10(A__FILTER(0, a__nats(mark(x1)), mark(x2)), MARK(0)) MARK(filter(s(z0), nats(x1), x2)) -> c10(A__FILTER(s(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(s(z0))) ---------------------------------------- (108) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, zprimes, x2)) -> c10(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x0)) MARK(filter(x0, s(z0), x2)) -> c10(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, x2)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, x2)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, x1, x2)) -> c10(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, x2)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x1)) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(x0, 0, filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, sieve(x2))) -> c10(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x0)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, sieve(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, sieve(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, sieve(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(0)), MARK(x0)) MARK(filter(x0, x1, sieve(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), sieve(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, sieve(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, sieve(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__sieve(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, sieve(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, sieve(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, sieve(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), a__sieve(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, sieve(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__sieve(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), sieve(x2))) -> c10(MARK(x0)) MARK(filter(x0, 0, sieve(x2))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, sieve(x2))) -> c10(MARK(s(z0))) MARK(filter(x0, 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MARK(filter(x0, sieve(s(z0)), x2)) -> c10(A__FILTER(mark(x0), a__sieve(s(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), sieve(x1), x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x1)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), sieve(x1), x2)) -> c10(A__FILTER(a__sieve(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), sieve(x1), x2)) -> c10(A__FILTER(a__nats(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, sieve(x1), x2)) -> c10(A__FILTER(a__zprimes, a__sieve(mark(x1)), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), sieve(x1), x2)) -> c10(A__FILTER(cons(mark(z0), z1), a__sieve(mark(x1)), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, sieve(x1), x2)) -> c10(MARK(x0)) MARK(filter(s(z0), sieve(x1), x2)) -> c10(MARK(s(z0))) MARK(filter(x0, nats(x1), x2)) -> c10(A__FILTER(mark(x0), cons(mark(mark(x1)), nats(s(mark(x1)))), mark(x2)), MARK(x0)) MARK(filter(x0, nats(x1), x2)) -> c10(A__FILTER(mark(x0), nats(mark(x1)), mark(x2)), MARK(x0)) MARK(filter(x0, nats(filter(z0, z1, z2)), x2)) -> c10(A__FILTER(mark(x0), a__nats(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(x0)) MARK(filter(x0, nats(sieve(z0)), x2)) -> c10(A__FILTER(mark(x0), a__nats(a__sieve(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, nats(nats(z0)), x2)) -> c10(A__FILTER(mark(x0), a__nats(a__nats(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, nats(zprimes), x2)) -> c10(A__FILTER(mark(x0), a__nats(a__zprimes), mark(x2)), MARK(x0)) MARK(filter(x0, nats(cons(z0, z1)), x2)) -> c10(A__FILTER(mark(x0), a__nats(cons(mark(z0), z1)), mark(x2)), MARK(x0)) MARK(filter(x0, nats(0), x2)) -> c10(A__FILTER(mark(x0), a__nats(0), mark(x2)), MARK(x0)) MARK(filter(x0, nats(s(z0)), x2)) -> c10(A__FILTER(mark(x0), a__nats(s(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), nats(x1), x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x1)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), nats(x1), x2)) -> c10(A__FILTER(a__sieve(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), nats(x1), x2)) -> c10(A__FILTER(a__nats(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, nats(x1), x2)) -> c10(A__FILTER(a__zprimes, a__nats(mark(x1)), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), nats(x1), x2)) -> c10(A__FILTER(cons(mark(z0), z1), a__nats(mark(x1)), mark(x2)), MARK(cons(z0, z1))) MARK(filter(0, nats(x1), x2)) -> c10(A__FILTER(0, a__nats(mark(x1)), mark(x2)), MARK(0)) MARK(filter(s(z0), nats(x1), x2)) -> c10(A__FILTER(s(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(s(z0))) S tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, zprimes, x2)) -> c10(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x0)) MARK(filter(x0, s(z0), x2)) -> c10(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, x2)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, x2)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, x1, x2)) -> c10(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, x2)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x1)) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, 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c10(A__FILTER(mark(x0), a__filter(mark(x1), a__filter(mark(z0), mark(z1), mark(z2)), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, sieve(z0), x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), a__sieve(mark(z0)), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, nats(z0), x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), a__nats(mark(z0)), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, zprimes, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), a__zprimes, mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, cons(z0, z1), x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), cons(mark(z0), z1), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, 0, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), 0, mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, s(z0), x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), s(mark(z0)), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(filter(z0, z1, z2), x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(sieve(z0), x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(a__sieve(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(nats(z0), x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(a__nats(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(zprimes, x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(a__zprimes, mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(cons(z0, z1), x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(cons(mark(z0), z1), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(0, x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(0, mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(s(z0), x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(s(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(filter(z0, z1, z2), filter(x1, x2, x3), x4)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), filter(x1, x2, x3), x4)) -> c10(A__FILTER(a__sieve(mark(z0)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(sieve(z0))) MARK(filter(nats(z0), filter(x1, x2, x3), x4)) -> c10(A__FILTER(a__nats(mark(z0)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(nats(z0))) MARK(filter(zprimes, filter(x1, x2, x3), x4)) -> c10(A__FILTER(a__zprimes, a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(zprimes)) MARK(filter(cons(z0, z1), filter(x1, x2, x3), x4)) -> c10(A__FILTER(cons(mark(z0), z1), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(cons(z0, z1))) MARK(filter(x0, filter(x1, x2, x3), x4)) -> c10(MARK(x0)) MARK(filter(s(z0), filter(x1, x2, x3), x4)) -> c10(MARK(s(z0))) MARK(filter(x0, sieve(filter(z0, z1, z2)), x2)) -> c10(A__FILTER(mark(x0), a__sieve(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(sieve(z0)), x2)) -> c10(A__FILTER(mark(x0), a__sieve(a__sieve(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(nats(z0)), x2)) -> c10(A__FILTER(mark(x0), a__sieve(a__nats(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(zprimes), x2)) -> c10(A__FILTER(mark(x0), a__sieve(a__zprimes), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(cons(z0, z1)), x2)) -> c10(A__FILTER(mark(x0), a__sieve(cons(mark(z0), z1)), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(0), x2)) -> c10(A__FILTER(mark(x0), a__sieve(0), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(s(z0)), x2)) -> c10(A__FILTER(mark(x0), a__sieve(s(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), sieve(x1), x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x1)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), sieve(x1), x2)) -> c10(A__FILTER(a__sieve(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), sieve(x1), x2)) -> c10(A__FILTER(a__nats(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, sieve(x1), x2)) -> c10(A__FILTER(a__zprimes, a__sieve(mark(x1)), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), sieve(x1), x2)) -> c10(A__FILTER(cons(mark(z0), z1), a__sieve(mark(x1)), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, sieve(x1), x2)) -> c10(MARK(x0)) MARK(filter(s(z0), sieve(x1), x2)) -> c10(MARK(s(z0))) MARK(filter(x0, nats(x1), x2)) -> c10(A__FILTER(mark(x0), cons(mark(mark(x1)), nats(s(mark(x1)))), mark(x2)), MARK(x0)) MARK(filter(x0, nats(x1), x2)) -> c10(A__FILTER(mark(x0), nats(mark(x1)), mark(x2)), MARK(x0)) MARK(filter(x0, nats(filter(z0, z1, z2)), x2)) -> c10(A__FILTER(mark(x0), a__nats(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(x0)) MARK(filter(x0, nats(sieve(z0)), x2)) -> c10(A__FILTER(mark(x0), a__nats(a__sieve(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, nats(nats(z0)), x2)) -> c10(A__FILTER(mark(x0), a__nats(a__nats(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, nats(zprimes), x2)) -> c10(A__FILTER(mark(x0), a__nats(a__zprimes), mark(x2)), MARK(x0)) MARK(filter(x0, nats(cons(z0, z1)), x2)) -> c10(A__FILTER(mark(x0), a__nats(cons(mark(z0), z1)), mark(x2)), MARK(x0)) MARK(filter(x0, nats(0), x2)) -> c10(A__FILTER(mark(x0), a__nats(0), mark(x2)), MARK(x0)) MARK(filter(x0, nats(s(z0)), x2)) -> c10(A__FILTER(mark(x0), a__nats(s(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), nats(x1), x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x1)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), nats(x1), x2)) -> c10(A__FILTER(a__sieve(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), nats(x1), x2)) -> c10(A__FILTER(a__nats(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, nats(x1), x2)) -> c10(A__FILTER(a__zprimes, a__nats(mark(x1)), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), nats(x1), x2)) -> c10(A__FILTER(cons(mark(z0), z1), a__nats(mark(x1)), mark(x2)), MARK(cons(z0, z1))) MARK(filter(0, nats(x1), x2)) -> c10(A__FILTER(0, a__nats(mark(x1)), mark(x2)), MARK(0)) MARK(filter(s(z0), nats(x1), x2)) -> c10(A__FILTER(s(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(s(z0))) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c14_2, c15_1, c16_1, c18_1, c8_1, c10_2, c10_1, c11_2, c11_1, c12_2, c12_1, c13_2, c13_1, c8_2 ---------------------------------------- (109) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(filter(0, nats(x1), x2)) -> c10(A__FILTER(0, a__nats(mark(x1)), mark(x2)), MARK(0)) ---------------------------------------- (110) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, zprimes, x2)) -> c10(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x0)) MARK(filter(x0, s(z0), x2)) -> c10(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, x2)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, x2)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, x1, x2)) -> c10(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, x2)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x1)) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), 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MARK(zprimes)) MARK(filter(cons(z0, z1), nats(x1), x2)) -> c10(A__FILTER(cons(mark(z0), z1), a__nats(mark(x1)), mark(x2)), MARK(cons(z0, z1))) MARK(filter(s(z0), nats(x1), x2)) -> c10(A__FILTER(s(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(s(z0))) S tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, zprimes, x2)) -> c10(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x0)) MARK(filter(x0, s(z0), x2)) -> c10(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, x2)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, x2)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, x1, x2)) -> c10(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, x2)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x1)) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> 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MARK(filter(cons(z0, z1), x1, cons(x2, x3))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), cons(mark(x2), x3)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), cons(x2, x3))) -> c10(MARK(x0)) MARK(filter(x0, 0, cons(x2, x3))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, cons(x2, x3))) -> c10(MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), 0)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), 0), MARK(x0)) MARK(filter(x0, sieve(z0), 0)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), 0), MARK(x0)) MARK(filter(x0, nats(z0), 0)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), 0), MARK(x0)) MARK(filter(x0, zprimes, 0)) -> c10(A__FILTER(mark(x0), a__zprimes, 0), MARK(x0)) MARK(filter(x0, s(z0), 0)) -> c10(A__FILTER(mark(x0), s(mark(z0)), 0), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, 0)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), 0), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, 0)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), 0), MARK(sieve(z0))) MARK(filter(nats(z0), x1, 0)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), 0), MARK(nats(z0))) MARK(filter(zprimes, x1, 0)) -> c10(A__FILTER(a__zprimes, mark(x1), 0), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, 0)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), 0), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), 0)) -> c10(MARK(x0)) MARK(filter(x0, 0, 0)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, 0)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, s(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), s(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, s(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), s(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, s(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), s(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, s(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), s(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, s(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), s(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, s(0))) -> c10(A__FILTER(mark(x0), mark(x1), s(0)), MARK(x0)) MARK(filter(x0, x1, s(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), s(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), s(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), s(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), s(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), s(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), s(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), s(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, s(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, s(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), s(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), s(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, s(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), s(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, s(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), s(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, s(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), s(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, s(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), s(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, s(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), s(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), s(x2))) -> c10(MARK(x0)) MARK(filter(x0, 0, s(x2))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, s(x2))) -> c10(MARK(s(z0))) MARK(filter(x0, filter(x1, x2, filter(z0, z1, z2)), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__filter(mark(z0), mark(z1), mark(z2))), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, sieve(z0)), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__sieve(mark(z0))), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, nats(z0)), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__nats(mark(z0))), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, zprimes), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__zprimes), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, cons(z0, z1)), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), cons(mark(z0), z1)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, 0), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), 0), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, s(z0)), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), s(mark(z0))), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, filter(z0, z1, z2), x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), a__filter(mark(z0), mark(z1), mark(z2)), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, sieve(z0), x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), a__sieve(mark(z0)), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, nats(z0), x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), a__nats(mark(z0)), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, zprimes, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), a__zprimes, mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, cons(z0, z1), x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), cons(mark(z0), z1), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, 0, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), 0, mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, s(z0), x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), s(mark(z0)), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(filter(z0, z1, z2), x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(sieve(z0), x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(a__sieve(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(nats(z0), x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(a__nats(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(zprimes, x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(a__zprimes, mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(cons(z0, z1), x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(cons(mark(z0), z1), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(0, x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(0, mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(s(z0), x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(s(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(filter(z0, z1, z2), filter(x1, x2, x3), x4)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), filter(x1, x2, x3), x4)) -> c10(A__FILTER(a__sieve(mark(z0)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(sieve(z0))) MARK(filter(nats(z0), filter(x1, x2, x3), x4)) -> c10(A__FILTER(a__nats(mark(z0)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(nats(z0))) MARK(filter(zprimes, filter(x1, x2, x3), x4)) -> c10(A__FILTER(a__zprimes, a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(zprimes)) MARK(filter(cons(z0, z1), filter(x1, x2, x3), x4)) -> c10(A__FILTER(cons(mark(z0), z1), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(cons(z0, z1))) MARK(filter(x0, filter(x1, x2, x3), x4)) -> c10(MARK(x0)) MARK(filter(s(z0), filter(x1, x2, x3), x4)) -> c10(MARK(s(z0))) MARK(filter(x0, sieve(filter(z0, z1, z2)), x2)) -> c10(A__FILTER(mark(x0), a__sieve(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(sieve(z0)), x2)) -> c10(A__FILTER(mark(x0), a__sieve(a__sieve(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(nats(z0)), x2)) -> c10(A__FILTER(mark(x0), a__sieve(a__nats(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(zprimes), x2)) -> c10(A__FILTER(mark(x0), a__sieve(a__zprimes), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(cons(z0, z1)), x2)) -> c10(A__FILTER(mark(x0), a__sieve(cons(mark(z0), z1)), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(0), x2)) -> c10(A__FILTER(mark(x0), a__sieve(0), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(s(z0)), x2)) -> c10(A__FILTER(mark(x0), a__sieve(s(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), sieve(x1), x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x1)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), sieve(x1), x2)) -> c10(A__FILTER(a__sieve(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), sieve(x1), x2)) -> c10(A__FILTER(a__nats(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, sieve(x1), x2)) -> c10(A__FILTER(a__zprimes, a__sieve(mark(x1)), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), sieve(x1), x2)) -> c10(A__FILTER(cons(mark(z0), z1), a__sieve(mark(x1)), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, sieve(x1), x2)) -> c10(MARK(x0)) MARK(filter(s(z0), sieve(x1), x2)) -> c10(MARK(s(z0))) MARK(filter(x0, nats(x1), x2)) -> c10(A__FILTER(mark(x0), cons(mark(mark(x1)), nats(s(mark(x1)))), mark(x2)), MARK(x0)) MARK(filter(x0, nats(x1), x2)) -> c10(A__FILTER(mark(x0), nats(mark(x1)), mark(x2)), MARK(x0)) MARK(filter(x0, nats(filter(z0, z1, z2)), x2)) -> c10(A__FILTER(mark(x0), a__nats(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(x0)) MARK(filter(x0, nats(sieve(z0)), x2)) -> c10(A__FILTER(mark(x0), a__nats(a__sieve(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, nats(nats(z0)), x2)) -> c10(A__FILTER(mark(x0), a__nats(a__nats(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, nats(zprimes), x2)) -> c10(A__FILTER(mark(x0), a__nats(a__zprimes), mark(x2)), MARK(x0)) MARK(filter(x0, nats(cons(z0, z1)), x2)) -> c10(A__FILTER(mark(x0), a__nats(cons(mark(z0), z1)), mark(x2)), MARK(x0)) MARK(filter(x0, nats(0), x2)) -> c10(A__FILTER(mark(x0), a__nats(0), mark(x2)), MARK(x0)) MARK(filter(x0, nats(s(z0)), x2)) -> c10(A__FILTER(mark(x0), a__nats(s(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), nats(x1), x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x1)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), nats(x1), x2)) -> c10(A__FILTER(a__sieve(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), nats(x1), x2)) -> c10(A__FILTER(a__nats(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, nats(x1), x2)) -> c10(A__FILTER(a__zprimes, a__nats(mark(x1)), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), nats(x1), x2)) -> c10(A__FILTER(cons(mark(z0), z1), a__nats(mark(x1)), mark(x2)), MARK(cons(z0, z1))) MARK(filter(s(z0), nats(x1), x2)) -> c10(A__FILTER(s(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(s(z0))) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c14_2, c15_1, c16_1, c18_1, c8_1, c10_2, c10_1, c11_2, c11_1, c12_2, c12_1, c13_2, c13_1, c8_2 ---------------------------------------- (111) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (112) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, zprimes, x2)) -> c10(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x0)) MARK(filter(x0, s(z0), x2)) -> c10(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, x2)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, x2)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, x1, x2)) -> c10(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, x2)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x1)) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), 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MARK(filter(s(z0), x1, zprimes)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, cons(filter(z0, z1, z2), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(a__filter(mark(z0), mark(z1), mark(z2)), x3)), MARK(x0)) MARK(filter(x0, x1, cons(sieve(z0), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(a__sieve(mark(z0)), x3)), MARK(x0)) MARK(filter(x0, x1, cons(nats(z0), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(a__nats(mark(z0)), x3)), MARK(x0)) MARK(filter(x0, x1, cons(zprimes, x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(a__zprimes, x3)), MARK(x0)) MARK(filter(x0, x1, cons(cons(z0, z1), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(cons(mark(z0), z1), x3)), MARK(x0)) MARK(filter(x0, x1, cons(0, x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(0, x3)), MARK(x0)) MARK(filter(x0, x1, cons(s(z0), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(s(mark(z0)), x3)), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), cons(x2, x3))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), cons(mark(x2), x3)), MARK(x0)) MARK(filter(x0, sieve(z0), cons(x2, x3))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), cons(mark(x2), x3)), MARK(x0)) MARK(filter(x0, nats(z0), cons(x2, x3))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), cons(mark(x2), x3)), MARK(x0)) MARK(filter(x0, zprimes, cons(x2, x3))) -> c10(A__FILTER(mark(x0), a__zprimes, cons(mark(x2), x3)), MARK(x0)) MARK(filter(x0, s(z0), cons(x2, x3))) -> c10(A__FILTER(mark(x0), s(mark(z0)), cons(mark(x2), x3)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, cons(x2, x3))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), cons(mark(x2), x3)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, cons(x2, x3))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), cons(mark(x2), x3)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, cons(x2, x3))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), cons(mark(x2), x3)), MARK(nats(z0))) MARK(filter(zprimes, x1, cons(x2, x3))) -> c10(A__FILTER(a__zprimes, mark(x1), cons(mark(x2), x3)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, cons(x2, x3))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), cons(mark(x2), x3)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), cons(x2, x3))) -> c10(MARK(x0)) MARK(filter(x0, 0, cons(x2, x3))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, cons(x2, x3))) -> c10(MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), 0)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), 0), MARK(x0)) MARK(filter(x0, sieve(z0), 0)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), 0), MARK(x0)) MARK(filter(x0, nats(z0), 0)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), 0), MARK(x0)) MARK(filter(x0, zprimes, 0)) -> c10(A__FILTER(mark(x0), a__zprimes, 0), MARK(x0)) MARK(filter(x0, s(z0), 0)) -> c10(A__FILTER(mark(x0), s(mark(z0)), 0), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, 0)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), 0), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, 0)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), 0), MARK(sieve(z0))) MARK(filter(nats(z0), x1, 0)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), 0), MARK(nats(z0))) MARK(filter(zprimes, x1, 0)) -> c10(A__FILTER(a__zprimes, mark(x1), 0), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, 0)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), 0), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), 0)) -> c10(MARK(x0)) MARK(filter(x0, 0, 0)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, 0)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, s(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), s(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, s(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), s(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, s(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), s(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, s(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), s(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, s(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), s(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, s(0))) -> c10(A__FILTER(mark(x0), mark(x1), s(0)), MARK(x0)) MARK(filter(x0, x1, s(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), s(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), s(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), s(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), s(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), s(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), s(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), s(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, s(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, s(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), s(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), s(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, s(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), s(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, s(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), s(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, s(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), s(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, s(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), s(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, s(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), s(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), s(x2))) -> c10(MARK(x0)) MARK(filter(x0, 0, s(x2))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, s(x2))) -> c10(MARK(s(z0))) MARK(filter(x0, filter(x1, x2, filter(z0, z1, z2)), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__filter(mark(z0), mark(z1), mark(z2))), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, sieve(z0)), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__sieve(mark(z0))), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, nats(z0)), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__nats(mark(z0))), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, zprimes), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__zprimes), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, cons(z0, z1)), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), cons(mark(z0), z1)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, 0), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), 0), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, s(z0)), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), s(mark(z0))), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, filter(z0, z1, z2), x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), a__filter(mark(z0), mark(z1), mark(z2)), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, sieve(z0), x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), a__sieve(mark(z0)), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, nats(z0), x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), a__nats(mark(z0)), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, zprimes, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), a__zprimes, mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, cons(z0, z1), x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), cons(mark(z0), z1), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, 0, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), 0, mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, s(z0), x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), s(mark(z0)), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(filter(z0, z1, z2), x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(sieve(z0), x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(a__sieve(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(nats(z0), x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(a__nats(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(zprimes, x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(a__zprimes, mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(cons(z0, z1), x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(cons(mark(z0), z1), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(0, x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(0, mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(s(z0), x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(s(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(filter(z0, z1, z2), filter(x1, x2, x3), x4)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), filter(x1, x2, x3), x4)) -> c10(A__FILTER(a__sieve(mark(z0)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(sieve(z0))) MARK(filter(nats(z0), filter(x1, x2, x3), x4)) -> c10(A__FILTER(a__nats(mark(z0)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(nats(z0))) MARK(filter(zprimes, filter(x1, x2, x3), x4)) -> c10(A__FILTER(a__zprimes, a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(zprimes)) MARK(filter(cons(z0, z1), filter(x1, x2, x3), x4)) -> c10(A__FILTER(cons(mark(z0), z1), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(cons(z0, z1))) MARK(filter(x0, filter(x1, x2, x3), x4)) -> c10(MARK(x0)) MARK(filter(s(z0), filter(x1, x2, x3), x4)) -> c10(MARK(s(z0))) MARK(filter(x0, sieve(filter(z0, z1, z2)), x2)) -> c10(A__FILTER(mark(x0), a__sieve(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(sieve(z0)), x2)) -> c10(A__FILTER(mark(x0), a__sieve(a__sieve(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(nats(z0)), x2)) -> c10(A__FILTER(mark(x0), a__sieve(a__nats(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(zprimes), x2)) -> c10(A__FILTER(mark(x0), a__sieve(a__zprimes), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(cons(z0, z1)), x2)) -> c10(A__FILTER(mark(x0), a__sieve(cons(mark(z0), z1)), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(0), x2)) -> c10(A__FILTER(mark(x0), a__sieve(0), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(s(z0)), x2)) -> c10(A__FILTER(mark(x0), a__sieve(s(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), sieve(x1), x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x1)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), sieve(x1), x2)) -> c10(A__FILTER(a__sieve(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), sieve(x1), x2)) -> c10(A__FILTER(a__nats(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, sieve(x1), x2)) -> c10(A__FILTER(a__zprimes, a__sieve(mark(x1)), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), sieve(x1), x2)) -> c10(A__FILTER(cons(mark(z0), z1), a__sieve(mark(x1)), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, sieve(x1), x2)) -> c10(MARK(x0)) MARK(filter(s(z0), sieve(x1), x2)) -> c10(MARK(s(z0))) MARK(filter(x0, nats(filter(z0, z1, z2)), x2)) -> c10(A__FILTER(mark(x0), a__nats(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(x0)) MARK(filter(x0, nats(sieve(z0)), x2)) -> c10(A__FILTER(mark(x0), a__nats(a__sieve(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, nats(nats(z0)), x2)) -> c10(A__FILTER(mark(x0), a__nats(a__nats(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, nats(zprimes), x2)) -> c10(A__FILTER(mark(x0), a__nats(a__zprimes), mark(x2)), MARK(x0)) MARK(filter(x0, nats(cons(z0, z1)), x2)) -> c10(A__FILTER(mark(x0), a__nats(cons(mark(z0), z1)), mark(x2)), MARK(x0)) MARK(filter(x0, nats(0), x2)) -> c10(A__FILTER(mark(x0), a__nats(0), mark(x2)), MARK(x0)) MARK(filter(x0, nats(s(z0)), x2)) -> c10(A__FILTER(mark(x0), a__nats(s(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), nats(x1), x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x1)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), nats(x1), x2)) -> c10(A__FILTER(a__sieve(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), nats(x1), x2)) -> c10(A__FILTER(a__nats(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, nats(x1), x2)) -> c10(A__FILTER(a__zprimes, a__nats(mark(x1)), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), nats(x1), x2)) -> c10(A__FILTER(cons(mark(z0), z1), a__nats(mark(x1)), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, nats(x1), x2)) -> c10(MARK(x0)) MARK(filter(s(z0), nats(x1), x2)) -> c10(MARK(s(z0))) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c14_2, c15_1, c16_1, c18_1, c8_1, c10_2, c10_1, c11_2, c11_1, c12_2, c12_1, c13_2, c13_1, c8_2 ---------------------------------------- (113) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(filter(x0, zprimes, x2)) -> c10(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x0)) by MARK(filter(x0, zprimes, filter(z0, z1, z2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(z0), mark(z1), mark(z2))), MARK(x0)) MARK(filter(x0, zprimes, sieve(z0))) -> c10(A__FILTER(mark(x0), a__zprimes, a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, zprimes, nats(z0))) -> c10(A__FILTER(mark(x0), a__zprimes, a__nats(mark(z0))), MARK(x0)) MARK(filter(x0, zprimes, zprimes)) -> c10(A__FILTER(mark(x0), a__zprimes, a__zprimes), MARK(x0)) MARK(filter(x0, zprimes, cons(z0, z1))) -> c10(A__FILTER(mark(x0), a__zprimes, cons(mark(z0), z1)), MARK(x0)) MARK(filter(x0, zprimes, 0)) -> c10(A__FILTER(mark(x0), a__zprimes, 0), MARK(x0)) MARK(filter(x0, zprimes, s(z0))) -> c10(A__FILTER(mark(x0), a__zprimes, s(mark(z0))), MARK(x0)) MARK(filter(x0, zprimes, x1)) -> c10(A__FILTER(mark(x0), a__sieve(a__nats(s(s(0)))), mark(x1)), MARK(x0)) MARK(filter(x0, zprimes, x1)) -> c10(A__FILTER(mark(x0), zprimes, mark(x1)), MARK(x0)) MARK(filter(filter(z0, z1, z2), zprimes, x1)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__zprimes, mark(x1)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), zprimes, x1)) -> c10(A__FILTER(a__sieve(mark(z0)), a__zprimes, mark(x1)), MARK(sieve(z0))) MARK(filter(nats(z0), zprimes, x1)) -> c10(A__FILTER(a__nats(mark(z0)), a__zprimes, mark(x1)), MARK(nats(z0))) MARK(filter(zprimes, zprimes, x1)) -> c10(A__FILTER(a__zprimes, a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(cons(z0, z1), zprimes, x1)) -> c10(A__FILTER(cons(mark(z0), z1), a__zprimes, mark(x1)), MARK(cons(z0, z1))) MARK(filter(0, zprimes, x1)) -> c10(A__FILTER(0, a__zprimes, mark(x1)), MARK(0)) MARK(filter(s(z0), zprimes, x1)) -> c10(A__FILTER(s(mark(z0)), a__zprimes, mark(x1)), MARK(s(z0))) ---------------------------------------- (114) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, s(z0), x2)) -> c10(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, x2)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, x2)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, x1, x2)) -> c10(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, x2)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x1)) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(x0, 0, filter(x2, x3, 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MARK(filter(x0, sieve(nats(z0)), x2)) -> c10(A__FILTER(mark(x0), a__sieve(a__nats(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(zprimes), x2)) -> c10(A__FILTER(mark(x0), a__sieve(a__zprimes), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(cons(z0, z1)), x2)) -> c10(A__FILTER(mark(x0), a__sieve(cons(mark(z0), z1)), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(0), x2)) -> c10(A__FILTER(mark(x0), a__sieve(0), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(s(z0)), x2)) -> c10(A__FILTER(mark(x0), a__sieve(s(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), sieve(x1), x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x1)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), sieve(x1), x2)) -> c10(A__FILTER(a__sieve(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), sieve(x1), x2)) -> c10(A__FILTER(a__nats(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, sieve(x1), x2)) -> c10(A__FILTER(a__zprimes, a__sieve(mark(x1)), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), sieve(x1), x2)) -> c10(A__FILTER(cons(mark(z0), z1), a__sieve(mark(x1)), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, sieve(x1), x2)) -> c10(MARK(x0)) MARK(filter(s(z0), sieve(x1), x2)) -> c10(MARK(s(z0))) MARK(filter(x0, nats(filter(z0, z1, z2)), x2)) -> c10(A__FILTER(mark(x0), a__nats(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(x0)) MARK(filter(x0, nats(sieve(z0)), x2)) -> c10(A__FILTER(mark(x0), a__nats(a__sieve(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, nats(nats(z0)), x2)) -> c10(A__FILTER(mark(x0), a__nats(a__nats(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, nats(zprimes), x2)) -> c10(A__FILTER(mark(x0), a__nats(a__zprimes), mark(x2)), MARK(x0)) MARK(filter(x0, nats(cons(z0, z1)), x2)) -> c10(A__FILTER(mark(x0), a__nats(cons(mark(z0), z1)), mark(x2)), MARK(x0)) MARK(filter(x0, nats(0), x2)) -> c10(A__FILTER(mark(x0), a__nats(0), mark(x2)), MARK(x0)) MARK(filter(x0, nats(s(z0)), x2)) -> c10(A__FILTER(mark(x0), a__nats(s(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), nats(x1), x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x1)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), nats(x1), x2)) -> c10(A__FILTER(a__sieve(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), nats(x1), x2)) -> c10(A__FILTER(a__nats(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, nats(x1), x2)) -> c10(A__FILTER(a__zprimes, a__nats(mark(x1)), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), nats(x1), x2)) -> c10(A__FILTER(cons(mark(z0), z1), a__nats(mark(x1)), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, nats(x1), x2)) -> c10(MARK(x0)) MARK(filter(s(z0), nats(x1), x2)) -> c10(MARK(s(z0))) MARK(filter(x0, zprimes, x1)) -> c10(A__FILTER(mark(x0), a__sieve(a__nats(s(s(0)))), mark(x1)), MARK(x0)) MARK(filter(x0, zprimes, x1)) -> c10(A__FILTER(mark(x0), zprimes, mark(x1)), MARK(x0)) MARK(filter(filter(z0, z1, z2), zprimes, x1)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__zprimes, mark(x1)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), zprimes, x1)) -> c10(A__FILTER(a__sieve(mark(z0)), a__zprimes, mark(x1)), MARK(sieve(z0))) MARK(filter(nats(z0), zprimes, x1)) -> c10(A__FILTER(a__nats(mark(z0)), a__zprimes, mark(x1)), MARK(nats(z0))) MARK(filter(zprimes, zprimes, x1)) -> c10(A__FILTER(a__zprimes, a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(cons(z0, z1), zprimes, x1)) -> c10(A__FILTER(cons(mark(z0), z1), a__zprimes, mark(x1)), MARK(cons(z0, z1))) MARK(filter(0, zprimes, x1)) -> c10(A__FILTER(0, a__zprimes, mark(x1)), MARK(0)) MARK(filter(s(z0), zprimes, x1)) -> c10(A__FILTER(s(mark(z0)), a__zprimes, mark(x1)), MARK(s(z0))) S tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, s(z0), x2)) -> c10(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, x2)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, x2)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, x1, x2)) -> c10(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, x2)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x1)) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) 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c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__filter(mark(z0), mark(z1), mark(z2))), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, sieve(z0)), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__sieve(mark(z0))), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, nats(z0)), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__nats(mark(z0))), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, zprimes), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__zprimes), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, cons(z0, z1)), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), cons(mark(z0), z1)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, 0), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), 0), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, s(z0)), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), s(mark(z0))), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, filter(z0, z1, z2), x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), a__filter(mark(z0), mark(z1), mark(z2)), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, sieve(z0), x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), a__sieve(mark(z0)), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, nats(z0), x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), a__nats(mark(z0)), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, zprimes, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), a__zprimes, mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, cons(z0, z1), x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), cons(mark(z0), z1), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, 0, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), 0, mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, s(z0), x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), s(mark(z0)), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(filter(z0, z1, z2), x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(sieve(z0), x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(a__sieve(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(nats(z0), x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(a__nats(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(zprimes, x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(a__zprimes, mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(cons(z0, z1), x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(cons(mark(z0), z1), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(0, x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(0, mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(s(z0), x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(s(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(filter(z0, z1, z2), filter(x1, x2, x3), x4)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), filter(x1, x2, x3), x4)) -> c10(A__FILTER(a__sieve(mark(z0)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(sieve(z0))) MARK(filter(nats(z0), filter(x1, x2, x3), x4)) -> c10(A__FILTER(a__nats(mark(z0)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(nats(z0))) MARK(filter(zprimes, filter(x1, x2, x3), x4)) -> c10(A__FILTER(a__zprimes, a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(zprimes)) MARK(filter(cons(z0, z1), filter(x1, x2, x3), x4)) -> c10(A__FILTER(cons(mark(z0), z1), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(cons(z0, z1))) MARK(filter(x0, filter(x1, x2, x3), x4)) -> c10(MARK(x0)) MARK(filter(s(z0), filter(x1, x2, x3), x4)) -> c10(MARK(s(z0))) MARK(filter(x0, sieve(filter(z0, z1, z2)), x2)) -> c10(A__FILTER(mark(x0), a__sieve(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(sieve(z0)), x2)) -> c10(A__FILTER(mark(x0), a__sieve(a__sieve(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(nats(z0)), x2)) -> c10(A__FILTER(mark(x0), a__sieve(a__nats(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(zprimes), x2)) -> c10(A__FILTER(mark(x0), a__sieve(a__zprimes), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(cons(z0, z1)), x2)) -> c10(A__FILTER(mark(x0), a__sieve(cons(mark(z0), z1)), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(0), x2)) -> c10(A__FILTER(mark(x0), a__sieve(0), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(s(z0)), x2)) -> c10(A__FILTER(mark(x0), a__sieve(s(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), sieve(x1), x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x1)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), sieve(x1), x2)) -> c10(A__FILTER(a__sieve(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), sieve(x1), x2)) -> c10(A__FILTER(a__nats(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, sieve(x1), x2)) -> c10(A__FILTER(a__zprimes, a__sieve(mark(x1)), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), sieve(x1), x2)) -> c10(A__FILTER(cons(mark(z0), z1), a__sieve(mark(x1)), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, sieve(x1), x2)) -> c10(MARK(x0)) MARK(filter(s(z0), sieve(x1), x2)) -> c10(MARK(s(z0))) MARK(filter(x0, nats(filter(z0, z1, z2)), x2)) -> c10(A__FILTER(mark(x0), a__nats(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(x0)) MARK(filter(x0, nats(sieve(z0)), x2)) -> c10(A__FILTER(mark(x0), a__nats(a__sieve(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, nats(nats(z0)), x2)) -> c10(A__FILTER(mark(x0), a__nats(a__nats(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, nats(zprimes), x2)) -> c10(A__FILTER(mark(x0), a__nats(a__zprimes), mark(x2)), MARK(x0)) MARK(filter(x0, nats(cons(z0, z1)), x2)) -> c10(A__FILTER(mark(x0), a__nats(cons(mark(z0), z1)), mark(x2)), MARK(x0)) MARK(filter(x0, nats(0), x2)) -> c10(A__FILTER(mark(x0), a__nats(0), mark(x2)), MARK(x0)) MARK(filter(x0, nats(s(z0)), x2)) -> c10(A__FILTER(mark(x0), a__nats(s(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), nats(x1), x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x1)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), nats(x1), x2)) -> c10(A__FILTER(a__sieve(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), nats(x1), x2)) -> c10(A__FILTER(a__nats(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, nats(x1), x2)) -> c10(A__FILTER(a__zprimes, a__nats(mark(x1)), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), nats(x1), x2)) -> c10(A__FILTER(cons(mark(z0), z1), a__nats(mark(x1)), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, nats(x1), x2)) -> c10(MARK(x0)) MARK(filter(s(z0), nats(x1), x2)) -> c10(MARK(s(z0))) MARK(filter(x0, zprimes, x1)) -> c10(A__FILTER(mark(x0), a__sieve(a__nats(s(s(0)))), mark(x1)), MARK(x0)) MARK(filter(x0, zprimes, x1)) -> c10(A__FILTER(mark(x0), zprimes, mark(x1)), MARK(x0)) MARK(filter(filter(z0, z1, z2), zprimes, x1)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__zprimes, mark(x1)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), zprimes, x1)) -> c10(A__FILTER(a__sieve(mark(z0)), a__zprimes, mark(x1)), MARK(sieve(z0))) MARK(filter(nats(z0), zprimes, x1)) -> c10(A__FILTER(a__nats(mark(z0)), a__zprimes, mark(x1)), MARK(nats(z0))) MARK(filter(zprimes, zprimes, x1)) -> c10(A__FILTER(a__zprimes, a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(cons(z0, z1), zprimes, x1)) -> c10(A__FILTER(cons(mark(z0), z1), a__zprimes, mark(x1)), MARK(cons(z0, z1))) MARK(filter(0, zprimes, x1)) -> c10(A__FILTER(0, a__zprimes, mark(x1)), MARK(0)) MARK(filter(s(z0), zprimes, x1)) -> c10(A__FILTER(s(mark(z0)), a__zprimes, mark(x1)), MARK(s(z0))) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c14_2, c15_1, c16_1, c18_1, c8_1, c10_2, c10_1, c11_2, c11_1, c12_2, c12_1, c13_2, c13_1, c8_2 ---------------------------------------- (115) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(filter(0, zprimes, x1)) -> c10(A__FILTER(0, a__zprimes, mark(x1)), MARK(0)) ---------------------------------------- (116) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, s(z0), x2)) -> c10(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, x2)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, x2)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, x1, x2)) -> c10(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, x2)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x1)) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) 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cons(x2, x3))) -> c10(A__FILTER(a__zprimes, mark(x1), cons(mark(x2), x3)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, cons(x2, x3))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), cons(mark(x2), x3)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), cons(x2, x3))) -> c10(MARK(x0)) MARK(filter(x0, 0, cons(x2, x3))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, cons(x2, x3))) -> c10(MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), 0)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), 0), MARK(x0)) MARK(filter(x0, sieve(z0), 0)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), 0), MARK(x0)) MARK(filter(x0, nats(z0), 0)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), 0), MARK(x0)) MARK(filter(x0, zprimes, 0)) -> c10(A__FILTER(mark(x0), a__zprimes, 0), MARK(x0)) MARK(filter(x0, s(z0), 0)) -> c10(A__FILTER(mark(x0), s(mark(z0)), 0), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, 0)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), 0), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, 0)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), 0), MARK(sieve(z0))) MARK(filter(nats(z0), x1, 0)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), 0), MARK(nats(z0))) MARK(filter(zprimes, x1, 0)) -> c10(A__FILTER(a__zprimes, mark(x1), 0), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, 0)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), 0), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), 0)) -> c10(MARK(x0)) MARK(filter(x0, 0, 0)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, 0)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, s(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), s(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, s(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), s(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, s(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), s(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, s(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), s(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, s(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), s(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, s(0))) -> c10(A__FILTER(mark(x0), mark(x1), s(0)), MARK(x0)) MARK(filter(x0, x1, s(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), s(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), s(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), s(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), s(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), s(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), s(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), s(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, s(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, s(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), s(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), s(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, s(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), s(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, s(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), s(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, s(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), s(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, s(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), s(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, s(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), s(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), s(x2))) -> c10(MARK(x0)) MARK(filter(x0, 0, s(x2))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, s(x2))) -> c10(MARK(s(z0))) MARK(filter(x0, filter(x1, x2, filter(z0, z1, z2)), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__filter(mark(z0), mark(z1), mark(z2))), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, sieve(z0)), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__sieve(mark(z0))), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, nats(z0)), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__nats(mark(z0))), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, zprimes), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__zprimes), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, cons(z0, z1)), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), cons(mark(z0), z1)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, 0), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), 0), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, s(z0)), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), s(mark(z0))), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, filter(z0, z1, z2), x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), a__filter(mark(z0), mark(z1), mark(z2)), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, sieve(z0), x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), a__sieve(mark(z0)), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, nats(z0), x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), a__nats(mark(z0)), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, zprimes, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), a__zprimes, mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, cons(z0, z1), x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), cons(mark(z0), z1), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, 0, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), 0, mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, s(z0), x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), s(mark(z0)), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(filter(z0, z1, z2), x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(sieve(z0), x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(a__sieve(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(nats(z0), x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(a__nats(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(zprimes, x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(a__zprimes, mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(cons(z0, z1), x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(cons(mark(z0), z1), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(0, x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(0, mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(s(z0), x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(s(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(filter(z0, z1, z2), filter(x1, x2, x3), x4)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), filter(x1, x2, x3), x4)) -> c10(A__FILTER(a__sieve(mark(z0)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(sieve(z0))) MARK(filter(nats(z0), filter(x1, x2, x3), x4)) -> c10(A__FILTER(a__nats(mark(z0)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(nats(z0))) MARK(filter(zprimes, filter(x1, x2, x3), x4)) -> c10(A__FILTER(a__zprimes, a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(zprimes)) MARK(filter(cons(z0, z1), filter(x1, x2, x3), x4)) -> c10(A__FILTER(cons(mark(z0), z1), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(cons(z0, z1))) MARK(filter(x0, filter(x1, x2, x3), x4)) -> c10(MARK(x0)) MARK(filter(s(z0), filter(x1, x2, x3), x4)) -> c10(MARK(s(z0))) MARK(filter(x0, sieve(filter(z0, z1, z2)), x2)) -> c10(A__FILTER(mark(x0), a__sieve(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(sieve(z0)), x2)) -> c10(A__FILTER(mark(x0), a__sieve(a__sieve(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(nats(z0)), x2)) -> c10(A__FILTER(mark(x0), a__sieve(a__nats(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(zprimes), x2)) -> c10(A__FILTER(mark(x0), a__sieve(a__zprimes), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(cons(z0, z1)), x2)) -> c10(A__FILTER(mark(x0), a__sieve(cons(mark(z0), z1)), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(0), x2)) -> c10(A__FILTER(mark(x0), a__sieve(0), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(s(z0)), x2)) -> c10(A__FILTER(mark(x0), a__sieve(s(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), sieve(x1), x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x1)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), sieve(x1), x2)) -> c10(A__FILTER(a__sieve(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), sieve(x1), x2)) -> c10(A__FILTER(a__nats(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, sieve(x1), x2)) -> c10(A__FILTER(a__zprimes, a__sieve(mark(x1)), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), sieve(x1), x2)) -> c10(A__FILTER(cons(mark(z0), z1), a__sieve(mark(x1)), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, sieve(x1), x2)) -> c10(MARK(x0)) MARK(filter(s(z0), sieve(x1), x2)) -> c10(MARK(s(z0))) MARK(filter(x0, nats(filter(z0, z1, z2)), x2)) -> c10(A__FILTER(mark(x0), a__nats(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(x0)) MARK(filter(x0, nats(sieve(z0)), x2)) -> c10(A__FILTER(mark(x0), a__nats(a__sieve(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, nats(nats(z0)), x2)) -> c10(A__FILTER(mark(x0), a__nats(a__nats(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, nats(zprimes), x2)) -> c10(A__FILTER(mark(x0), a__nats(a__zprimes), mark(x2)), MARK(x0)) MARK(filter(x0, nats(cons(z0, z1)), x2)) -> c10(A__FILTER(mark(x0), a__nats(cons(mark(z0), z1)), mark(x2)), MARK(x0)) MARK(filter(x0, nats(0), x2)) -> c10(A__FILTER(mark(x0), a__nats(0), mark(x2)), MARK(x0)) MARK(filter(x0, nats(s(z0)), x2)) -> c10(A__FILTER(mark(x0), a__nats(s(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), nats(x1), x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x1)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), nats(x1), x2)) -> c10(A__FILTER(a__sieve(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), nats(x1), x2)) -> c10(A__FILTER(a__nats(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, nats(x1), x2)) -> c10(A__FILTER(a__zprimes, a__nats(mark(x1)), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), nats(x1), x2)) -> c10(A__FILTER(cons(mark(z0), z1), a__nats(mark(x1)), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, nats(x1), x2)) -> c10(MARK(x0)) MARK(filter(s(z0), nats(x1), x2)) -> c10(MARK(s(z0))) MARK(filter(x0, zprimes, x1)) -> c10(A__FILTER(mark(x0), a__sieve(a__nats(s(s(0)))), mark(x1)), MARK(x0)) MARK(filter(x0, zprimes, x1)) -> c10(A__FILTER(mark(x0), zprimes, mark(x1)), MARK(x0)) MARK(filter(filter(z0, z1, z2), zprimes, x1)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__zprimes, mark(x1)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), zprimes, x1)) -> c10(A__FILTER(a__sieve(mark(z0)), a__zprimes, mark(x1)), MARK(sieve(z0))) MARK(filter(nats(z0), zprimes, x1)) -> c10(A__FILTER(a__nats(mark(z0)), a__zprimes, mark(x1)), MARK(nats(z0))) MARK(filter(zprimes, zprimes, x1)) -> c10(A__FILTER(a__zprimes, a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(cons(z0, z1), zprimes, x1)) -> c10(A__FILTER(cons(mark(z0), z1), a__zprimes, mark(x1)), MARK(cons(z0, z1))) MARK(filter(s(z0), zprimes, x1)) -> c10(A__FILTER(s(mark(z0)), a__zprimes, mark(x1)), MARK(s(z0))) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c14_2, c15_1, c16_1, c18_1, c8_1, c10_2, c10_1, c11_2, c11_1, c12_2, c12_1, c13_2, c13_1, c8_2 ---------------------------------------- (117) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (118) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, s(z0), x2)) -> c10(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, x2)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, x2)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, x1, x2)) -> c10(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, x2)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x1)) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(x0, 0, filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, sieve(x2))) -> c10(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x0)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, sieve(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, sieve(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, sieve(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(0)), MARK(x0)) MARK(filter(x0, x1, sieve(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), sieve(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, sieve(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, sieve(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__sieve(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, sieve(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, sieve(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, sieve(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), a__sieve(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, sieve(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__sieve(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), sieve(x2))) -> c10(MARK(x0)) MARK(filter(x0, 0, sieve(x2))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, sieve(x2))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(mark(x2)), nats(s(mark(x2))))), MARK(x0)) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), nats(mark(x2))), MARK(x0)) MARK(filter(x0, x1, nats(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, nats(sieve(z0)))) -> c10(A__FILTER(mark(x0), 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c10(MARK(s(z0))) MARK(filter(x0, zprimes, x1)) -> c10(A__FILTER(mark(x0), a__sieve(a__nats(s(s(0)))), mark(x1)), MARK(x0)) MARK(filter(filter(z0, z1, z2), zprimes, x1)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__zprimes, mark(x1)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), zprimes, x1)) -> c10(A__FILTER(a__sieve(mark(z0)), a__zprimes, mark(x1)), MARK(sieve(z0))) MARK(filter(nats(z0), zprimes, x1)) -> c10(A__FILTER(a__nats(mark(z0)), a__zprimes, mark(x1)), MARK(nats(z0))) MARK(filter(zprimes, zprimes, x1)) -> c10(A__FILTER(a__zprimes, a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(cons(z0, z1), zprimes, x1)) -> c10(A__FILTER(cons(mark(z0), z1), a__zprimes, mark(x1)), MARK(cons(z0, z1))) MARK(filter(x0, zprimes, x1)) -> c10(MARK(x0)) MARK(filter(s(z0), zprimes, x1)) -> c10(MARK(s(z0))) S tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, s(z0), x2)) -> c10(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, x2)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, x2)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, x1, x2)) -> c10(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, x2)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x1)) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(x0, 0, filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, sieve(x2))) -> c10(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x0)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, sieve(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, sieve(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, sieve(0))) -> 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filter(x1, x2, x3), x4)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), filter(x1, x2, x3), x4)) -> c10(A__FILTER(a__sieve(mark(z0)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(sieve(z0))) MARK(filter(nats(z0), filter(x1, x2, x3), x4)) -> c10(A__FILTER(a__nats(mark(z0)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(nats(z0))) MARK(filter(zprimes, filter(x1, x2, x3), x4)) -> c10(A__FILTER(a__zprimes, a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(zprimes)) MARK(filter(cons(z0, z1), filter(x1, x2, x3), x4)) -> c10(A__FILTER(cons(mark(z0), z1), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(cons(z0, z1))) MARK(filter(x0, filter(x1, x2, x3), x4)) -> c10(MARK(x0)) MARK(filter(s(z0), filter(x1, x2, x3), x4)) -> c10(MARK(s(z0))) MARK(filter(x0, sieve(filter(z0, z1, z2)), x2)) -> c10(A__FILTER(mark(x0), a__sieve(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(sieve(z0)), x2)) -> c10(A__FILTER(mark(x0), a__sieve(a__sieve(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(nats(z0)), x2)) -> c10(A__FILTER(mark(x0), a__sieve(a__nats(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(zprimes), x2)) -> c10(A__FILTER(mark(x0), a__sieve(a__zprimes), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(cons(z0, z1)), x2)) -> c10(A__FILTER(mark(x0), a__sieve(cons(mark(z0), z1)), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(0), x2)) -> c10(A__FILTER(mark(x0), a__sieve(0), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(s(z0)), x2)) -> c10(A__FILTER(mark(x0), a__sieve(s(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), sieve(x1), x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x1)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), sieve(x1), x2)) -> c10(A__FILTER(a__sieve(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), sieve(x1), x2)) -> c10(A__FILTER(a__nats(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, sieve(x1), x2)) -> c10(A__FILTER(a__zprimes, a__sieve(mark(x1)), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), sieve(x1), x2)) -> c10(A__FILTER(cons(mark(z0), z1), a__sieve(mark(x1)), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, sieve(x1), x2)) -> c10(MARK(x0)) MARK(filter(s(z0), sieve(x1), x2)) -> c10(MARK(s(z0))) MARK(filter(x0, nats(filter(z0, z1, z2)), x2)) -> c10(A__FILTER(mark(x0), a__nats(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(x0)) MARK(filter(x0, nats(sieve(z0)), x2)) -> c10(A__FILTER(mark(x0), a__nats(a__sieve(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, nats(nats(z0)), x2)) -> c10(A__FILTER(mark(x0), a__nats(a__nats(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, nats(zprimes), x2)) -> c10(A__FILTER(mark(x0), a__nats(a__zprimes), mark(x2)), MARK(x0)) MARK(filter(x0, nats(cons(z0, z1)), x2)) -> c10(A__FILTER(mark(x0), a__nats(cons(mark(z0), z1)), mark(x2)), MARK(x0)) MARK(filter(x0, nats(0), x2)) -> c10(A__FILTER(mark(x0), a__nats(0), mark(x2)), MARK(x0)) MARK(filter(x0, nats(s(z0)), x2)) -> c10(A__FILTER(mark(x0), a__nats(s(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), nats(x1), x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x1)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), nats(x1), x2)) -> c10(A__FILTER(a__sieve(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), nats(x1), x2)) -> c10(A__FILTER(a__nats(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, nats(x1), x2)) -> c10(A__FILTER(a__zprimes, a__nats(mark(x1)), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), nats(x1), x2)) -> c10(A__FILTER(cons(mark(z0), z1), a__nats(mark(x1)), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, nats(x1), x2)) -> c10(MARK(x0)) MARK(filter(s(z0), nats(x1), x2)) -> c10(MARK(s(z0))) MARK(filter(x0, zprimes, x1)) -> c10(A__FILTER(mark(x0), a__sieve(a__nats(s(s(0)))), mark(x1)), MARK(x0)) MARK(filter(filter(z0, z1, z2), zprimes, x1)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__zprimes, mark(x1)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), zprimes, x1)) -> c10(A__FILTER(a__sieve(mark(z0)), a__zprimes, mark(x1)), MARK(sieve(z0))) MARK(filter(nats(z0), zprimes, x1)) -> c10(A__FILTER(a__nats(mark(z0)), a__zprimes, mark(x1)), MARK(nats(z0))) MARK(filter(zprimes, zprimes, x1)) -> c10(A__FILTER(a__zprimes, a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(cons(z0, z1), zprimes, x1)) -> c10(A__FILTER(cons(mark(z0), z1), a__zprimes, mark(x1)), MARK(cons(z0, z1))) MARK(filter(x0, zprimes, x1)) -> c10(MARK(x0)) MARK(filter(s(z0), zprimes, x1)) -> c10(MARK(s(z0))) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c14_2, c15_1, c16_1, c18_1, c8_1, c10_2, c10_1, c11_2, c11_1, c12_2, c12_1, c13_2, c13_1, c8_2 ---------------------------------------- (119) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(filter(x0, s(z0), x2)) -> c10(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x0)) by MARK(filter(x0, s(x1), filter(z0, z1, z2))) -> c10(A__FILTER(mark(x0), s(mark(x1)), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x0)) MARK(filter(x0, s(x1), sieve(z0))) -> c10(A__FILTER(mark(x0), s(mark(x1)), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, s(x1), nats(z0))) -> c10(A__FILTER(mark(x0), s(mark(x1)), a__nats(mark(z0))), MARK(x0)) MARK(filter(x0, s(x1), zprimes)) -> c10(A__FILTER(mark(x0), s(mark(x1)), a__zprimes), MARK(x0)) MARK(filter(x0, s(x1), cons(z0, z1))) -> c10(A__FILTER(mark(x0), s(mark(x1)), cons(mark(z0), z1)), MARK(x0)) MARK(filter(x0, s(x1), 0)) -> c10(A__FILTER(mark(x0), s(mark(x1)), 0), MARK(x0)) MARK(filter(x0, s(x1), s(z0))) -> c10(A__FILTER(mark(x0), s(mark(x1)), s(mark(z0))), MARK(x0)) MARK(filter(x0, s(filter(z0, z1, z2)), x2)) -> c10(A__FILTER(mark(x0), s(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(x0)) MARK(filter(x0, s(sieve(z0)), x2)) -> c10(A__FILTER(mark(x0), s(a__sieve(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, s(nats(z0)), x2)) -> c10(A__FILTER(mark(x0), s(a__nats(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, s(zprimes), x2)) -> c10(A__FILTER(mark(x0), s(a__zprimes), mark(x2)), MARK(x0)) MARK(filter(x0, s(cons(z0, z1)), x2)) -> c10(A__FILTER(mark(x0), s(cons(mark(z0), z1)), mark(x2)), MARK(x0)) MARK(filter(x0, s(0), x2)) -> c10(A__FILTER(mark(x0), s(0), mark(x2)), MARK(x0)) MARK(filter(x0, s(s(z0)), x2)) -> c10(A__FILTER(mark(x0), s(s(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), s(x1), x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), s(mark(x1)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), s(x1), x2)) -> c10(A__FILTER(a__sieve(mark(z0)), s(mark(x1)), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), s(x1), x2)) -> c10(A__FILTER(a__nats(mark(z0)), s(mark(x1)), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, s(x1), x2)) -> c10(A__FILTER(a__zprimes, s(mark(x1)), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), s(x1), x2)) -> c10(A__FILTER(cons(mark(z0), z1), s(mark(x1)), mark(x2)), MARK(cons(z0, z1))) MARK(filter(0, s(x1), x2)) -> c10(A__FILTER(0, s(mark(x1)), mark(x2)), MARK(0)) MARK(filter(s(z0), s(x1), x2)) -> c10(A__FILTER(s(mark(z0)), s(mark(x1)), mark(x2)), MARK(s(z0))) ---------------------------------------- (120) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, x2)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, x2)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, x1, x2)) -> c10(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, x2)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x1)) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), 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c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), 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MARK(nats(z0))) MARK(filter(zprimes, x1, s(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), s(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, s(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), s(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), s(x2))) -> c10(MARK(x0)) MARK(filter(x0, 0, s(x2))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, s(x2))) -> c10(MARK(s(z0))) MARK(filter(x0, filter(x1, x2, filter(z0, z1, z2)), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__filter(mark(z0), mark(z1), mark(z2))), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, sieve(z0)), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__sieve(mark(z0))), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, nats(z0)), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__nats(mark(z0))), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, zprimes), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__zprimes), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, cons(z0, z1)), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), cons(mark(z0), z1)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, 0), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), 0), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, s(z0)), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), s(mark(z0))), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, filter(z0, z1, z2), x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), a__filter(mark(z0), mark(z1), mark(z2)), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, sieve(z0), x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), a__sieve(mark(z0)), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, nats(z0), x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), a__nats(mark(z0)), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, zprimes, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), a__zprimes, mark(x3)), 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c10(A__FILTER(a__zprimes, a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(zprimes)) MARK(filter(cons(z0, z1), filter(x1, x2, x3), x4)) -> c10(A__FILTER(cons(mark(z0), z1), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(cons(z0, z1))) MARK(filter(x0, filter(x1, x2, x3), x4)) -> c10(MARK(x0)) MARK(filter(s(z0), filter(x1, x2, x3), x4)) -> c10(MARK(s(z0))) MARK(filter(x0, sieve(filter(z0, z1, z2)), x2)) -> c10(A__FILTER(mark(x0), a__sieve(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(sieve(z0)), x2)) -> c10(A__FILTER(mark(x0), a__sieve(a__sieve(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(nats(z0)), x2)) -> c10(A__FILTER(mark(x0), a__sieve(a__nats(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(zprimes), x2)) -> c10(A__FILTER(mark(x0), a__sieve(a__zprimes), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(cons(z0, z1)), x2)) -> c10(A__FILTER(mark(x0), a__sieve(cons(mark(z0), z1)), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(0), x2)) -> c10(A__FILTER(mark(x0), a__sieve(0), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(s(z0)), x2)) -> c10(A__FILTER(mark(x0), a__sieve(s(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), sieve(x1), x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x1)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), sieve(x1), x2)) -> c10(A__FILTER(a__sieve(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), sieve(x1), x2)) -> c10(A__FILTER(a__nats(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, sieve(x1), x2)) -> c10(A__FILTER(a__zprimes, a__sieve(mark(x1)), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), sieve(x1), x2)) -> c10(A__FILTER(cons(mark(z0), z1), a__sieve(mark(x1)), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, sieve(x1), x2)) -> c10(MARK(x0)) MARK(filter(s(z0), sieve(x1), x2)) -> c10(MARK(s(z0))) MARK(filter(x0, nats(filter(z0, z1, z2)), x2)) -> c10(A__FILTER(mark(x0), a__nats(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(x0)) MARK(filter(x0, nats(sieve(z0)), x2)) -> c10(A__FILTER(mark(x0), a__nats(a__sieve(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, nats(nats(z0)), x2)) -> c10(A__FILTER(mark(x0), a__nats(a__nats(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, nats(zprimes), x2)) -> c10(A__FILTER(mark(x0), a__nats(a__zprimes), mark(x2)), MARK(x0)) MARK(filter(x0, nats(cons(z0, z1)), x2)) -> c10(A__FILTER(mark(x0), a__nats(cons(mark(z0), z1)), mark(x2)), MARK(x0)) MARK(filter(x0, nats(0), x2)) -> c10(A__FILTER(mark(x0), a__nats(0), mark(x2)), MARK(x0)) MARK(filter(x0, nats(s(z0)), x2)) -> c10(A__FILTER(mark(x0), a__nats(s(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), nats(x1), x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x1)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), nats(x1), x2)) -> c10(A__FILTER(a__sieve(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), nats(x1), x2)) -> c10(A__FILTER(a__nats(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, nats(x1), x2)) -> c10(A__FILTER(a__zprimes, a__nats(mark(x1)), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), nats(x1), x2)) -> c10(A__FILTER(cons(mark(z0), z1), a__nats(mark(x1)), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, nats(x1), x2)) -> c10(MARK(x0)) MARK(filter(s(z0), nats(x1), x2)) -> c10(MARK(s(z0))) MARK(filter(x0, zprimes, x1)) -> c10(A__FILTER(mark(x0), a__sieve(a__nats(s(s(0)))), mark(x1)), MARK(x0)) MARK(filter(filter(z0, z1, z2), zprimes, x1)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__zprimes, mark(x1)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), zprimes, x1)) -> c10(A__FILTER(a__sieve(mark(z0)), a__zprimes, mark(x1)), MARK(sieve(z0))) MARK(filter(nats(z0), zprimes, x1)) -> c10(A__FILTER(a__nats(mark(z0)), a__zprimes, mark(x1)), MARK(nats(z0))) MARK(filter(zprimes, zprimes, x1)) -> c10(A__FILTER(a__zprimes, a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(cons(z0, z1), zprimes, x1)) -> c10(A__FILTER(cons(mark(z0), z1), a__zprimes, mark(x1)), MARK(cons(z0, z1))) MARK(filter(x0, zprimes, x1)) -> c10(MARK(x0)) MARK(filter(s(z0), zprimes, x1)) -> c10(MARK(s(z0))) MARK(filter(x0, s(filter(z0, z1, z2)), x2)) -> c10(A__FILTER(mark(x0), s(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(x0)) MARK(filter(x0, s(sieve(z0)), x2)) -> c10(A__FILTER(mark(x0), s(a__sieve(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, s(nats(z0)), x2)) -> c10(A__FILTER(mark(x0), s(a__nats(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, s(zprimes), x2)) -> c10(A__FILTER(mark(x0), s(a__zprimes), mark(x2)), MARK(x0)) MARK(filter(x0, s(cons(z0, z1)), x2)) -> c10(A__FILTER(mark(x0), s(cons(mark(z0), z1)), mark(x2)), MARK(x0)) MARK(filter(x0, s(0), x2)) -> c10(A__FILTER(mark(x0), s(0), mark(x2)), MARK(x0)) MARK(filter(x0, s(s(z0)), x2)) -> c10(A__FILTER(mark(x0), s(s(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), s(x1), x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), s(mark(x1)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), s(x1), x2)) -> c10(A__FILTER(a__sieve(mark(z0)), s(mark(x1)), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), s(x1), x2)) -> c10(A__FILTER(a__nats(mark(z0)), s(mark(x1)), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, s(x1), x2)) -> c10(A__FILTER(a__zprimes, s(mark(x1)), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), s(x1), x2)) -> c10(A__FILTER(cons(mark(z0), z1), s(mark(x1)), mark(x2)), MARK(cons(z0, z1))) MARK(filter(0, s(x1), x2)) -> c10(A__FILTER(0, s(mark(x1)), mark(x2)), MARK(0)) MARK(filter(s(z0), s(x1), x2)) -> c10(A__FILTER(s(mark(z0)), s(mark(x1)), mark(x2)), MARK(s(z0))) S tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, x2)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, x2)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, x1, x2)) -> c10(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, x2)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x1)) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(x0, 0, filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, sieve(x2))) -> c10(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x0)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, sieve(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, sieve(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, sieve(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(0)), MARK(x0)) MARK(filter(x0, x1, sieve(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), sieve(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, sieve(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, sieve(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__sieve(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, sieve(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, sieve(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, sieve(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), a__sieve(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, sieve(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__sieve(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), sieve(x2))) -> c10(MARK(x0)) MARK(filter(x0, 0, sieve(x2))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, sieve(x2))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(mark(x2)), nats(s(mark(x2))))), MARK(x0)) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), nats(mark(x2))), MARK(x0)) MARK(filter(x0, x1, nats(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, nats(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, nats(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, nats(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, nats(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, nats(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(0)), MARK(x0)) MARK(filter(x0, x1, nats(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), nats(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), nats(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), nats(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, nats(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), nats(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, nats(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__nats(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, nats(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, nats(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, nats(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), a__nats(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, nats(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__nats(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), nats(x2))) -> c10(MARK(x0)) MARK(filter(x0, 0, nats(x2))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, nats(x2))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, zprimes)) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(s(s(0))))), MARK(x0)) MARK(filter(x0, x1, zprimes)) -> c10(A__FILTER(mark(x0), mark(x1), zprimes), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), zprimes)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__zprimes), MARK(x0)) MARK(filter(x0, sieve(z0), zprimes)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__zprimes), MARK(x0)) MARK(filter(x0, nats(z0), zprimes)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__zprimes), MARK(x0)) MARK(filter(x0, zprimes, zprimes)) -> c10(A__FILTER(mark(x0), a__zprimes, a__zprimes), MARK(x0)) MARK(filter(x0, s(z0), zprimes)) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__zprimes), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, zprimes)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__zprimes), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, zprimes)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__zprimes), MARK(sieve(z0))) MARK(filter(nats(z0), x1, zprimes)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__zprimes), MARK(nats(z0))) MARK(filter(zprimes, x1, zprimes)) -> c10(A__FILTER(a__zprimes, mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, zprimes)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__zprimes), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), zprimes)) -> c10(MARK(x0)) MARK(filter(x0, 0, zprimes)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, zprimes)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, cons(filter(z0, z1, z2), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(a__filter(mark(z0), mark(z1), mark(z2)), x3)), MARK(x0)) MARK(filter(x0, x1, cons(sieve(z0), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(a__sieve(mark(z0)), x3)), MARK(x0)) MARK(filter(x0, x1, cons(nats(z0), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(a__nats(mark(z0)), x3)), MARK(x0)) MARK(filter(x0, x1, cons(zprimes, x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(a__zprimes, x3)), MARK(x0)) MARK(filter(x0, x1, cons(cons(z0, z1), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(cons(mark(z0), z1), x3)), MARK(x0)) MARK(filter(x0, x1, cons(0, x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(0, x3)), MARK(x0)) MARK(filter(x0, x1, cons(s(z0), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(s(mark(z0)), x3)), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), cons(x2, x3))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), cons(mark(x2), x3)), MARK(x0)) MARK(filter(x0, sieve(z0), cons(x2, x3))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), cons(mark(x2), x3)), MARK(x0)) MARK(filter(x0, nats(z0), cons(x2, x3))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), cons(mark(x2), x3)), MARK(x0)) MARK(filter(x0, zprimes, cons(x2, x3))) -> c10(A__FILTER(mark(x0), a__zprimes, cons(mark(x2), x3)), MARK(x0)) MARK(filter(x0, s(z0), cons(x2, x3))) -> c10(A__FILTER(mark(x0), s(mark(z0)), cons(mark(x2), x3)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, cons(x2, x3))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), cons(mark(x2), x3)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, cons(x2, x3))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), cons(mark(x2), x3)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, cons(x2, x3))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), cons(mark(x2), x3)), MARK(nats(z0))) MARK(filter(zprimes, x1, cons(x2, x3))) -> c10(A__FILTER(a__zprimes, mark(x1), cons(mark(x2), x3)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, cons(x2, x3))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), cons(mark(x2), x3)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), cons(x2, x3))) -> c10(MARK(x0)) MARK(filter(x0, 0, cons(x2, x3))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, cons(x2, x3))) -> c10(MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), 0)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), 0), MARK(x0)) MARK(filter(x0, sieve(z0), 0)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), 0), MARK(x0)) MARK(filter(x0, nats(z0), 0)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), 0), MARK(x0)) MARK(filter(x0, zprimes, 0)) -> c10(A__FILTER(mark(x0), a__zprimes, 0), MARK(x0)) MARK(filter(x0, s(z0), 0)) -> c10(A__FILTER(mark(x0), s(mark(z0)), 0), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, 0)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), 0), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, 0)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), 0), MARK(sieve(z0))) MARK(filter(nats(z0), x1, 0)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), 0), MARK(nats(z0))) MARK(filter(zprimes, x1, 0)) -> c10(A__FILTER(a__zprimes, mark(x1), 0), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, 0)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), 0), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), 0)) -> c10(MARK(x0)) MARK(filter(x0, 0, 0)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, 0)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, s(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), s(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, s(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), s(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, s(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), s(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, s(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), s(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, s(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), s(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, s(0))) -> c10(A__FILTER(mark(x0), mark(x1), s(0)), MARK(x0)) MARK(filter(x0, x1, s(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), s(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), s(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), s(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), s(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), s(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), s(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), s(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, s(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, s(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), s(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), s(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, s(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), s(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, s(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), s(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, s(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), s(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, s(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), s(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, s(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), s(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), s(x2))) -> c10(MARK(x0)) MARK(filter(x0, 0, s(x2))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, s(x2))) -> c10(MARK(s(z0))) MARK(filter(x0, filter(x1, x2, filter(z0, z1, z2)), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__filter(mark(z0), mark(z1), mark(z2))), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, sieve(z0)), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__sieve(mark(z0))), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, nats(z0)), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__nats(mark(z0))), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, zprimes), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__zprimes), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, cons(z0, z1)), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), cons(mark(z0), z1)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, 0), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), 0), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, s(z0)), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), s(mark(z0))), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, filter(z0, z1, z2), x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), a__filter(mark(z0), mark(z1), mark(z2)), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, sieve(z0), x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), a__sieve(mark(z0)), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, nats(z0), x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), a__nats(mark(z0)), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, zprimes, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), a__zprimes, mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, cons(z0, z1), x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), cons(mark(z0), z1), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, 0, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), 0, mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, s(z0), x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), s(mark(z0)), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(filter(z0, z1, z2), x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(sieve(z0), x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(a__sieve(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(nats(z0), x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(a__nats(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(zprimes, x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(a__zprimes, mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(cons(z0, z1), x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(cons(mark(z0), z1), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(0, x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(0, mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(s(z0), x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(s(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(filter(z0, z1, z2), filter(x1, x2, x3), x4)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), filter(x1, x2, x3), x4)) -> c10(A__FILTER(a__sieve(mark(z0)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(sieve(z0))) MARK(filter(nats(z0), filter(x1, x2, x3), x4)) -> c10(A__FILTER(a__nats(mark(z0)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(nats(z0))) MARK(filter(zprimes, filter(x1, x2, x3), x4)) -> c10(A__FILTER(a__zprimes, a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(zprimes)) MARK(filter(cons(z0, z1), filter(x1, x2, x3), x4)) -> c10(A__FILTER(cons(mark(z0), z1), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(cons(z0, z1))) MARK(filter(x0, filter(x1, x2, x3), x4)) -> c10(MARK(x0)) MARK(filter(s(z0), filter(x1, x2, x3), x4)) -> c10(MARK(s(z0))) MARK(filter(x0, sieve(filter(z0, z1, z2)), x2)) -> c10(A__FILTER(mark(x0), a__sieve(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(sieve(z0)), x2)) -> c10(A__FILTER(mark(x0), a__sieve(a__sieve(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(nats(z0)), x2)) -> c10(A__FILTER(mark(x0), a__sieve(a__nats(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(zprimes), x2)) -> c10(A__FILTER(mark(x0), a__sieve(a__zprimes), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(cons(z0, z1)), x2)) -> c10(A__FILTER(mark(x0), a__sieve(cons(mark(z0), z1)), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(0), x2)) -> c10(A__FILTER(mark(x0), a__sieve(0), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(s(z0)), x2)) -> c10(A__FILTER(mark(x0), a__sieve(s(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), sieve(x1), x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x1)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), sieve(x1), x2)) -> c10(A__FILTER(a__sieve(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), sieve(x1), x2)) -> c10(A__FILTER(a__nats(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, sieve(x1), x2)) -> c10(A__FILTER(a__zprimes, a__sieve(mark(x1)), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), sieve(x1), x2)) -> c10(A__FILTER(cons(mark(z0), z1), a__sieve(mark(x1)), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, sieve(x1), x2)) -> c10(MARK(x0)) MARK(filter(s(z0), sieve(x1), x2)) -> c10(MARK(s(z0))) MARK(filter(x0, nats(filter(z0, z1, z2)), x2)) -> c10(A__FILTER(mark(x0), a__nats(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(x0)) MARK(filter(x0, nats(sieve(z0)), x2)) -> c10(A__FILTER(mark(x0), a__nats(a__sieve(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, nats(nats(z0)), x2)) -> c10(A__FILTER(mark(x0), a__nats(a__nats(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, nats(zprimes), x2)) -> c10(A__FILTER(mark(x0), a__nats(a__zprimes), mark(x2)), MARK(x0)) MARK(filter(x0, nats(cons(z0, z1)), x2)) -> c10(A__FILTER(mark(x0), a__nats(cons(mark(z0), z1)), mark(x2)), MARK(x0)) MARK(filter(x0, nats(0), x2)) -> c10(A__FILTER(mark(x0), a__nats(0), mark(x2)), MARK(x0)) MARK(filter(x0, nats(s(z0)), x2)) -> c10(A__FILTER(mark(x0), a__nats(s(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), nats(x1), x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x1)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), nats(x1), x2)) -> c10(A__FILTER(a__sieve(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), nats(x1), x2)) -> c10(A__FILTER(a__nats(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, nats(x1), x2)) -> c10(A__FILTER(a__zprimes, a__nats(mark(x1)), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), nats(x1), x2)) -> c10(A__FILTER(cons(mark(z0), z1), a__nats(mark(x1)), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, nats(x1), x2)) -> c10(MARK(x0)) MARK(filter(s(z0), nats(x1), x2)) -> c10(MARK(s(z0))) MARK(filter(x0, zprimes, x1)) -> c10(A__FILTER(mark(x0), a__sieve(a__nats(s(s(0)))), mark(x1)), MARK(x0)) MARK(filter(filter(z0, z1, z2), zprimes, x1)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__zprimes, mark(x1)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), zprimes, x1)) -> c10(A__FILTER(a__sieve(mark(z0)), a__zprimes, mark(x1)), MARK(sieve(z0))) MARK(filter(nats(z0), zprimes, x1)) -> c10(A__FILTER(a__nats(mark(z0)), a__zprimes, mark(x1)), MARK(nats(z0))) MARK(filter(zprimes, zprimes, x1)) -> c10(A__FILTER(a__zprimes, a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(cons(z0, z1), zprimes, x1)) -> c10(A__FILTER(cons(mark(z0), z1), a__zprimes, mark(x1)), MARK(cons(z0, z1))) MARK(filter(x0, zprimes, x1)) -> c10(MARK(x0)) MARK(filter(s(z0), zprimes, x1)) -> c10(MARK(s(z0))) MARK(filter(x0, s(filter(z0, z1, z2)), x2)) -> c10(A__FILTER(mark(x0), s(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(x0)) MARK(filter(x0, s(sieve(z0)), x2)) -> c10(A__FILTER(mark(x0), s(a__sieve(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, s(nats(z0)), x2)) -> c10(A__FILTER(mark(x0), s(a__nats(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, s(zprimes), x2)) -> c10(A__FILTER(mark(x0), s(a__zprimes), mark(x2)), MARK(x0)) MARK(filter(x0, s(cons(z0, z1)), x2)) -> c10(A__FILTER(mark(x0), s(cons(mark(z0), z1)), mark(x2)), MARK(x0)) MARK(filter(x0, s(0), x2)) -> c10(A__FILTER(mark(x0), s(0), mark(x2)), MARK(x0)) MARK(filter(x0, s(s(z0)), x2)) -> c10(A__FILTER(mark(x0), s(s(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), s(x1), x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), s(mark(x1)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), s(x1), x2)) -> c10(A__FILTER(a__sieve(mark(z0)), s(mark(x1)), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), s(x1), x2)) -> c10(A__FILTER(a__nats(mark(z0)), s(mark(x1)), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, s(x1), x2)) -> c10(A__FILTER(a__zprimes, s(mark(x1)), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), s(x1), x2)) -> c10(A__FILTER(cons(mark(z0), z1), s(mark(x1)), mark(x2)), MARK(cons(z0, z1))) MARK(filter(0, s(x1), x2)) -> c10(A__FILTER(0, s(mark(x1)), mark(x2)), MARK(0)) MARK(filter(s(z0), s(x1), x2)) -> c10(A__FILTER(s(mark(z0)), s(mark(x1)), mark(x2)), MARK(s(z0))) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c14_2, c15_1, c16_1, c18_1, c8_1, c10_2, c10_1, c11_2, c11_1, c12_2, c12_1, c13_2, c13_1, c8_2 ---------------------------------------- (121) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(filter(0, s(x1), x2)) -> c10(A__FILTER(0, s(mark(x1)), mark(x2)), MARK(0)) ---------------------------------------- (122) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, x2)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, x2)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, x1, x2)) -> c10(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, x2)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x1)) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(x0, 0, filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, sieve(x2))) -> c10(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x0)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, sieve(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, sieve(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, sieve(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(0)), MARK(x0)) MARK(filter(x0, x1, sieve(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), sieve(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__sieve(mark(x2))), MARK(x0)) 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sieve(x1), x2)) -> c10(MARK(s(z0))) MARK(filter(x0, nats(filter(z0, z1, z2)), x2)) -> c10(A__FILTER(mark(x0), a__nats(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(x0)) MARK(filter(x0, nats(sieve(z0)), x2)) -> c10(A__FILTER(mark(x0), a__nats(a__sieve(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, nats(nats(z0)), x2)) -> c10(A__FILTER(mark(x0), a__nats(a__nats(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, nats(zprimes), x2)) -> c10(A__FILTER(mark(x0), a__nats(a__zprimes), mark(x2)), MARK(x0)) MARK(filter(x0, nats(cons(z0, z1)), x2)) -> c10(A__FILTER(mark(x0), a__nats(cons(mark(z0), z1)), mark(x2)), MARK(x0)) MARK(filter(x0, nats(0), x2)) -> c10(A__FILTER(mark(x0), a__nats(0), mark(x2)), MARK(x0)) MARK(filter(x0, nats(s(z0)), x2)) -> c10(A__FILTER(mark(x0), a__nats(s(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), nats(x1), x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x1)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), nats(x1), x2)) -> c10(A__FILTER(a__sieve(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), nats(x1), x2)) -> c10(A__FILTER(a__nats(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, nats(x1), x2)) -> c10(A__FILTER(a__zprimes, a__nats(mark(x1)), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), nats(x1), x2)) -> c10(A__FILTER(cons(mark(z0), z1), a__nats(mark(x1)), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, nats(x1), x2)) -> c10(MARK(x0)) MARK(filter(s(z0), nats(x1), x2)) -> c10(MARK(s(z0))) MARK(filter(x0, zprimes, x1)) -> c10(A__FILTER(mark(x0), a__sieve(a__nats(s(s(0)))), mark(x1)), MARK(x0)) MARK(filter(filter(z0, z1, z2), zprimes, x1)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__zprimes, mark(x1)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), zprimes, x1)) -> c10(A__FILTER(a__sieve(mark(z0)), a__zprimes, mark(x1)), MARK(sieve(z0))) MARK(filter(nats(z0), zprimes, x1)) -> c10(A__FILTER(a__nats(mark(z0)), a__zprimes, mark(x1)), MARK(nats(z0))) MARK(filter(zprimes, zprimes, x1)) -> c10(A__FILTER(a__zprimes, a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(cons(z0, z1), zprimes, x1)) -> c10(A__FILTER(cons(mark(z0), z1), a__zprimes, mark(x1)), MARK(cons(z0, z1))) MARK(filter(x0, zprimes, x1)) -> c10(MARK(x0)) MARK(filter(s(z0), zprimes, x1)) -> c10(MARK(s(z0))) MARK(filter(x0, s(filter(z0, z1, z2)), x2)) -> c10(A__FILTER(mark(x0), s(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(x0)) MARK(filter(x0, s(sieve(z0)), x2)) -> c10(A__FILTER(mark(x0), s(a__sieve(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, s(nats(z0)), x2)) -> c10(A__FILTER(mark(x0), s(a__nats(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, s(zprimes), x2)) -> c10(A__FILTER(mark(x0), s(a__zprimes), mark(x2)), MARK(x0)) MARK(filter(x0, s(cons(z0, z1)), x2)) -> c10(A__FILTER(mark(x0), s(cons(mark(z0), z1)), mark(x2)), MARK(x0)) MARK(filter(x0, s(0), x2)) -> c10(A__FILTER(mark(x0), s(0), mark(x2)), MARK(x0)) MARK(filter(x0, s(s(z0)), x2)) -> c10(A__FILTER(mark(x0), s(s(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), s(x1), x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), s(mark(x1)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), s(x1), x2)) -> c10(A__FILTER(a__sieve(mark(z0)), s(mark(x1)), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), s(x1), x2)) -> c10(A__FILTER(a__nats(mark(z0)), s(mark(x1)), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, s(x1), x2)) -> c10(A__FILTER(a__zprimes, s(mark(x1)), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), s(x1), x2)) -> c10(A__FILTER(cons(mark(z0), z1), s(mark(x1)), mark(x2)), MARK(cons(z0, z1))) MARK(filter(s(z0), s(x1), x2)) -> c10(A__FILTER(s(mark(z0)), s(mark(x1)), mark(x2)), MARK(s(z0))) S tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, x2)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, x2)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, x1, x2)) -> c10(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, x2)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x1)) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, 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MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> 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c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__sieve(mark(z0))), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, nats(z0)), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__nats(mark(z0))), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, zprimes), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__zprimes), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, cons(z0, z1)), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), cons(mark(z0), z1)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, 0), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), 0), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, s(z0)), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), s(mark(z0))), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, filter(z0, z1, z2), x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), a__filter(mark(z0), mark(z1), mark(z2)), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, sieve(z0), x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), a__sieve(mark(z0)), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, nats(z0), x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), a__nats(mark(z0)), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, zprimes, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), a__zprimes, mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, cons(z0, z1), x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), cons(mark(z0), z1), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, 0, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), 0, mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, s(z0), x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), s(mark(z0)), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(filter(z0, z1, z2), x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(sieve(z0), x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(a__sieve(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(nats(z0), x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(a__nats(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(zprimes, x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(a__zprimes, mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(cons(z0, z1), x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(cons(mark(z0), z1), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(0, x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(0, mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(s(z0), x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(s(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(filter(z0, z1, z2), filter(x1, x2, x3), x4)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), filter(x1, x2, x3), x4)) -> c10(A__FILTER(a__sieve(mark(z0)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(sieve(z0))) MARK(filter(nats(z0), filter(x1, x2, x3), x4)) -> c10(A__FILTER(a__nats(mark(z0)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(nats(z0))) MARK(filter(zprimes, filter(x1, x2, x3), x4)) -> c10(A__FILTER(a__zprimes, a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(zprimes)) MARK(filter(cons(z0, z1), filter(x1, x2, x3), x4)) -> c10(A__FILTER(cons(mark(z0), z1), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(cons(z0, z1))) MARK(filter(x0, filter(x1, x2, x3), x4)) -> c10(MARK(x0)) MARK(filter(s(z0), filter(x1, x2, x3), x4)) -> c10(MARK(s(z0))) MARK(filter(x0, sieve(filter(z0, z1, z2)), x2)) -> c10(A__FILTER(mark(x0), a__sieve(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(sieve(z0)), x2)) -> c10(A__FILTER(mark(x0), a__sieve(a__sieve(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(nats(z0)), x2)) -> c10(A__FILTER(mark(x0), a__sieve(a__nats(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(zprimes), x2)) -> c10(A__FILTER(mark(x0), a__sieve(a__zprimes), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(cons(z0, z1)), x2)) -> c10(A__FILTER(mark(x0), a__sieve(cons(mark(z0), z1)), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(0), x2)) -> c10(A__FILTER(mark(x0), a__sieve(0), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(s(z0)), x2)) -> c10(A__FILTER(mark(x0), a__sieve(s(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), sieve(x1), x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x1)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), sieve(x1), x2)) -> c10(A__FILTER(a__sieve(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), sieve(x1), x2)) -> c10(A__FILTER(a__nats(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, sieve(x1), x2)) -> c10(A__FILTER(a__zprimes, a__sieve(mark(x1)), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), sieve(x1), x2)) -> c10(A__FILTER(cons(mark(z0), z1), a__sieve(mark(x1)), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, sieve(x1), x2)) -> c10(MARK(x0)) MARK(filter(s(z0), sieve(x1), x2)) -> c10(MARK(s(z0))) MARK(filter(x0, nats(filter(z0, z1, z2)), x2)) -> c10(A__FILTER(mark(x0), a__nats(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(x0)) MARK(filter(x0, nats(sieve(z0)), x2)) -> c10(A__FILTER(mark(x0), a__nats(a__sieve(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, nats(nats(z0)), x2)) -> c10(A__FILTER(mark(x0), a__nats(a__nats(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, nats(zprimes), x2)) -> c10(A__FILTER(mark(x0), a__nats(a__zprimes), mark(x2)), MARK(x0)) MARK(filter(x0, nats(cons(z0, z1)), x2)) -> c10(A__FILTER(mark(x0), a__nats(cons(mark(z0), z1)), mark(x2)), MARK(x0)) MARK(filter(x0, nats(0), x2)) -> c10(A__FILTER(mark(x0), a__nats(0), mark(x2)), MARK(x0)) MARK(filter(x0, nats(s(z0)), x2)) -> c10(A__FILTER(mark(x0), a__nats(s(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), nats(x1), x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x1)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), nats(x1), x2)) -> c10(A__FILTER(a__sieve(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), nats(x1), x2)) -> c10(A__FILTER(a__nats(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, nats(x1), x2)) -> c10(A__FILTER(a__zprimes, a__nats(mark(x1)), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), nats(x1), x2)) -> c10(A__FILTER(cons(mark(z0), z1), a__nats(mark(x1)), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, nats(x1), x2)) -> c10(MARK(x0)) MARK(filter(s(z0), nats(x1), x2)) -> c10(MARK(s(z0))) MARK(filter(x0, zprimes, x1)) -> c10(A__FILTER(mark(x0), a__sieve(a__nats(s(s(0)))), mark(x1)), MARK(x0)) MARK(filter(filter(z0, z1, z2), zprimes, x1)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__zprimes, mark(x1)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), zprimes, x1)) -> c10(A__FILTER(a__sieve(mark(z0)), a__zprimes, mark(x1)), MARK(sieve(z0))) MARK(filter(nats(z0), zprimes, x1)) -> c10(A__FILTER(a__nats(mark(z0)), a__zprimes, mark(x1)), MARK(nats(z0))) MARK(filter(zprimes, zprimes, x1)) -> c10(A__FILTER(a__zprimes, a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(cons(z0, z1), zprimes, x1)) -> c10(A__FILTER(cons(mark(z0), z1), a__zprimes, mark(x1)), MARK(cons(z0, z1))) MARK(filter(x0, zprimes, x1)) -> c10(MARK(x0)) MARK(filter(s(z0), zprimes, x1)) -> c10(MARK(s(z0))) MARK(filter(x0, s(filter(z0, z1, z2)), x2)) -> c10(A__FILTER(mark(x0), s(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(x0)) MARK(filter(x0, s(sieve(z0)), x2)) -> c10(A__FILTER(mark(x0), s(a__sieve(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, s(nats(z0)), x2)) -> c10(A__FILTER(mark(x0), s(a__nats(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, s(zprimes), x2)) -> c10(A__FILTER(mark(x0), s(a__zprimes), mark(x2)), MARK(x0)) MARK(filter(x0, s(cons(z0, z1)), x2)) -> c10(A__FILTER(mark(x0), s(cons(mark(z0), z1)), mark(x2)), MARK(x0)) MARK(filter(x0, s(0), x2)) -> c10(A__FILTER(mark(x0), s(0), mark(x2)), MARK(x0)) MARK(filter(x0, s(s(z0)), x2)) -> c10(A__FILTER(mark(x0), s(s(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), s(x1), x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), s(mark(x1)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), s(x1), x2)) -> c10(A__FILTER(a__sieve(mark(z0)), s(mark(x1)), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), s(x1), x2)) -> c10(A__FILTER(a__nats(mark(z0)), s(mark(x1)), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, s(x1), x2)) -> c10(A__FILTER(a__zprimes, s(mark(x1)), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), s(x1), x2)) -> c10(A__FILTER(cons(mark(z0), z1), s(mark(x1)), mark(x2)), MARK(cons(z0, z1))) MARK(filter(s(z0), s(x1), x2)) -> c10(A__FILTER(s(mark(z0)), s(mark(x1)), mark(x2)), MARK(s(z0))) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c14_2, c15_1, c16_1, c18_1, c8_1, c10_2, c10_1, c11_2, c11_1, c12_2, c12_1, c13_2, c13_1, c8_2 ---------------------------------------- (123) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (124) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, x2)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, x2)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, x1, x2)) -> c10(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, x2)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x1)) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), 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MARK(filter(filter(z0, z1, z2), nats(x1), x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x1)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), nats(x1), x2)) -> c10(A__FILTER(a__sieve(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), nats(x1), x2)) -> c10(A__FILTER(a__nats(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, nats(x1), x2)) -> c10(A__FILTER(a__zprimes, a__nats(mark(x1)), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), nats(x1), x2)) -> c10(A__FILTER(cons(mark(z0), z1), a__nats(mark(x1)), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, nats(x1), x2)) -> c10(MARK(x0)) MARK(filter(s(z0), nats(x1), x2)) -> c10(MARK(s(z0))) MARK(filter(x0, zprimes, x1)) -> c10(A__FILTER(mark(x0), a__sieve(a__nats(s(s(0)))), mark(x1)), MARK(x0)) MARK(filter(filter(z0, z1, z2), zprimes, x1)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__zprimes, mark(x1)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), zprimes, x1)) -> c10(A__FILTER(a__sieve(mark(z0)), a__zprimes, mark(x1)), MARK(sieve(z0))) MARK(filter(nats(z0), zprimes, x1)) -> c10(A__FILTER(a__nats(mark(z0)), a__zprimes, mark(x1)), MARK(nats(z0))) MARK(filter(zprimes, zprimes, x1)) -> c10(A__FILTER(a__zprimes, a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(cons(z0, z1), zprimes, x1)) -> c10(A__FILTER(cons(mark(z0), z1), a__zprimes, mark(x1)), MARK(cons(z0, z1))) MARK(filter(x0, zprimes, x1)) -> c10(MARK(x0)) MARK(filter(s(z0), zprimes, x1)) -> c10(MARK(s(z0))) MARK(filter(x0, s(filter(z0, z1, z2)), x2)) -> c10(A__FILTER(mark(x0), s(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(x0)) MARK(filter(x0, s(sieve(z0)), x2)) -> c10(A__FILTER(mark(x0), s(a__sieve(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, s(nats(z0)), x2)) -> c10(A__FILTER(mark(x0), s(a__nats(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, s(zprimes), x2)) -> c10(A__FILTER(mark(x0), s(a__zprimes), mark(x2)), MARK(x0)) MARK(filter(x0, s(cons(z0, z1)), x2)) -> c10(A__FILTER(mark(x0), s(cons(mark(z0), z1)), mark(x2)), MARK(x0)) MARK(filter(x0, s(0), x2)) -> c10(A__FILTER(mark(x0), s(0), mark(x2)), MARK(x0)) MARK(filter(x0, s(s(z0)), x2)) -> c10(A__FILTER(mark(x0), s(s(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), s(x1), x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), s(mark(x1)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), s(x1), x2)) -> c10(A__FILTER(a__sieve(mark(z0)), s(mark(x1)), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), s(x1), x2)) -> c10(A__FILTER(a__nats(mark(z0)), s(mark(x1)), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, s(x1), x2)) -> c10(A__FILTER(a__zprimes, s(mark(x1)), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), s(x1), x2)) -> c10(A__FILTER(cons(mark(z0), z1), s(mark(x1)), mark(x2)), MARK(cons(z0, z1))) MARK(filter(s(z0), s(x1), x2)) -> c10(MARK(s(z0))) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c14_2, c15_1, c16_1, c18_1, c8_1, c10_2, c10_1, c11_2, c11_1, c12_2, c12_1, c13_2, c13_1, c8_2 ---------------------------------------- (125) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(filter(filter(z0, z1, z2), x1, x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(filter(z0, z1, z2))) by MARK(filter(filter(x0, x1, x2), x3, filter(z0, z1, z2))) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), mark(x2)), mark(x3), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(x0, x1, x2))) MARK(filter(filter(x0, x1, x2), x3, sieve(z0))) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), mark(x2)), mark(x3), a__sieve(mark(z0))), MARK(filter(x0, x1, x2))) MARK(filter(filter(x0, x1, x2), x3, nats(z0))) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), mark(x2)), mark(x3), a__nats(mark(z0))), MARK(filter(x0, x1, x2))) MARK(filter(filter(x0, x1, x2), x3, zprimes)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), mark(x2)), mark(x3), a__zprimes), MARK(filter(x0, x1, x2))) MARK(filter(filter(x0, x1, x2), x3, cons(z0, z1))) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), mark(x2)), mark(x3), cons(mark(z0), z1)), MARK(filter(x0, x1, x2))) MARK(filter(filter(x0, x1, x2), x3, 0)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), mark(x2)), mark(x3), 0), MARK(filter(x0, x1, x2))) MARK(filter(filter(x0, x1, x2), x3, s(z0))) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), mark(x2)), mark(x3), s(mark(z0))), MARK(filter(x0, x1, x2))) MARK(filter(filter(x0, x1, x2), filter(z0, z1, z2), x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), mark(x2)), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4)), MARK(filter(x0, x1, x2))) MARK(filter(filter(x0, x1, x2), sieve(z0), x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), mark(x2)), a__sieve(mark(z0)), mark(x4)), MARK(filter(x0, x1, x2))) MARK(filter(filter(x0, x1, x2), nats(z0), x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), mark(x2)), a__nats(mark(z0)), mark(x4)), MARK(filter(x0, x1, x2))) MARK(filter(filter(x0, x1, x2), zprimes, x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), mark(x2)), a__zprimes, mark(x4)), MARK(filter(x0, x1, x2))) MARK(filter(filter(x0, x1, x2), cons(z0, z1), x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), mark(x2)), cons(mark(z0), z1), mark(x4)), MARK(filter(x0, x1, x2))) MARK(filter(filter(x0, x1, x2), 0, x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), mark(x2)), 0, mark(x4)), MARK(filter(x0, x1, x2))) MARK(filter(filter(x0, x1, x2), s(z0), x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), mark(x2)), s(mark(z0)), mark(x4)), MARK(filter(x0, x1, x2))) MARK(filter(filter(x0, x1, x2), x3, x4)) -> c10(A__FILTER(filter(mark(x0), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, x1, x2))) MARK(filter(filter(x0, x1, filter(z0, z1, z2)), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), mark(x3), mark(x4)), MARK(filter(x0, x1, filter(z0, z1, z2)))) MARK(filter(filter(x0, x1, sieve(z0)), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), a__sieve(mark(z0))), mark(x3), mark(x4)), MARK(filter(x0, x1, sieve(z0)))) MARK(filter(filter(x0, x1, nats(z0)), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), a__nats(mark(z0))), mark(x3), mark(x4)), MARK(filter(x0, x1, nats(z0)))) MARK(filter(filter(x0, x1, zprimes), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), a__zprimes), mark(x3), mark(x4)), MARK(filter(x0, x1, zprimes))) MARK(filter(filter(x0, x1, cons(z0, z1)), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), cons(mark(z0), z1)), mark(x3), mark(x4)), MARK(filter(x0, x1, cons(z0, z1)))) MARK(filter(filter(x0, x1, 0), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), 0), mark(x3), mark(x4)), MARK(filter(x0, x1, 0))) MARK(filter(filter(x0, x1, s(z0)), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), s(mark(z0))), mark(x3), mark(x4)), MARK(filter(x0, x1, s(z0)))) MARK(filter(filter(x0, filter(z0, z1, z2), x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, filter(z0, z1, z2), x2))) MARK(filter(filter(x0, sieve(z0), x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), a__sieve(mark(z0)), mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, sieve(z0), x2))) MARK(filter(filter(x0, nats(z0), x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), a__nats(mark(z0)), mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, nats(z0), x2))) MARK(filter(filter(x0, zprimes, x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), a__zprimes, mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, zprimes, x2))) MARK(filter(filter(x0, cons(z0, z1), x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), cons(mark(z0), z1), mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, cons(z0, z1), x2))) MARK(filter(filter(x0, 0, x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), 0, mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, 0, x2))) MARK(filter(filter(x0, s(z0), x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), s(mark(z0)), mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, s(z0), x2))) MARK(filter(filter(filter(z0, z1, z2), x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(filter(z0, z1, z2), x1, x2))) MARK(filter(filter(sieve(z0), x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(a__sieve(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(sieve(z0), x1, x2))) MARK(filter(filter(nats(z0), x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(a__nats(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(nats(z0), x1, x2))) MARK(filter(filter(zprimes, x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(a__zprimes, mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(zprimes, x1, x2))) MARK(filter(filter(cons(z0, z1), x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(cons(mark(z0), z1), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(cons(z0, z1), x1, x2))) MARK(filter(filter(0, x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(0, mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(0, x1, x2))) MARK(filter(filter(s(z0), x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(s(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(s(z0), x1, x2))) ---------------------------------------- (126) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(sieve(z0), x1, x2)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, x2)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, x1, x2)) -> c10(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, x2)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x1)) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), 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MARK(zprimes)) MARK(filter(cons(z0, z1), filter(x1, x2, x3), x4)) -> c10(A__FILTER(cons(mark(z0), z1), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(cons(z0, z1))) MARK(filter(x0, filter(x1, x2, x3), x4)) -> c10(MARK(x0)) MARK(filter(s(z0), filter(x1, x2, x3), x4)) -> c10(MARK(s(z0))) MARK(filter(x0, sieve(filter(z0, z1, z2)), x2)) -> c10(A__FILTER(mark(x0), a__sieve(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(sieve(z0)), x2)) -> c10(A__FILTER(mark(x0), a__sieve(a__sieve(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(nats(z0)), x2)) -> c10(A__FILTER(mark(x0), a__sieve(a__nats(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(zprimes), x2)) -> c10(A__FILTER(mark(x0), a__sieve(a__zprimes), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(cons(z0, z1)), x2)) -> c10(A__FILTER(mark(x0), a__sieve(cons(mark(z0), z1)), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(0), x2)) -> c10(A__FILTER(mark(x0), a__sieve(0), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(s(z0)), x2)) -> c10(A__FILTER(mark(x0), a__sieve(s(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), sieve(x1), x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x1)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), sieve(x1), x2)) -> c10(A__FILTER(a__sieve(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), sieve(x1), x2)) -> c10(A__FILTER(a__nats(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, sieve(x1), x2)) -> c10(A__FILTER(a__zprimes, a__sieve(mark(x1)), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), sieve(x1), x2)) -> c10(A__FILTER(cons(mark(z0), z1), a__sieve(mark(x1)), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, sieve(x1), x2)) -> c10(MARK(x0)) MARK(filter(s(z0), sieve(x1), x2)) -> c10(MARK(s(z0))) MARK(filter(x0, nats(filter(z0, z1, z2)), x2)) -> c10(A__FILTER(mark(x0), a__nats(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(x0)) MARK(filter(x0, nats(sieve(z0)), x2)) -> c10(A__FILTER(mark(x0), a__nats(a__sieve(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, nats(nats(z0)), x2)) -> c10(A__FILTER(mark(x0), a__nats(a__nats(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, nats(zprimes), x2)) -> c10(A__FILTER(mark(x0), a__nats(a__zprimes), mark(x2)), MARK(x0)) MARK(filter(x0, nats(cons(z0, z1)), x2)) -> c10(A__FILTER(mark(x0), a__nats(cons(mark(z0), z1)), mark(x2)), MARK(x0)) MARK(filter(x0, nats(0), x2)) -> c10(A__FILTER(mark(x0), a__nats(0), mark(x2)), MARK(x0)) MARK(filter(x0, nats(s(z0)), x2)) -> c10(A__FILTER(mark(x0), a__nats(s(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), nats(x1), x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x1)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), nats(x1), x2)) -> c10(A__FILTER(a__sieve(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), nats(x1), x2)) -> c10(A__FILTER(a__nats(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, nats(x1), x2)) -> c10(A__FILTER(a__zprimes, a__nats(mark(x1)), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), nats(x1), x2)) -> c10(A__FILTER(cons(mark(z0), z1), a__nats(mark(x1)), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, nats(x1), x2)) -> c10(MARK(x0)) MARK(filter(s(z0), nats(x1), x2)) -> c10(MARK(s(z0))) MARK(filter(x0, zprimes, x1)) -> c10(A__FILTER(mark(x0), a__sieve(a__nats(s(s(0)))), mark(x1)), MARK(x0)) MARK(filter(filter(z0, z1, z2), zprimes, x1)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__zprimes, mark(x1)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), zprimes, x1)) -> c10(A__FILTER(a__sieve(mark(z0)), a__zprimes, mark(x1)), MARK(sieve(z0))) MARK(filter(nats(z0), zprimes, x1)) -> c10(A__FILTER(a__nats(mark(z0)), a__zprimes, mark(x1)), MARK(nats(z0))) MARK(filter(zprimes, zprimes, x1)) -> c10(A__FILTER(a__zprimes, a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(cons(z0, z1), zprimes, x1)) -> c10(A__FILTER(cons(mark(z0), z1), a__zprimes, mark(x1)), MARK(cons(z0, z1))) MARK(filter(x0, zprimes, x1)) -> c10(MARK(x0)) MARK(filter(s(z0), zprimes, x1)) -> c10(MARK(s(z0))) MARK(filter(x0, s(filter(z0, z1, z2)), x2)) -> c10(A__FILTER(mark(x0), s(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(x0)) MARK(filter(x0, s(sieve(z0)), x2)) -> c10(A__FILTER(mark(x0), s(a__sieve(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, s(nats(z0)), x2)) -> c10(A__FILTER(mark(x0), s(a__nats(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, s(zprimes), x2)) -> c10(A__FILTER(mark(x0), s(a__zprimes), mark(x2)), MARK(x0)) MARK(filter(x0, s(cons(z0, z1)), x2)) -> c10(A__FILTER(mark(x0), s(cons(mark(z0), z1)), mark(x2)), MARK(x0)) MARK(filter(x0, s(0), x2)) -> c10(A__FILTER(mark(x0), s(0), mark(x2)), MARK(x0)) MARK(filter(x0, s(s(z0)), x2)) -> c10(A__FILTER(mark(x0), s(s(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), s(x1), x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), s(mark(x1)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), s(x1), x2)) -> c10(A__FILTER(a__sieve(mark(z0)), s(mark(x1)), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), s(x1), x2)) -> c10(A__FILTER(a__nats(mark(z0)), s(mark(x1)), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, s(x1), x2)) -> c10(A__FILTER(a__zprimes, s(mark(x1)), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), s(x1), x2)) -> c10(A__FILTER(cons(mark(z0), z1), s(mark(x1)), mark(x2)), MARK(cons(z0, z1))) MARK(filter(s(z0), s(x1), x2)) -> c10(MARK(s(z0))) MARK(filter(filter(x0, x1, x2), cons(z0, z1), x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), mark(x2)), cons(mark(z0), z1), mark(x4)), MARK(filter(x0, x1, x2))) MARK(filter(filter(x0, x1, x2), 0, x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), mark(x2)), 0, mark(x4)), MARK(filter(x0, x1, x2))) MARK(filter(filter(x0, x1, x2), x3, x4)) -> c10(A__FILTER(filter(mark(x0), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, x1, x2))) MARK(filter(filter(x0, x1, filter(z0, z1, z2)), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), mark(x3), mark(x4)), MARK(filter(x0, x1, filter(z0, z1, z2)))) MARK(filter(filter(x0, x1, sieve(z0)), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), a__sieve(mark(z0))), mark(x3), mark(x4)), MARK(filter(x0, x1, sieve(z0)))) MARK(filter(filter(x0, x1, nats(z0)), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), a__nats(mark(z0))), mark(x3), mark(x4)), MARK(filter(x0, x1, nats(z0)))) MARK(filter(filter(x0, x1, zprimes), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), a__zprimes), mark(x3), mark(x4)), MARK(filter(x0, x1, zprimes))) MARK(filter(filter(x0, x1, cons(z0, z1)), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), cons(mark(z0), z1)), mark(x3), mark(x4)), MARK(filter(x0, x1, cons(z0, z1)))) MARK(filter(filter(x0, x1, 0), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), 0), mark(x3), mark(x4)), MARK(filter(x0, x1, 0))) MARK(filter(filter(x0, x1, s(z0)), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), s(mark(z0))), mark(x3), mark(x4)), MARK(filter(x0, x1, s(z0)))) MARK(filter(filter(x0, filter(z0, z1, z2), x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, filter(z0, z1, z2), x2))) MARK(filter(filter(x0, sieve(z0), x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), a__sieve(mark(z0)), mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, sieve(z0), x2))) MARK(filter(filter(x0, nats(z0), x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), a__nats(mark(z0)), mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, nats(z0), x2))) MARK(filter(filter(x0, zprimes, x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), a__zprimes, mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, zprimes, x2))) MARK(filter(filter(x0, cons(z0, z1), x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), cons(mark(z0), z1), mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, cons(z0, z1), x2))) MARK(filter(filter(x0, 0, x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), 0, mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, 0, x2))) MARK(filter(filter(x0, s(z0), x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), s(mark(z0)), mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, s(z0), x2))) MARK(filter(filter(filter(z0, z1, z2), x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(filter(z0, z1, z2), x1, x2))) MARK(filter(filter(sieve(z0), x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(a__sieve(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(sieve(z0), x1, x2))) MARK(filter(filter(nats(z0), x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(a__nats(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(nats(z0), x1, x2))) MARK(filter(filter(zprimes, x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(a__zprimes, mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(zprimes, x1, x2))) MARK(filter(filter(cons(z0, z1), x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(cons(mark(z0), z1), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(cons(z0, z1), x1, x2))) MARK(filter(filter(0, x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(0, mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(0, x1, x2))) MARK(filter(filter(s(z0), x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(s(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(s(z0), x1, x2))) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c14_2, c15_1, c16_1, c18_1, c8_1, c10_2, c10_1, c11_2, c11_1, c12_2, c12_1, c13_2, c13_1, c8_2 ---------------------------------------- (127) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (128) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(sieve(z0), x1, x2)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, x2)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, x1, x2)) -> c10(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, x2)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x1)) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, 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0), mark(x3), mark(x4)), MARK(filter(x0, x1, 0))) MARK(filter(filter(x0, x1, s(z0)), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), s(mark(z0))), mark(x3), mark(x4)), MARK(filter(x0, x1, s(z0)))) MARK(filter(filter(x0, filter(z0, z1, z2), x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, filter(z0, z1, z2), x2))) MARK(filter(filter(x0, sieve(z0), x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), a__sieve(mark(z0)), mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, sieve(z0), x2))) MARK(filter(filter(x0, nats(z0), x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), a__nats(mark(z0)), mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, nats(z0), x2))) MARK(filter(filter(x0, zprimes, x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), a__zprimes, mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, zprimes, x2))) MARK(filter(filter(x0, cons(z0, z1), x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), cons(mark(z0), z1), mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, cons(z0, z1), x2))) MARK(filter(filter(x0, 0, x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), 0, mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, 0, x2))) MARK(filter(filter(x0, s(z0), x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), s(mark(z0)), mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, s(z0), x2))) MARK(filter(filter(filter(z0, z1, z2), x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(filter(z0, z1, z2), x1, x2))) MARK(filter(filter(sieve(z0), x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(a__sieve(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(sieve(z0), x1, x2))) MARK(filter(filter(nats(z0), x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(a__nats(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(nats(z0), x1, x2))) MARK(filter(filter(zprimes, x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(a__zprimes, mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(zprimes, x1, x2))) MARK(filter(filter(cons(z0, z1), x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(cons(mark(z0), z1), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(cons(z0, z1), x1, x2))) MARK(filter(filter(0, x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(0, mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(0, x1, x2))) MARK(filter(filter(s(z0), x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(s(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(s(z0), x1, x2))) MARK(filter(filter(x0, x1, x2), cons(z0, z1), x4)) -> c10(MARK(filter(x0, x1, x2))) MARK(filter(filter(x0, x1, x2), 0, x4)) -> c10(MARK(filter(x0, x1, x2))) MARK(filter(filter(x0, x1, x2), x3, x4)) -> c10(MARK(filter(x0, x1, x2))) S tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(sieve(z0), x1, x2)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, x2)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, x1, x2)) -> c10(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, x2)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x1)) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), 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x4)) -> c10(A__FILTER(a__sieve(mark(z0)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(sieve(z0))) MARK(filter(nats(z0), filter(x1, x2, x3), x4)) -> c10(A__FILTER(a__nats(mark(z0)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(nats(z0))) MARK(filter(zprimes, filter(x1, x2, x3), x4)) -> c10(A__FILTER(a__zprimes, a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(zprimes)) MARK(filter(cons(z0, z1), filter(x1, x2, x3), x4)) -> c10(A__FILTER(cons(mark(z0), z1), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(cons(z0, z1))) MARK(filter(x0, filter(x1, x2, x3), x4)) -> c10(MARK(x0)) MARK(filter(s(z0), filter(x1, x2, x3), x4)) -> c10(MARK(s(z0))) MARK(filter(x0, sieve(filter(z0, z1, z2)), x2)) -> c10(A__FILTER(mark(x0), a__sieve(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(sieve(z0)), x2)) -> c10(A__FILTER(mark(x0), a__sieve(a__sieve(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(nats(z0)), x2)) -> c10(A__FILTER(mark(x0), a__sieve(a__nats(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(zprimes), x2)) -> c10(A__FILTER(mark(x0), a__sieve(a__zprimes), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(cons(z0, z1)), x2)) -> c10(A__FILTER(mark(x0), a__sieve(cons(mark(z0), z1)), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(0), x2)) -> c10(A__FILTER(mark(x0), a__sieve(0), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(s(z0)), x2)) -> c10(A__FILTER(mark(x0), a__sieve(s(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), sieve(x1), x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x1)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), sieve(x1), x2)) -> c10(A__FILTER(a__sieve(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), sieve(x1), x2)) -> c10(A__FILTER(a__nats(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, sieve(x1), x2)) -> c10(A__FILTER(a__zprimes, a__sieve(mark(x1)), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), sieve(x1), x2)) -> c10(A__FILTER(cons(mark(z0), z1), a__sieve(mark(x1)), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, sieve(x1), x2)) -> c10(MARK(x0)) MARK(filter(s(z0), sieve(x1), x2)) -> c10(MARK(s(z0))) MARK(filter(x0, nats(filter(z0, z1, z2)), x2)) -> c10(A__FILTER(mark(x0), a__nats(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(x0)) MARK(filter(x0, nats(sieve(z0)), x2)) -> c10(A__FILTER(mark(x0), a__nats(a__sieve(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, nats(nats(z0)), x2)) -> c10(A__FILTER(mark(x0), a__nats(a__nats(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, nats(zprimes), x2)) -> c10(A__FILTER(mark(x0), a__nats(a__zprimes), mark(x2)), MARK(x0)) MARK(filter(x0, nats(cons(z0, z1)), x2)) -> c10(A__FILTER(mark(x0), a__nats(cons(mark(z0), z1)), mark(x2)), MARK(x0)) MARK(filter(x0, nats(0), x2)) -> c10(A__FILTER(mark(x0), a__nats(0), mark(x2)), MARK(x0)) MARK(filter(x0, nats(s(z0)), x2)) -> c10(A__FILTER(mark(x0), a__nats(s(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), nats(x1), x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x1)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), nats(x1), x2)) -> c10(A__FILTER(a__sieve(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), nats(x1), x2)) -> c10(A__FILTER(a__nats(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, nats(x1), x2)) -> c10(A__FILTER(a__zprimes, a__nats(mark(x1)), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), nats(x1), x2)) -> c10(A__FILTER(cons(mark(z0), z1), a__nats(mark(x1)), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, nats(x1), x2)) -> c10(MARK(x0)) MARK(filter(s(z0), nats(x1), x2)) -> c10(MARK(s(z0))) MARK(filter(x0, zprimes, x1)) -> c10(A__FILTER(mark(x0), a__sieve(a__nats(s(s(0)))), mark(x1)), MARK(x0)) MARK(filter(filter(z0, z1, z2), zprimes, x1)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__zprimes, mark(x1)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), zprimes, x1)) -> c10(A__FILTER(a__sieve(mark(z0)), a__zprimes, mark(x1)), MARK(sieve(z0))) MARK(filter(nats(z0), zprimes, x1)) -> c10(A__FILTER(a__nats(mark(z0)), a__zprimes, mark(x1)), MARK(nats(z0))) MARK(filter(zprimes, zprimes, x1)) -> c10(A__FILTER(a__zprimes, a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(cons(z0, z1), zprimes, x1)) -> c10(A__FILTER(cons(mark(z0), z1), a__zprimes, mark(x1)), MARK(cons(z0, z1))) MARK(filter(x0, zprimes, x1)) -> c10(MARK(x0)) MARK(filter(s(z0), zprimes, x1)) -> c10(MARK(s(z0))) MARK(filter(x0, s(filter(z0, z1, z2)), x2)) -> c10(A__FILTER(mark(x0), s(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(x0)) MARK(filter(x0, s(sieve(z0)), x2)) -> c10(A__FILTER(mark(x0), s(a__sieve(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, s(nats(z0)), x2)) -> c10(A__FILTER(mark(x0), s(a__nats(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, s(zprimes), x2)) -> c10(A__FILTER(mark(x0), s(a__zprimes), mark(x2)), MARK(x0)) MARK(filter(x0, s(cons(z0, z1)), x2)) -> c10(A__FILTER(mark(x0), s(cons(mark(z0), z1)), mark(x2)), MARK(x0)) MARK(filter(x0, s(0), x2)) -> c10(A__FILTER(mark(x0), s(0), mark(x2)), MARK(x0)) MARK(filter(x0, s(s(z0)), x2)) -> c10(A__FILTER(mark(x0), s(s(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), s(x1), x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), s(mark(x1)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), s(x1), x2)) -> c10(A__FILTER(a__sieve(mark(z0)), s(mark(x1)), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), s(x1), x2)) -> c10(A__FILTER(a__nats(mark(z0)), s(mark(x1)), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, s(x1), x2)) -> c10(A__FILTER(a__zprimes, s(mark(x1)), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), s(x1), x2)) -> c10(A__FILTER(cons(mark(z0), z1), s(mark(x1)), mark(x2)), MARK(cons(z0, z1))) MARK(filter(s(z0), s(x1), x2)) -> c10(MARK(s(z0))) MARK(filter(filter(x0, x1, filter(z0, z1, z2)), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), mark(x3), mark(x4)), MARK(filter(x0, x1, filter(z0, z1, z2)))) MARK(filter(filter(x0, x1, sieve(z0)), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), a__sieve(mark(z0))), mark(x3), mark(x4)), MARK(filter(x0, x1, sieve(z0)))) MARK(filter(filter(x0, x1, nats(z0)), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), a__nats(mark(z0))), mark(x3), mark(x4)), MARK(filter(x0, x1, nats(z0)))) MARK(filter(filter(x0, x1, zprimes), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), a__zprimes), mark(x3), mark(x4)), MARK(filter(x0, x1, zprimes))) MARK(filter(filter(x0, x1, cons(z0, z1)), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), cons(mark(z0), z1)), mark(x3), mark(x4)), MARK(filter(x0, x1, cons(z0, z1)))) MARK(filter(filter(x0, x1, 0), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), 0), mark(x3), mark(x4)), MARK(filter(x0, x1, 0))) MARK(filter(filter(x0, x1, s(z0)), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), s(mark(z0))), mark(x3), mark(x4)), MARK(filter(x0, x1, s(z0)))) MARK(filter(filter(x0, filter(z0, z1, z2), x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, filter(z0, z1, z2), x2))) MARK(filter(filter(x0, sieve(z0), x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), a__sieve(mark(z0)), mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, sieve(z0), x2))) MARK(filter(filter(x0, nats(z0), x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), a__nats(mark(z0)), mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, nats(z0), x2))) MARK(filter(filter(x0, zprimes, x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), a__zprimes, mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, zprimes, x2))) MARK(filter(filter(x0, cons(z0, z1), x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), cons(mark(z0), z1), mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, cons(z0, z1), x2))) MARK(filter(filter(x0, 0, x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), 0, mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, 0, x2))) MARK(filter(filter(x0, s(z0), x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), s(mark(z0)), mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, s(z0), x2))) MARK(filter(filter(filter(z0, z1, z2), x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(filter(z0, z1, z2), x1, x2))) MARK(filter(filter(sieve(z0), x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(a__sieve(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(sieve(z0), x1, x2))) MARK(filter(filter(nats(z0), x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(a__nats(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(nats(z0), x1, x2))) MARK(filter(filter(zprimes, x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(a__zprimes, mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(zprimes, x1, x2))) MARK(filter(filter(cons(z0, z1), x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(cons(mark(z0), z1), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(cons(z0, z1), x1, x2))) MARK(filter(filter(0, x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(0, mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(0, x1, x2))) MARK(filter(filter(s(z0), x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(s(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(s(z0), x1, x2))) MARK(filter(filter(x0, x1, x2), cons(z0, z1), x4)) -> c10(MARK(filter(x0, x1, x2))) MARK(filter(filter(x0, x1, x2), 0, x4)) -> c10(MARK(filter(x0, x1, x2))) MARK(filter(filter(x0, x1, x2), x3, x4)) -> c10(MARK(filter(x0, x1, x2))) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c14_2, c15_1, c16_1, c18_1, c8_1, c10_2, c10_1, c11_2, c11_1, c12_2, c12_1, c13_2, c13_1, c8_2 ---------------------------------------- (129) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(filter(sieve(z0), x1, x2)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(sieve(z0))) by MARK(filter(sieve(x0), x1, filter(z0, z1, z2))) -> c10(A__FILTER(a__sieve(mark(x0)), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(sieve(x0))) MARK(filter(sieve(x0), x1, sieve(z0))) -> c10(A__FILTER(a__sieve(mark(x0)), mark(x1), a__sieve(mark(z0))), MARK(sieve(x0))) MARK(filter(sieve(x0), x1, nats(z0))) -> c10(A__FILTER(a__sieve(mark(x0)), mark(x1), a__nats(mark(z0))), MARK(sieve(x0))) MARK(filter(sieve(x0), x1, zprimes)) -> c10(A__FILTER(a__sieve(mark(x0)), mark(x1), a__zprimes), MARK(sieve(x0))) MARK(filter(sieve(x0), x1, cons(z0, z1))) -> c10(A__FILTER(a__sieve(mark(x0)), mark(x1), cons(mark(z0), z1)), MARK(sieve(x0))) MARK(filter(sieve(x0), x1, 0)) -> c10(A__FILTER(a__sieve(mark(x0)), mark(x1), 0), MARK(sieve(x0))) MARK(filter(sieve(x0), x1, s(z0))) -> c10(A__FILTER(a__sieve(mark(x0)), mark(x1), s(mark(z0))), MARK(sieve(x0))) MARK(filter(sieve(x0), filter(z0, z1, z2), x2)) -> c10(A__FILTER(a__sieve(mark(x0)), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(sieve(x0))) MARK(filter(sieve(x0), sieve(z0), x2)) -> c10(A__FILTER(a__sieve(mark(x0)), a__sieve(mark(z0)), mark(x2)), MARK(sieve(x0))) MARK(filter(sieve(x0), nats(z0), x2)) -> c10(A__FILTER(a__sieve(mark(x0)), a__nats(mark(z0)), mark(x2)), MARK(sieve(x0))) MARK(filter(sieve(x0), zprimes, x2)) -> c10(A__FILTER(a__sieve(mark(x0)), a__zprimes, mark(x2)), MARK(sieve(x0))) MARK(filter(sieve(x0), cons(z0, z1), x2)) -> c10(A__FILTER(a__sieve(mark(x0)), cons(mark(z0), z1), mark(x2)), MARK(sieve(x0))) MARK(filter(sieve(x0), 0, x2)) -> c10(A__FILTER(a__sieve(mark(x0)), 0, mark(x2)), MARK(sieve(x0))) MARK(filter(sieve(x0), s(z0), x2)) -> c10(A__FILTER(a__sieve(mark(x0)), s(mark(z0)), mark(x2)), MARK(sieve(x0))) MARK(filter(sieve(x0), x1, x2)) -> c10(A__FILTER(sieve(mark(x0)), mark(x1), mark(x2)), MARK(sieve(x0))) MARK(filter(sieve(filter(z0, z1, z2)), x1, x2)) -> c10(A__FILTER(a__sieve(a__filter(mark(z0), mark(z1), mark(z2))), mark(x1), mark(x2)), MARK(sieve(filter(z0, z1, z2)))) MARK(filter(sieve(sieve(z0)), x1, x2)) -> c10(A__FILTER(a__sieve(a__sieve(mark(z0))), mark(x1), mark(x2)), MARK(sieve(sieve(z0)))) MARK(filter(sieve(nats(z0)), x1, x2)) -> c10(A__FILTER(a__sieve(a__nats(mark(z0))), mark(x1), mark(x2)), MARK(sieve(nats(z0)))) MARK(filter(sieve(zprimes), x1, x2)) -> c10(A__FILTER(a__sieve(a__zprimes), mark(x1), mark(x2)), MARK(sieve(zprimes))) MARK(filter(sieve(cons(z0, z1)), x1, x2)) -> c10(A__FILTER(a__sieve(cons(mark(z0), z1)), mark(x1), mark(x2)), MARK(sieve(cons(z0, z1)))) MARK(filter(sieve(0), x1, x2)) -> c10(A__FILTER(a__sieve(0), mark(x1), mark(x2)), MARK(sieve(0))) MARK(filter(sieve(s(z0)), x1, x2)) -> c10(A__FILTER(a__sieve(s(mark(z0))), mark(x1), mark(x2)), MARK(sieve(s(z0)))) ---------------------------------------- (130) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(nats(z0), x1, x2)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, x1, x2)) -> c10(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, x2)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x1)) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(x0, 0, filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, sieve(x2))) -> c10(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x0)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, sieve(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, sieve(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, sieve(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(0)), MARK(x0)) MARK(filter(x0, x1, sieve(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), sieve(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, sieve(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, sieve(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__sieve(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, sieve(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, sieve(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, sieve(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), a__sieve(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, sieve(x2))) -> 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c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x1)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), nats(x1), x2)) -> c10(A__FILTER(a__sieve(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), nats(x1), x2)) -> c10(A__FILTER(a__nats(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, nats(x1), x2)) -> c10(A__FILTER(a__zprimes, a__nats(mark(x1)), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), nats(x1), x2)) -> c10(A__FILTER(cons(mark(z0), z1), a__nats(mark(x1)), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, nats(x1), x2)) -> c10(MARK(x0)) MARK(filter(s(z0), nats(x1), x2)) -> c10(MARK(s(z0))) MARK(filter(x0, zprimes, x1)) -> c10(A__FILTER(mark(x0), a__sieve(a__nats(s(s(0)))), mark(x1)), MARK(x0)) MARK(filter(filter(z0, z1, z2), zprimes, x1)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__zprimes, mark(x1)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), zprimes, x1)) -> c10(A__FILTER(a__sieve(mark(z0)), a__zprimes, mark(x1)), MARK(sieve(z0))) MARK(filter(nats(z0), zprimes, x1)) -> c10(A__FILTER(a__nats(mark(z0)), a__zprimes, mark(x1)), MARK(nats(z0))) MARK(filter(zprimes, zprimes, x1)) -> c10(A__FILTER(a__zprimes, a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(cons(z0, z1), zprimes, x1)) -> c10(A__FILTER(cons(mark(z0), z1), a__zprimes, mark(x1)), MARK(cons(z0, z1))) MARK(filter(x0, zprimes, x1)) -> c10(MARK(x0)) MARK(filter(s(z0), zprimes, x1)) -> c10(MARK(s(z0))) MARK(filter(x0, s(filter(z0, z1, z2)), x2)) -> c10(A__FILTER(mark(x0), s(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(x0)) MARK(filter(x0, s(sieve(z0)), x2)) -> c10(A__FILTER(mark(x0), s(a__sieve(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, s(nats(z0)), x2)) -> c10(A__FILTER(mark(x0), s(a__nats(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, s(zprimes), x2)) -> c10(A__FILTER(mark(x0), s(a__zprimes), mark(x2)), MARK(x0)) MARK(filter(x0, s(cons(z0, z1)), x2)) -> c10(A__FILTER(mark(x0), s(cons(mark(z0), z1)), mark(x2)), MARK(x0)) MARK(filter(x0, s(0), x2)) -> c10(A__FILTER(mark(x0), s(0), mark(x2)), MARK(x0)) MARK(filter(x0, s(s(z0)), x2)) -> c10(A__FILTER(mark(x0), s(s(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), s(x1), x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), s(mark(x1)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), s(x1), x2)) -> c10(A__FILTER(a__sieve(mark(z0)), s(mark(x1)), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), s(x1), x2)) -> c10(A__FILTER(a__nats(mark(z0)), s(mark(x1)), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, s(x1), x2)) -> c10(A__FILTER(a__zprimes, s(mark(x1)), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), s(x1), x2)) -> c10(A__FILTER(cons(mark(z0), z1), s(mark(x1)), mark(x2)), MARK(cons(z0, z1))) MARK(filter(s(z0), s(x1), x2)) -> c10(MARK(s(z0))) MARK(filter(filter(x0, x1, filter(z0, z1, z2)), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), mark(x3), mark(x4)), MARK(filter(x0, x1, filter(z0, z1, z2)))) MARK(filter(filter(x0, x1, sieve(z0)), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), a__sieve(mark(z0))), mark(x3), mark(x4)), MARK(filter(x0, x1, sieve(z0)))) MARK(filter(filter(x0, x1, nats(z0)), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), a__nats(mark(z0))), mark(x3), mark(x4)), MARK(filter(x0, x1, nats(z0)))) MARK(filter(filter(x0, x1, zprimes), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), a__zprimes), mark(x3), mark(x4)), MARK(filter(x0, x1, zprimes))) MARK(filter(filter(x0, x1, cons(z0, z1)), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), cons(mark(z0), z1)), mark(x3), mark(x4)), MARK(filter(x0, x1, cons(z0, z1)))) MARK(filter(filter(x0, x1, 0), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), 0), mark(x3), mark(x4)), MARK(filter(x0, x1, 0))) MARK(filter(filter(x0, x1, s(z0)), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), s(mark(z0))), mark(x3), mark(x4)), MARK(filter(x0, x1, s(z0)))) MARK(filter(filter(x0, filter(z0, z1, z2), x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, filter(z0, z1, z2), x2))) MARK(filter(filter(x0, sieve(z0), x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), a__sieve(mark(z0)), mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, sieve(z0), x2))) MARK(filter(filter(x0, nats(z0), x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), a__nats(mark(z0)), mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, nats(z0), x2))) MARK(filter(filter(x0, zprimes, x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), a__zprimes, mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, zprimes, x2))) MARK(filter(filter(x0, cons(z0, z1), x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), cons(mark(z0), z1), mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, cons(z0, z1), x2))) MARK(filter(filter(x0, 0, x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), 0, mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, 0, x2))) MARK(filter(filter(x0, s(z0), x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), s(mark(z0)), mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, s(z0), x2))) MARK(filter(filter(filter(z0, z1, z2), x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(filter(z0, z1, z2), x1, x2))) MARK(filter(filter(sieve(z0), x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(a__sieve(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(sieve(z0), x1, x2))) MARK(filter(filter(nats(z0), x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(a__nats(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(nats(z0), x1, x2))) MARK(filter(filter(zprimes, x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(a__zprimes, mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(zprimes, x1, x2))) MARK(filter(filter(cons(z0, z1), x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(cons(mark(z0), z1), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(cons(z0, z1), x1, x2))) MARK(filter(filter(0, x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(0, mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(0, x1, x2))) MARK(filter(filter(s(z0), x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(s(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(s(z0), x1, x2))) MARK(filter(filter(x0, x1, x2), cons(z0, z1), x4)) -> c10(MARK(filter(x0, x1, x2))) MARK(filter(filter(x0, x1, x2), 0, x4)) -> c10(MARK(filter(x0, x1, x2))) MARK(filter(filter(x0, x1, x2), x3, x4)) -> c10(MARK(filter(x0, x1, x2))) MARK(filter(sieve(x0), cons(z0, z1), x2)) -> c10(A__FILTER(a__sieve(mark(x0)), cons(mark(z0), z1), mark(x2)), MARK(sieve(x0))) MARK(filter(sieve(x0), 0, x2)) -> c10(A__FILTER(a__sieve(mark(x0)), 0, mark(x2)), MARK(sieve(x0))) MARK(filter(sieve(x0), x1, x2)) -> c10(A__FILTER(sieve(mark(x0)), mark(x1), mark(x2)), MARK(sieve(x0))) MARK(filter(sieve(filter(z0, z1, z2)), x1, x2)) -> c10(A__FILTER(a__sieve(a__filter(mark(z0), mark(z1), mark(z2))), mark(x1), mark(x2)), MARK(sieve(filter(z0, z1, z2)))) MARK(filter(sieve(sieve(z0)), x1, x2)) -> c10(A__FILTER(a__sieve(a__sieve(mark(z0))), mark(x1), mark(x2)), MARK(sieve(sieve(z0)))) MARK(filter(sieve(nats(z0)), x1, x2)) -> c10(A__FILTER(a__sieve(a__nats(mark(z0))), mark(x1), mark(x2)), MARK(sieve(nats(z0)))) MARK(filter(sieve(zprimes), x1, x2)) -> c10(A__FILTER(a__sieve(a__zprimes), mark(x1), mark(x2)), MARK(sieve(zprimes))) MARK(filter(sieve(cons(z0, z1)), x1, x2)) -> c10(A__FILTER(a__sieve(cons(mark(z0), z1)), mark(x1), mark(x2)), MARK(sieve(cons(z0, z1)))) MARK(filter(sieve(0), x1, x2)) -> c10(A__FILTER(a__sieve(0), mark(x1), mark(x2)), MARK(sieve(0))) MARK(filter(sieve(s(z0)), x1, x2)) -> c10(A__FILTER(a__sieve(s(mark(z0))), mark(x1), mark(x2)), MARK(sieve(s(z0)))) S tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(nats(z0), x1, x2)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, x1, x2)) -> c10(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, x2)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x1)) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> 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mark(x2)), mark(x3), mark(x4)), MARK(filter(zprimes, x1, x2))) MARK(filter(filter(cons(z0, z1), x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(cons(mark(z0), z1), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(cons(z0, z1), x1, x2))) MARK(filter(filter(0, x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(0, mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(0, x1, x2))) MARK(filter(filter(s(z0), x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(s(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(s(z0), x1, x2))) MARK(filter(filter(x0, x1, x2), cons(z0, z1), x4)) -> c10(MARK(filter(x0, x1, x2))) MARK(filter(filter(x0, x1, x2), 0, x4)) -> c10(MARK(filter(x0, x1, x2))) MARK(filter(filter(x0, x1, x2), x3, x4)) -> c10(MARK(filter(x0, x1, x2))) MARK(filter(sieve(x0), cons(z0, z1), x2)) -> c10(A__FILTER(a__sieve(mark(x0)), cons(mark(z0), z1), mark(x2)), MARK(sieve(x0))) MARK(filter(sieve(x0), 0, x2)) -> c10(A__FILTER(a__sieve(mark(x0)), 0, mark(x2)), MARK(sieve(x0))) MARK(filter(sieve(x0), x1, x2)) -> c10(A__FILTER(sieve(mark(x0)), mark(x1), mark(x2)), MARK(sieve(x0))) MARK(filter(sieve(filter(z0, z1, z2)), x1, x2)) -> c10(A__FILTER(a__sieve(a__filter(mark(z0), mark(z1), mark(z2))), mark(x1), mark(x2)), MARK(sieve(filter(z0, z1, z2)))) MARK(filter(sieve(sieve(z0)), x1, x2)) -> c10(A__FILTER(a__sieve(a__sieve(mark(z0))), mark(x1), mark(x2)), MARK(sieve(sieve(z0)))) MARK(filter(sieve(nats(z0)), x1, x2)) -> c10(A__FILTER(a__sieve(a__nats(mark(z0))), mark(x1), mark(x2)), MARK(sieve(nats(z0)))) MARK(filter(sieve(zprimes), x1, x2)) -> c10(A__FILTER(a__sieve(a__zprimes), mark(x1), mark(x2)), MARK(sieve(zprimes))) MARK(filter(sieve(cons(z0, z1)), x1, x2)) -> c10(A__FILTER(a__sieve(cons(mark(z0), z1)), mark(x1), mark(x2)), MARK(sieve(cons(z0, z1)))) MARK(filter(sieve(0), x1, x2)) -> c10(A__FILTER(a__sieve(0), mark(x1), mark(x2)), MARK(sieve(0))) MARK(filter(sieve(s(z0)), x1, x2)) -> c10(A__FILTER(a__sieve(s(mark(z0))), mark(x1), mark(x2)), MARK(sieve(s(z0)))) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c14_2, c15_1, c16_1, c18_1, c8_1, c10_2, c10_1, c11_2, c11_1, c12_2, c12_1, c13_2, c13_1, c8_2 ---------------------------------------- (131) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing tuple parts ---------------------------------------- (132) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(nats(z0), x1, x2)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, x1, x2)) -> c10(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, x2)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x1)) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, 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MARK(filter(sieve(z0), sieve(x1), x2)) -> c10(A__FILTER(a__sieve(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), sieve(x1), x2)) -> c10(A__FILTER(a__nats(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, sieve(x1), x2)) -> c10(A__FILTER(a__zprimes, a__sieve(mark(x1)), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), sieve(x1), x2)) -> c10(A__FILTER(cons(mark(z0), z1), a__sieve(mark(x1)), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, sieve(x1), x2)) -> c10(MARK(x0)) MARK(filter(s(z0), sieve(x1), x2)) -> c10(MARK(s(z0))) MARK(filter(x0, nats(filter(z0, z1, z2)), x2)) -> c10(A__FILTER(mark(x0), a__nats(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(x0)) MARK(filter(x0, nats(sieve(z0)), x2)) -> c10(A__FILTER(mark(x0), a__nats(a__sieve(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, nats(nats(z0)), x2)) -> c10(A__FILTER(mark(x0), a__nats(a__nats(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, nats(zprimes), x2)) -> c10(A__FILTER(mark(x0), a__nats(a__zprimes), mark(x2)), MARK(x0)) MARK(filter(x0, nats(cons(z0, z1)), x2)) -> c10(A__FILTER(mark(x0), a__nats(cons(mark(z0), z1)), mark(x2)), MARK(x0)) MARK(filter(x0, nats(0), x2)) -> c10(A__FILTER(mark(x0), a__nats(0), mark(x2)), MARK(x0)) MARK(filter(x0, nats(s(z0)), x2)) -> c10(A__FILTER(mark(x0), a__nats(s(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), nats(x1), x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x1)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), nats(x1), x2)) -> c10(A__FILTER(a__sieve(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), nats(x1), x2)) -> c10(A__FILTER(a__nats(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, nats(x1), x2)) -> c10(A__FILTER(a__zprimes, a__nats(mark(x1)), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), nats(x1), x2)) -> c10(A__FILTER(cons(mark(z0), z1), a__nats(mark(x1)), mark(x2)), 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s(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(x0)) MARK(filter(x0, s(sieve(z0)), x2)) -> c10(A__FILTER(mark(x0), s(a__sieve(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, s(nats(z0)), x2)) -> c10(A__FILTER(mark(x0), s(a__nats(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, s(zprimes), x2)) -> c10(A__FILTER(mark(x0), s(a__zprimes), mark(x2)), MARK(x0)) MARK(filter(x0, s(cons(z0, z1)), x2)) -> c10(A__FILTER(mark(x0), s(cons(mark(z0), z1)), mark(x2)), MARK(x0)) MARK(filter(x0, s(0), x2)) -> c10(A__FILTER(mark(x0), s(0), mark(x2)), MARK(x0)) MARK(filter(x0, s(s(z0)), x2)) -> c10(A__FILTER(mark(x0), s(s(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), s(x1), x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), s(mark(x1)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), s(x1), x2)) -> c10(A__FILTER(a__sieve(mark(z0)), s(mark(x1)), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), s(x1), x2)) -> c10(A__FILTER(a__nats(mark(z0)), s(mark(x1)), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, s(x1), x2)) -> c10(A__FILTER(a__zprimes, s(mark(x1)), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), s(x1), x2)) -> c10(A__FILTER(cons(mark(z0), z1), s(mark(x1)), mark(x2)), MARK(cons(z0, z1))) MARK(filter(s(z0), s(x1), x2)) -> c10(MARK(s(z0))) MARK(filter(filter(x0, x1, filter(z0, z1, z2)), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), mark(x3), mark(x4)), MARK(filter(x0, x1, filter(z0, z1, z2)))) MARK(filter(filter(x0, x1, sieve(z0)), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), a__sieve(mark(z0))), mark(x3), mark(x4)), MARK(filter(x0, x1, sieve(z0)))) MARK(filter(filter(x0, x1, nats(z0)), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), a__nats(mark(z0))), mark(x3), mark(x4)), MARK(filter(x0, x1, nats(z0)))) MARK(filter(filter(x0, x1, zprimes), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), a__zprimes), mark(x3), mark(x4)), MARK(filter(x0, 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MARK(filter(filter(x0, x1, x2), x3, x4)) -> c10(MARK(filter(x0, x1, x2))) MARK(filter(sieve(filter(z0, z1, z2)), x1, x2)) -> c10(A__FILTER(a__sieve(a__filter(mark(z0), mark(z1), mark(z2))), mark(x1), mark(x2)), MARK(sieve(filter(z0, z1, z2)))) MARK(filter(sieve(sieve(z0)), x1, x2)) -> c10(A__FILTER(a__sieve(a__sieve(mark(z0))), mark(x1), mark(x2)), MARK(sieve(sieve(z0)))) MARK(filter(sieve(nats(z0)), x1, x2)) -> c10(A__FILTER(a__sieve(a__nats(mark(z0))), mark(x1), mark(x2)), MARK(sieve(nats(z0)))) MARK(filter(sieve(zprimes), x1, x2)) -> c10(A__FILTER(a__sieve(a__zprimes), mark(x1), mark(x2)), MARK(sieve(zprimes))) MARK(filter(sieve(cons(z0, z1)), x1, x2)) -> c10(A__FILTER(a__sieve(cons(mark(z0), z1)), mark(x1), mark(x2)), MARK(sieve(cons(z0, z1)))) MARK(filter(sieve(s(z0)), x1, x2)) -> c10(A__FILTER(a__sieve(s(mark(z0))), mark(x1), mark(x2)), MARK(sieve(s(z0)))) MARK(filter(sieve(x0), cons(z0, z1), x2)) -> c10(MARK(sieve(x0))) MARK(filter(sieve(x0), 0, x2)) -> c10(MARK(sieve(x0))) MARK(filter(sieve(x0), x1, x2)) -> c10(MARK(sieve(x0))) MARK(filter(sieve(0), x1, x2)) -> c10(A__FILTER(a__sieve(0), mark(x1), mark(x2))) S tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(nats(z0), x1, x2)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, x1, x2)) -> c10(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, x2)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x1)) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> 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a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(x0, 0, filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, sieve(x2))) -> c10(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x0)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, sieve(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, sieve(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, sieve(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(0)), MARK(x0)) MARK(filter(x0, x1, sieve(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), sieve(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, sieve(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, sieve(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__sieve(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, sieve(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, sieve(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, sieve(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), a__sieve(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, sieve(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__sieve(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), sieve(x2))) -> c10(MARK(x0)) MARK(filter(x0, 0, sieve(x2))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, sieve(x2))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(mark(x2)), nats(s(mark(x2))))), MARK(x0)) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), nats(mark(x2))), MARK(x0)) MARK(filter(x0, x1, nats(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, nats(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, nats(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, nats(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, nats(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, nats(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(0)), MARK(x0)) MARK(filter(x0, x1, nats(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), nats(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), nats(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), nats(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, nats(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), nats(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, nats(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__nats(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, nats(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, nats(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, nats(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), a__nats(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, nats(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__nats(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), nats(x2))) -> c10(MARK(x0)) MARK(filter(x0, 0, nats(x2))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, nats(x2))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, zprimes)) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(s(s(0))))), MARK(x0)) MARK(filter(x0, x1, zprimes)) -> c10(A__FILTER(mark(x0), mark(x1), zprimes), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), zprimes)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__zprimes), MARK(x0)) MARK(filter(x0, sieve(z0), zprimes)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__zprimes), MARK(x0)) MARK(filter(x0, nats(z0), zprimes)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__zprimes), MARK(x0)) MARK(filter(x0, zprimes, zprimes)) -> c10(A__FILTER(mark(x0), a__zprimes, a__zprimes), MARK(x0)) MARK(filter(x0, s(z0), zprimes)) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__zprimes), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, zprimes)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__zprimes), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, zprimes)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__zprimes), MARK(sieve(z0))) MARK(filter(nats(z0), x1, zprimes)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__zprimes), MARK(nats(z0))) MARK(filter(zprimes, x1, zprimes)) -> c10(A__FILTER(a__zprimes, mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, zprimes)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__zprimes), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), zprimes)) -> c10(MARK(x0)) MARK(filter(x0, 0, zprimes)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, zprimes)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, cons(filter(z0, z1, z2), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(a__filter(mark(z0), mark(z1), mark(z2)), x3)), MARK(x0)) MARK(filter(x0, x1, cons(sieve(z0), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(a__sieve(mark(z0)), x3)), MARK(x0)) MARK(filter(x0, x1, cons(nats(z0), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(a__nats(mark(z0)), x3)), MARK(x0)) MARK(filter(x0, x1, cons(zprimes, x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(a__zprimes, x3)), MARK(x0)) MARK(filter(x0, x1, cons(cons(z0, z1), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(cons(mark(z0), z1), x3)), MARK(x0)) MARK(filter(x0, x1, cons(0, x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(0, x3)), MARK(x0)) MARK(filter(x0, x1, cons(s(z0), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(s(mark(z0)), x3)), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), cons(x2, x3))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), cons(mark(x2), x3)), MARK(x0)) MARK(filter(x0, sieve(z0), cons(x2, x3))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), cons(mark(x2), x3)), MARK(x0)) MARK(filter(x0, nats(z0), cons(x2, x3))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), cons(mark(x2), x3)), MARK(x0)) MARK(filter(x0, zprimes, cons(x2, x3))) -> c10(A__FILTER(mark(x0), a__zprimes, cons(mark(x2), x3)), MARK(x0)) MARK(filter(x0, s(z0), cons(x2, x3))) -> c10(A__FILTER(mark(x0), s(mark(z0)), cons(mark(x2), x3)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, cons(x2, x3))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), cons(mark(x2), x3)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, cons(x2, x3))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), cons(mark(x2), x3)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, cons(x2, x3))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), cons(mark(x2), x3)), MARK(nats(z0))) MARK(filter(zprimes, x1, cons(x2, x3))) -> c10(A__FILTER(a__zprimes, mark(x1), cons(mark(x2), x3)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, cons(x2, x3))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), cons(mark(x2), x3)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), cons(x2, x3))) -> c10(MARK(x0)) MARK(filter(x0, 0, cons(x2, x3))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, cons(x2, x3))) -> c10(MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), 0)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), 0), MARK(x0)) MARK(filter(x0, sieve(z0), 0)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), 0), MARK(x0)) MARK(filter(x0, nats(z0), 0)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), 0), MARK(x0)) MARK(filter(x0, zprimes, 0)) -> c10(A__FILTER(mark(x0), a__zprimes, 0), MARK(x0)) MARK(filter(x0, s(z0), 0)) -> c10(A__FILTER(mark(x0), s(mark(z0)), 0), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, 0)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), 0), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, 0)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), 0), MARK(sieve(z0))) MARK(filter(nats(z0), x1, 0)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), 0), MARK(nats(z0))) MARK(filter(zprimes, x1, 0)) -> c10(A__FILTER(a__zprimes, mark(x1), 0), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, 0)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), 0), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), 0)) -> c10(MARK(x0)) MARK(filter(x0, 0, 0)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, 0)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, s(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), s(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, s(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), s(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, s(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), s(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, s(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), s(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, s(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), s(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, s(0))) -> c10(A__FILTER(mark(x0), mark(x1), s(0)), MARK(x0)) MARK(filter(x0, x1, s(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), s(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), s(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), s(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), s(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), s(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), s(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), s(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, s(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, s(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), s(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), s(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, s(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), s(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, s(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), s(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, s(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), s(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, s(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), s(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, s(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), s(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), s(x2))) -> c10(MARK(x0)) MARK(filter(x0, 0, s(x2))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, s(x2))) -> c10(MARK(s(z0))) MARK(filter(x0, filter(x1, x2, filter(z0, z1, z2)), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__filter(mark(z0), mark(z1), mark(z2))), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, sieve(z0)), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__sieve(mark(z0))), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, nats(z0)), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__nats(mark(z0))), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, zprimes), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__zprimes), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, cons(z0, z1)), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), cons(mark(z0), z1)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, 0), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), 0), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, s(z0)), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), s(mark(z0))), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, filter(z0, z1, z2), x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), a__filter(mark(z0), mark(z1), mark(z2)), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, sieve(z0), x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), a__sieve(mark(z0)), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, nats(z0), x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), a__nats(mark(z0)), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, zprimes, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), a__zprimes, mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, cons(z0, z1), x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), cons(mark(z0), z1), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, 0, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), 0, mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, s(z0), x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), s(mark(z0)), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(filter(z0, z1, z2), x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(sieve(z0), x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(a__sieve(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(nats(z0), x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(a__nats(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(zprimes, x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(a__zprimes, mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(cons(z0, z1), x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(cons(mark(z0), z1), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(0, x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(0, mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(s(z0), x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(s(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(filter(z0, z1, z2), filter(x1, x2, x3), x4)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), filter(x1, x2, x3), x4)) -> c10(A__FILTER(a__sieve(mark(z0)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(sieve(z0))) MARK(filter(nats(z0), filter(x1, x2, x3), x4)) -> c10(A__FILTER(a__nats(mark(z0)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(nats(z0))) MARK(filter(zprimes, filter(x1, x2, x3), x4)) -> c10(A__FILTER(a__zprimes, a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(zprimes)) MARK(filter(cons(z0, z1), filter(x1, x2, x3), x4)) -> c10(A__FILTER(cons(mark(z0), z1), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(cons(z0, z1))) MARK(filter(x0, filter(x1, x2, x3), x4)) -> c10(MARK(x0)) MARK(filter(s(z0), filter(x1, x2, x3), x4)) -> c10(MARK(s(z0))) MARK(filter(x0, sieve(filter(z0, z1, z2)), x2)) -> c10(A__FILTER(mark(x0), a__sieve(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(sieve(z0)), x2)) -> c10(A__FILTER(mark(x0), a__sieve(a__sieve(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(nats(z0)), x2)) -> c10(A__FILTER(mark(x0), a__sieve(a__nats(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(zprimes), x2)) -> c10(A__FILTER(mark(x0), a__sieve(a__zprimes), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(cons(z0, z1)), x2)) -> c10(A__FILTER(mark(x0), a__sieve(cons(mark(z0), z1)), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(0), x2)) -> c10(A__FILTER(mark(x0), a__sieve(0), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(s(z0)), x2)) -> c10(A__FILTER(mark(x0), a__sieve(s(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), sieve(x1), x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x1)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), sieve(x1), x2)) -> c10(A__FILTER(a__sieve(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), sieve(x1), x2)) -> c10(A__FILTER(a__nats(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, sieve(x1), x2)) -> c10(A__FILTER(a__zprimes, a__sieve(mark(x1)), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), sieve(x1), x2)) -> c10(A__FILTER(cons(mark(z0), z1), a__sieve(mark(x1)), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, sieve(x1), x2)) -> c10(MARK(x0)) MARK(filter(s(z0), sieve(x1), x2)) -> c10(MARK(s(z0))) MARK(filter(x0, nats(filter(z0, z1, z2)), x2)) -> c10(A__FILTER(mark(x0), a__nats(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(x0)) MARK(filter(x0, nats(sieve(z0)), x2)) -> c10(A__FILTER(mark(x0), a__nats(a__sieve(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, nats(nats(z0)), x2)) -> c10(A__FILTER(mark(x0), a__nats(a__nats(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, nats(zprimes), x2)) -> c10(A__FILTER(mark(x0), a__nats(a__zprimes), mark(x2)), MARK(x0)) MARK(filter(x0, nats(cons(z0, z1)), x2)) -> c10(A__FILTER(mark(x0), a__nats(cons(mark(z0), z1)), mark(x2)), MARK(x0)) MARK(filter(x0, nats(0), x2)) -> c10(A__FILTER(mark(x0), a__nats(0), mark(x2)), MARK(x0)) MARK(filter(x0, nats(s(z0)), x2)) -> c10(A__FILTER(mark(x0), a__nats(s(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), nats(x1), x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x1)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), nats(x1), x2)) -> c10(A__FILTER(a__sieve(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), nats(x1), x2)) -> c10(A__FILTER(a__nats(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, nats(x1), x2)) -> c10(A__FILTER(a__zprimes, a__nats(mark(x1)), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), nats(x1), x2)) -> c10(A__FILTER(cons(mark(z0), z1), a__nats(mark(x1)), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, nats(x1), x2)) -> c10(MARK(x0)) MARK(filter(s(z0), nats(x1), x2)) -> c10(MARK(s(z0))) MARK(filter(x0, zprimes, x1)) -> c10(A__FILTER(mark(x0), a__sieve(a__nats(s(s(0)))), mark(x1)), MARK(x0)) MARK(filter(filter(z0, z1, z2), zprimes, x1)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__zprimes, mark(x1)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), zprimes, x1)) -> c10(A__FILTER(a__sieve(mark(z0)), a__zprimes, mark(x1)), MARK(sieve(z0))) MARK(filter(nats(z0), zprimes, x1)) -> c10(A__FILTER(a__nats(mark(z0)), a__zprimes, mark(x1)), MARK(nats(z0))) MARK(filter(zprimes, zprimes, x1)) -> c10(A__FILTER(a__zprimes, a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(cons(z0, z1), zprimes, x1)) -> c10(A__FILTER(cons(mark(z0), z1), a__zprimes, mark(x1)), MARK(cons(z0, z1))) MARK(filter(x0, zprimes, x1)) -> c10(MARK(x0)) MARK(filter(s(z0), zprimes, x1)) -> c10(MARK(s(z0))) MARK(filter(x0, s(filter(z0, z1, z2)), x2)) -> c10(A__FILTER(mark(x0), s(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(x0)) MARK(filter(x0, s(sieve(z0)), x2)) -> c10(A__FILTER(mark(x0), s(a__sieve(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, s(nats(z0)), x2)) -> c10(A__FILTER(mark(x0), s(a__nats(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, s(zprimes), x2)) -> c10(A__FILTER(mark(x0), s(a__zprimes), mark(x2)), MARK(x0)) MARK(filter(x0, s(cons(z0, z1)), x2)) -> c10(A__FILTER(mark(x0), s(cons(mark(z0), z1)), mark(x2)), MARK(x0)) MARK(filter(x0, s(0), x2)) -> c10(A__FILTER(mark(x0), s(0), mark(x2)), MARK(x0)) MARK(filter(x0, s(s(z0)), x2)) -> c10(A__FILTER(mark(x0), s(s(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), s(x1), x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), s(mark(x1)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), s(x1), x2)) -> c10(A__FILTER(a__sieve(mark(z0)), s(mark(x1)), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), s(x1), x2)) -> c10(A__FILTER(a__nats(mark(z0)), s(mark(x1)), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, s(x1), x2)) -> c10(A__FILTER(a__zprimes, s(mark(x1)), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), s(x1), x2)) -> c10(A__FILTER(cons(mark(z0), z1), s(mark(x1)), mark(x2)), MARK(cons(z0, z1))) MARK(filter(s(z0), s(x1), x2)) -> c10(MARK(s(z0))) MARK(filter(filter(x0, x1, filter(z0, z1, z2)), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), mark(x3), mark(x4)), MARK(filter(x0, x1, filter(z0, z1, z2)))) MARK(filter(filter(x0, x1, sieve(z0)), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), a__sieve(mark(z0))), mark(x3), mark(x4)), MARK(filter(x0, x1, sieve(z0)))) MARK(filter(filter(x0, x1, nats(z0)), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), a__nats(mark(z0))), mark(x3), mark(x4)), MARK(filter(x0, x1, nats(z0)))) MARK(filter(filter(x0, x1, zprimes), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), a__zprimes), mark(x3), mark(x4)), MARK(filter(x0, x1, zprimes))) MARK(filter(filter(x0, x1, cons(z0, z1)), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), cons(mark(z0), z1)), mark(x3), mark(x4)), MARK(filter(x0, x1, cons(z0, z1)))) MARK(filter(filter(x0, x1, 0), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), 0), mark(x3), mark(x4)), MARK(filter(x0, x1, 0))) MARK(filter(filter(x0, x1, s(z0)), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), s(mark(z0))), mark(x3), mark(x4)), MARK(filter(x0, x1, s(z0)))) MARK(filter(filter(x0, filter(z0, z1, z2), x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, filter(z0, z1, z2), x2))) MARK(filter(filter(x0, sieve(z0), x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), a__sieve(mark(z0)), mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, sieve(z0), x2))) MARK(filter(filter(x0, nats(z0), x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), a__nats(mark(z0)), mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, nats(z0), x2))) MARK(filter(filter(x0, zprimes, x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), a__zprimes, mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, zprimes, x2))) MARK(filter(filter(x0, cons(z0, z1), x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), cons(mark(z0), z1), mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, cons(z0, z1), x2))) MARK(filter(filter(x0, 0, x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), 0, mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, 0, x2))) MARK(filter(filter(x0, s(z0), x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), s(mark(z0)), mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, s(z0), x2))) MARK(filter(filter(filter(z0, z1, z2), x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(filter(z0, z1, z2), x1, x2))) MARK(filter(filter(sieve(z0), x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(a__sieve(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(sieve(z0), x1, x2))) MARK(filter(filter(nats(z0), x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(a__nats(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(nats(z0), x1, x2))) MARK(filter(filter(zprimes, x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(a__zprimes, mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(zprimes, x1, x2))) MARK(filter(filter(cons(z0, z1), x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(cons(mark(z0), z1), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(cons(z0, z1), x1, x2))) MARK(filter(filter(0, x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(0, mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(0, x1, x2))) MARK(filter(filter(s(z0), x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(s(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(s(z0), x1, x2))) MARK(filter(filter(x0, x1, x2), cons(z0, z1), x4)) -> c10(MARK(filter(x0, x1, x2))) MARK(filter(filter(x0, x1, x2), 0, x4)) -> c10(MARK(filter(x0, x1, x2))) MARK(filter(filter(x0, x1, x2), x3, x4)) -> c10(MARK(filter(x0, x1, x2))) MARK(filter(sieve(filter(z0, z1, z2)), x1, x2)) -> c10(A__FILTER(a__sieve(a__filter(mark(z0), mark(z1), mark(z2))), mark(x1), mark(x2)), MARK(sieve(filter(z0, z1, z2)))) MARK(filter(sieve(sieve(z0)), x1, x2)) -> c10(A__FILTER(a__sieve(a__sieve(mark(z0))), mark(x1), mark(x2)), MARK(sieve(sieve(z0)))) MARK(filter(sieve(nats(z0)), x1, x2)) -> c10(A__FILTER(a__sieve(a__nats(mark(z0))), mark(x1), mark(x2)), MARK(sieve(nats(z0)))) MARK(filter(sieve(zprimes), x1, x2)) -> c10(A__FILTER(a__sieve(a__zprimes), mark(x1), mark(x2)), MARK(sieve(zprimes))) MARK(filter(sieve(cons(z0, z1)), x1, x2)) -> c10(A__FILTER(a__sieve(cons(mark(z0), z1)), mark(x1), mark(x2)), MARK(sieve(cons(z0, z1)))) MARK(filter(sieve(s(z0)), x1, x2)) -> c10(A__FILTER(a__sieve(s(mark(z0))), mark(x1), mark(x2)), MARK(sieve(s(z0)))) MARK(filter(sieve(x0), cons(z0, z1), x2)) -> c10(MARK(sieve(x0))) MARK(filter(sieve(x0), 0, x2)) -> c10(MARK(sieve(x0))) MARK(filter(sieve(x0), x1, x2)) -> c10(MARK(sieve(x0))) MARK(filter(sieve(0), x1, x2)) -> c10(A__FILTER(a__sieve(0), mark(x1), mark(x2))) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c14_2, c15_1, c16_1, c18_1, c8_1, c10_2, c10_1, c11_2, c11_1, c12_2, c12_1, c13_2, c13_1, c8_2 ---------------------------------------- (133) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(filter(nats(z0), x1, x2)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(nats(z0))) by MARK(filter(nats(x0), x1, filter(z0, z1, z2))) -> c10(A__FILTER(a__nats(mark(x0)), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(nats(x0))) MARK(filter(nats(x0), x1, sieve(z0))) -> c10(A__FILTER(a__nats(mark(x0)), mark(x1), a__sieve(mark(z0))), MARK(nats(x0))) MARK(filter(nats(x0), x1, nats(z0))) -> c10(A__FILTER(a__nats(mark(x0)), mark(x1), a__nats(mark(z0))), MARK(nats(x0))) MARK(filter(nats(x0), x1, zprimes)) -> c10(A__FILTER(a__nats(mark(x0)), mark(x1), a__zprimes), MARK(nats(x0))) MARK(filter(nats(x0), x1, cons(z0, z1))) -> c10(A__FILTER(a__nats(mark(x0)), mark(x1), cons(mark(z0), z1)), MARK(nats(x0))) MARK(filter(nats(x0), x1, 0)) -> c10(A__FILTER(a__nats(mark(x0)), mark(x1), 0), MARK(nats(x0))) MARK(filter(nats(x0), x1, s(z0))) -> c10(A__FILTER(a__nats(mark(x0)), mark(x1), s(mark(z0))), MARK(nats(x0))) MARK(filter(nats(x0), filter(z0, z1, z2), x2)) -> c10(A__FILTER(a__nats(mark(x0)), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(nats(x0))) MARK(filter(nats(x0), sieve(z0), x2)) -> c10(A__FILTER(a__nats(mark(x0)), a__sieve(mark(z0)), mark(x2)), MARK(nats(x0))) MARK(filter(nats(x0), nats(z0), x2)) -> c10(A__FILTER(a__nats(mark(x0)), a__nats(mark(z0)), mark(x2)), MARK(nats(x0))) MARK(filter(nats(x0), zprimes, x2)) -> c10(A__FILTER(a__nats(mark(x0)), a__zprimes, mark(x2)), MARK(nats(x0))) MARK(filter(nats(x0), cons(z0, z1), x2)) -> c10(A__FILTER(a__nats(mark(x0)), cons(mark(z0), z1), mark(x2)), MARK(nats(x0))) MARK(filter(nats(x0), 0, x2)) -> c10(A__FILTER(a__nats(mark(x0)), 0, mark(x2)), MARK(nats(x0))) MARK(filter(nats(x0), s(z0), x2)) -> c10(A__FILTER(a__nats(mark(x0)), s(mark(z0)), mark(x2)), MARK(nats(x0))) MARK(filter(nats(x0), x1, x2)) -> c10(A__FILTER(cons(mark(mark(x0)), nats(s(mark(x0)))), mark(x1), mark(x2)), MARK(nats(x0))) MARK(filter(nats(x0), x1, x2)) -> c10(A__FILTER(nats(mark(x0)), mark(x1), mark(x2)), MARK(nats(x0))) MARK(filter(nats(filter(z0, z1, z2)), x1, x2)) -> c10(A__FILTER(a__nats(a__filter(mark(z0), mark(z1), mark(z2))), mark(x1), mark(x2)), MARK(nats(filter(z0, z1, z2)))) MARK(filter(nats(sieve(z0)), x1, x2)) -> c10(A__FILTER(a__nats(a__sieve(mark(z0))), mark(x1), mark(x2)), MARK(nats(sieve(z0)))) MARK(filter(nats(nats(z0)), x1, x2)) -> c10(A__FILTER(a__nats(a__nats(mark(z0))), mark(x1), mark(x2)), MARK(nats(nats(z0)))) MARK(filter(nats(zprimes), x1, x2)) -> c10(A__FILTER(a__nats(a__zprimes), mark(x1), mark(x2)), MARK(nats(zprimes))) MARK(filter(nats(cons(z0, z1)), x1, x2)) -> c10(A__FILTER(a__nats(cons(mark(z0), z1)), mark(x1), mark(x2)), MARK(nats(cons(z0, z1)))) MARK(filter(nats(0), x1, x2)) -> c10(A__FILTER(a__nats(0), mark(x1), mark(x2)), MARK(nats(0))) MARK(filter(nats(s(z0)), x1, x2)) -> c10(A__FILTER(a__nats(s(mark(z0))), mark(x1), mark(x2)), MARK(nats(s(z0)))) MARK(filter(nats(x0), x1, x2)) -> c10(MARK(nats(x0))) ---------------------------------------- (134) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(zprimes, x1, x2)) -> c10(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, x2)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x1)) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> 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c10(A__FILTER(a__filter(mark(x0), a__zprimes, mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, zprimes, x2))) MARK(filter(filter(x0, cons(z0, z1), x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), cons(mark(z0), z1), mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, cons(z0, z1), x2))) MARK(filter(filter(x0, 0, x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), 0, mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, 0, x2))) MARK(filter(filter(x0, s(z0), x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), s(mark(z0)), mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, s(z0), x2))) MARK(filter(filter(filter(z0, z1, z2), x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(filter(z0, z1, z2), x1, x2))) MARK(filter(filter(sieve(z0), x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(a__sieve(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(sieve(z0), x1, x2))) MARK(filter(filter(nats(z0), x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(a__nats(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(nats(z0), x1, x2))) MARK(filter(filter(zprimes, x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(a__zprimes, mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(zprimes, x1, x2))) MARK(filter(filter(cons(z0, z1), x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(cons(mark(z0), z1), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(cons(z0, z1), x1, x2))) MARK(filter(filter(0, x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(0, mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(0, x1, x2))) MARK(filter(filter(s(z0), x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(s(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(s(z0), x1, x2))) MARK(filter(filter(x0, x1, x2), cons(z0, z1), x4)) -> c10(MARK(filter(x0, x1, x2))) MARK(filter(filter(x0, x1, x2), 0, x4)) -> c10(MARK(filter(x0, x1, x2))) MARK(filter(filter(x0, x1, x2), x3, x4)) -> c10(MARK(filter(x0, x1, x2))) MARK(filter(sieve(filter(z0, z1, z2)), x1, x2)) -> c10(A__FILTER(a__sieve(a__filter(mark(z0), mark(z1), mark(z2))), mark(x1), mark(x2)), MARK(sieve(filter(z0, z1, z2)))) MARK(filter(sieve(sieve(z0)), x1, x2)) -> c10(A__FILTER(a__sieve(a__sieve(mark(z0))), mark(x1), mark(x2)), MARK(sieve(sieve(z0)))) MARK(filter(sieve(nats(z0)), x1, x2)) -> c10(A__FILTER(a__sieve(a__nats(mark(z0))), mark(x1), mark(x2)), MARK(sieve(nats(z0)))) MARK(filter(sieve(zprimes), x1, x2)) -> c10(A__FILTER(a__sieve(a__zprimes), mark(x1), mark(x2)), MARK(sieve(zprimes))) MARK(filter(sieve(cons(z0, z1)), x1, x2)) -> c10(A__FILTER(a__sieve(cons(mark(z0), z1)), mark(x1), mark(x2)), MARK(sieve(cons(z0, z1)))) MARK(filter(sieve(s(z0)), x1, x2)) -> c10(A__FILTER(a__sieve(s(mark(z0))), mark(x1), mark(x2)), MARK(sieve(s(z0)))) MARK(filter(sieve(x0), cons(z0, z1), x2)) -> c10(MARK(sieve(x0))) MARK(filter(sieve(x0), 0, x2)) -> c10(MARK(sieve(x0))) MARK(filter(sieve(x0), x1, x2)) -> c10(MARK(sieve(x0))) MARK(filter(sieve(0), x1, x2)) -> c10(A__FILTER(a__sieve(0), mark(x1), mark(x2))) MARK(filter(nats(x0), cons(z0, z1), x2)) -> c10(A__FILTER(a__nats(mark(x0)), cons(mark(z0), z1), mark(x2)), MARK(nats(x0))) MARK(filter(nats(x0), 0, x2)) -> c10(A__FILTER(a__nats(mark(x0)), 0, mark(x2)), MARK(nats(x0))) MARK(filter(nats(x0), x1, x2)) -> c10(A__FILTER(cons(mark(mark(x0)), nats(s(mark(x0)))), mark(x1), mark(x2)), MARK(nats(x0))) MARK(filter(nats(x0), x1, x2)) -> c10(A__FILTER(nats(mark(x0)), mark(x1), mark(x2)), MARK(nats(x0))) MARK(filter(nats(filter(z0, z1, z2)), x1, x2)) -> c10(A__FILTER(a__nats(a__filter(mark(z0), mark(z1), mark(z2))), mark(x1), mark(x2)), MARK(nats(filter(z0, z1, z2)))) MARK(filter(nats(sieve(z0)), x1, x2)) -> c10(A__FILTER(a__nats(a__sieve(mark(z0))), mark(x1), mark(x2)), MARK(nats(sieve(z0)))) MARK(filter(nats(nats(z0)), x1, x2)) -> c10(A__FILTER(a__nats(a__nats(mark(z0))), mark(x1), mark(x2)), MARK(nats(nats(z0)))) MARK(filter(nats(zprimes), x1, x2)) -> c10(A__FILTER(a__nats(a__zprimes), mark(x1), mark(x2)), MARK(nats(zprimes))) MARK(filter(nats(cons(z0, z1)), x1, x2)) -> c10(A__FILTER(a__nats(cons(mark(z0), z1)), mark(x1), mark(x2)), MARK(nats(cons(z0, z1)))) MARK(filter(nats(0), x1, x2)) -> c10(A__FILTER(a__nats(0), mark(x1), mark(x2)), MARK(nats(0))) MARK(filter(nats(s(z0)), x1, x2)) -> c10(A__FILTER(a__nats(s(mark(z0))), mark(x1), mark(x2)), MARK(nats(s(z0)))) MARK(filter(nats(x0), x1, x2)) -> c10(MARK(nats(x0))) S tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(zprimes, x1, x2)) -> c10(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, x2)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x1)) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, 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x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), s(mark(z0)), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(filter(z0, z1, z2), x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(sieve(z0), x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(a__sieve(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(nats(z0), x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(a__nats(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(zprimes, x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(a__zprimes, mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(cons(z0, z1), x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(cons(mark(z0), z1), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(0, x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(0, mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(s(z0), x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(s(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(filter(z0, z1, z2), filter(x1, x2, x3), x4)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), filter(x1, x2, x3), x4)) -> c10(A__FILTER(a__sieve(mark(z0)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(sieve(z0))) MARK(filter(nats(z0), filter(x1, x2, x3), x4)) -> c10(A__FILTER(a__nats(mark(z0)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(nats(z0))) MARK(filter(zprimes, filter(x1, x2, x3), x4)) -> c10(A__FILTER(a__zprimes, a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(zprimes)) MARK(filter(cons(z0, z1), filter(x1, x2, x3), x4)) -> c10(A__FILTER(cons(mark(z0), z1), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(cons(z0, z1))) MARK(filter(x0, filter(x1, x2, x3), x4)) -> c10(MARK(x0)) MARK(filter(s(z0), filter(x1, x2, x3), x4)) -> c10(MARK(s(z0))) MARK(filter(x0, sieve(filter(z0, z1, z2)), x2)) -> c10(A__FILTER(mark(x0), a__sieve(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(sieve(z0)), x2)) -> c10(A__FILTER(mark(x0), a__sieve(a__sieve(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(nats(z0)), x2)) -> c10(A__FILTER(mark(x0), a__sieve(a__nats(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(zprimes), x2)) -> c10(A__FILTER(mark(x0), a__sieve(a__zprimes), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(cons(z0, z1)), x2)) -> c10(A__FILTER(mark(x0), a__sieve(cons(mark(z0), z1)), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(0), x2)) -> c10(A__FILTER(mark(x0), a__sieve(0), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(s(z0)), x2)) -> c10(A__FILTER(mark(x0), a__sieve(s(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), sieve(x1), x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x1)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), sieve(x1), x2)) -> c10(A__FILTER(a__sieve(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), sieve(x1), x2)) -> c10(A__FILTER(a__nats(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, sieve(x1), x2)) -> c10(A__FILTER(a__zprimes, a__sieve(mark(x1)), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), sieve(x1), x2)) -> c10(A__FILTER(cons(mark(z0), z1), a__sieve(mark(x1)), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, sieve(x1), x2)) -> c10(MARK(x0)) MARK(filter(s(z0), sieve(x1), x2)) -> c10(MARK(s(z0))) MARK(filter(x0, nats(filter(z0, z1, z2)), x2)) -> c10(A__FILTER(mark(x0), a__nats(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(x0)) MARK(filter(x0, nats(sieve(z0)), x2)) -> c10(A__FILTER(mark(x0), a__nats(a__sieve(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, nats(nats(z0)), x2)) -> c10(A__FILTER(mark(x0), a__nats(a__nats(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, nats(zprimes), x2)) -> c10(A__FILTER(mark(x0), a__nats(a__zprimes), mark(x2)), MARK(x0)) MARK(filter(x0, nats(cons(z0, z1)), x2)) -> c10(A__FILTER(mark(x0), a__nats(cons(mark(z0), z1)), mark(x2)), MARK(x0)) MARK(filter(x0, nats(0), x2)) -> c10(A__FILTER(mark(x0), a__nats(0), mark(x2)), MARK(x0)) MARK(filter(x0, nats(s(z0)), x2)) -> c10(A__FILTER(mark(x0), a__nats(s(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), nats(x1), x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x1)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), nats(x1), x2)) -> c10(A__FILTER(a__sieve(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), nats(x1), x2)) -> c10(A__FILTER(a__nats(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, nats(x1), x2)) -> c10(A__FILTER(a__zprimes, a__nats(mark(x1)), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), nats(x1), x2)) -> c10(A__FILTER(cons(mark(z0), z1), a__nats(mark(x1)), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, nats(x1), x2)) -> c10(MARK(x0)) MARK(filter(s(z0), nats(x1), x2)) -> c10(MARK(s(z0))) MARK(filter(x0, zprimes, x1)) -> c10(A__FILTER(mark(x0), a__sieve(a__nats(s(s(0)))), mark(x1)), MARK(x0)) MARK(filter(filter(z0, z1, z2), zprimes, x1)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__zprimes, mark(x1)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), zprimes, x1)) -> c10(A__FILTER(a__sieve(mark(z0)), a__zprimes, mark(x1)), MARK(sieve(z0))) MARK(filter(nats(z0), zprimes, x1)) -> c10(A__FILTER(a__nats(mark(z0)), a__zprimes, mark(x1)), MARK(nats(z0))) MARK(filter(zprimes, zprimes, x1)) -> c10(A__FILTER(a__zprimes, a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(cons(z0, z1), zprimes, x1)) -> c10(A__FILTER(cons(mark(z0), z1), a__zprimes, mark(x1)), MARK(cons(z0, z1))) MARK(filter(x0, zprimes, x1)) -> c10(MARK(x0)) MARK(filter(s(z0), zprimes, x1)) -> c10(MARK(s(z0))) MARK(filter(x0, s(filter(z0, z1, z2)), x2)) -> c10(A__FILTER(mark(x0), s(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(x0)) MARK(filter(x0, s(sieve(z0)), x2)) -> c10(A__FILTER(mark(x0), s(a__sieve(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, s(nats(z0)), x2)) -> c10(A__FILTER(mark(x0), s(a__nats(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, s(zprimes), x2)) -> c10(A__FILTER(mark(x0), s(a__zprimes), mark(x2)), MARK(x0)) MARK(filter(x0, s(cons(z0, z1)), x2)) -> c10(A__FILTER(mark(x0), s(cons(mark(z0), z1)), mark(x2)), MARK(x0)) MARK(filter(x0, s(0), x2)) -> c10(A__FILTER(mark(x0), s(0), mark(x2)), MARK(x0)) MARK(filter(x0, s(s(z0)), x2)) -> c10(A__FILTER(mark(x0), s(s(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), s(x1), x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), s(mark(x1)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), s(x1), x2)) -> c10(A__FILTER(a__sieve(mark(z0)), s(mark(x1)), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), s(x1), x2)) -> c10(A__FILTER(a__nats(mark(z0)), s(mark(x1)), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, s(x1), x2)) -> c10(A__FILTER(a__zprimes, s(mark(x1)), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), s(x1), x2)) -> c10(A__FILTER(cons(mark(z0), z1), s(mark(x1)), mark(x2)), MARK(cons(z0, z1))) MARK(filter(s(z0), s(x1), x2)) -> c10(MARK(s(z0))) MARK(filter(filter(x0, x1, filter(z0, z1, z2)), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), mark(x3), mark(x4)), MARK(filter(x0, x1, filter(z0, z1, z2)))) MARK(filter(filter(x0, x1, sieve(z0)), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), a__sieve(mark(z0))), mark(x3), mark(x4)), MARK(filter(x0, x1, sieve(z0)))) MARK(filter(filter(x0, x1, nats(z0)), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), a__nats(mark(z0))), mark(x3), mark(x4)), MARK(filter(x0, x1, nats(z0)))) MARK(filter(filter(x0, x1, zprimes), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), a__zprimes), mark(x3), mark(x4)), MARK(filter(x0, x1, zprimes))) MARK(filter(filter(x0, x1, cons(z0, z1)), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), cons(mark(z0), z1)), mark(x3), mark(x4)), MARK(filter(x0, x1, cons(z0, z1)))) MARK(filter(filter(x0, x1, 0), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), 0), mark(x3), mark(x4)), MARK(filter(x0, x1, 0))) MARK(filter(filter(x0, x1, s(z0)), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), s(mark(z0))), mark(x3), mark(x4)), MARK(filter(x0, x1, s(z0)))) MARK(filter(filter(x0, filter(z0, z1, z2), x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, filter(z0, z1, z2), x2))) MARK(filter(filter(x0, sieve(z0), x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), a__sieve(mark(z0)), mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, sieve(z0), x2))) MARK(filter(filter(x0, nats(z0), x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), a__nats(mark(z0)), mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, nats(z0), x2))) MARK(filter(filter(x0, zprimes, x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), a__zprimes, mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, zprimes, x2))) MARK(filter(filter(x0, cons(z0, z1), x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), cons(mark(z0), z1), mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, cons(z0, z1), x2))) MARK(filter(filter(x0, 0, x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), 0, mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, 0, x2))) MARK(filter(filter(x0, s(z0), x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), s(mark(z0)), mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, s(z0), x2))) MARK(filter(filter(filter(z0, z1, z2), x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(filter(z0, z1, z2), x1, x2))) MARK(filter(filter(sieve(z0), x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(a__sieve(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(sieve(z0), x1, x2))) MARK(filter(filter(nats(z0), x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(a__nats(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(nats(z0), x1, x2))) MARK(filter(filter(zprimes, x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(a__zprimes, mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(zprimes, x1, x2))) MARK(filter(filter(cons(z0, z1), x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(cons(mark(z0), z1), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(cons(z0, z1), x1, x2))) MARK(filter(filter(0, x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(0, mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(0, x1, x2))) MARK(filter(filter(s(z0), x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(s(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(s(z0), x1, x2))) MARK(filter(filter(x0, x1, x2), cons(z0, z1), x4)) -> c10(MARK(filter(x0, x1, x2))) MARK(filter(filter(x0, x1, x2), 0, x4)) -> c10(MARK(filter(x0, x1, x2))) MARK(filter(filter(x0, x1, x2), x3, x4)) -> c10(MARK(filter(x0, x1, x2))) MARK(filter(sieve(filter(z0, z1, z2)), x1, x2)) -> c10(A__FILTER(a__sieve(a__filter(mark(z0), mark(z1), mark(z2))), mark(x1), mark(x2)), MARK(sieve(filter(z0, z1, z2)))) MARK(filter(sieve(sieve(z0)), x1, x2)) -> c10(A__FILTER(a__sieve(a__sieve(mark(z0))), mark(x1), mark(x2)), MARK(sieve(sieve(z0)))) MARK(filter(sieve(nats(z0)), x1, x2)) -> c10(A__FILTER(a__sieve(a__nats(mark(z0))), mark(x1), mark(x2)), MARK(sieve(nats(z0)))) MARK(filter(sieve(zprimes), x1, x2)) -> c10(A__FILTER(a__sieve(a__zprimes), mark(x1), mark(x2)), MARK(sieve(zprimes))) MARK(filter(sieve(cons(z0, z1)), x1, x2)) -> c10(A__FILTER(a__sieve(cons(mark(z0), z1)), mark(x1), mark(x2)), MARK(sieve(cons(z0, z1)))) MARK(filter(sieve(s(z0)), x1, x2)) -> c10(A__FILTER(a__sieve(s(mark(z0))), mark(x1), mark(x2)), MARK(sieve(s(z0)))) MARK(filter(sieve(x0), cons(z0, z1), x2)) -> c10(MARK(sieve(x0))) MARK(filter(sieve(x0), 0, x2)) -> c10(MARK(sieve(x0))) MARK(filter(sieve(x0), x1, x2)) -> c10(MARK(sieve(x0))) MARK(filter(sieve(0), x1, x2)) -> c10(A__FILTER(a__sieve(0), mark(x1), mark(x2))) MARK(filter(nats(x0), cons(z0, z1), x2)) -> c10(A__FILTER(a__nats(mark(x0)), cons(mark(z0), z1), mark(x2)), MARK(nats(x0))) MARK(filter(nats(x0), 0, x2)) -> c10(A__FILTER(a__nats(mark(x0)), 0, mark(x2)), MARK(nats(x0))) MARK(filter(nats(x0), x1, x2)) -> c10(A__FILTER(cons(mark(mark(x0)), nats(s(mark(x0)))), mark(x1), mark(x2)), MARK(nats(x0))) MARK(filter(nats(x0), x1, x2)) -> c10(A__FILTER(nats(mark(x0)), mark(x1), mark(x2)), MARK(nats(x0))) MARK(filter(nats(filter(z0, z1, z2)), x1, x2)) -> c10(A__FILTER(a__nats(a__filter(mark(z0), mark(z1), mark(z2))), mark(x1), mark(x2)), MARK(nats(filter(z0, z1, z2)))) MARK(filter(nats(sieve(z0)), x1, x2)) -> c10(A__FILTER(a__nats(a__sieve(mark(z0))), mark(x1), mark(x2)), MARK(nats(sieve(z0)))) MARK(filter(nats(nats(z0)), x1, x2)) -> c10(A__FILTER(a__nats(a__nats(mark(z0))), mark(x1), mark(x2)), MARK(nats(nats(z0)))) MARK(filter(nats(zprimes), x1, x2)) -> c10(A__FILTER(a__nats(a__zprimes), mark(x1), mark(x2)), MARK(nats(zprimes))) MARK(filter(nats(cons(z0, z1)), x1, x2)) -> c10(A__FILTER(a__nats(cons(mark(z0), z1)), mark(x1), mark(x2)), MARK(nats(cons(z0, z1)))) MARK(filter(nats(0), x1, x2)) -> c10(A__FILTER(a__nats(0), mark(x1), mark(x2)), MARK(nats(0))) MARK(filter(nats(s(z0)), x1, x2)) -> c10(A__FILTER(a__nats(s(mark(z0))), mark(x1), mark(x2)), MARK(nats(s(z0)))) MARK(filter(nats(x0), x1, x2)) -> c10(MARK(nats(x0))) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c14_2, c15_1, c16_1, c18_1, c8_1, c10_2, c10_1, c11_2, c11_1, c12_2, c12_1, c13_2, c13_1, c8_2 ---------------------------------------- (135) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (136) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(zprimes, x1, x2)) -> c10(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, x2)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x1)) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(x0, 0, filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, sieve(x2))) -> c10(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x0)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, sieve(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, sieve(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, sieve(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(0)), MARK(x0)) MARK(filter(x0, x1, sieve(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), sieve(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__sieve(mark(x2))), MARK(x0)) 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MARK(nats(sieve(z0)))) MARK(filter(nats(nats(z0)), x1, x2)) -> c10(A__FILTER(a__nats(a__nats(mark(z0))), mark(x1), mark(x2)), MARK(nats(nats(z0)))) MARK(filter(nats(zprimes), x1, x2)) -> c10(A__FILTER(a__nats(a__zprimes), mark(x1), mark(x2)), MARK(nats(zprimes))) MARK(filter(nats(cons(z0, z1)), x1, x2)) -> c10(A__FILTER(a__nats(cons(mark(z0), z1)), mark(x1), mark(x2)), MARK(nats(cons(z0, z1)))) MARK(filter(nats(0), x1, x2)) -> c10(A__FILTER(a__nats(0), mark(x1), mark(x2)), MARK(nats(0))) MARK(filter(nats(s(z0)), x1, x2)) -> c10(A__FILTER(a__nats(s(mark(z0))), mark(x1), mark(x2)), MARK(nats(s(z0)))) MARK(filter(nats(x0), x1, x2)) -> c10(MARK(nats(x0))) MARK(filter(nats(x0), cons(z0, z1), x2)) -> c10(MARK(nats(x0))) MARK(filter(nats(x0), 0, x2)) -> c10(MARK(nats(x0))) S tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) 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MARK(filter(x0, s(s(z0)), x2)) -> c10(A__FILTER(mark(x0), s(s(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), s(x1), x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), s(mark(x1)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), s(x1), x2)) -> c10(A__FILTER(a__sieve(mark(z0)), s(mark(x1)), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), s(x1), x2)) -> c10(A__FILTER(a__nats(mark(z0)), s(mark(x1)), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, s(x1), x2)) -> c10(A__FILTER(a__zprimes, s(mark(x1)), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), s(x1), x2)) -> c10(A__FILTER(cons(mark(z0), z1), s(mark(x1)), mark(x2)), MARK(cons(z0, z1))) MARK(filter(s(z0), s(x1), x2)) -> c10(MARK(s(z0))) MARK(filter(filter(x0, x1, filter(z0, z1, z2)), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), mark(x3), mark(x4)), MARK(filter(x0, x1, filter(z0, z1, z2)))) MARK(filter(filter(x0, x1, sieve(z0)), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), a__sieve(mark(z0))), mark(x3), mark(x4)), MARK(filter(x0, x1, sieve(z0)))) MARK(filter(filter(x0, x1, nats(z0)), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), a__nats(mark(z0))), mark(x3), mark(x4)), MARK(filter(x0, x1, nats(z0)))) MARK(filter(filter(x0, x1, zprimes), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), a__zprimes), mark(x3), mark(x4)), MARK(filter(x0, x1, zprimes))) MARK(filter(filter(x0, x1, cons(z0, z1)), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), cons(mark(z0), z1)), mark(x3), mark(x4)), MARK(filter(x0, x1, cons(z0, z1)))) MARK(filter(filter(x0, x1, 0), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), 0), mark(x3), mark(x4)), MARK(filter(x0, x1, 0))) MARK(filter(filter(x0, x1, s(z0)), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), s(mark(z0))), mark(x3), mark(x4)), MARK(filter(x0, x1, s(z0)))) MARK(filter(filter(x0, filter(z0, z1, z2), x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, filter(z0, z1, z2), x2))) MARK(filter(filter(x0, sieve(z0), x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), a__sieve(mark(z0)), mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, sieve(z0), x2))) MARK(filter(filter(x0, nats(z0), x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), a__nats(mark(z0)), mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, nats(z0), x2))) MARK(filter(filter(x0, zprimes, x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), a__zprimes, mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, zprimes, x2))) MARK(filter(filter(x0, cons(z0, z1), x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), cons(mark(z0), z1), mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, cons(z0, z1), x2))) MARK(filter(filter(x0, 0, x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), 0, mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, 0, x2))) MARK(filter(filter(x0, s(z0), x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), s(mark(z0)), mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, s(z0), x2))) MARK(filter(filter(filter(z0, z1, z2), x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(filter(z0, z1, z2), x1, x2))) MARK(filter(filter(sieve(z0), x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(a__sieve(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(sieve(z0), x1, x2))) MARK(filter(filter(nats(z0), x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(a__nats(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(nats(z0), x1, x2))) MARK(filter(filter(zprimes, x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(a__zprimes, mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(zprimes, x1, x2))) MARK(filter(filter(cons(z0, z1), x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(cons(mark(z0), z1), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(cons(z0, z1), x1, x2))) MARK(filter(filter(0, x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(0, mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(0, x1, x2))) MARK(filter(filter(s(z0), x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(s(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(s(z0), x1, x2))) MARK(filter(filter(x0, x1, x2), cons(z0, z1), x4)) -> c10(MARK(filter(x0, x1, x2))) MARK(filter(filter(x0, x1, x2), 0, x4)) -> c10(MARK(filter(x0, x1, x2))) MARK(filter(filter(x0, x1, x2), x3, x4)) -> c10(MARK(filter(x0, x1, x2))) MARK(filter(sieve(filter(z0, z1, z2)), x1, x2)) -> c10(A__FILTER(a__sieve(a__filter(mark(z0), mark(z1), mark(z2))), mark(x1), mark(x2)), MARK(sieve(filter(z0, z1, z2)))) MARK(filter(sieve(sieve(z0)), x1, x2)) -> c10(A__FILTER(a__sieve(a__sieve(mark(z0))), mark(x1), mark(x2)), MARK(sieve(sieve(z0)))) MARK(filter(sieve(nats(z0)), x1, x2)) -> c10(A__FILTER(a__sieve(a__nats(mark(z0))), mark(x1), mark(x2)), MARK(sieve(nats(z0)))) MARK(filter(sieve(zprimes), x1, x2)) -> c10(A__FILTER(a__sieve(a__zprimes), mark(x1), mark(x2)), MARK(sieve(zprimes))) MARK(filter(sieve(cons(z0, z1)), x1, x2)) -> c10(A__FILTER(a__sieve(cons(mark(z0), z1)), mark(x1), mark(x2)), MARK(sieve(cons(z0, z1)))) MARK(filter(sieve(s(z0)), x1, x2)) -> c10(A__FILTER(a__sieve(s(mark(z0))), mark(x1), mark(x2)), MARK(sieve(s(z0)))) MARK(filter(sieve(x0), cons(z0, z1), x2)) -> c10(MARK(sieve(x0))) MARK(filter(sieve(x0), 0, x2)) -> c10(MARK(sieve(x0))) MARK(filter(sieve(x0), x1, x2)) -> c10(MARK(sieve(x0))) MARK(filter(sieve(0), x1, x2)) -> c10(A__FILTER(a__sieve(0), mark(x1), mark(x2))) MARK(filter(nats(x0), x1, x2)) -> c10(A__FILTER(cons(mark(mark(x0)), nats(s(mark(x0)))), mark(x1), mark(x2)), MARK(nats(x0))) MARK(filter(nats(filter(z0, z1, z2)), x1, x2)) -> c10(A__FILTER(a__nats(a__filter(mark(z0), mark(z1), mark(z2))), mark(x1), mark(x2)), MARK(nats(filter(z0, z1, z2)))) MARK(filter(nats(sieve(z0)), x1, x2)) -> c10(A__FILTER(a__nats(a__sieve(mark(z0))), mark(x1), mark(x2)), MARK(nats(sieve(z0)))) MARK(filter(nats(nats(z0)), x1, x2)) -> c10(A__FILTER(a__nats(a__nats(mark(z0))), mark(x1), mark(x2)), MARK(nats(nats(z0)))) MARK(filter(nats(zprimes), x1, x2)) -> c10(A__FILTER(a__nats(a__zprimes), mark(x1), mark(x2)), MARK(nats(zprimes))) MARK(filter(nats(cons(z0, z1)), x1, x2)) -> c10(A__FILTER(a__nats(cons(mark(z0), z1)), mark(x1), mark(x2)), MARK(nats(cons(z0, z1)))) MARK(filter(nats(0), x1, x2)) -> c10(A__FILTER(a__nats(0), mark(x1), mark(x2)), MARK(nats(0))) MARK(filter(nats(s(z0)), x1, x2)) -> c10(A__FILTER(a__nats(s(mark(z0))), mark(x1), mark(x2)), MARK(nats(s(z0)))) MARK(filter(nats(x0), x1, x2)) -> c10(MARK(nats(x0))) MARK(filter(nats(x0), cons(z0, z1), x2)) -> c10(MARK(nats(x0))) MARK(filter(nats(x0), 0, x2)) -> c10(MARK(nats(x0))) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c14_2, c15_1, c16_1, c18_1, c8_1, c10_2, c10_1, c11_2, c11_1, c12_2, c12_1, c13_2, c13_1, c8_2 ---------------------------------------- (137) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(filter(zprimes, x1, x2)) -> c10(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(zprimes)) by MARK(filter(zprimes, x0, filter(z0, z1, z2))) -> c10(A__FILTER(a__zprimes, mark(x0), a__filter(mark(z0), mark(z1), mark(z2))), MARK(zprimes)) MARK(filter(zprimes, x0, sieve(z0))) -> c10(A__FILTER(a__zprimes, mark(x0), a__sieve(mark(z0))), MARK(zprimes)) MARK(filter(zprimes, x0, nats(z0))) -> c10(A__FILTER(a__zprimes, mark(x0), a__nats(mark(z0))), MARK(zprimes)) MARK(filter(zprimes, x0, zprimes)) -> c10(A__FILTER(a__zprimes, mark(x0), a__zprimes), MARK(zprimes)) MARK(filter(zprimes, x0, cons(z0, z1))) -> c10(A__FILTER(a__zprimes, mark(x0), cons(mark(z0), z1)), MARK(zprimes)) MARK(filter(zprimes, x0, 0)) -> c10(A__FILTER(a__zprimes, mark(x0), 0), MARK(zprimes)) MARK(filter(zprimes, x0, s(z0))) -> c10(A__FILTER(a__zprimes, mark(x0), s(mark(z0))), MARK(zprimes)) MARK(filter(zprimes, filter(z0, z1, z2), x1)) -> c10(A__FILTER(a__zprimes, a__filter(mark(z0), mark(z1), mark(z2)), mark(x1)), MARK(zprimes)) MARK(filter(zprimes, sieve(z0), x1)) -> c10(A__FILTER(a__zprimes, a__sieve(mark(z0)), mark(x1)), MARK(zprimes)) MARK(filter(zprimes, nats(z0), x1)) -> c10(A__FILTER(a__zprimes, a__nats(mark(z0)), mark(x1)), MARK(zprimes)) MARK(filter(zprimes, zprimes, x1)) -> c10(A__FILTER(a__zprimes, a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(zprimes, cons(z0, z1), x1)) -> c10(A__FILTER(a__zprimes, cons(mark(z0), z1), mark(x1)), MARK(zprimes)) MARK(filter(zprimes, 0, x1)) -> c10(A__FILTER(a__zprimes, 0, mark(x1)), MARK(zprimes)) MARK(filter(zprimes, s(z0), x1)) -> c10(A__FILTER(a__zprimes, s(mark(z0)), mark(x1)), MARK(zprimes)) MARK(filter(zprimes, x0, x1)) -> c10(A__FILTER(a__sieve(a__nats(s(s(0)))), mark(x0), mark(x1)), MARK(zprimes)) MARK(filter(zprimes, x0, x1)) -> c10(A__FILTER(zprimes, mark(x0), mark(x1)), MARK(zprimes)) ---------------------------------------- (138) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(cons(z0, z1), x1, x2)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x1)) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, 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a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), 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MARK(filter(filter(x0, x1, filter(z0, z1, z2)), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), mark(x3), mark(x4)), MARK(filter(x0, x1, filter(z0, z1, z2)))) MARK(filter(filter(x0, x1, sieve(z0)), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), a__sieve(mark(z0))), mark(x3), mark(x4)), MARK(filter(x0, x1, sieve(z0)))) MARK(filter(filter(x0, x1, nats(z0)), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), a__nats(mark(z0))), mark(x3), mark(x4)), MARK(filter(x0, x1, nats(z0)))) MARK(filter(filter(x0, x1, zprimes), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), a__zprimes), mark(x3), mark(x4)), MARK(filter(x0, x1, zprimes))) MARK(filter(filter(x0, x1, cons(z0, z1)), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), cons(mark(z0), z1)), mark(x3), mark(x4)), MARK(filter(x0, x1, cons(z0, z1)))) MARK(filter(filter(x0, x1, 0), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), 0), mark(x3), mark(x4)), 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c10(A__FILTER(a__sieve(a__sieve(mark(z0))), mark(x1), mark(x2)), MARK(sieve(sieve(z0)))) MARK(filter(sieve(nats(z0)), x1, x2)) -> c10(A__FILTER(a__sieve(a__nats(mark(z0))), mark(x1), mark(x2)), MARK(sieve(nats(z0)))) MARK(filter(sieve(zprimes), x1, x2)) -> c10(A__FILTER(a__sieve(a__zprimes), mark(x1), mark(x2)), MARK(sieve(zprimes))) MARK(filter(sieve(cons(z0, z1)), x1, x2)) -> c10(A__FILTER(a__sieve(cons(mark(z0), z1)), mark(x1), mark(x2)), MARK(sieve(cons(z0, z1)))) MARK(filter(sieve(s(z0)), x1, x2)) -> c10(A__FILTER(a__sieve(s(mark(z0))), mark(x1), mark(x2)), MARK(sieve(s(z0)))) MARK(filter(sieve(x0), cons(z0, z1), x2)) -> c10(MARK(sieve(x0))) MARK(filter(sieve(x0), 0, x2)) -> c10(MARK(sieve(x0))) MARK(filter(sieve(x0), x1, x2)) -> c10(MARK(sieve(x0))) MARK(filter(sieve(0), x1, x2)) -> c10(A__FILTER(a__sieve(0), mark(x1), mark(x2))) MARK(filter(nats(x0), x1, x2)) -> c10(A__FILTER(cons(mark(mark(x0)), nats(s(mark(x0)))), mark(x1), mark(x2)), MARK(nats(x0))) MARK(filter(nats(filter(z0, z1, z2)), x1, x2)) -> c10(A__FILTER(a__nats(a__filter(mark(z0), mark(z1), mark(z2))), mark(x1), mark(x2)), MARK(nats(filter(z0, z1, z2)))) MARK(filter(nats(sieve(z0)), x1, x2)) -> c10(A__FILTER(a__nats(a__sieve(mark(z0))), mark(x1), mark(x2)), MARK(nats(sieve(z0)))) MARK(filter(nats(nats(z0)), x1, x2)) -> c10(A__FILTER(a__nats(a__nats(mark(z0))), mark(x1), mark(x2)), MARK(nats(nats(z0)))) MARK(filter(nats(zprimes), x1, x2)) -> c10(A__FILTER(a__nats(a__zprimes), mark(x1), mark(x2)), MARK(nats(zprimes))) MARK(filter(nats(cons(z0, z1)), x1, x2)) -> c10(A__FILTER(a__nats(cons(mark(z0), z1)), mark(x1), mark(x2)), MARK(nats(cons(z0, z1)))) MARK(filter(nats(0), x1, x2)) -> c10(A__FILTER(a__nats(0), mark(x1), mark(x2)), MARK(nats(0))) MARK(filter(nats(s(z0)), x1, x2)) -> c10(A__FILTER(a__nats(s(mark(z0))), mark(x1), mark(x2)), MARK(nats(s(z0)))) MARK(filter(nats(x0), x1, x2)) -> c10(MARK(nats(x0))) MARK(filter(nats(x0), cons(z0, z1), x2)) -> c10(MARK(nats(x0))) MARK(filter(nats(x0), 0, x2)) -> c10(MARK(nats(x0))) MARK(filter(zprimes, cons(z0, z1), x1)) -> c10(A__FILTER(a__zprimes, cons(mark(z0), z1), mark(x1)), MARK(zprimes)) MARK(filter(zprimes, 0, x1)) -> c10(A__FILTER(a__zprimes, 0, mark(x1)), MARK(zprimes)) MARK(filter(zprimes, x0, x1)) -> c10(A__FILTER(a__sieve(a__nats(s(s(0)))), mark(x0), mark(x1)), MARK(zprimes)) MARK(filter(zprimes, x0, x1)) -> c10(A__FILTER(zprimes, mark(x0), mark(x1)), MARK(zprimes)) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c14_2, c15_1, c16_1, c18_1, c8_1, c10_2, c10_1, c11_2, c11_1, c12_2, c12_1, c13_2, c13_1, c8_2 ---------------------------------------- (139) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (140) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(cons(z0, z1), x1, x2)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x1)) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(x0, 0, filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, sieve(x2))) -> c10(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x0)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, sieve(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, sieve(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, sieve(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(0)), MARK(x0)) MARK(filter(x0, x1, sieve(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), sieve(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, sieve(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, sieve(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__sieve(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, sieve(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, sieve(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, sieve(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), a__sieve(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, sieve(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__sieve(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), sieve(x2))) -> c10(MARK(x0)) MARK(filter(x0, 0, sieve(x2))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, sieve(x2))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(mark(x2)), nats(s(mark(x2))))), MARK(x0)) 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z2), x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(filter(z0, z1, z2), x1, x2))) MARK(filter(filter(sieve(z0), x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(a__sieve(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(sieve(z0), x1, x2))) MARK(filter(filter(nats(z0), x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(a__nats(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(nats(z0), x1, x2))) MARK(filter(filter(zprimes, x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(a__zprimes, mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(zprimes, x1, x2))) MARK(filter(filter(cons(z0, z1), x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(cons(mark(z0), z1), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(cons(z0, z1), x1, x2))) MARK(filter(filter(0, x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(0, mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(0, x1, x2))) MARK(filter(filter(s(z0), x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(s(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(s(z0), x1, x2))) MARK(filter(filter(x0, x1, x2), cons(z0, z1), x4)) -> c10(MARK(filter(x0, x1, x2))) MARK(filter(filter(x0, x1, x2), 0, x4)) -> c10(MARK(filter(x0, x1, x2))) MARK(filter(filter(x0, x1, x2), x3, x4)) -> c10(MARK(filter(x0, x1, x2))) MARK(filter(sieve(filter(z0, z1, z2)), x1, x2)) -> c10(A__FILTER(a__sieve(a__filter(mark(z0), mark(z1), mark(z2))), mark(x1), mark(x2)), MARK(sieve(filter(z0, z1, z2)))) MARK(filter(sieve(sieve(z0)), x1, x2)) -> c10(A__FILTER(a__sieve(a__sieve(mark(z0))), mark(x1), mark(x2)), MARK(sieve(sieve(z0)))) MARK(filter(sieve(nats(z0)), x1, x2)) -> c10(A__FILTER(a__sieve(a__nats(mark(z0))), mark(x1), mark(x2)), MARK(sieve(nats(z0)))) MARK(filter(sieve(zprimes), x1, x2)) -> c10(A__FILTER(a__sieve(a__zprimes), mark(x1), mark(x2)), MARK(sieve(zprimes))) MARK(filter(sieve(cons(z0, z1)), x1, x2)) -> c10(A__FILTER(a__sieve(cons(mark(z0), z1)), mark(x1), mark(x2)), MARK(sieve(cons(z0, z1)))) MARK(filter(sieve(s(z0)), x1, x2)) -> c10(A__FILTER(a__sieve(s(mark(z0))), mark(x1), mark(x2)), MARK(sieve(s(z0)))) MARK(filter(sieve(x0), cons(z0, z1), x2)) -> c10(MARK(sieve(x0))) MARK(filter(sieve(x0), 0, x2)) -> c10(MARK(sieve(x0))) MARK(filter(sieve(x0), x1, x2)) -> c10(MARK(sieve(x0))) MARK(filter(sieve(0), x1, x2)) -> c10(A__FILTER(a__sieve(0), mark(x1), mark(x2))) MARK(filter(nats(x0), x1, x2)) -> c10(A__FILTER(cons(mark(mark(x0)), nats(s(mark(x0)))), mark(x1), mark(x2)), MARK(nats(x0))) MARK(filter(nats(filter(z0, z1, z2)), x1, x2)) -> c10(A__FILTER(a__nats(a__filter(mark(z0), mark(z1), mark(z2))), mark(x1), mark(x2)), MARK(nats(filter(z0, z1, z2)))) MARK(filter(nats(sieve(z0)), x1, x2)) -> c10(A__FILTER(a__nats(a__sieve(mark(z0))), mark(x1), mark(x2)), MARK(nats(sieve(z0)))) MARK(filter(nats(nats(z0)), x1, x2)) -> c10(A__FILTER(a__nats(a__nats(mark(z0))), mark(x1), mark(x2)), MARK(nats(nats(z0)))) MARK(filter(nats(zprimes), x1, x2)) -> c10(A__FILTER(a__nats(a__zprimes), mark(x1), mark(x2)), MARK(nats(zprimes))) MARK(filter(nats(cons(z0, z1)), x1, x2)) -> c10(A__FILTER(a__nats(cons(mark(z0), z1)), mark(x1), mark(x2)), MARK(nats(cons(z0, z1)))) MARK(filter(nats(0), x1, x2)) -> c10(A__FILTER(a__nats(0), mark(x1), mark(x2)), MARK(nats(0))) MARK(filter(nats(s(z0)), x1, x2)) -> c10(A__FILTER(a__nats(s(mark(z0))), mark(x1), mark(x2)), MARK(nats(s(z0)))) MARK(filter(nats(x0), x1, x2)) -> c10(MARK(nats(x0))) MARK(filter(nats(x0), cons(z0, z1), x2)) -> c10(MARK(nats(x0))) MARK(filter(nats(x0), 0, x2)) -> c10(MARK(nats(x0))) MARK(filter(zprimes, x0, x1)) -> c10(A__FILTER(a__sieve(a__nats(s(s(0)))), mark(x0), mark(x1)), MARK(zprimes)) MARK(filter(zprimes, cons(z0, z1), x1)) -> c10(MARK(zprimes)) MARK(filter(zprimes, 0, x1)) -> c10(MARK(zprimes)) MARK(filter(zprimes, x0, x1)) -> c10(MARK(zprimes)) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c14_2, c15_1, c16_1, c18_1, c8_1, c10_2, c10_1, c11_2, c11_1, c12_2, c12_1, c13_2, c13_1, c8_2 ---------------------------------------- (141) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(filter(cons(z0, z1), x1, x2)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(cons(z0, z1))) by MARK(filter(cons(x0, x1), x2, filter(z0, z1, z2))) -> c10(A__FILTER(cons(mark(x0), x1), mark(x2), a__filter(mark(z0), mark(z1), mark(z2))), MARK(cons(x0, x1))) MARK(filter(cons(x0, x1), x2, sieve(z0))) -> c10(A__FILTER(cons(mark(x0), x1), mark(x2), a__sieve(mark(z0))), MARK(cons(x0, x1))) MARK(filter(cons(x0, x1), x2, nats(z0))) -> c10(A__FILTER(cons(mark(x0), x1), mark(x2), a__nats(mark(z0))), MARK(cons(x0, x1))) MARK(filter(cons(x0, x1), x2, zprimes)) -> c10(A__FILTER(cons(mark(x0), x1), mark(x2), a__zprimes), MARK(cons(x0, x1))) MARK(filter(cons(x0, x1), x2, cons(z0, z1))) -> c10(A__FILTER(cons(mark(x0), x1), mark(x2), cons(mark(z0), z1)), MARK(cons(x0, x1))) MARK(filter(cons(x0, x1), x2, 0)) -> c10(A__FILTER(cons(mark(x0), x1), mark(x2), 0), MARK(cons(x0, x1))) MARK(filter(cons(x0, x1), x2, s(z0))) -> c10(A__FILTER(cons(mark(x0), x1), mark(x2), s(mark(z0))), MARK(cons(x0, x1))) MARK(filter(cons(x0, x1), filter(z0, z1, z2), x3)) -> c10(A__FILTER(cons(mark(x0), x1), a__filter(mark(z0), mark(z1), mark(z2)), mark(x3)), MARK(cons(x0, x1))) MARK(filter(cons(x0, x1), sieve(z0), x3)) -> c10(A__FILTER(cons(mark(x0), x1), a__sieve(mark(z0)), mark(x3)), MARK(cons(x0, x1))) MARK(filter(cons(x0, x1), nats(z0), x3)) -> c10(A__FILTER(cons(mark(x0), x1), a__nats(mark(z0)), mark(x3)), MARK(cons(x0, x1))) MARK(filter(cons(x0, x1), zprimes, x3)) -> c10(A__FILTER(cons(mark(x0), x1), a__zprimes, mark(x3)), MARK(cons(x0, x1))) MARK(filter(cons(x0, x1), cons(z0, z1), x3)) -> c10(A__FILTER(cons(mark(x0), x1), cons(mark(z0), z1), mark(x3)), MARK(cons(x0, x1))) MARK(filter(cons(x0, x1), 0, x3)) -> c10(A__FILTER(cons(mark(x0), x1), 0, mark(x3)), MARK(cons(x0, x1))) MARK(filter(cons(x0, x1), s(z0), x3)) -> c10(A__FILTER(cons(mark(x0), x1), s(mark(z0)), mark(x3)), MARK(cons(x0, x1))) MARK(filter(cons(filter(z0, z1, z2), x1), x2, x3)) -> c10(A__FILTER(cons(a__filter(mark(z0), mark(z1), mark(z2)), x1), mark(x2), mark(x3)), MARK(cons(filter(z0, z1, z2), x1))) MARK(filter(cons(sieve(z0), x1), x2, x3)) -> c10(A__FILTER(cons(a__sieve(mark(z0)), x1), mark(x2), mark(x3)), MARK(cons(sieve(z0), x1))) MARK(filter(cons(nats(z0), x1), x2, x3)) -> c10(A__FILTER(cons(a__nats(mark(z0)), x1), mark(x2), mark(x3)), MARK(cons(nats(z0), x1))) MARK(filter(cons(zprimes, x1), x2, x3)) -> c10(A__FILTER(cons(a__zprimes, x1), mark(x2), mark(x3)), MARK(cons(zprimes, x1))) MARK(filter(cons(cons(z0, z1), x1), x2, x3)) -> c10(A__FILTER(cons(cons(mark(z0), z1), x1), mark(x2), mark(x3)), MARK(cons(cons(z0, z1), x1))) MARK(filter(cons(0, x1), x2, x3)) -> c10(A__FILTER(cons(0, x1), mark(x2), mark(x3)), MARK(cons(0, x1))) MARK(filter(cons(s(z0), x1), x2, x3)) -> c10(A__FILTER(cons(s(mark(z0)), x1), mark(x2), mark(x3)), MARK(cons(s(z0), x1))) MARK(filter(cons(x0, x1), x2, x3)) -> c10(MARK(cons(x0, x1))) ---------------------------------------- (142) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x1)) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) 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MARK(filter(filter(0, x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(0, mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(0, x1, x2))) MARK(filter(filter(s(z0), x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(s(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(s(z0), x1, x2))) MARK(filter(filter(x0, x1, x2), cons(z0, z1), x4)) -> c10(MARK(filter(x0, x1, x2))) MARK(filter(filter(x0, x1, x2), 0, x4)) -> c10(MARK(filter(x0, x1, x2))) MARK(filter(filter(x0, x1, x2), x3, x4)) -> c10(MARK(filter(x0, x1, x2))) MARK(filter(sieve(filter(z0, z1, z2)), x1, x2)) -> c10(A__FILTER(a__sieve(a__filter(mark(z0), mark(z1), mark(z2))), mark(x1), mark(x2)), MARK(sieve(filter(z0, z1, z2)))) MARK(filter(sieve(sieve(z0)), x1, x2)) -> c10(A__FILTER(a__sieve(a__sieve(mark(z0))), mark(x1), mark(x2)), MARK(sieve(sieve(z0)))) MARK(filter(sieve(nats(z0)), x1, x2)) -> c10(A__FILTER(a__sieve(a__nats(mark(z0))), mark(x1), mark(x2)), MARK(sieve(nats(z0)))) MARK(filter(sieve(zprimes), x1, x2)) -> c10(A__FILTER(a__sieve(a__zprimes), mark(x1), mark(x2)), MARK(sieve(zprimes))) MARK(filter(sieve(cons(z0, z1)), x1, x2)) -> c10(A__FILTER(a__sieve(cons(mark(z0), z1)), mark(x1), mark(x2)), MARK(sieve(cons(z0, z1)))) MARK(filter(sieve(s(z0)), x1, x2)) -> c10(A__FILTER(a__sieve(s(mark(z0))), mark(x1), mark(x2)), MARK(sieve(s(z0)))) MARK(filter(sieve(x0), cons(z0, z1), x2)) -> c10(MARK(sieve(x0))) MARK(filter(sieve(x0), 0, x2)) -> c10(MARK(sieve(x0))) MARK(filter(sieve(x0), x1, x2)) -> c10(MARK(sieve(x0))) MARK(filter(sieve(0), x1, x2)) -> c10(A__FILTER(a__sieve(0), mark(x1), mark(x2))) MARK(filter(nats(x0), x1, x2)) -> c10(A__FILTER(cons(mark(mark(x0)), nats(s(mark(x0)))), mark(x1), mark(x2)), MARK(nats(x0))) MARK(filter(nats(filter(z0, z1, z2)), x1, x2)) -> c10(A__FILTER(a__nats(a__filter(mark(z0), mark(z1), mark(z2))), mark(x1), mark(x2)), MARK(nats(filter(z0, z1, z2)))) MARK(filter(nats(sieve(z0)), x1, x2)) -> c10(A__FILTER(a__nats(a__sieve(mark(z0))), mark(x1), mark(x2)), MARK(nats(sieve(z0)))) MARK(filter(nats(nats(z0)), x1, x2)) -> c10(A__FILTER(a__nats(a__nats(mark(z0))), mark(x1), mark(x2)), MARK(nats(nats(z0)))) MARK(filter(nats(zprimes), x1, x2)) -> c10(A__FILTER(a__nats(a__zprimes), mark(x1), mark(x2)), MARK(nats(zprimes))) MARK(filter(nats(cons(z0, z1)), x1, x2)) -> c10(A__FILTER(a__nats(cons(mark(z0), z1)), mark(x1), mark(x2)), MARK(nats(cons(z0, z1)))) MARK(filter(nats(0), x1, x2)) -> c10(A__FILTER(a__nats(0), mark(x1), mark(x2)), MARK(nats(0))) MARK(filter(nats(s(z0)), x1, x2)) -> c10(A__FILTER(a__nats(s(mark(z0))), mark(x1), mark(x2)), MARK(nats(s(z0)))) MARK(filter(nats(x0), x1, x2)) -> c10(MARK(nats(x0))) MARK(filter(nats(x0), cons(z0, z1), x2)) -> c10(MARK(nats(x0))) MARK(filter(nats(x0), 0, x2)) -> c10(MARK(nats(x0))) MARK(filter(zprimes, x0, x1)) -> c10(A__FILTER(a__sieve(a__nats(s(s(0)))), mark(x0), mark(x1)), MARK(zprimes)) MARK(filter(zprimes, cons(z0, z1), x1)) -> c10(MARK(zprimes)) MARK(filter(zprimes, 0, x1)) -> c10(MARK(zprimes)) MARK(filter(zprimes, x0, x1)) -> c10(MARK(zprimes)) MARK(filter(cons(x0, x1), cons(z0, z1), x3)) -> c10(A__FILTER(cons(mark(x0), x1), cons(mark(z0), z1), mark(x3)), MARK(cons(x0, x1))) MARK(filter(cons(x0, x1), 0, x3)) -> c10(A__FILTER(cons(mark(x0), x1), 0, mark(x3)), MARK(cons(x0, x1))) MARK(filter(cons(filter(z0, z1, z2), x1), x2, x3)) -> c10(A__FILTER(cons(a__filter(mark(z0), mark(z1), mark(z2)), x1), mark(x2), mark(x3)), MARK(cons(filter(z0, z1, z2), x1))) MARK(filter(cons(sieve(z0), x1), x2, x3)) -> c10(A__FILTER(cons(a__sieve(mark(z0)), x1), mark(x2), mark(x3)), MARK(cons(sieve(z0), x1))) MARK(filter(cons(nats(z0), x1), x2, x3)) -> c10(A__FILTER(cons(a__nats(mark(z0)), x1), mark(x2), mark(x3)), MARK(cons(nats(z0), x1))) MARK(filter(cons(zprimes, x1), x2, x3)) -> c10(A__FILTER(cons(a__zprimes, x1), mark(x2), mark(x3)), MARK(cons(zprimes, x1))) MARK(filter(cons(cons(z0, z1), x1), x2, x3)) -> c10(A__FILTER(cons(cons(mark(z0), z1), x1), mark(x2), mark(x3)), MARK(cons(cons(z0, z1), x1))) MARK(filter(cons(0, x1), x2, x3)) -> c10(A__FILTER(cons(0, x1), mark(x2), mark(x3)), MARK(cons(0, x1))) MARK(filter(cons(s(z0), x1), x2, x3)) -> c10(A__FILTER(cons(s(mark(z0)), x1), mark(x2), mark(x3)), MARK(cons(s(z0), x1))) MARK(filter(cons(x0, x1), x2, x3)) -> c10(MARK(cons(x0, x1))) S tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x1)) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> 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mark(x3)), MARK(cons(nats(z0), x1))) MARK(filter(cons(zprimes, x1), x2, x3)) -> c10(A__FILTER(cons(a__zprimes, x1), mark(x2), mark(x3)), MARK(cons(zprimes, x1))) MARK(filter(cons(cons(z0, z1), x1), x2, x3)) -> c10(A__FILTER(cons(cons(mark(z0), z1), x1), mark(x2), mark(x3)), MARK(cons(cons(z0, z1), x1))) MARK(filter(cons(0, x1), x2, x3)) -> c10(A__FILTER(cons(0, x1), mark(x2), mark(x3)), MARK(cons(0, x1))) MARK(filter(cons(s(z0), x1), x2, x3)) -> c10(A__FILTER(cons(s(mark(z0)), x1), mark(x2), mark(x3)), MARK(cons(s(z0), x1))) MARK(filter(cons(x0, x1), x2, x3)) -> c10(MARK(cons(x0, x1))) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c14_2, c15_1, c16_1, c18_1, c8_1, c10_1, c11_2, c11_1, c12_2, c12_1, c13_2, c13_1, c8_2, c10_2 ---------------------------------------- (143) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (144) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x1)) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(x0, 0, filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, sieve(x2))) -> c10(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x0)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, sieve(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, 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c10(A__FILTER(mark(x0), a__sieve(s(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), sieve(x1), x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x1)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), sieve(x1), x2)) -> c10(A__FILTER(a__sieve(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), sieve(x1), x2)) -> c10(A__FILTER(a__nats(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, sieve(x1), x2)) -> c10(A__FILTER(a__zprimes, a__sieve(mark(x1)), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), sieve(x1), x2)) -> c10(A__FILTER(cons(mark(z0), z1), a__sieve(mark(x1)), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, sieve(x1), x2)) -> c10(MARK(x0)) MARK(filter(s(z0), sieve(x1), x2)) -> c10(MARK(s(z0))) MARK(filter(x0, nats(filter(z0, z1, z2)), x2)) -> c10(A__FILTER(mark(x0), a__nats(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(x0)) MARK(filter(x0, nats(sieve(z0)), x2)) -> c10(A__FILTER(mark(x0), a__nats(a__sieve(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, nats(nats(z0)), x2)) -> c10(A__FILTER(mark(x0), a__nats(a__nats(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, nats(zprimes), x2)) -> c10(A__FILTER(mark(x0), a__nats(a__zprimes), mark(x2)), MARK(x0)) MARK(filter(x0, nats(cons(z0, z1)), x2)) -> c10(A__FILTER(mark(x0), a__nats(cons(mark(z0), z1)), mark(x2)), MARK(x0)) MARK(filter(x0, nats(0), x2)) -> c10(A__FILTER(mark(x0), a__nats(0), mark(x2)), MARK(x0)) MARK(filter(x0, nats(s(z0)), x2)) -> c10(A__FILTER(mark(x0), a__nats(s(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), nats(x1), x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x1)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), nats(x1), x2)) -> c10(A__FILTER(a__sieve(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), nats(x1), x2)) -> c10(A__FILTER(a__nats(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, nats(x1), x2)) -> c10(A__FILTER(a__zprimes, a__nats(mark(x1)), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), nats(x1), x2)) -> c10(A__FILTER(cons(mark(z0), z1), a__nats(mark(x1)), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, nats(x1), x2)) -> c10(MARK(x0)) MARK(filter(s(z0), nats(x1), x2)) -> c10(MARK(s(z0))) MARK(filter(x0, zprimes, x1)) -> c10(A__FILTER(mark(x0), a__sieve(a__nats(s(s(0)))), mark(x1)), MARK(x0)) MARK(filter(filter(z0, z1, z2), zprimes, x1)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__zprimes, mark(x1)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), zprimes, x1)) -> c10(A__FILTER(a__sieve(mark(z0)), a__zprimes, mark(x1)), MARK(sieve(z0))) MARK(filter(nats(z0), zprimes, x1)) -> c10(A__FILTER(a__nats(mark(z0)), a__zprimes, mark(x1)), MARK(nats(z0))) MARK(filter(zprimes, zprimes, x1)) -> c10(A__FILTER(a__zprimes, a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(cons(z0, z1), zprimes, x1)) -> c10(A__FILTER(cons(mark(z0), z1), a__zprimes, mark(x1)), MARK(cons(z0, z1))) MARK(filter(x0, zprimes, x1)) -> c10(MARK(x0)) MARK(filter(s(z0), zprimes, x1)) -> c10(MARK(s(z0))) MARK(filter(x0, s(filter(z0, z1, z2)), x2)) -> c10(A__FILTER(mark(x0), s(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(x0)) MARK(filter(x0, s(sieve(z0)), x2)) -> c10(A__FILTER(mark(x0), s(a__sieve(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, s(nats(z0)), x2)) -> c10(A__FILTER(mark(x0), s(a__nats(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, s(zprimes), x2)) -> c10(A__FILTER(mark(x0), s(a__zprimes), mark(x2)), MARK(x0)) MARK(filter(x0, s(cons(z0, z1)), x2)) -> c10(A__FILTER(mark(x0), s(cons(mark(z0), z1)), mark(x2)), MARK(x0)) MARK(filter(x0, s(0), x2)) -> c10(A__FILTER(mark(x0), s(0), mark(x2)), MARK(x0)) MARK(filter(x0, s(s(z0)), x2)) -> c10(A__FILTER(mark(x0), s(s(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), s(x1), x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), s(mark(x1)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), s(x1), x2)) -> c10(A__FILTER(a__sieve(mark(z0)), s(mark(x1)), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), s(x1), x2)) -> c10(A__FILTER(a__nats(mark(z0)), s(mark(x1)), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, s(x1), x2)) -> c10(A__FILTER(a__zprimes, s(mark(x1)), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), s(x1), x2)) -> c10(A__FILTER(cons(mark(z0), z1), s(mark(x1)), mark(x2)), MARK(cons(z0, z1))) MARK(filter(s(z0), s(x1), x2)) -> c10(MARK(s(z0))) MARK(filter(filter(x0, x1, filter(z0, z1, z2)), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), mark(x3), mark(x4)), MARK(filter(x0, x1, filter(z0, z1, z2)))) MARK(filter(filter(x0, x1, sieve(z0)), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), a__sieve(mark(z0))), mark(x3), mark(x4)), MARK(filter(x0, x1, sieve(z0)))) MARK(filter(filter(x0, x1, nats(z0)), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), a__nats(mark(z0))), mark(x3), mark(x4)), MARK(filter(x0, x1, nats(z0)))) MARK(filter(filter(x0, x1, zprimes), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), a__zprimes), mark(x3), mark(x4)), MARK(filter(x0, x1, zprimes))) MARK(filter(filter(x0, x1, cons(z0, z1)), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), cons(mark(z0), z1)), mark(x3), mark(x4)), MARK(filter(x0, x1, cons(z0, z1)))) MARK(filter(filter(x0, x1, 0), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), 0), mark(x3), mark(x4)), MARK(filter(x0, x1, 0))) MARK(filter(filter(x0, x1, s(z0)), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), s(mark(z0))), mark(x3), mark(x4)), MARK(filter(x0, x1, s(z0)))) MARK(filter(filter(x0, filter(z0, z1, z2), x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, filter(z0, z1, z2), x2))) MARK(filter(filter(x0, sieve(z0), x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), a__sieve(mark(z0)), mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, sieve(z0), x2))) MARK(filter(filter(x0, nats(z0), x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), a__nats(mark(z0)), mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, nats(z0), x2))) MARK(filter(filter(x0, zprimes, x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), a__zprimes, mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, zprimes, x2))) MARK(filter(filter(x0, cons(z0, z1), x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), cons(mark(z0), z1), mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, cons(z0, z1), x2))) MARK(filter(filter(x0, 0, x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), 0, mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, 0, x2))) MARK(filter(filter(x0, s(z0), x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), s(mark(z0)), mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, s(z0), x2))) MARK(filter(filter(filter(z0, z1, z2), x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(filter(z0, z1, z2), x1, x2))) MARK(filter(filter(sieve(z0), x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(a__sieve(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(sieve(z0), x1, x2))) MARK(filter(filter(nats(z0), x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(a__nats(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(nats(z0), x1, x2))) MARK(filter(filter(zprimes, x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(a__zprimes, mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(zprimes, x1, x2))) MARK(filter(filter(cons(z0, z1), x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(cons(mark(z0), z1), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(cons(z0, z1), x1, x2))) MARK(filter(filter(0, x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(0, mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(0, x1, x2))) MARK(filter(filter(s(z0), x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(s(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(s(z0), x1, x2))) MARK(filter(filter(x0, x1, x2), cons(z0, z1), x4)) -> c10(MARK(filter(x0, x1, x2))) MARK(filter(filter(x0, x1, x2), 0, x4)) -> c10(MARK(filter(x0, x1, x2))) MARK(filter(filter(x0, x1, x2), x3, x4)) -> c10(MARK(filter(x0, x1, x2))) MARK(filter(sieve(filter(z0, z1, z2)), x1, x2)) -> c10(A__FILTER(a__sieve(a__filter(mark(z0), mark(z1), mark(z2))), mark(x1), mark(x2)), MARK(sieve(filter(z0, z1, z2)))) MARK(filter(sieve(sieve(z0)), x1, x2)) -> c10(A__FILTER(a__sieve(a__sieve(mark(z0))), mark(x1), mark(x2)), MARK(sieve(sieve(z0)))) MARK(filter(sieve(nats(z0)), x1, x2)) -> c10(A__FILTER(a__sieve(a__nats(mark(z0))), mark(x1), mark(x2)), MARK(sieve(nats(z0)))) MARK(filter(sieve(zprimes), x1, x2)) -> c10(A__FILTER(a__sieve(a__zprimes), mark(x1), mark(x2)), MARK(sieve(zprimes))) MARK(filter(sieve(cons(z0, z1)), x1, x2)) -> c10(A__FILTER(a__sieve(cons(mark(z0), z1)), mark(x1), mark(x2)), MARK(sieve(cons(z0, z1)))) MARK(filter(sieve(s(z0)), x1, x2)) -> c10(A__FILTER(a__sieve(s(mark(z0))), mark(x1), mark(x2)), MARK(sieve(s(z0)))) MARK(filter(sieve(x0), cons(z0, z1), x2)) -> c10(MARK(sieve(x0))) MARK(filter(sieve(x0), 0, x2)) -> c10(MARK(sieve(x0))) MARK(filter(sieve(x0), x1, x2)) -> c10(MARK(sieve(x0))) MARK(filter(sieve(0), x1, x2)) -> c10(A__FILTER(a__sieve(0), mark(x1), mark(x2))) MARK(filter(nats(x0), x1, x2)) -> c10(A__FILTER(cons(mark(mark(x0)), nats(s(mark(x0)))), mark(x1), mark(x2)), MARK(nats(x0))) MARK(filter(nats(filter(z0, z1, z2)), x1, x2)) -> c10(A__FILTER(a__nats(a__filter(mark(z0), mark(z1), mark(z2))), mark(x1), mark(x2)), MARK(nats(filter(z0, z1, z2)))) MARK(filter(nats(sieve(z0)), x1, x2)) -> c10(A__FILTER(a__nats(a__sieve(mark(z0))), mark(x1), mark(x2)), MARK(nats(sieve(z0)))) MARK(filter(nats(nats(z0)), x1, x2)) -> c10(A__FILTER(a__nats(a__nats(mark(z0))), mark(x1), mark(x2)), MARK(nats(nats(z0)))) MARK(filter(nats(zprimes), x1, x2)) -> c10(A__FILTER(a__nats(a__zprimes), mark(x1), mark(x2)), MARK(nats(zprimes))) MARK(filter(nats(cons(z0, z1)), x1, x2)) -> c10(A__FILTER(a__nats(cons(mark(z0), z1)), mark(x1), mark(x2)), MARK(nats(cons(z0, z1)))) MARK(filter(nats(0), x1, x2)) -> c10(A__FILTER(a__nats(0), mark(x1), mark(x2)), MARK(nats(0))) MARK(filter(nats(s(z0)), x1, x2)) -> c10(A__FILTER(a__nats(s(mark(z0))), mark(x1), mark(x2)), MARK(nats(s(z0)))) MARK(filter(nats(x0), x1, x2)) -> c10(MARK(nats(x0))) MARK(filter(nats(x0), cons(z0, z1), x2)) -> c10(MARK(nats(x0))) MARK(filter(nats(x0), 0, x2)) -> c10(MARK(nats(x0))) MARK(filter(zprimes, x0, x1)) -> c10(A__FILTER(a__sieve(a__nats(s(s(0)))), mark(x0), mark(x1)), MARK(zprimes)) MARK(filter(zprimes, cons(z0, z1), x1)) -> c10(MARK(zprimes)) MARK(filter(zprimes, 0, x1)) -> c10(MARK(zprimes)) MARK(filter(zprimes, x0, x1)) -> c10(MARK(zprimes)) MARK(filter(cons(filter(z0, z1, z2), x1), x2, x3)) -> c10(A__FILTER(cons(a__filter(mark(z0), mark(z1), mark(z2)), x1), mark(x2), mark(x3)), MARK(cons(filter(z0, z1, z2), x1))) MARK(filter(cons(sieve(z0), x1), x2, x3)) -> c10(A__FILTER(cons(a__sieve(mark(z0)), x1), mark(x2), mark(x3)), MARK(cons(sieve(z0), x1))) MARK(filter(cons(nats(z0), x1), x2, x3)) -> c10(A__FILTER(cons(a__nats(mark(z0)), x1), mark(x2), mark(x3)), MARK(cons(nats(z0), x1))) MARK(filter(cons(zprimes, x1), x2, x3)) -> c10(A__FILTER(cons(a__zprimes, x1), mark(x2), mark(x3)), MARK(cons(zprimes, x1))) MARK(filter(cons(cons(z0, z1), x1), x2, x3)) -> c10(A__FILTER(cons(cons(mark(z0), z1), x1), mark(x2), mark(x3)), MARK(cons(cons(z0, z1), x1))) MARK(filter(cons(0, x1), x2, x3)) -> c10(A__FILTER(cons(0, x1), mark(x2), mark(x3)), MARK(cons(0, x1))) MARK(filter(cons(s(z0), x1), x2, x3)) -> c10(A__FILTER(cons(s(mark(z0)), x1), mark(x2), mark(x3)), MARK(cons(s(z0), x1))) MARK(filter(cons(x0, x1), x2, x3)) -> c10(MARK(cons(x0, x1))) MARK(filter(cons(x0, x1), cons(z0, z1), x3)) -> c10(MARK(cons(x0, x1))) MARK(filter(cons(x0, x1), 0, x3)) -> c10(MARK(cons(x0, x1))) S tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x1)) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(x0, 0, filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, sieve(x2))) -> c10(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x0)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, sieve(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, sieve(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, sieve(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(0)), MARK(x0)) MARK(filter(x0, x1, sieve(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), sieve(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, sieve(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, sieve(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__sieve(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, sieve(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, sieve(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, sieve(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), a__sieve(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, sieve(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__sieve(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), sieve(x2))) -> c10(MARK(x0)) MARK(filter(x0, 0, sieve(x2))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, sieve(x2))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(mark(x2)), nats(s(mark(x2))))), MARK(x0)) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), nats(mark(x2))), MARK(x0)) MARK(filter(x0, x1, nats(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, nats(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, nats(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, nats(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, nats(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, nats(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(0)), MARK(x0)) MARK(filter(x0, x1, nats(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), nats(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), nats(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), nats(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, nats(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), nats(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, nats(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__nats(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, nats(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__nats(mark(x2))), 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x2)) -> c10(A__FILTER(a__sieve(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), nats(x1), x2)) -> c10(A__FILTER(a__nats(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, nats(x1), x2)) -> c10(A__FILTER(a__zprimes, a__nats(mark(x1)), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), nats(x1), x2)) -> c10(A__FILTER(cons(mark(z0), z1), a__nats(mark(x1)), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, nats(x1), x2)) -> c10(MARK(x0)) MARK(filter(s(z0), nats(x1), x2)) -> c10(MARK(s(z0))) MARK(filter(x0, zprimes, x1)) -> c10(A__FILTER(mark(x0), a__sieve(a__nats(s(s(0)))), mark(x1)), MARK(x0)) MARK(filter(filter(z0, z1, z2), zprimes, x1)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__zprimes, mark(x1)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), zprimes, x1)) -> c10(A__FILTER(a__sieve(mark(z0)), a__zprimes, mark(x1)), MARK(sieve(z0))) MARK(filter(nats(z0), zprimes, x1)) -> c10(A__FILTER(a__nats(mark(z0)), 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MARK(filter(x0, s(s(z0)), x2)) -> c10(A__FILTER(mark(x0), s(s(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), s(x1), x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), s(mark(x1)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), s(x1), x2)) -> c10(A__FILTER(a__sieve(mark(z0)), s(mark(x1)), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), s(x1), x2)) -> c10(A__FILTER(a__nats(mark(z0)), s(mark(x1)), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, s(x1), x2)) -> c10(A__FILTER(a__zprimes, s(mark(x1)), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), s(x1), x2)) -> c10(A__FILTER(cons(mark(z0), z1), s(mark(x1)), mark(x2)), MARK(cons(z0, z1))) MARK(filter(s(z0), s(x1), x2)) -> c10(MARK(s(z0))) MARK(filter(filter(x0, x1, filter(z0, z1, z2)), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), mark(x3), mark(x4)), MARK(filter(x0, x1, filter(z0, z1, z2)))) MARK(filter(filter(x0, x1, sieve(z0)), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), a__sieve(mark(z0))), mark(x3), mark(x4)), MARK(filter(x0, x1, sieve(z0)))) MARK(filter(filter(x0, x1, nats(z0)), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), a__nats(mark(z0))), mark(x3), mark(x4)), MARK(filter(x0, x1, nats(z0)))) MARK(filter(filter(x0, x1, zprimes), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), a__zprimes), mark(x3), mark(x4)), MARK(filter(x0, x1, zprimes))) MARK(filter(filter(x0, x1, cons(z0, z1)), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), cons(mark(z0), z1)), mark(x3), mark(x4)), MARK(filter(x0, x1, cons(z0, z1)))) MARK(filter(filter(x0, x1, 0), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), 0), mark(x3), mark(x4)), MARK(filter(x0, x1, 0))) MARK(filter(filter(x0, x1, s(z0)), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), s(mark(z0))), mark(x3), mark(x4)), MARK(filter(x0, x1, s(z0)))) MARK(filter(filter(x0, filter(z0, z1, z2), x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, filter(z0, z1, z2), x2))) MARK(filter(filter(x0, sieve(z0), x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), a__sieve(mark(z0)), mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, sieve(z0), x2))) MARK(filter(filter(x0, nats(z0), x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), a__nats(mark(z0)), mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, nats(z0), x2))) MARK(filter(filter(x0, zprimes, x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), a__zprimes, mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, zprimes, x2))) MARK(filter(filter(x0, cons(z0, z1), x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), cons(mark(z0), z1), mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, cons(z0, z1), x2))) MARK(filter(filter(x0, 0, x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), 0, mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, 0, x2))) MARK(filter(filter(x0, s(z0), x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), s(mark(z0)), mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, s(z0), x2))) MARK(filter(filter(filter(z0, z1, z2), x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(filter(z0, z1, z2), x1, x2))) MARK(filter(filter(sieve(z0), x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(a__sieve(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(sieve(z0), x1, x2))) MARK(filter(filter(nats(z0), x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(a__nats(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(nats(z0), x1, x2))) MARK(filter(filter(zprimes, x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(a__zprimes, mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(zprimes, x1, x2))) MARK(filter(filter(cons(z0, z1), x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(cons(mark(z0), z1), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(cons(z0, z1), x1, x2))) MARK(filter(filter(0, x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(0, mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(0, x1, x2))) MARK(filter(filter(s(z0), x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(s(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(s(z0), x1, x2))) MARK(filter(filter(x0, x1, x2), cons(z0, z1), x4)) -> c10(MARK(filter(x0, x1, x2))) MARK(filter(filter(x0, x1, x2), 0, x4)) -> c10(MARK(filter(x0, x1, x2))) MARK(filter(filter(x0, x1, x2), x3, x4)) -> c10(MARK(filter(x0, x1, x2))) MARK(filter(sieve(filter(z0, z1, z2)), x1, x2)) -> c10(A__FILTER(a__sieve(a__filter(mark(z0), mark(z1), mark(z2))), mark(x1), mark(x2)), MARK(sieve(filter(z0, z1, z2)))) MARK(filter(sieve(sieve(z0)), x1, x2)) -> c10(A__FILTER(a__sieve(a__sieve(mark(z0))), mark(x1), mark(x2)), MARK(sieve(sieve(z0)))) MARK(filter(sieve(nats(z0)), x1, x2)) -> c10(A__FILTER(a__sieve(a__nats(mark(z0))), mark(x1), mark(x2)), MARK(sieve(nats(z0)))) MARK(filter(sieve(zprimes), x1, x2)) -> c10(A__FILTER(a__sieve(a__zprimes), mark(x1), mark(x2)), MARK(sieve(zprimes))) MARK(filter(sieve(cons(z0, z1)), x1, x2)) -> c10(A__FILTER(a__sieve(cons(mark(z0), z1)), mark(x1), mark(x2)), MARK(sieve(cons(z0, z1)))) MARK(filter(sieve(s(z0)), x1, x2)) -> c10(A__FILTER(a__sieve(s(mark(z0))), mark(x1), mark(x2)), MARK(sieve(s(z0)))) MARK(filter(sieve(x0), cons(z0, z1), x2)) -> c10(MARK(sieve(x0))) MARK(filter(sieve(x0), 0, x2)) -> c10(MARK(sieve(x0))) MARK(filter(sieve(x0), x1, x2)) -> c10(MARK(sieve(x0))) MARK(filter(sieve(0), x1, x2)) -> c10(A__FILTER(a__sieve(0), mark(x1), mark(x2))) MARK(filter(nats(x0), x1, x2)) -> c10(A__FILTER(cons(mark(mark(x0)), nats(s(mark(x0)))), mark(x1), mark(x2)), MARK(nats(x0))) MARK(filter(nats(filter(z0, z1, z2)), x1, x2)) -> c10(A__FILTER(a__nats(a__filter(mark(z0), mark(z1), mark(z2))), mark(x1), mark(x2)), MARK(nats(filter(z0, z1, z2)))) MARK(filter(nats(sieve(z0)), x1, x2)) -> c10(A__FILTER(a__nats(a__sieve(mark(z0))), mark(x1), mark(x2)), MARK(nats(sieve(z0)))) MARK(filter(nats(nats(z0)), x1, x2)) -> c10(A__FILTER(a__nats(a__nats(mark(z0))), mark(x1), mark(x2)), MARK(nats(nats(z0)))) MARK(filter(nats(zprimes), x1, x2)) -> c10(A__FILTER(a__nats(a__zprimes), mark(x1), mark(x2)), MARK(nats(zprimes))) MARK(filter(nats(cons(z0, z1)), x1, x2)) -> c10(A__FILTER(a__nats(cons(mark(z0), z1)), mark(x1), mark(x2)), MARK(nats(cons(z0, z1)))) MARK(filter(nats(0), x1, x2)) -> c10(A__FILTER(a__nats(0), mark(x1), mark(x2)), MARK(nats(0))) MARK(filter(nats(s(z0)), x1, x2)) -> c10(A__FILTER(a__nats(s(mark(z0))), mark(x1), mark(x2)), MARK(nats(s(z0)))) MARK(filter(nats(x0), x1, x2)) -> c10(MARK(nats(x0))) MARK(filter(nats(x0), cons(z0, z1), x2)) -> c10(MARK(nats(x0))) MARK(filter(nats(x0), 0, x2)) -> c10(MARK(nats(x0))) MARK(filter(zprimes, x0, x1)) -> c10(A__FILTER(a__sieve(a__nats(s(s(0)))), mark(x0), mark(x1)), MARK(zprimes)) MARK(filter(zprimes, cons(z0, z1), x1)) -> c10(MARK(zprimes)) MARK(filter(zprimes, 0, x1)) -> c10(MARK(zprimes)) MARK(filter(zprimes, x0, x1)) -> c10(MARK(zprimes)) MARK(filter(cons(filter(z0, z1, z2), x1), x2, x3)) -> c10(A__FILTER(cons(a__filter(mark(z0), mark(z1), mark(z2)), x1), mark(x2), mark(x3)), MARK(cons(filter(z0, z1, z2), x1))) MARK(filter(cons(sieve(z0), x1), x2, x3)) -> c10(A__FILTER(cons(a__sieve(mark(z0)), x1), mark(x2), mark(x3)), MARK(cons(sieve(z0), x1))) MARK(filter(cons(nats(z0), x1), x2, x3)) -> c10(A__FILTER(cons(a__nats(mark(z0)), x1), mark(x2), mark(x3)), MARK(cons(nats(z0), x1))) MARK(filter(cons(zprimes, x1), x2, x3)) -> c10(A__FILTER(cons(a__zprimes, x1), mark(x2), mark(x3)), MARK(cons(zprimes, x1))) MARK(filter(cons(cons(z0, z1), x1), x2, x3)) -> c10(A__FILTER(cons(cons(mark(z0), z1), x1), mark(x2), mark(x3)), MARK(cons(cons(z0, z1), x1))) MARK(filter(cons(0, x1), x2, x3)) -> c10(A__FILTER(cons(0, x1), mark(x2), mark(x3)), MARK(cons(0, x1))) MARK(filter(cons(s(z0), x1), x2, x3)) -> c10(A__FILTER(cons(s(mark(z0)), x1), mark(x2), mark(x3)), MARK(cons(s(z0), x1))) MARK(filter(cons(x0, x1), x2, x3)) -> c10(MARK(cons(x0, x1))) MARK(filter(cons(x0, x1), cons(z0, z1), x3)) -> c10(MARK(cons(x0, x1))) MARK(filter(cons(x0, x1), 0, x3)) -> c10(MARK(cons(x0, x1))) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c14_2, c15_1, c16_1, c18_1, c8_1, c10_1, c11_2, c11_1, c12_2, c12_1, c13_2, c13_1, c8_2, c10_2 ---------------------------------------- (145) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(filter(x0, x1, filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x1)) by MARK(filter(x0, x1, filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), cons(mark(z0), z1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(cons(z0, z1))) MARK(filter(x0, 0, filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), 0, a__filter(mark(x2), mark(x3), mark(x4))), MARK(0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c11(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(0, x1, filter(x2, x3, x4))) -> c11(A__FILTER(0, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c11(A__FILTER(s(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) ---------------------------------------- (146) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(x0, 0, filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, sieve(x2))) -> c10(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x0)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, sieve(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, sieve(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, sieve(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(0)), MARK(x0)) MARK(filter(x0, x1, sieve(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), sieve(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, sieve(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__sieve(mark(x2))), MARK(x0)) 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x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), cons(mark(z0), z1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(cons(z0, z1))) MARK(filter(x0, 0, filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), 0, a__filter(mark(x2), mark(x3), mark(x4))), MARK(0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c11(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(0, x1, filter(x2, x3, x4))) -> c11(A__FILTER(0, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c11(A__FILTER(s(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) S tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(x0, 0, filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, sieve(x2))) -> c10(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x0)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), 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MARK(filter(cons(x0, x1), x2, x3)) -> c10(MARK(cons(x0, x1))) MARK(filter(cons(x0, x1), cons(z0, z1), x3)) -> c10(MARK(cons(x0, x1))) MARK(filter(cons(x0, x1), 0, x3)) -> c10(MARK(cons(x0, x1))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), cons(mark(z0), z1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(cons(z0, z1))) MARK(filter(x0, 0, filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), 0, a__filter(mark(x2), mark(x3), mark(x4))), MARK(0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c11(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(0, x1, filter(x2, x3, x4))) -> c11(A__FILTER(0, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c11(A__FILTER(s(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c14_2, c15_1, c16_1, c18_1, c8_1, c10_1, c11_2, c11_1, c12_2, c12_1, c13_2, c13_1, c8_2, c10_2 ---------------------------------------- (147) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(filter(x0, 0, filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), 0, a__filter(mark(x2), mark(x3), mark(x4))), MARK(0)) ---------------------------------------- (148) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), 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c10(A__FILTER(cons(a__sieve(mark(z0)), x1), mark(x2), mark(x3)), MARK(cons(sieve(z0), x1))) MARK(filter(cons(nats(z0), x1), x2, x3)) -> c10(A__FILTER(cons(a__nats(mark(z0)), x1), mark(x2), mark(x3)), MARK(cons(nats(z0), x1))) MARK(filter(cons(zprimes, x1), x2, x3)) -> c10(A__FILTER(cons(a__zprimes, x1), mark(x2), mark(x3)), MARK(cons(zprimes, x1))) MARK(filter(cons(cons(z0, z1), x1), x2, x3)) -> c10(A__FILTER(cons(cons(mark(z0), z1), x1), mark(x2), mark(x3)), MARK(cons(cons(z0, z1), x1))) MARK(filter(cons(0, x1), x2, x3)) -> c10(A__FILTER(cons(0, x1), mark(x2), mark(x3)), MARK(cons(0, x1))) MARK(filter(cons(s(z0), x1), x2, x3)) -> c10(A__FILTER(cons(s(mark(z0)), x1), mark(x2), mark(x3)), MARK(cons(s(z0), x1))) MARK(filter(cons(x0, x1), x2, x3)) -> c10(MARK(cons(x0, x1))) MARK(filter(cons(x0, x1), cons(z0, z1), x3)) -> c10(MARK(cons(x0, x1))) MARK(filter(cons(x0, x1), 0, x3)) -> c10(MARK(cons(x0, x1))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), cons(mark(z0), z1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(cons(z0, z1))) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c11(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(0, x1, filter(x2, x3, x4))) -> c11(A__FILTER(0, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c11(A__FILTER(s(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) S tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> 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c10(A__FILTER(a__nats(a__nats(mark(z0))), mark(x1), mark(x2)), MARK(nats(nats(z0)))) MARK(filter(nats(zprimes), x1, x2)) -> c10(A__FILTER(a__nats(a__zprimes), mark(x1), mark(x2)), MARK(nats(zprimes))) MARK(filter(nats(cons(z0, z1)), x1, x2)) -> c10(A__FILTER(a__nats(cons(mark(z0), z1)), mark(x1), mark(x2)), MARK(nats(cons(z0, z1)))) MARK(filter(nats(0), x1, x2)) -> c10(A__FILTER(a__nats(0), mark(x1), mark(x2)), MARK(nats(0))) MARK(filter(nats(s(z0)), x1, x2)) -> c10(A__FILTER(a__nats(s(mark(z0))), mark(x1), mark(x2)), MARK(nats(s(z0)))) MARK(filter(nats(x0), x1, x2)) -> c10(MARK(nats(x0))) MARK(filter(nats(x0), cons(z0, z1), x2)) -> c10(MARK(nats(x0))) MARK(filter(nats(x0), 0, x2)) -> c10(MARK(nats(x0))) MARK(filter(zprimes, x0, x1)) -> c10(A__FILTER(a__sieve(a__nats(s(s(0)))), mark(x0), mark(x1)), MARK(zprimes)) MARK(filter(zprimes, cons(z0, z1), x1)) -> c10(MARK(zprimes)) MARK(filter(zprimes, 0, x1)) -> c10(MARK(zprimes)) MARK(filter(zprimes, x0, x1)) -> c10(MARK(zprimes)) MARK(filter(cons(filter(z0, z1, z2), x1), x2, x3)) -> c10(A__FILTER(cons(a__filter(mark(z0), mark(z1), mark(z2)), x1), mark(x2), mark(x3)), MARK(cons(filter(z0, z1, z2), x1))) MARK(filter(cons(sieve(z0), x1), x2, x3)) -> c10(A__FILTER(cons(a__sieve(mark(z0)), x1), mark(x2), mark(x3)), MARK(cons(sieve(z0), x1))) MARK(filter(cons(nats(z0), x1), x2, x3)) -> c10(A__FILTER(cons(a__nats(mark(z0)), x1), mark(x2), mark(x3)), MARK(cons(nats(z0), x1))) MARK(filter(cons(zprimes, x1), x2, x3)) -> c10(A__FILTER(cons(a__zprimes, x1), mark(x2), mark(x3)), MARK(cons(zprimes, x1))) MARK(filter(cons(cons(z0, z1), x1), x2, x3)) -> c10(A__FILTER(cons(cons(mark(z0), z1), x1), mark(x2), mark(x3)), MARK(cons(cons(z0, z1), x1))) MARK(filter(cons(0, x1), x2, x3)) -> c10(A__FILTER(cons(0, x1), mark(x2), mark(x3)), MARK(cons(0, x1))) MARK(filter(cons(s(z0), x1), x2, x3)) -> c10(A__FILTER(cons(s(mark(z0)), x1), mark(x2), mark(x3)), MARK(cons(s(z0), x1))) MARK(filter(cons(x0, x1), x2, x3)) -> c10(MARK(cons(x0, x1))) MARK(filter(cons(x0, x1), cons(z0, z1), x3)) -> c10(MARK(cons(x0, x1))) MARK(filter(cons(x0, x1), 0, x3)) -> c10(MARK(cons(x0, x1))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), cons(mark(z0), z1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(cons(z0, z1))) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c11(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(0, x1, filter(x2, x3, x4))) -> c11(A__FILTER(0, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c11(A__FILTER(s(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c14_2, c15_1, c16_1, c18_1, c8_1, c10_1, c11_2, c11_1, c12_2, c12_1, c13_2, c13_1, c8_2, c10_2 ---------------------------------------- (149) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (150) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), 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x4)) -> c10(A__FILTER(a__filter(a__nats(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(nats(z0), x1, x2))) MARK(filter(filter(zprimes, x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(a__zprimes, mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(zprimes, x1, x2))) MARK(filter(filter(cons(z0, z1), x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(cons(mark(z0), z1), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(cons(z0, z1), x1, x2))) MARK(filter(filter(0, x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(0, mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(0, x1, x2))) MARK(filter(filter(s(z0), x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(s(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(s(z0), x1, x2))) MARK(filter(filter(x0, x1, x2), cons(z0, z1), x4)) -> c10(MARK(filter(x0, x1, x2))) MARK(filter(filter(x0, x1, x2), 0, x4)) -> c10(MARK(filter(x0, x1, x2))) MARK(filter(filter(x0, x1, x2), x3, x4)) -> c10(MARK(filter(x0, x1, x2))) MARK(filter(sieve(filter(z0, z1, z2)), x1, x2)) -> c10(A__FILTER(a__sieve(a__filter(mark(z0), mark(z1), mark(z2))), mark(x1), mark(x2)), MARK(sieve(filter(z0, z1, z2)))) MARK(filter(sieve(sieve(z0)), x1, x2)) -> c10(A__FILTER(a__sieve(a__sieve(mark(z0))), mark(x1), mark(x2)), MARK(sieve(sieve(z0)))) MARK(filter(sieve(nats(z0)), x1, x2)) -> c10(A__FILTER(a__sieve(a__nats(mark(z0))), mark(x1), mark(x2)), MARK(sieve(nats(z0)))) MARK(filter(sieve(zprimes), x1, x2)) -> c10(A__FILTER(a__sieve(a__zprimes), mark(x1), mark(x2)), MARK(sieve(zprimes))) MARK(filter(sieve(cons(z0, z1)), x1, x2)) -> c10(A__FILTER(a__sieve(cons(mark(z0), z1)), mark(x1), mark(x2)), MARK(sieve(cons(z0, z1)))) MARK(filter(sieve(s(z0)), x1, x2)) -> c10(A__FILTER(a__sieve(s(mark(z0))), mark(x1), mark(x2)), MARK(sieve(s(z0)))) MARK(filter(sieve(x0), cons(z0, z1), x2)) -> c10(MARK(sieve(x0))) MARK(filter(sieve(x0), 0, x2)) -> c10(MARK(sieve(x0))) MARK(filter(sieve(x0), x1, x2)) -> c10(MARK(sieve(x0))) MARK(filter(sieve(0), x1, x2)) -> c10(A__FILTER(a__sieve(0), mark(x1), mark(x2))) MARK(filter(nats(x0), x1, x2)) -> c10(A__FILTER(cons(mark(mark(x0)), nats(s(mark(x0)))), mark(x1), mark(x2)), MARK(nats(x0))) MARK(filter(nats(filter(z0, z1, z2)), x1, x2)) -> c10(A__FILTER(a__nats(a__filter(mark(z0), mark(z1), mark(z2))), mark(x1), mark(x2)), MARK(nats(filter(z0, z1, z2)))) MARK(filter(nats(sieve(z0)), x1, x2)) -> c10(A__FILTER(a__nats(a__sieve(mark(z0))), mark(x1), mark(x2)), MARK(nats(sieve(z0)))) MARK(filter(nats(nats(z0)), x1, x2)) -> c10(A__FILTER(a__nats(a__nats(mark(z0))), mark(x1), mark(x2)), MARK(nats(nats(z0)))) MARK(filter(nats(zprimes), x1, x2)) -> c10(A__FILTER(a__nats(a__zprimes), mark(x1), mark(x2)), MARK(nats(zprimes))) MARK(filter(nats(cons(z0, z1)), x1, x2)) -> c10(A__FILTER(a__nats(cons(mark(z0), z1)), mark(x1), mark(x2)), MARK(nats(cons(z0, z1)))) MARK(filter(nats(0), x1, x2)) -> c10(A__FILTER(a__nats(0), mark(x1), mark(x2)), MARK(nats(0))) MARK(filter(nats(s(z0)), x1, x2)) -> c10(A__FILTER(a__nats(s(mark(z0))), mark(x1), mark(x2)), MARK(nats(s(z0)))) MARK(filter(nats(x0), x1, x2)) -> c10(MARK(nats(x0))) MARK(filter(nats(x0), cons(z0, z1), x2)) -> c10(MARK(nats(x0))) MARK(filter(nats(x0), 0, x2)) -> c10(MARK(nats(x0))) MARK(filter(zprimes, x0, x1)) -> c10(A__FILTER(a__sieve(a__nats(s(s(0)))), mark(x0), mark(x1)), MARK(zprimes)) MARK(filter(zprimes, cons(z0, z1), x1)) -> c10(MARK(zprimes)) MARK(filter(zprimes, 0, x1)) -> c10(MARK(zprimes)) MARK(filter(zprimes, x0, x1)) -> c10(MARK(zprimes)) MARK(filter(cons(filter(z0, z1, z2), x1), x2, x3)) -> c10(A__FILTER(cons(a__filter(mark(z0), mark(z1), mark(z2)), x1), mark(x2), mark(x3)), MARK(cons(filter(z0, z1, z2), x1))) MARK(filter(cons(sieve(z0), x1), x2, x3)) -> c10(A__FILTER(cons(a__sieve(mark(z0)), x1), mark(x2), mark(x3)), MARK(cons(sieve(z0), x1))) MARK(filter(cons(nats(z0), x1), x2, x3)) -> c10(A__FILTER(cons(a__nats(mark(z0)), x1), mark(x2), mark(x3)), MARK(cons(nats(z0), x1))) MARK(filter(cons(zprimes, x1), x2, x3)) -> c10(A__FILTER(cons(a__zprimes, x1), mark(x2), mark(x3)), MARK(cons(zprimes, x1))) MARK(filter(cons(cons(z0, z1), x1), x2, x3)) -> c10(A__FILTER(cons(cons(mark(z0), z1), x1), mark(x2), mark(x3)), MARK(cons(cons(z0, z1), x1))) MARK(filter(cons(0, x1), x2, x3)) -> c10(A__FILTER(cons(0, x1), mark(x2), mark(x3)), MARK(cons(0, x1))) MARK(filter(cons(s(z0), x1), x2, x3)) -> c10(A__FILTER(cons(s(mark(z0)), x1), mark(x2), mark(x3)), MARK(cons(s(z0), x1))) MARK(filter(cons(x0, x1), x2, x3)) -> c10(MARK(cons(x0, x1))) MARK(filter(cons(x0, x1), cons(z0, z1), x3)) -> c10(MARK(cons(x0, x1))) MARK(filter(cons(x0, x1), 0, x3)) -> c10(MARK(cons(x0, x1))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c11(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, filter(x2, x3, x4))) -> c11(MARK(x1)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c11(MARK(x1)) S tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(x0, 0, filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, sieve(x2))) -> c10(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x0)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, sieve(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, sieve(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, sieve(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(0)), MARK(x0)) MARK(filter(x0, x1, sieve(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), sieve(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, sieve(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, sieve(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__sieve(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, sieve(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, sieve(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, sieve(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), a__sieve(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, sieve(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__sieve(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), sieve(x2))) -> c10(MARK(x0)) MARK(filter(x0, 0, sieve(x2))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, sieve(x2))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(mark(x2)), nats(s(mark(x2))))), MARK(x0)) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), nats(mark(x2))), MARK(x0)) MARK(filter(x0, x1, nats(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, nats(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, nats(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, nats(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, nats(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, nats(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(0)), MARK(x0)) MARK(filter(x0, x1, nats(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), nats(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), nats(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), nats(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, nats(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), nats(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, nats(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__nats(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, nats(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, nats(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, nats(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), a__nats(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, nats(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__nats(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), nats(x2))) -> c10(MARK(x0)) MARK(filter(x0, 0, nats(x2))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, nats(x2))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, zprimes)) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(s(s(0))))), MARK(x0)) MARK(filter(x0, x1, zprimes)) -> c10(A__FILTER(mark(x0), mark(x1), zprimes), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), zprimes)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__zprimes), MARK(x0)) MARK(filter(x0, sieve(z0), zprimes)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__zprimes), MARK(x0)) MARK(filter(x0, nats(z0), zprimes)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__zprimes), MARK(x0)) MARK(filter(x0, zprimes, zprimes)) -> c10(A__FILTER(mark(x0), a__zprimes, a__zprimes), MARK(x0)) MARK(filter(x0, s(z0), zprimes)) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__zprimes), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, zprimes)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__zprimes), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, zprimes)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__zprimes), MARK(sieve(z0))) MARK(filter(nats(z0), x1, zprimes)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__zprimes), MARK(nats(z0))) MARK(filter(zprimes, x1, zprimes)) -> c10(A__FILTER(a__zprimes, mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, zprimes)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__zprimes), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), zprimes)) -> c10(MARK(x0)) MARK(filter(x0, 0, zprimes)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, zprimes)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, cons(filter(z0, z1, z2), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(a__filter(mark(z0), mark(z1), mark(z2)), x3)), MARK(x0)) MARK(filter(x0, x1, cons(sieve(z0), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(a__sieve(mark(z0)), x3)), MARK(x0)) MARK(filter(x0, x1, cons(nats(z0), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(a__nats(mark(z0)), x3)), MARK(x0)) MARK(filter(x0, x1, cons(zprimes, x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(a__zprimes, x3)), MARK(x0)) MARK(filter(x0, x1, cons(cons(z0, z1), 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c10(A__FILTER(a__filter(mark(x0), a__zprimes, mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, zprimes, x2))) MARK(filter(filter(x0, cons(z0, z1), x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), cons(mark(z0), z1), mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, cons(z0, z1), x2))) MARK(filter(filter(x0, 0, x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), 0, mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, 0, x2))) MARK(filter(filter(x0, s(z0), x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), s(mark(z0)), mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, s(z0), x2))) MARK(filter(filter(filter(z0, z1, z2), x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(filter(z0, z1, z2), x1, x2))) MARK(filter(filter(sieve(z0), x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(a__sieve(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(sieve(z0), x1, x2))) MARK(filter(filter(nats(z0), x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(a__nats(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(nats(z0), x1, x2))) MARK(filter(filter(zprimes, x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(a__zprimes, mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(zprimes, x1, x2))) MARK(filter(filter(cons(z0, z1), x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(cons(mark(z0), z1), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(cons(z0, z1), x1, x2))) MARK(filter(filter(0, x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(0, mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(0, x1, x2))) MARK(filter(filter(s(z0), x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(s(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(s(z0), x1, x2))) MARK(filter(filter(x0, x1, x2), cons(z0, z1), x4)) -> c10(MARK(filter(x0, x1, x2))) MARK(filter(filter(x0, x1, x2), 0, x4)) -> c10(MARK(filter(x0, x1, x2))) MARK(filter(filter(x0, x1, x2), x3, x4)) -> c10(MARK(filter(x0, x1, x2))) MARK(filter(sieve(filter(z0, z1, z2)), x1, x2)) -> c10(A__FILTER(a__sieve(a__filter(mark(z0), mark(z1), mark(z2))), mark(x1), mark(x2)), MARK(sieve(filter(z0, z1, z2)))) MARK(filter(sieve(sieve(z0)), x1, x2)) -> c10(A__FILTER(a__sieve(a__sieve(mark(z0))), mark(x1), mark(x2)), MARK(sieve(sieve(z0)))) MARK(filter(sieve(nats(z0)), x1, x2)) -> c10(A__FILTER(a__sieve(a__nats(mark(z0))), mark(x1), mark(x2)), MARK(sieve(nats(z0)))) MARK(filter(sieve(zprimes), x1, x2)) -> c10(A__FILTER(a__sieve(a__zprimes), mark(x1), mark(x2)), MARK(sieve(zprimes))) MARK(filter(sieve(cons(z0, z1)), x1, x2)) -> c10(A__FILTER(a__sieve(cons(mark(z0), z1)), mark(x1), mark(x2)), MARK(sieve(cons(z0, z1)))) MARK(filter(sieve(s(z0)), x1, x2)) -> c10(A__FILTER(a__sieve(s(mark(z0))), mark(x1), mark(x2)), MARK(sieve(s(z0)))) MARK(filter(sieve(x0), cons(z0, z1), x2)) -> c10(MARK(sieve(x0))) MARK(filter(sieve(x0), 0, x2)) -> c10(MARK(sieve(x0))) MARK(filter(sieve(x0), x1, x2)) -> c10(MARK(sieve(x0))) MARK(filter(sieve(0), x1, x2)) -> c10(A__FILTER(a__sieve(0), mark(x1), mark(x2))) MARK(filter(nats(x0), x1, x2)) -> c10(A__FILTER(cons(mark(mark(x0)), nats(s(mark(x0)))), mark(x1), mark(x2)), MARK(nats(x0))) MARK(filter(nats(filter(z0, z1, z2)), x1, x2)) -> c10(A__FILTER(a__nats(a__filter(mark(z0), mark(z1), mark(z2))), mark(x1), mark(x2)), MARK(nats(filter(z0, z1, z2)))) MARK(filter(nats(sieve(z0)), x1, x2)) -> c10(A__FILTER(a__nats(a__sieve(mark(z0))), mark(x1), mark(x2)), MARK(nats(sieve(z0)))) MARK(filter(nats(nats(z0)), x1, x2)) -> c10(A__FILTER(a__nats(a__nats(mark(z0))), mark(x1), mark(x2)), MARK(nats(nats(z0)))) MARK(filter(nats(zprimes), x1, x2)) -> c10(A__FILTER(a__nats(a__zprimes), mark(x1), mark(x2)), MARK(nats(zprimes))) MARK(filter(nats(cons(z0, z1)), x1, x2)) -> c10(A__FILTER(a__nats(cons(mark(z0), z1)), mark(x1), mark(x2)), MARK(nats(cons(z0, z1)))) MARK(filter(nats(0), x1, x2)) -> c10(A__FILTER(a__nats(0), mark(x1), mark(x2)), MARK(nats(0))) MARK(filter(nats(s(z0)), x1, x2)) -> c10(A__FILTER(a__nats(s(mark(z0))), mark(x1), mark(x2)), MARK(nats(s(z0)))) MARK(filter(nats(x0), x1, x2)) -> c10(MARK(nats(x0))) MARK(filter(nats(x0), cons(z0, z1), x2)) -> c10(MARK(nats(x0))) MARK(filter(nats(x0), 0, x2)) -> c10(MARK(nats(x0))) MARK(filter(zprimes, x0, x1)) -> c10(A__FILTER(a__sieve(a__nats(s(s(0)))), mark(x0), mark(x1)), MARK(zprimes)) MARK(filter(zprimes, cons(z0, z1), x1)) -> c10(MARK(zprimes)) MARK(filter(zprimes, 0, x1)) -> c10(MARK(zprimes)) MARK(filter(zprimes, x0, x1)) -> c10(MARK(zprimes)) MARK(filter(cons(filter(z0, z1, z2), x1), x2, x3)) -> c10(A__FILTER(cons(a__filter(mark(z0), mark(z1), mark(z2)), x1), mark(x2), mark(x3)), MARK(cons(filter(z0, z1, z2), x1))) MARK(filter(cons(sieve(z0), x1), x2, x3)) -> c10(A__FILTER(cons(a__sieve(mark(z0)), x1), mark(x2), mark(x3)), MARK(cons(sieve(z0), x1))) MARK(filter(cons(nats(z0), x1), x2, x3)) -> c10(A__FILTER(cons(a__nats(mark(z0)), x1), mark(x2), mark(x3)), MARK(cons(nats(z0), x1))) MARK(filter(cons(zprimes, x1), x2, x3)) -> c10(A__FILTER(cons(a__zprimes, x1), mark(x2), mark(x3)), MARK(cons(zprimes, x1))) MARK(filter(cons(cons(z0, z1), x1), x2, x3)) -> c10(A__FILTER(cons(cons(mark(z0), z1), x1), mark(x2), mark(x3)), MARK(cons(cons(z0, z1), x1))) MARK(filter(cons(0, x1), x2, x3)) -> c10(A__FILTER(cons(0, x1), mark(x2), mark(x3)), MARK(cons(0, x1))) MARK(filter(cons(s(z0), x1), x2, x3)) -> c10(A__FILTER(cons(s(mark(z0)), x1), mark(x2), mark(x3)), MARK(cons(s(z0), x1))) MARK(filter(cons(x0, x1), x2, x3)) -> c10(MARK(cons(x0, x1))) MARK(filter(cons(x0, x1), cons(z0, z1), x3)) -> c10(MARK(cons(x0, x1))) MARK(filter(cons(x0, x1), 0, x3)) -> c10(MARK(cons(x0, x1))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c11(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, filter(x2, x3, x4))) -> c11(MARK(x1)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c11(MARK(x1)) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c14_2, c15_1, c16_1, c18_1, c8_1, c10_1, c11_2, c11_1, c12_2, c12_1, c13_2, c13_1, c8_2, c10_2 ---------------------------------------- (151) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(filter(x0, x1, sieve(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(x1)) by MARK(filter(x0, x1, sieve(x2))) -> c11(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x1)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x1)) MARK(filter(x0, x1, sieve(sieve(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__sieve(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, sieve(nats(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, sieve(zprimes))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__zprimes)), MARK(x1)) MARK(filter(x0, x1, sieve(cons(z0, z1)))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(cons(mark(z0), z1))), MARK(x1)) MARK(filter(x0, x1, sieve(0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(0)), MARK(x1)) MARK(filter(x0, x1, sieve(s(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(s(mark(z0)))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), sieve(x2))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), sieve(x2))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), a__sieve(mark(x2))), MARK(sieve(z0))) MARK(filter(x0, nats(z0), sieve(x2))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), a__sieve(mark(x2))), MARK(nats(z0))) MARK(filter(x0, zprimes, sieve(x2))) -> c11(A__FILTER(mark(x0), a__zprimes, a__sieve(mark(x2))), MARK(zprimes)) MARK(filter(x0, cons(z0, z1), sieve(x2))) -> c11(A__FILTER(mark(x0), cons(mark(z0), z1), a__sieve(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, 0, sieve(x2))) -> c11(A__FILTER(mark(x0), 0, a__sieve(mark(x2))), MARK(0)) MARK(filter(x0, s(z0), sieve(x2))) -> c11(A__FILTER(mark(x0), s(mark(z0)), a__sieve(mark(x2))), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, sieve(x2))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(sieve(z0), x1, sieve(x2))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(nats(z0), x1, sieve(x2))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(zprimes, x1, sieve(x2))) -> c11(A__FILTER(a__zprimes, mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(cons(z0, z1), x1, sieve(x2))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(0, x1, sieve(x2))) -> c11(A__FILTER(0, mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(s(z0), x1, sieve(x2))) -> c11(A__FILTER(s(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(x1)) ---------------------------------------- (152) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(x0, 0, filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, sieve(x2))) -> c10(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x0)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, sieve(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, sieve(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, sieve(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(0)), MARK(x0)) MARK(filter(x0, x1, sieve(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), sieve(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, sieve(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, sieve(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__sieve(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, sieve(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, sieve(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, sieve(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), 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c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(nats(z0), 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MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c11(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, filter(x2, x3, x4))) -> c11(MARK(x1)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c11(MARK(x1)) MARK(filter(x0, x1, sieve(x2))) -> c11(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x1)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x1)) MARK(filter(x0, x1, sieve(sieve(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__sieve(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, sieve(nats(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, sieve(zprimes))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__zprimes)), MARK(x1)) MARK(filter(x0, x1, sieve(cons(z0, z1)))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(cons(mark(z0), z1))), MARK(x1)) MARK(filter(x0, x1, sieve(0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(0)), MARK(x1)) MARK(filter(x0, x1, sieve(s(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(s(mark(z0)))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), sieve(x2))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), sieve(x2))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), a__sieve(mark(x2))), MARK(sieve(z0))) MARK(filter(x0, nats(z0), sieve(x2))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), a__sieve(mark(x2))), MARK(nats(z0))) MARK(filter(x0, zprimes, sieve(x2))) -> c11(A__FILTER(mark(x0), a__zprimes, a__sieve(mark(x2))), MARK(zprimes)) MARK(filter(x0, cons(z0, z1), sieve(x2))) -> c11(A__FILTER(mark(x0), cons(mark(z0), z1), a__sieve(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, 0, sieve(x2))) -> c11(A__FILTER(mark(x0), 0, a__sieve(mark(x2))), MARK(0)) MARK(filter(x0, s(z0), sieve(x2))) -> c11(A__FILTER(mark(x0), s(mark(z0)), a__sieve(mark(x2))), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, sieve(x2))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(sieve(z0), x1, sieve(x2))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(nats(z0), x1, sieve(x2))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(zprimes, x1, sieve(x2))) -> c11(A__FILTER(a__zprimes, mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(cons(z0, z1), x1, sieve(x2))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(0, x1, sieve(x2))) -> c11(A__FILTER(0, mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(s(z0), x1, sieve(x2))) -> c11(A__FILTER(s(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(x1)) S tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, 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filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c11(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, filter(x2, x3, x4))) -> c11(MARK(x1)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c11(MARK(x1)) MARK(filter(x0, x1, sieve(x2))) -> c11(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x1)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x1)) MARK(filter(x0, x1, sieve(sieve(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__sieve(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, sieve(nats(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, sieve(zprimes))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__zprimes)), MARK(x1)) MARK(filter(x0, x1, sieve(cons(z0, z1)))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(cons(mark(z0), z1))), MARK(x1)) MARK(filter(x0, x1, sieve(0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(0)), MARK(x1)) MARK(filter(x0, x1, sieve(s(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(s(mark(z0)))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), sieve(x2))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), sieve(x2))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), a__sieve(mark(x2))), MARK(sieve(z0))) MARK(filter(x0, nats(z0), sieve(x2))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), a__sieve(mark(x2))), MARK(nats(z0))) MARK(filter(x0, zprimes, sieve(x2))) -> c11(A__FILTER(mark(x0), a__zprimes, a__sieve(mark(x2))), MARK(zprimes)) MARK(filter(x0, cons(z0, z1), sieve(x2))) -> c11(A__FILTER(mark(x0), cons(mark(z0), z1), a__sieve(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, 0, sieve(x2))) -> c11(A__FILTER(mark(x0), 0, a__sieve(mark(x2))), MARK(0)) MARK(filter(x0, s(z0), sieve(x2))) -> c11(A__FILTER(mark(x0), s(mark(z0)), a__sieve(mark(x2))), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, sieve(x2))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(sieve(z0), x1, sieve(x2))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(nats(z0), x1, sieve(x2))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(zprimes, x1, sieve(x2))) -> c11(A__FILTER(a__zprimes, mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(cons(z0, z1), x1, sieve(x2))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(0, x1, sieve(x2))) -> c11(A__FILTER(0, mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(s(z0), x1, sieve(x2))) -> c11(A__FILTER(s(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(x1)) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c14_2, c15_1, c16_1, c18_1, c8_1, c10_1, c11_2, c11_1, c12_2, c12_1, c13_2, c13_1, c8_2, c10_2 ---------------------------------------- (153) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(filter(x0, 0, sieve(x2))) -> c11(A__FILTER(mark(x0), 0, a__sieve(mark(x2))), MARK(0)) ---------------------------------------- (154) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 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MARK(filter(x0, nats(z0), 0)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), 0), MARK(x0)) MARK(filter(x0, zprimes, 0)) -> c10(A__FILTER(mark(x0), a__zprimes, 0), MARK(x0)) MARK(filter(x0, s(z0), 0)) -> c10(A__FILTER(mark(x0), s(mark(z0)), 0), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, 0)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), 0), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, 0)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), 0), MARK(sieve(z0))) MARK(filter(nats(z0), x1, 0)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), 0), MARK(nats(z0))) MARK(filter(zprimes, x1, 0)) -> c10(A__FILTER(a__zprimes, mark(x1), 0), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, 0)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), 0), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), 0)) -> c10(MARK(x0)) MARK(filter(x0, 0, 0)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, 0)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, s(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), s(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, s(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), s(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, s(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), s(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, s(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), s(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, s(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), s(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, s(0))) -> c10(A__FILTER(mark(x0), mark(x1), s(0)), MARK(x0)) MARK(filter(x0, x1, s(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), s(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), s(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), s(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), s(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), s(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), s(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), s(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, s(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, s(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), s(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), s(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, s(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), s(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, s(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), s(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, s(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), s(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, s(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), s(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, s(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), s(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), s(x2))) -> c10(MARK(x0)) MARK(filter(x0, 0, s(x2))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, s(x2))) -> c10(MARK(s(z0))) MARK(filter(x0, filter(x1, x2, filter(z0, z1, z2)), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__filter(mark(z0), mark(z1), mark(z2))), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, sieve(z0)), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__sieve(mark(z0))), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, nats(z0)), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__nats(mark(z0))), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, zprimes), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__zprimes), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, cons(z0, z1)), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), cons(mark(z0), z1)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, 0), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), 0), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, x2, s(z0)), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), s(mark(z0))), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, filter(z0, z1, z2), x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), a__filter(mark(z0), mark(z1), mark(z2)), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, sieve(z0), x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), a__sieve(mark(z0)), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, nats(z0), x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), a__nats(mark(z0)), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, zprimes, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), a__zprimes, mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, cons(z0, z1), x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), cons(mark(z0), z1), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, 0, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), 0, mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(x1, s(z0), x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(mark(x1), s(mark(z0)), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(filter(z0, z1, z2), x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(sieve(z0), x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(a__sieve(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(nats(z0), x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(a__nats(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(zprimes, x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(a__zprimes, mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(cons(z0, z1), x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(cons(mark(z0), z1), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(0, x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(0, mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(x0, filter(s(z0), x2, x3), x4)) -> c10(A__FILTER(mark(x0), a__filter(s(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(x0)) MARK(filter(filter(z0, z1, z2), filter(x1, x2, x3), x4)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), filter(x1, x2, x3), x4)) -> c10(A__FILTER(a__sieve(mark(z0)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(sieve(z0))) MARK(filter(nats(z0), filter(x1, x2, x3), x4)) -> c10(A__FILTER(a__nats(mark(z0)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(nats(z0))) MARK(filter(zprimes, filter(x1, x2, x3), x4)) -> c10(A__FILTER(a__zprimes, a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(zprimes)) MARK(filter(cons(z0, z1), filter(x1, x2, x3), x4)) -> c10(A__FILTER(cons(mark(z0), z1), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(cons(z0, z1))) MARK(filter(x0, filter(x1, x2, x3), x4)) -> c10(MARK(x0)) MARK(filter(s(z0), filter(x1, x2, x3), x4)) -> c10(MARK(s(z0))) MARK(filter(x0, sieve(filter(z0, z1, z2)), x2)) -> c10(A__FILTER(mark(x0), a__sieve(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(sieve(z0)), x2)) -> c10(A__FILTER(mark(x0), a__sieve(a__sieve(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(nats(z0)), x2)) -> c10(A__FILTER(mark(x0), a__sieve(a__nats(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(zprimes), x2)) -> c10(A__FILTER(mark(x0), a__sieve(a__zprimes), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(cons(z0, z1)), x2)) -> c10(A__FILTER(mark(x0), a__sieve(cons(mark(z0), z1)), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(0), x2)) -> c10(A__FILTER(mark(x0), a__sieve(0), mark(x2)), MARK(x0)) MARK(filter(x0, sieve(s(z0)), x2)) -> c10(A__FILTER(mark(x0), a__sieve(s(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), sieve(x1), x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x1)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), sieve(x1), x2)) -> c10(A__FILTER(a__sieve(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), sieve(x1), x2)) -> c10(A__FILTER(a__nats(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, sieve(x1), x2)) -> c10(A__FILTER(a__zprimes, a__sieve(mark(x1)), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), sieve(x1), x2)) -> c10(A__FILTER(cons(mark(z0), z1), a__sieve(mark(x1)), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, sieve(x1), x2)) -> c10(MARK(x0)) MARK(filter(s(z0), sieve(x1), x2)) -> c10(MARK(s(z0))) MARK(filter(x0, nats(filter(z0, z1, z2)), x2)) -> c10(A__FILTER(mark(x0), a__nats(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(x0)) MARK(filter(x0, nats(sieve(z0)), x2)) -> c10(A__FILTER(mark(x0), a__nats(a__sieve(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, nats(nats(z0)), x2)) -> c10(A__FILTER(mark(x0), a__nats(a__nats(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, nats(zprimes), x2)) -> c10(A__FILTER(mark(x0), a__nats(a__zprimes), mark(x2)), MARK(x0)) MARK(filter(x0, nats(cons(z0, z1)), x2)) -> c10(A__FILTER(mark(x0), a__nats(cons(mark(z0), z1)), mark(x2)), MARK(x0)) MARK(filter(x0, nats(0), x2)) -> c10(A__FILTER(mark(x0), a__nats(0), mark(x2)), MARK(x0)) MARK(filter(x0, nats(s(z0)), x2)) -> c10(A__FILTER(mark(x0), a__nats(s(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), nats(x1), x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x1)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), nats(x1), x2)) -> c10(A__FILTER(a__sieve(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), nats(x1), x2)) -> c10(A__FILTER(a__nats(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, nats(x1), x2)) -> c10(A__FILTER(a__zprimes, a__nats(mark(x1)), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), nats(x1), x2)) -> c10(A__FILTER(cons(mark(z0), z1), a__nats(mark(x1)), mark(x2)), MARK(cons(z0, z1))) MARK(filter(x0, nats(x1), x2)) -> c10(MARK(x0)) MARK(filter(s(z0), nats(x1), x2)) -> c10(MARK(s(z0))) MARK(filter(x0, zprimes, x1)) -> c10(A__FILTER(mark(x0), a__sieve(a__nats(s(s(0)))), mark(x1)), MARK(x0)) MARK(filter(filter(z0, z1, z2), zprimes, x1)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__zprimes, mark(x1)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), zprimes, x1)) -> c10(A__FILTER(a__sieve(mark(z0)), a__zprimes, mark(x1)), MARK(sieve(z0))) MARK(filter(nats(z0), zprimes, x1)) -> c10(A__FILTER(a__nats(mark(z0)), a__zprimes, mark(x1)), MARK(nats(z0))) MARK(filter(zprimes, zprimes, x1)) -> c10(A__FILTER(a__zprimes, a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(cons(z0, z1), zprimes, x1)) -> c10(A__FILTER(cons(mark(z0), z1), a__zprimes, mark(x1)), MARK(cons(z0, z1))) MARK(filter(x0, zprimes, x1)) -> c10(MARK(x0)) MARK(filter(s(z0), zprimes, x1)) -> c10(MARK(s(z0))) MARK(filter(x0, s(filter(z0, z1, z2)), x2)) -> c10(A__FILTER(mark(x0), s(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(x0)) MARK(filter(x0, s(sieve(z0)), x2)) -> c10(A__FILTER(mark(x0), s(a__sieve(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, s(nats(z0)), x2)) -> c10(A__FILTER(mark(x0), s(a__nats(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, s(zprimes), x2)) -> c10(A__FILTER(mark(x0), s(a__zprimes), mark(x2)), MARK(x0)) MARK(filter(x0, s(cons(z0, z1)), x2)) -> c10(A__FILTER(mark(x0), s(cons(mark(z0), z1)), mark(x2)), MARK(x0)) MARK(filter(x0, s(0), x2)) -> c10(A__FILTER(mark(x0), s(0), mark(x2)), MARK(x0)) MARK(filter(x0, s(s(z0)), x2)) -> c10(A__FILTER(mark(x0), s(s(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), s(x1), x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), s(mark(x1)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), s(x1), x2)) -> c10(A__FILTER(a__sieve(mark(z0)), s(mark(x1)), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), s(x1), x2)) -> c10(A__FILTER(a__nats(mark(z0)), s(mark(x1)), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, s(x1), x2)) -> c10(A__FILTER(a__zprimes, s(mark(x1)), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), s(x1), x2)) -> c10(A__FILTER(cons(mark(z0), z1), s(mark(x1)), mark(x2)), MARK(cons(z0, z1))) MARK(filter(s(z0), s(x1), x2)) -> c10(MARK(s(z0))) MARK(filter(filter(x0, x1, filter(z0, z1, z2)), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), mark(x3), mark(x4)), MARK(filter(x0, x1, filter(z0, z1, z2)))) MARK(filter(filter(x0, x1, sieve(z0)), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), a__sieve(mark(z0))), mark(x3), mark(x4)), MARK(filter(x0, x1, sieve(z0)))) MARK(filter(filter(x0, x1, nats(z0)), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), a__nats(mark(z0))), mark(x3), mark(x4)), MARK(filter(x0, x1, nats(z0)))) MARK(filter(filter(x0, x1, zprimes), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), a__zprimes), mark(x3), mark(x4)), MARK(filter(x0, x1, zprimes))) MARK(filter(filter(x0, x1, cons(z0, z1)), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), 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sieve(x2))) -> c11(A__FILTER(s(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(x1)) S tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(x0, 0, filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, sieve(x2))) -> c10(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x0)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, sieve(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, sieve(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, sieve(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(0)), MARK(x0)) MARK(filter(x0, x1, sieve(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), sieve(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, sieve(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, sieve(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__sieve(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, sieve(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, sieve(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, sieve(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), a__sieve(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, sieve(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__sieve(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), sieve(x2))) -> c10(MARK(x0)) MARK(filter(x0, 0, sieve(x2))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, sieve(x2))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(mark(x2)), nats(s(mark(x2))))), MARK(x0)) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), nats(mark(x2))), MARK(x0)) MARK(filter(x0, x1, nats(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, nats(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, nats(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, nats(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, nats(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, nats(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(0)), MARK(x0)) MARK(filter(x0, x1, nats(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), nats(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), nats(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), nats(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, nats(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), nats(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, nats(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__nats(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, nats(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, nats(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, nats(x2))) -> c10(A__FILTER(a__zprimes, 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-> c10(A__FILTER(a__filter(mark(x0), a__sieve(mark(z0)), mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, sieve(z0), x2))) MARK(filter(filter(x0, nats(z0), x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), a__nats(mark(z0)), mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, nats(z0), x2))) MARK(filter(filter(x0, zprimes, x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), a__zprimes, mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, zprimes, x2))) MARK(filter(filter(x0, cons(z0, z1), x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), cons(mark(z0), z1), mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, cons(z0, z1), x2))) MARK(filter(filter(x0, 0, x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), 0, mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, 0, x2))) MARK(filter(filter(x0, s(z0), x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), s(mark(z0)), mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, s(z0), x2))) MARK(filter(filter(filter(z0, z1, z2), x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(filter(z0, z1, z2), x1, x2))) MARK(filter(filter(sieve(z0), x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(a__sieve(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(sieve(z0), x1, x2))) MARK(filter(filter(nats(z0), x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(a__nats(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(nats(z0), x1, x2))) MARK(filter(filter(zprimes, x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(a__zprimes, mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(zprimes, x1, x2))) MARK(filter(filter(cons(z0, z1), x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(cons(mark(z0), z1), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(cons(z0, z1), x1, x2))) MARK(filter(filter(0, x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(0, mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(0, x1, x2))) MARK(filter(filter(s(z0), x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(s(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(s(z0), x1, x2))) MARK(filter(filter(x0, x1, x2), cons(z0, z1), x4)) -> c10(MARK(filter(x0, x1, x2))) MARK(filter(filter(x0, x1, x2), 0, x4)) -> c10(MARK(filter(x0, x1, x2))) MARK(filter(filter(x0, x1, x2), x3, x4)) -> c10(MARK(filter(x0, x1, x2))) MARK(filter(sieve(filter(z0, z1, z2)), x1, x2)) -> c10(A__FILTER(a__sieve(a__filter(mark(z0), mark(z1), mark(z2))), mark(x1), mark(x2)), MARK(sieve(filter(z0, z1, z2)))) MARK(filter(sieve(sieve(z0)), x1, x2)) -> c10(A__FILTER(a__sieve(a__sieve(mark(z0))), mark(x1), mark(x2)), MARK(sieve(sieve(z0)))) MARK(filter(sieve(nats(z0)), x1, x2)) -> c10(A__FILTER(a__sieve(a__nats(mark(z0))), mark(x1), mark(x2)), MARK(sieve(nats(z0)))) MARK(filter(sieve(zprimes), x1, x2)) -> c10(A__FILTER(a__sieve(a__zprimes), mark(x1), mark(x2)), MARK(sieve(zprimes))) MARK(filter(sieve(cons(z0, z1)), x1, x2)) -> c10(A__FILTER(a__sieve(cons(mark(z0), z1)), mark(x1), mark(x2)), 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filter(x2, x3, x4))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c11(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, filter(x2, x3, x4))) -> c11(MARK(x1)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c11(MARK(x1)) MARK(filter(x0, x1, sieve(x2))) -> c11(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x1)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x1)) MARK(filter(x0, x1, sieve(sieve(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__sieve(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, sieve(nats(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, sieve(zprimes))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__zprimes)), MARK(x1)) MARK(filter(x0, x1, sieve(cons(z0, z1)))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(cons(mark(z0), z1))), MARK(x1)) MARK(filter(x0, x1, sieve(0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(0)), MARK(x1)) MARK(filter(x0, x1, sieve(s(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(s(mark(z0)))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), sieve(x2))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), sieve(x2))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), a__sieve(mark(x2))), MARK(sieve(z0))) MARK(filter(x0, nats(z0), sieve(x2))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), a__sieve(mark(x2))), MARK(nats(z0))) MARK(filter(x0, zprimes, sieve(x2))) -> c11(A__FILTER(mark(x0), a__zprimes, a__sieve(mark(x2))), MARK(zprimes)) MARK(filter(x0, cons(z0, z1), sieve(x2))) -> c11(A__FILTER(mark(x0), cons(mark(z0), z1), a__sieve(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, s(z0), sieve(x2))) -> c11(A__FILTER(mark(x0), s(mark(z0)), a__sieve(mark(x2))), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, sieve(x2))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(sieve(z0), x1, sieve(x2))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(nats(z0), x1, sieve(x2))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(zprimes, x1, sieve(x2))) -> c11(A__FILTER(a__zprimes, mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(cons(z0, z1), x1, sieve(x2))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(0, x1, sieve(x2))) -> c11(A__FILTER(0, mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(s(z0), x1, sieve(x2))) -> c11(A__FILTER(s(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(x1)) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c14_2, c15_1, c16_1, c18_1, c8_1, c10_1, c11_2, c11_1, c12_2, c12_1, c13_2, c13_1, c8_2, c10_2 ---------------------------------------- (155) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (156) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(x0, 0, filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, sieve(x2))) -> c10(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x0)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, sieve(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, sieve(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, sieve(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(0)), MARK(x0)) MARK(filter(x0, x1, sieve(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), sieve(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, sieve(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, sieve(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__sieve(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, sieve(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__sieve(mark(x2))), 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filter(x2, sieve(z0), x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c11(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, filter(x2, x3, x4))) -> c11(MARK(x1)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c11(MARK(x1)) MARK(filter(x0, x1, sieve(x2))) -> c11(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x1)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x1)) MARK(filter(x0, x1, sieve(sieve(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__sieve(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, sieve(nats(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, sieve(zprimes))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__zprimes)), MARK(x1)) MARK(filter(x0, x1, sieve(cons(z0, z1)))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(cons(mark(z0), z1))), MARK(x1)) MARK(filter(x0, x1, sieve(0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(0)), MARK(x1)) MARK(filter(x0, x1, sieve(s(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(s(mark(z0)))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), sieve(x2))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), sieve(x2))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), a__sieve(mark(x2))), MARK(sieve(z0))) MARK(filter(x0, nats(z0), sieve(x2))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), a__sieve(mark(x2))), MARK(nats(z0))) MARK(filter(x0, zprimes, sieve(x2))) -> c11(A__FILTER(mark(x0), a__zprimes, a__sieve(mark(x2))), MARK(zprimes)) MARK(filter(x0, s(z0), sieve(x2))) -> c11(A__FILTER(mark(x0), s(mark(z0)), a__sieve(mark(x2))), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, sieve(x2))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(sieve(z0), x1, sieve(x2))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(nats(z0), x1, sieve(x2))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(zprimes, x1, sieve(x2))) -> c11(A__FILTER(a__zprimes, mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(cons(z0, z1), x1, sieve(x2))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(x0, cons(z0, z1), sieve(x2))) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, sieve(x2))) -> c11(MARK(x1)) MARK(filter(s(z0), x1, sieve(x2))) -> c11(MARK(x1)) S tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 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MARK(filter(filter(x0, x1, x2), cons(z0, z1), x4)) -> c10(MARK(filter(x0, x1, x2))) MARK(filter(filter(x0, x1, x2), 0, x4)) -> c10(MARK(filter(x0, x1, x2))) MARK(filter(filter(x0, x1, x2), x3, x4)) -> c10(MARK(filter(x0, x1, x2))) MARK(filter(sieve(filter(z0, z1, z2)), x1, x2)) -> c10(A__FILTER(a__sieve(a__filter(mark(z0), mark(z1), mark(z2))), mark(x1), mark(x2)), MARK(sieve(filter(z0, z1, z2)))) MARK(filter(sieve(sieve(z0)), x1, x2)) -> c10(A__FILTER(a__sieve(a__sieve(mark(z0))), mark(x1), mark(x2)), MARK(sieve(sieve(z0)))) MARK(filter(sieve(nats(z0)), x1, x2)) -> c10(A__FILTER(a__sieve(a__nats(mark(z0))), mark(x1), mark(x2)), MARK(sieve(nats(z0)))) MARK(filter(sieve(zprimes), x1, x2)) -> c10(A__FILTER(a__sieve(a__zprimes), mark(x1), mark(x2)), MARK(sieve(zprimes))) MARK(filter(sieve(cons(z0, z1)), x1, x2)) -> c10(A__FILTER(a__sieve(cons(mark(z0), z1)), mark(x1), mark(x2)), MARK(sieve(cons(z0, z1)))) MARK(filter(sieve(s(z0)), x1, x2)) -> c10(A__FILTER(a__sieve(s(mark(z0))), mark(x1), mark(x2)), MARK(sieve(s(z0)))) MARK(filter(sieve(x0), cons(z0, z1), x2)) -> c10(MARK(sieve(x0))) MARK(filter(sieve(x0), 0, x2)) -> c10(MARK(sieve(x0))) MARK(filter(sieve(x0), x1, x2)) -> c10(MARK(sieve(x0))) MARK(filter(sieve(0), x1, x2)) -> c10(A__FILTER(a__sieve(0), mark(x1), mark(x2))) MARK(filter(nats(x0), x1, x2)) -> c10(A__FILTER(cons(mark(mark(x0)), nats(s(mark(x0)))), mark(x1), mark(x2)), MARK(nats(x0))) MARK(filter(nats(filter(z0, z1, z2)), x1, x2)) -> c10(A__FILTER(a__nats(a__filter(mark(z0), mark(z1), mark(z2))), mark(x1), mark(x2)), MARK(nats(filter(z0, z1, z2)))) MARK(filter(nats(sieve(z0)), x1, x2)) -> c10(A__FILTER(a__nats(a__sieve(mark(z0))), mark(x1), mark(x2)), MARK(nats(sieve(z0)))) MARK(filter(nats(nats(z0)), x1, x2)) -> c10(A__FILTER(a__nats(a__nats(mark(z0))), mark(x1), mark(x2)), MARK(nats(nats(z0)))) MARK(filter(nats(zprimes), x1, x2)) -> c10(A__FILTER(a__nats(a__zprimes), mark(x1), mark(x2)), MARK(nats(zprimes))) MARK(filter(nats(cons(z0, z1)), x1, x2)) -> c10(A__FILTER(a__nats(cons(mark(z0), z1)), mark(x1), mark(x2)), MARK(nats(cons(z0, z1)))) MARK(filter(nats(0), x1, x2)) -> c10(A__FILTER(a__nats(0), mark(x1), mark(x2)), MARK(nats(0))) MARK(filter(nats(s(z0)), x1, x2)) -> c10(A__FILTER(a__nats(s(mark(z0))), mark(x1), mark(x2)), MARK(nats(s(z0)))) MARK(filter(nats(x0), x1, x2)) -> c10(MARK(nats(x0))) MARK(filter(nats(x0), cons(z0, z1), x2)) -> c10(MARK(nats(x0))) MARK(filter(nats(x0), 0, x2)) -> c10(MARK(nats(x0))) MARK(filter(zprimes, x0, x1)) -> c10(A__FILTER(a__sieve(a__nats(s(s(0)))), mark(x0), mark(x1)), MARK(zprimes)) MARK(filter(zprimes, cons(z0, z1), x1)) -> c10(MARK(zprimes)) MARK(filter(zprimes, 0, x1)) -> c10(MARK(zprimes)) MARK(filter(zprimes, x0, x1)) -> c10(MARK(zprimes)) MARK(filter(cons(filter(z0, z1, z2), x1), x2, x3)) -> c10(A__FILTER(cons(a__filter(mark(z0), mark(z1), mark(z2)), x1), mark(x2), mark(x3)), MARK(cons(filter(z0, z1, z2), x1))) MARK(filter(cons(sieve(z0), x1), x2, x3)) -> c10(A__FILTER(cons(a__sieve(mark(z0)), x1), mark(x2), mark(x3)), MARK(cons(sieve(z0), x1))) MARK(filter(cons(nats(z0), x1), x2, x3)) -> c10(A__FILTER(cons(a__nats(mark(z0)), x1), mark(x2), mark(x3)), MARK(cons(nats(z0), x1))) MARK(filter(cons(zprimes, x1), x2, x3)) -> c10(A__FILTER(cons(a__zprimes, x1), mark(x2), mark(x3)), MARK(cons(zprimes, x1))) MARK(filter(cons(cons(z0, z1), x1), x2, x3)) -> c10(A__FILTER(cons(cons(mark(z0), z1), x1), mark(x2), mark(x3)), MARK(cons(cons(z0, z1), x1))) MARK(filter(cons(0, x1), x2, x3)) -> c10(A__FILTER(cons(0, x1), mark(x2), mark(x3)), MARK(cons(0, x1))) MARK(filter(cons(s(z0), x1), x2, x3)) -> c10(A__FILTER(cons(s(mark(z0)), x1), mark(x2), mark(x3)), MARK(cons(s(z0), x1))) MARK(filter(cons(x0, x1), x2, x3)) -> c10(MARK(cons(x0, x1))) MARK(filter(cons(x0, x1), cons(z0, z1), x3)) -> c10(MARK(cons(x0, x1))) MARK(filter(cons(x0, x1), 0, x3)) -> c10(MARK(cons(x0, x1))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c11(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, filter(x2, x3, x4))) -> c11(MARK(x1)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c11(MARK(x1)) MARK(filter(x0, x1, sieve(x2))) -> c11(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x1)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x1)) MARK(filter(x0, x1, sieve(sieve(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__sieve(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, sieve(nats(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, sieve(zprimes))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__zprimes)), MARK(x1)) MARK(filter(x0, x1, sieve(cons(z0, z1)))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(cons(mark(z0), z1))), MARK(x1)) MARK(filter(x0, x1, sieve(0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(0)), MARK(x1)) MARK(filter(x0, x1, sieve(s(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(s(mark(z0)))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), sieve(x2))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), sieve(x2))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), a__sieve(mark(x2))), MARK(sieve(z0))) MARK(filter(x0, nats(z0), sieve(x2))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), a__sieve(mark(x2))), MARK(nats(z0))) MARK(filter(x0, zprimes, sieve(x2))) -> c11(A__FILTER(mark(x0), a__zprimes, a__sieve(mark(x2))), MARK(zprimes)) MARK(filter(x0, s(z0), sieve(x2))) -> c11(A__FILTER(mark(x0), s(mark(z0)), a__sieve(mark(x2))), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, sieve(x2))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(sieve(z0), x1, sieve(x2))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(nats(z0), x1, sieve(x2))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(zprimes, x1, sieve(x2))) -> c11(A__FILTER(a__zprimes, mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(cons(z0, z1), x1, sieve(x2))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(x0, cons(z0, z1), sieve(x2))) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, sieve(x2))) -> c11(MARK(x1)) MARK(filter(s(z0), x1, sieve(x2))) -> c11(MARK(x1)) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c14_2, c15_1, c16_1, c18_1, c8_1, c10_1, c11_2, c11_1, c12_2, c12_1, c13_2, c13_1, c8_2, c10_2 ---------------------------------------- (157) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(filter(x0, x1, nats(z0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(x1)) by MARK(filter(x0, x1, nats(x2))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(mark(x2)), nats(s(mark(x2))))), MARK(x1)) MARK(filter(x0, x1, nats(x2))) -> c11(A__FILTER(mark(x0), mark(x1), nats(mark(x2))), MARK(x1)) MARK(filter(x0, x1, nats(filter(z0, z1, z2)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x1)) MARK(filter(x0, x1, nats(sieve(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__sieve(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, nats(nats(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__nats(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, nats(zprimes))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__zprimes)), MARK(x1)) MARK(filter(x0, x1, nats(cons(z0, z1)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(cons(mark(z0), z1))), MARK(x1)) MARK(filter(x0, x1, nats(0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(0)), MARK(x1)) MARK(filter(x0, x1, nats(s(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(s(mark(z0)))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), nats(x2))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), nats(x2))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), a__nats(mark(x2))), MARK(sieve(z0))) MARK(filter(x0, nats(z0), nats(x2))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), a__nats(mark(x2))), MARK(nats(z0))) MARK(filter(x0, zprimes, nats(x2))) -> c11(A__FILTER(mark(x0), a__zprimes, a__nats(mark(x2))), MARK(zprimes)) MARK(filter(x0, cons(z0, z1), nats(x2))) -> c11(A__FILTER(mark(x0), cons(mark(z0), z1), a__nats(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, 0, nats(x2))) -> c11(A__FILTER(mark(x0), 0, a__nats(mark(x2))), MARK(0)) MARK(filter(x0, s(z0), nats(x2))) -> c11(A__FILTER(mark(x0), s(mark(z0)), a__nats(mark(x2))), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, nats(x2))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(sieve(z0), x1, nats(x2))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(nats(z0), x1, nats(x2))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(zprimes, x1, nats(x2))) -> c11(A__FILTER(a__zprimes, mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(cons(z0, z1), x1, nats(x2))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(0, x1, nats(x2))) -> c11(A__FILTER(0, mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(s(z0), x1, nats(x2))) -> c11(A__FILTER(s(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(x1)) ---------------------------------------- (158) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(x0, 0, filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, sieve(x2))) -> c10(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x0)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, sieve(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, sieve(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, sieve(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(0)), MARK(x0)) MARK(filter(x0, x1, sieve(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), sieve(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, sieve(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, sieve(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__sieve(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, sieve(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, sieve(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, sieve(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), a__sieve(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, sieve(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__sieve(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), sieve(x2))) -> c10(MARK(x0)) MARK(filter(x0, 0, sieve(x2))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, sieve(x2))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(mark(x2)), nats(s(mark(x2))))), MARK(x0)) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), nats(mark(x2))), MARK(x0)) MARK(filter(x0, x1, nats(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, nats(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, nats(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, nats(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, nats(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, nats(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(0)), MARK(x0)) MARK(filter(x0, x1, nats(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), nats(x2))) -> 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c10(A__FILTER(cons(mark(z0), z1), a__zprimes, mark(x1)), MARK(cons(z0, z1))) MARK(filter(x0, zprimes, x1)) -> c10(MARK(x0)) MARK(filter(s(z0), zprimes, x1)) -> c10(MARK(s(z0))) MARK(filter(x0, s(filter(z0, z1, z2)), x2)) -> c10(A__FILTER(mark(x0), s(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(x0)) MARK(filter(x0, s(sieve(z0)), x2)) -> c10(A__FILTER(mark(x0), s(a__sieve(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, s(nats(z0)), x2)) -> c10(A__FILTER(mark(x0), s(a__nats(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, s(zprimes), x2)) -> c10(A__FILTER(mark(x0), s(a__zprimes), mark(x2)), MARK(x0)) MARK(filter(x0, s(cons(z0, z1)), x2)) -> c10(A__FILTER(mark(x0), s(cons(mark(z0), z1)), mark(x2)), MARK(x0)) MARK(filter(x0, s(0), x2)) -> c10(A__FILTER(mark(x0), s(0), mark(x2)), MARK(x0)) MARK(filter(x0, s(s(z0)), x2)) -> c10(A__FILTER(mark(x0), s(s(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), s(x1), x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), s(mark(x1)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), s(x1), x2)) -> c10(A__FILTER(a__sieve(mark(z0)), s(mark(x1)), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), s(x1), x2)) -> c10(A__FILTER(a__nats(mark(z0)), s(mark(x1)), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, s(x1), x2)) -> c10(A__FILTER(a__zprimes, s(mark(x1)), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), s(x1), x2)) -> c10(A__FILTER(cons(mark(z0), z1), s(mark(x1)), mark(x2)), MARK(cons(z0, z1))) MARK(filter(s(z0), s(x1), x2)) -> c10(MARK(s(z0))) MARK(filter(filter(x0, x1, filter(z0, z1, z2)), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), mark(x3), mark(x4)), MARK(filter(x0, x1, filter(z0, z1, z2)))) MARK(filter(filter(x0, x1, sieve(z0)), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), a__sieve(mark(z0))), mark(x3), mark(x4)), MARK(filter(x0, x1, sieve(z0)))) MARK(filter(filter(x0, x1, nats(z0)), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), a__nats(mark(z0))), mark(x3), mark(x4)), MARK(filter(x0, x1, nats(z0)))) MARK(filter(filter(x0, x1, zprimes), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), a__zprimes), mark(x3), mark(x4)), MARK(filter(x0, x1, zprimes))) MARK(filter(filter(x0, x1, cons(z0, z1)), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), cons(mark(z0), z1)), mark(x3), mark(x4)), MARK(filter(x0, x1, cons(z0, z1)))) MARK(filter(filter(x0, x1, 0), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), 0), mark(x3), mark(x4)), MARK(filter(x0, x1, 0))) MARK(filter(filter(x0, x1, s(z0)), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), s(mark(z0))), mark(x3), mark(x4)), MARK(filter(x0, x1, s(z0)))) MARK(filter(filter(x0, filter(z0, z1, z2), x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, filter(z0, z1, z2), x2))) MARK(filter(filter(x0, sieve(z0), x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), a__sieve(mark(z0)), mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, sieve(z0), x2))) MARK(filter(filter(x0, nats(z0), x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), a__nats(mark(z0)), mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, nats(z0), x2))) MARK(filter(filter(x0, zprimes, x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), a__zprimes, mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, zprimes, x2))) MARK(filter(filter(x0, cons(z0, z1), x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), cons(mark(z0), z1), mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, cons(z0, z1), x2))) MARK(filter(filter(x0, 0, x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), 0, mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, 0, x2))) MARK(filter(filter(x0, s(z0), x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), s(mark(z0)), mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, s(z0), x2))) MARK(filter(filter(filter(z0, z1, z2), x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(filter(z0, z1, z2), x1, x2))) MARK(filter(filter(sieve(z0), x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(a__sieve(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(sieve(z0), x1, x2))) MARK(filter(filter(nats(z0), x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(a__nats(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(nats(z0), x1, x2))) MARK(filter(filter(zprimes, x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(a__zprimes, mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(zprimes, x1, x2))) MARK(filter(filter(cons(z0, z1), x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(cons(mark(z0), z1), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(cons(z0, z1), x1, x2))) MARK(filter(filter(0, x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(0, mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(0, x1, x2))) MARK(filter(filter(s(z0), x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(s(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(s(z0), x1, x2))) MARK(filter(filter(x0, x1, x2), cons(z0, z1), x4)) -> c10(MARK(filter(x0, x1, x2))) MARK(filter(filter(x0, x1, x2), 0, x4)) -> c10(MARK(filter(x0, x1, x2))) MARK(filter(filter(x0, x1, x2), x3, x4)) -> c10(MARK(filter(x0, x1, x2))) MARK(filter(sieve(filter(z0, z1, z2)), x1, x2)) -> c10(A__FILTER(a__sieve(a__filter(mark(z0), mark(z1), mark(z2))), mark(x1), mark(x2)), MARK(sieve(filter(z0, z1, z2)))) MARK(filter(sieve(sieve(z0)), x1, x2)) -> c10(A__FILTER(a__sieve(a__sieve(mark(z0))), mark(x1), mark(x2)), MARK(sieve(sieve(z0)))) MARK(filter(sieve(nats(z0)), x1, x2)) -> c10(A__FILTER(a__sieve(a__nats(mark(z0))), mark(x1), mark(x2)), MARK(sieve(nats(z0)))) MARK(filter(sieve(zprimes), x1, x2)) -> c10(A__FILTER(a__sieve(a__zprimes), mark(x1), mark(x2)), MARK(sieve(zprimes))) MARK(filter(sieve(cons(z0, z1)), x1, x2)) -> c10(A__FILTER(a__sieve(cons(mark(z0), z1)), mark(x1), mark(x2)), MARK(sieve(cons(z0, z1)))) MARK(filter(sieve(s(z0)), x1, x2)) -> c10(A__FILTER(a__sieve(s(mark(z0))), mark(x1), mark(x2)), MARK(sieve(s(z0)))) MARK(filter(sieve(x0), cons(z0, z1), x2)) -> c10(MARK(sieve(x0))) MARK(filter(sieve(x0), 0, x2)) -> c10(MARK(sieve(x0))) MARK(filter(sieve(x0), x1, x2)) -> c10(MARK(sieve(x0))) MARK(filter(sieve(0), x1, x2)) -> c10(A__FILTER(a__sieve(0), mark(x1), mark(x2))) MARK(filter(nats(x0), x1, x2)) -> c10(A__FILTER(cons(mark(mark(x0)), nats(s(mark(x0)))), mark(x1), mark(x2)), MARK(nats(x0))) MARK(filter(nats(filter(z0, z1, z2)), x1, x2)) -> c10(A__FILTER(a__nats(a__filter(mark(z0), mark(z1), mark(z2))), mark(x1), mark(x2)), MARK(nats(filter(z0, z1, z2)))) MARK(filter(nats(sieve(z0)), x1, x2)) -> c10(A__FILTER(a__nats(a__sieve(mark(z0))), mark(x1), mark(x2)), MARK(nats(sieve(z0)))) MARK(filter(nats(nats(z0)), x1, x2)) -> c10(A__FILTER(a__nats(a__nats(mark(z0))), mark(x1), mark(x2)), MARK(nats(nats(z0)))) MARK(filter(nats(zprimes), x1, x2)) -> c10(A__FILTER(a__nats(a__zprimes), mark(x1), mark(x2)), MARK(nats(zprimes))) MARK(filter(nats(cons(z0, z1)), x1, x2)) -> c10(A__FILTER(a__nats(cons(mark(z0), z1)), mark(x1), mark(x2)), MARK(nats(cons(z0, z1)))) MARK(filter(nats(0), x1, x2)) -> c10(A__FILTER(a__nats(0), mark(x1), mark(x2)), MARK(nats(0))) MARK(filter(nats(s(z0)), x1, x2)) -> c10(A__FILTER(a__nats(s(mark(z0))), mark(x1), mark(x2)), MARK(nats(s(z0)))) MARK(filter(nats(x0), x1, x2)) -> c10(MARK(nats(x0))) MARK(filter(nats(x0), cons(z0, z1), x2)) -> c10(MARK(nats(x0))) MARK(filter(nats(x0), 0, x2)) -> c10(MARK(nats(x0))) MARK(filter(zprimes, x0, x1)) -> c10(A__FILTER(a__sieve(a__nats(s(s(0)))), mark(x0), mark(x1)), MARK(zprimes)) MARK(filter(zprimes, cons(z0, z1), x1)) -> c10(MARK(zprimes)) MARK(filter(zprimes, 0, x1)) -> c10(MARK(zprimes)) MARK(filter(zprimes, x0, x1)) -> c10(MARK(zprimes)) MARK(filter(cons(filter(z0, z1, z2), x1), x2, x3)) -> c10(A__FILTER(cons(a__filter(mark(z0), mark(z1), mark(z2)), x1), mark(x2), mark(x3)), MARK(cons(filter(z0, z1, z2), x1))) MARK(filter(cons(sieve(z0), x1), x2, x3)) -> c10(A__FILTER(cons(a__sieve(mark(z0)), x1), mark(x2), mark(x3)), MARK(cons(sieve(z0), x1))) MARK(filter(cons(nats(z0), x1), x2, x3)) -> c10(A__FILTER(cons(a__nats(mark(z0)), x1), mark(x2), mark(x3)), MARK(cons(nats(z0), x1))) MARK(filter(cons(zprimes, x1), x2, x3)) -> c10(A__FILTER(cons(a__zprimes, x1), mark(x2), mark(x3)), MARK(cons(zprimes, x1))) MARK(filter(cons(cons(z0, z1), x1), x2, x3)) -> c10(A__FILTER(cons(cons(mark(z0), z1), x1), mark(x2), mark(x3)), MARK(cons(cons(z0, z1), x1))) MARK(filter(cons(0, x1), x2, x3)) -> c10(A__FILTER(cons(0, x1), mark(x2), mark(x3)), MARK(cons(0, x1))) MARK(filter(cons(s(z0), x1), x2, x3)) -> c10(A__FILTER(cons(s(mark(z0)), x1), mark(x2), mark(x3)), MARK(cons(s(z0), x1))) MARK(filter(cons(x0, x1), x2, x3)) -> c10(MARK(cons(x0, x1))) MARK(filter(cons(x0, x1), cons(z0, z1), x3)) -> c10(MARK(cons(x0, x1))) MARK(filter(cons(x0, x1), 0, x3)) -> c10(MARK(cons(x0, x1))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c11(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, filter(x2, x3, x4))) -> c11(MARK(x1)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c11(MARK(x1)) MARK(filter(x0, x1, sieve(x2))) -> c11(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x1)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x1)) MARK(filter(x0, x1, sieve(sieve(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__sieve(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, sieve(nats(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, sieve(zprimes))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__zprimes)), MARK(x1)) MARK(filter(x0, x1, sieve(cons(z0, z1)))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(cons(mark(z0), z1))), MARK(x1)) MARK(filter(x0, x1, sieve(0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(0)), MARK(x1)) MARK(filter(x0, x1, sieve(s(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(s(mark(z0)))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), sieve(x2))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), sieve(x2))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), a__sieve(mark(x2))), MARK(sieve(z0))) MARK(filter(x0, nats(z0), sieve(x2))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), a__sieve(mark(x2))), MARK(nats(z0))) MARK(filter(x0, zprimes, sieve(x2))) -> c11(A__FILTER(mark(x0), a__zprimes, a__sieve(mark(x2))), MARK(zprimes)) MARK(filter(x0, s(z0), sieve(x2))) -> c11(A__FILTER(mark(x0), s(mark(z0)), a__sieve(mark(x2))), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, sieve(x2))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(sieve(z0), x1, sieve(x2))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(nats(z0), x1, sieve(x2))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(zprimes, x1, sieve(x2))) -> c11(A__FILTER(a__zprimes, mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(cons(z0, z1), x1, sieve(x2))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(x0, cons(z0, z1), sieve(x2))) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, sieve(x2))) -> c11(MARK(x1)) MARK(filter(s(z0), x1, sieve(x2))) -> c11(MARK(x1)) MARK(filter(x0, x1, nats(x2))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(mark(x2)), nats(s(mark(x2))))), MARK(x1)) MARK(filter(x0, x1, nats(x2))) -> c11(A__FILTER(mark(x0), mark(x1), nats(mark(x2))), MARK(x1)) MARK(filter(x0, x1, nats(filter(z0, z1, z2)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x1)) MARK(filter(x0, x1, nats(sieve(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__sieve(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, nats(nats(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__nats(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, nats(zprimes))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__zprimes)), MARK(x1)) MARK(filter(x0, x1, nats(cons(z0, z1)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(cons(mark(z0), z1))), MARK(x1)) MARK(filter(x0, x1, nats(0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(0)), MARK(x1)) MARK(filter(x0, x1, nats(s(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(s(mark(z0)))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), nats(x2))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), nats(x2))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), a__nats(mark(x2))), MARK(sieve(z0))) MARK(filter(x0, nats(z0), nats(x2))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), a__nats(mark(x2))), MARK(nats(z0))) MARK(filter(x0, zprimes, nats(x2))) -> c11(A__FILTER(mark(x0), a__zprimes, a__nats(mark(x2))), MARK(zprimes)) MARK(filter(x0, cons(z0, z1), nats(x2))) -> c11(A__FILTER(mark(x0), cons(mark(z0), z1), a__nats(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, 0, nats(x2))) -> c11(A__FILTER(mark(x0), 0, a__nats(mark(x2))), MARK(0)) MARK(filter(x0, s(z0), nats(x2))) -> c11(A__FILTER(mark(x0), s(mark(z0)), a__nats(mark(x2))), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, nats(x2))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(sieve(z0), x1, nats(x2))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(nats(z0), x1, nats(x2))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(zprimes, x1, nats(x2))) -> c11(A__FILTER(a__zprimes, mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(cons(z0, z1), x1, nats(x2))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(0, x1, nats(x2))) -> c11(A__FILTER(0, mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(s(z0), x1, nats(x2))) -> c11(A__FILTER(s(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(x1)) S tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), 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MARK(filter(sieve(x0), x1, x2)) -> c10(MARK(sieve(x0))) MARK(filter(sieve(0), x1, x2)) -> c10(A__FILTER(a__sieve(0), mark(x1), mark(x2))) MARK(filter(nats(x0), x1, x2)) -> c10(A__FILTER(cons(mark(mark(x0)), nats(s(mark(x0)))), mark(x1), mark(x2)), MARK(nats(x0))) MARK(filter(nats(filter(z0, z1, z2)), x1, x2)) -> c10(A__FILTER(a__nats(a__filter(mark(z0), mark(z1), mark(z2))), mark(x1), mark(x2)), MARK(nats(filter(z0, z1, z2)))) MARK(filter(nats(sieve(z0)), x1, x2)) -> c10(A__FILTER(a__nats(a__sieve(mark(z0))), mark(x1), mark(x2)), MARK(nats(sieve(z0)))) MARK(filter(nats(nats(z0)), x1, x2)) -> c10(A__FILTER(a__nats(a__nats(mark(z0))), mark(x1), mark(x2)), MARK(nats(nats(z0)))) MARK(filter(nats(zprimes), x1, x2)) -> c10(A__FILTER(a__nats(a__zprimes), mark(x1), mark(x2)), MARK(nats(zprimes))) MARK(filter(nats(cons(z0, z1)), x1, x2)) -> c10(A__FILTER(a__nats(cons(mark(z0), z1)), mark(x1), mark(x2)), MARK(nats(cons(z0, z1)))) MARK(filter(nats(0), x1, x2)) -> c10(A__FILTER(a__nats(0), mark(x1), mark(x2)), MARK(nats(0))) MARK(filter(nats(s(z0)), x1, x2)) -> c10(A__FILTER(a__nats(s(mark(z0))), mark(x1), mark(x2)), MARK(nats(s(z0)))) MARK(filter(nats(x0), x1, x2)) -> c10(MARK(nats(x0))) MARK(filter(nats(x0), cons(z0, z1), x2)) -> c10(MARK(nats(x0))) MARK(filter(nats(x0), 0, x2)) -> c10(MARK(nats(x0))) MARK(filter(zprimes, x0, x1)) -> c10(A__FILTER(a__sieve(a__nats(s(s(0)))), mark(x0), mark(x1)), MARK(zprimes)) MARK(filter(zprimes, cons(z0, z1), x1)) -> c10(MARK(zprimes)) MARK(filter(zprimes, 0, x1)) -> c10(MARK(zprimes)) MARK(filter(zprimes, x0, x1)) -> c10(MARK(zprimes)) MARK(filter(cons(filter(z0, z1, z2), x1), x2, x3)) -> c10(A__FILTER(cons(a__filter(mark(z0), mark(z1), mark(z2)), x1), mark(x2), mark(x3)), MARK(cons(filter(z0, z1, z2), x1))) MARK(filter(cons(sieve(z0), x1), x2, x3)) -> c10(A__FILTER(cons(a__sieve(mark(z0)), x1), mark(x2), mark(x3)), MARK(cons(sieve(z0), x1))) MARK(filter(cons(nats(z0), x1), x2, x3)) -> c10(A__FILTER(cons(a__nats(mark(z0)), x1), mark(x2), mark(x3)), MARK(cons(nats(z0), x1))) MARK(filter(cons(zprimes, x1), x2, x3)) -> c10(A__FILTER(cons(a__zprimes, x1), mark(x2), mark(x3)), MARK(cons(zprimes, x1))) MARK(filter(cons(cons(z0, z1), x1), x2, x3)) -> c10(A__FILTER(cons(cons(mark(z0), z1), x1), mark(x2), mark(x3)), MARK(cons(cons(z0, z1), x1))) MARK(filter(cons(0, x1), x2, x3)) -> c10(A__FILTER(cons(0, x1), mark(x2), mark(x3)), MARK(cons(0, x1))) MARK(filter(cons(s(z0), x1), x2, x3)) -> c10(A__FILTER(cons(s(mark(z0)), x1), mark(x2), mark(x3)), MARK(cons(s(z0), x1))) MARK(filter(cons(x0, x1), x2, x3)) -> c10(MARK(cons(x0, x1))) MARK(filter(cons(x0, x1), cons(z0, z1), x3)) -> c10(MARK(cons(x0, x1))) MARK(filter(cons(x0, x1), 0, x3)) -> c10(MARK(cons(x0, x1))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c11(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, filter(x2, x3, x4))) -> c11(MARK(x1)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c11(MARK(x1)) MARK(filter(x0, x1, sieve(x2))) -> c11(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x1)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x1)) MARK(filter(x0, x1, sieve(sieve(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__sieve(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, sieve(nats(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, sieve(zprimes))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__zprimes)), MARK(x1)) MARK(filter(x0, x1, sieve(cons(z0, z1)))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(cons(mark(z0), z1))), MARK(x1)) MARK(filter(x0, x1, sieve(0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(0)), MARK(x1)) MARK(filter(x0, x1, sieve(s(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(s(mark(z0)))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), sieve(x2))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), sieve(x2))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), a__sieve(mark(x2))), MARK(sieve(z0))) MARK(filter(x0, nats(z0), sieve(x2))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), a__sieve(mark(x2))), MARK(nats(z0))) MARK(filter(x0, zprimes, sieve(x2))) -> c11(A__FILTER(mark(x0), a__zprimes, a__sieve(mark(x2))), MARK(zprimes)) MARK(filter(x0, s(z0), sieve(x2))) -> c11(A__FILTER(mark(x0), s(mark(z0)), a__sieve(mark(x2))), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, sieve(x2))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(sieve(z0), x1, sieve(x2))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(nats(z0), x1, sieve(x2))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(zprimes, x1, sieve(x2))) -> c11(A__FILTER(a__zprimes, mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(cons(z0, z1), x1, sieve(x2))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(x0, cons(z0, z1), sieve(x2))) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, sieve(x2))) -> c11(MARK(x1)) MARK(filter(s(z0), x1, sieve(x2))) -> c11(MARK(x1)) MARK(filter(x0, x1, nats(x2))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(mark(x2)), nats(s(mark(x2))))), MARK(x1)) MARK(filter(x0, x1, nats(x2))) -> c11(A__FILTER(mark(x0), mark(x1), nats(mark(x2))), MARK(x1)) MARK(filter(x0, x1, nats(filter(z0, z1, z2)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x1)) MARK(filter(x0, x1, nats(sieve(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__sieve(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, nats(nats(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__nats(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, nats(zprimes))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__zprimes)), MARK(x1)) MARK(filter(x0, x1, nats(cons(z0, z1)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(cons(mark(z0), z1))), MARK(x1)) MARK(filter(x0, x1, nats(0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(0)), MARK(x1)) MARK(filter(x0, x1, nats(s(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(s(mark(z0)))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), nats(x2))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), nats(x2))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), a__nats(mark(x2))), MARK(sieve(z0))) MARK(filter(x0, nats(z0), nats(x2))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), a__nats(mark(x2))), MARK(nats(z0))) MARK(filter(x0, zprimes, nats(x2))) -> c11(A__FILTER(mark(x0), a__zprimes, a__nats(mark(x2))), MARK(zprimes)) MARK(filter(x0, cons(z0, z1), nats(x2))) -> c11(A__FILTER(mark(x0), cons(mark(z0), z1), a__nats(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, 0, nats(x2))) -> c11(A__FILTER(mark(x0), 0, a__nats(mark(x2))), MARK(0)) MARK(filter(x0, s(z0), nats(x2))) -> c11(A__FILTER(mark(x0), s(mark(z0)), a__nats(mark(x2))), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, nats(x2))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(sieve(z0), x1, nats(x2))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(nats(z0), x1, nats(x2))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(zprimes, x1, nats(x2))) -> c11(A__FILTER(a__zprimes, mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(cons(z0, z1), x1, nats(x2))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(0, x1, nats(x2))) -> c11(A__FILTER(0, mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(s(z0), x1, nats(x2))) -> c11(A__FILTER(s(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(x1)) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c14_2, c15_1, c16_1, c18_1, c8_1, c10_1, c11_2, c11_1, c12_2, c12_1, c13_2, c13_1, c8_2, c10_2 ---------------------------------------- (159) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(filter(x0, 0, nats(x2))) -> c11(A__FILTER(mark(x0), 0, a__nats(mark(x2))), MARK(0)) ---------------------------------------- (160) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> 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a__nats(mark(z0)), a__sieve(mark(x2))), MARK(nats(z0))) MARK(filter(x0, zprimes, sieve(x2))) -> c11(A__FILTER(mark(x0), a__zprimes, a__sieve(mark(x2))), MARK(zprimes)) MARK(filter(x0, s(z0), sieve(x2))) -> c11(A__FILTER(mark(x0), s(mark(z0)), a__sieve(mark(x2))), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, sieve(x2))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(sieve(z0), x1, sieve(x2))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(nats(z0), x1, sieve(x2))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(zprimes, x1, sieve(x2))) -> c11(A__FILTER(a__zprimes, mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(cons(z0, z1), x1, sieve(x2))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(x0, cons(z0, z1), sieve(x2))) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, sieve(x2))) -> c11(MARK(x1)) MARK(filter(s(z0), x1, sieve(x2))) -> c11(MARK(x1)) MARK(filter(x0, x1, nats(x2))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(mark(x2)), nats(s(mark(x2))))), MARK(x1)) MARK(filter(x0, x1, nats(x2))) -> c11(A__FILTER(mark(x0), mark(x1), nats(mark(x2))), MARK(x1)) MARK(filter(x0, x1, nats(filter(z0, z1, z2)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x1)) MARK(filter(x0, x1, nats(sieve(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__sieve(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, nats(nats(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__nats(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, nats(zprimes))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__zprimes)), MARK(x1)) MARK(filter(x0, x1, nats(cons(z0, z1)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(cons(mark(z0), z1))), MARK(x1)) MARK(filter(x0, x1, nats(0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(0)), MARK(x1)) MARK(filter(x0, x1, nats(s(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(s(mark(z0)))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), nats(x2))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), nats(x2))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), a__nats(mark(x2))), MARK(sieve(z0))) MARK(filter(x0, nats(z0), nats(x2))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), a__nats(mark(x2))), MARK(nats(z0))) MARK(filter(x0, zprimes, nats(x2))) -> c11(A__FILTER(mark(x0), a__zprimes, a__nats(mark(x2))), MARK(zprimes)) MARK(filter(x0, cons(z0, z1), nats(x2))) -> c11(A__FILTER(mark(x0), cons(mark(z0), z1), a__nats(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, s(z0), nats(x2))) -> c11(A__FILTER(mark(x0), s(mark(z0)), a__nats(mark(x2))), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, nats(x2))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(sieve(z0), x1, nats(x2))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(nats(z0), x1, nats(x2))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(zprimes, x1, nats(x2))) -> c11(A__FILTER(a__zprimes, mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(cons(z0, z1), x1, nats(x2))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(0, x1, nats(x2))) -> c11(A__FILTER(0, mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(s(z0), x1, nats(x2))) -> c11(A__FILTER(s(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(x1)) S tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, 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x3, nats(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c11(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, filter(x2, x3, x4))) -> c11(MARK(x1)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c11(MARK(x1)) MARK(filter(x0, x1, sieve(x2))) -> c11(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x1)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x1)) MARK(filter(x0, x1, sieve(sieve(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__sieve(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, sieve(nats(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, sieve(zprimes))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__zprimes)), MARK(x1)) MARK(filter(x0, x1, sieve(cons(z0, z1)))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(cons(mark(z0), z1))), MARK(x1)) MARK(filter(x0, x1, sieve(0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(0)), MARK(x1)) MARK(filter(x0, x1, sieve(s(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(s(mark(z0)))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), sieve(x2))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), sieve(x2))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), a__sieve(mark(x2))), MARK(sieve(z0))) MARK(filter(x0, nats(z0), sieve(x2))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), a__sieve(mark(x2))), MARK(nats(z0))) MARK(filter(x0, zprimes, sieve(x2))) -> c11(A__FILTER(mark(x0), a__zprimes, a__sieve(mark(x2))), MARK(zprimes)) MARK(filter(x0, s(z0), sieve(x2))) -> c11(A__FILTER(mark(x0), s(mark(z0)), a__sieve(mark(x2))), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, sieve(x2))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(sieve(z0), x1, sieve(x2))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(nats(z0), x1, sieve(x2))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(zprimes, x1, sieve(x2))) -> c11(A__FILTER(a__zprimes, mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(cons(z0, z1), x1, sieve(x2))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(x0, cons(z0, z1), sieve(x2))) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, sieve(x2))) -> c11(MARK(x1)) MARK(filter(s(z0), x1, sieve(x2))) -> c11(MARK(x1)) MARK(filter(x0, x1, nats(x2))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(mark(x2)), nats(s(mark(x2))))), MARK(x1)) MARK(filter(x0, x1, nats(x2))) -> c11(A__FILTER(mark(x0), mark(x1), nats(mark(x2))), MARK(x1)) MARK(filter(x0, x1, nats(filter(z0, z1, z2)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x1)) MARK(filter(x0, x1, nats(sieve(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__sieve(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, nats(nats(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__nats(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, nats(zprimes))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__zprimes)), MARK(x1)) MARK(filter(x0, x1, nats(cons(z0, z1)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(cons(mark(z0), z1))), MARK(x1)) MARK(filter(x0, x1, nats(0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(0)), MARK(x1)) MARK(filter(x0, x1, nats(s(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(s(mark(z0)))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), nats(x2))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), nats(x2))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), a__nats(mark(x2))), MARK(sieve(z0))) MARK(filter(x0, nats(z0), nats(x2))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), a__nats(mark(x2))), MARK(nats(z0))) MARK(filter(x0, zprimes, nats(x2))) -> c11(A__FILTER(mark(x0), a__zprimes, a__nats(mark(x2))), MARK(zprimes)) MARK(filter(x0, cons(z0, z1), nats(x2))) -> c11(A__FILTER(mark(x0), cons(mark(z0), z1), a__nats(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, s(z0), nats(x2))) -> c11(A__FILTER(mark(x0), s(mark(z0)), a__nats(mark(x2))), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, nats(x2))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(sieve(z0), x1, nats(x2))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(nats(z0), x1, nats(x2))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(zprimes, x1, nats(x2))) -> c11(A__FILTER(a__zprimes, mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(cons(z0, z1), x1, nats(x2))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(0, x1, nats(x2))) -> c11(A__FILTER(0, mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(s(z0), x1, nats(x2))) -> c11(A__FILTER(s(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(x1)) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c14_2, c15_1, c16_1, c18_1, c8_1, c10_1, c11_2, c11_1, c12_2, c12_1, c13_2, c13_1, c8_2, c10_2 ---------------------------------------- (161) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (162) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), 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c11(A__FILTER(mark(x0), mark(x1), a__nats(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x1)) MARK(filter(x0, x1, nats(sieve(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__sieve(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, nats(nats(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__nats(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, nats(zprimes))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__zprimes)), MARK(x1)) MARK(filter(x0, x1, nats(cons(z0, z1)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(cons(mark(z0), z1))), MARK(x1)) MARK(filter(x0, x1, nats(0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(0)), MARK(x1)) MARK(filter(x0, x1, nats(s(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(s(mark(z0)))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), nats(x2))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), nats(x2))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), 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nats(x2))) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, nats(x2))) -> c11(MARK(x1)) MARK(filter(s(z0), x1, nats(x2))) -> c11(MARK(x1)) S tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 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x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(x0, 0, filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, sieve(x2))) -> c10(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x0)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, sieve(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), 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filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c11(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, filter(x2, x3, x4))) -> c11(MARK(x1)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c11(MARK(x1)) MARK(filter(x0, x1, sieve(x2))) -> c11(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x1)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x1)) MARK(filter(x0, x1, sieve(sieve(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__sieve(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, sieve(nats(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, sieve(zprimes))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__zprimes)), MARK(x1)) MARK(filter(x0, x1, sieve(cons(z0, z1)))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(cons(mark(z0), z1))), MARK(x1)) MARK(filter(x0, x1, sieve(0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(0)), MARK(x1)) MARK(filter(x0, x1, sieve(s(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(s(mark(z0)))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), sieve(x2))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), sieve(x2))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), a__sieve(mark(x2))), MARK(sieve(z0))) MARK(filter(x0, nats(z0), sieve(x2))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), a__sieve(mark(x2))), MARK(nats(z0))) MARK(filter(x0, zprimes, sieve(x2))) -> c11(A__FILTER(mark(x0), a__zprimes, a__sieve(mark(x2))), MARK(zprimes)) MARK(filter(x0, s(z0), sieve(x2))) -> c11(A__FILTER(mark(x0), s(mark(z0)), a__sieve(mark(x2))), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, sieve(x2))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(sieve(z0), x1, sieve(x2))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(nats(z0), x1, sieve(x2))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(zprimes, x1, sieve(x2))) -> c11(A__FILTER(a__zprimes, mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(cons(z0, z1), x1, sieve(x2))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(x0, cons(z0, z1), sieve(x2))) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, sieve(x2))) -> c11(MARK(x1)) MARK(filter(s(z0), x1, sieve(x2))) -> c11(MARK(x1)) MARK(filter(x0, x1, nats(x2))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(mark(x2)), nats(s(mark(x2))))), MARK(x1)) MARK(filter(x0, x1, nats(x2))) -> c11(A__FILTER(mark(x0), mark(x1), nats(mark(x2))), MARK(x1)) MARK(filter(x0, x1, nats(filter(z0, z1, z2)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x1)) MARK(filter(x0, x1, nats(sieve(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__sieve(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, nats(nats(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__nats(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, nats(zprimes))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__zprimes)), MARK(x1)) MARK(filter(x0, x1, nats(cons(z0, z1)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(cons(mark(z0), z1))), MARK(x1)) MARK(filter(x0, x1, nats(0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(0)), MARK(x1)) MARK(filter(x0, x1, nats(s(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(s(mark(z0)))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), nats(x2))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), nats(x2))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), a__nats(mark(x2))), MARK(sieve(z0))) MARK(filter(x0, nats(z0), nats(x2))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), a__nats(mark(x2))), MARK(nats(z0))) MARK(filter(x0, zprimes, nats(x2))) -> c11(A__FILTER(mark(x0), a__zprimes, a__nats(mark(x2))), MARK(zprimes)) MARK(filter(x0, s(z0), nats(x2))) -> c11(A__FILTER(mark(x0), s(mark(z0)), a__nats(mark(x2))), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, nats(x2))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(sieve(z0), x1, nats(x2))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(nats(z0), x1, nats(x2))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(zprimes, x1, nats(x2))) -> c11(A__FILTER(a__zprimes, mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(cons(z0, z1), x1, nats(x2))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(x0, cons(z0, z1), nats(x2))) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, nats(x2))) -> c11(MARK(x1)) MARK(filter(s(z0), x1, nats(x2))) -> c11(MARK(x1)) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c14_2, c15_1, c16_1, c18_1, c8_1, c10_1, c11_2, c11_1, c12_2, c12_1, c13_2, c13_1, c8_2, c10_2 ---------------------------------------- (163) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(x1)) by MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(s(s(0))))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), zprimes), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), zprimes)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__zprimes), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), zprimes)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), a__zprimes), MARK(sieve(z0))) MARK(filter(x0, nats(z0), zprimes)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), a__zprimes), MARK(nats(z0))) MARK(filter(x0, zprimes, zprimes)) -> c11(A__FILTER(mark(x0), a__zprimes, a__zprimes), MARK(zprimes)) MARK(filter(x0, cons(z0, z1), zprimes)) -> c11(A__FILTER(mark(x0), cons(mark(z0), z1), a__zprimes), MARK(cons(z0, z1))) MARK(filter(x0, 0, zprimes)) -> c11(A__FILTER(mark(x0), 0, a__zprimes), MARK(0)) MARK(filter(x0, s(z0), zprimes)) -> c11(A__FILTER(mark(x0), s(mark(z0)), a__zprimes), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, zprimes)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__zprimes), MARK(x1)) MARK(filter(sieve(z0), x1, zprimes)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__zprimes), MARK(x1)) MARK(filter(nats(z0), x1, zprimes)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__zprimes), MARK(x1)) MARK(filter(zprimes, x1, zprimes)) -> c11(A__FILTER(a__zprimes, mark(x1), a__zprimes), MARK(x1)) MARK(filter(cons(z0, z1), x1, zprimes)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__zprimes), MARK(x1)) MARK(filter(0, x1, zprimes)) -> c11(A__FILTER(0, mark(x1), a__zprimes), MARK(x1)) MARK(filter(s(z0), x1, zprimes)) -> c11(A__FILTER(s(mark(z0)), mark(x1), a__zprimes), MARK(x1)) ---------------------------------------- (164) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> 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mark(z2)))), MARK(x1)) MARK(filter(x0, x1, nats(sieve(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__sieve(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, nats(nats(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__nats(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, nats(zprimes))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__zprimes)), MARK(x1)) MARK(filter(x0, x1, nats(cons(z0, z1)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(cons(mark(z0), z1))), MARK(x1)) MARK(filter(x0, x1, nats(0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(0)), MARK(x1)) MARK(filter(x0, x1, nats(s(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(s(mark(z0)))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), nats(x2))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), nats(x2))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), a__nats(mark(x2))), MARK(sieve(z0))) MARK(filter(x0, nats(z0), nats(x2))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), a__nats(mark(x2))), MARK(nats(z0))) MARK(filter(x0, zprimes, nats(x2))) -> c11(A__FILTER(mark(x0), a__zprimes, a__nats(mark(x2))), MARK(zprimes)) MARK(filter(x0, s(z0), nats(x2))) -> c11(A__FILTER(mark(x0), s(mark(z0)), a__nats(mark(x2))), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, nats(x2))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(sieve(z0), x1, nats(x2))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(nats(z0), x1, nats(x2))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(zprimes, x1, nats(x2))) -> c11(A__FILTER(a__zprimes, mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(cons(z0, z1), x1, nats(x2))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(x0, cons(z0, z1), nats(x2))) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, nats(x2))) -> c11(MARK(x1)) MARK(filter(s(z0), x1, nats(x2))) -> c11(MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(s(s(0))))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), zprimes), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), zprimes)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__zprimes), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), zprimes)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), a__zprimes), MARK(sieve(z0))) MARK(filter(x0, nats(z0), zprimes)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), a__zprimes), MARK(nats(z0))) MARK(filter(x0, zprimes, zprimes)) -> c11(A__FILTER(mark(x0), a__zprimes, a__zprimes), MARK(zprimes)) MARK(filter(x0, cons(z0, z1), zprimes)) -> c11(A__FILTER(mark(x0), cons(mark(z0), z1), a__zprimes), MARK(cons(z0, z1))) MARK(filter(x0, 0, zprimes)) -> c11(A__FILTER(mark(x0), 0, a__zprimes), MARK(0)) MARK(filter(x0, s(z0), zprimes)) -> c11(A__FILTER(mark(x0), s(mark(z0)), a__zprimes), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, zprimes)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__zprimes), MARK(x1)) MARK(filter(sieve(z0), x1, zprimes)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__zprimes), MARK(x1)) MARK(filter(nats(z0), x1, zprimes)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__zprimes), MARK(x1)) MARK(filter(zprimes, x1, zprimes)) -> c11(A__FILTER(a__zprimes, mark(x1), a__zprimes), MARK(x1)) MARK(filter(cons(z0, z1), x1, zprimes)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__zprimes), MARK(x1)) MARK(filter(0, x1, zprimes)) -> c11(A__FILTER(0, mark(x1), a__zprimes), MARK(x1)) MARK(filter(s(z0), x1, zprimes)) -> c11(A__FILTER(s(mark(z0)), mark(x1), a__zprimes), MARK(x1)) S tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(x0, 0, filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, sieve(x2))) -> c10(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x0)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, sieve(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, sieve(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, sieve(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(0)), MARK(x0)) MARK(filter(x0, x1, sieve(s(z0)))) -> c10(A__FILTER(mark(x0), 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x3, nats(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c11(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, filter(x2, x3, x4))) -> c11(MARK(x1)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c11(MARK(x1)) MARK(filter(x0, x1, sieve(x2))) -> c11(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x1)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x1)) MARK(filter(x0, x1, sieve(sieve(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__sieve(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, sieve(nats(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, sieve(zprimes))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__zprimes)), MARK(x1)) MARK(filter(x0, x1, sieve(cons(z0, z1)))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(cons(mark(z0), z1))), MARK(x1)) MARK(filter(x0, x1, sieve(0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(0)), MARK(x1)) MARK(filter(x0, x1, sieve(s(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(s(mark(z0)))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), sieve(x2))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), sieve(x2))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), a__sieve(mark(x2))), MARK(sieve(z0))) MARK(filter(x0, nats(z0), sieve(x2))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), a__sieve(mark(x2))), MARK(nats(z0))) MARK(filter(x0, zprimes, sieve(x2))) -> c11(A__FILTER(mark(x0), a__zprimes, a__sieve(mark(x2))), MARK(zprimes)) MARK(filter(x0, s(z0), sieve(x2))) -> c11(A__FILTER(mark(x0), s(mark(z0)), a__sieve(mark(x2))), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, sieve(x2))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(sieve(z0), x1, sieve(x2))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(nats(z0), x1, sieve(x2))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(zprimes, x1, sieve(x2))) -> c11(A__FILTER(a__zprimes, mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(cons(z0, z1), x1, sieve(x2))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(x0, cons(z0, z1), sieve(x2))) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, sieve(x2))) -> c11(MARK(x1)) MARK(filter(s(z0), x1, sieve(x2))) -> c11(MARK(x1)) MARK(filter(x0, x1, nats(x2))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(mark(x2)), nats(s(mark(x2))))), MARK(x1)) MARK(filter(x0, x1, nats(x2))) -> c11(A__FILTER(mark(x0), mark(x1), nats(mark(x2))), MARK(x1)) MARK(filter(x0, x1, nats(filter(z0, z1, z2)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x1)) MARK(filter(x0, x1, nats(sieve(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__sieve(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, nats(nats(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__nats(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, nats(zprimes))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__zprimes)), MARK(x1)) MARK(filter(x0, x1, nats(cons(z0, z1)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(cons(mark(z0), z1))), MARK(x1)) MARK(filter(x0, x1, nats(0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(0)), MARK(x1)) MARK(filter(x0, x1, nats(s(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(s(mark(z0)))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), nats(x2))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), nats(x2))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), a__nats(mark(x2))), MARK(sieve(z0))) MARK(filter(x0, nats(z0), nats(x2))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), a__nats(mark(x2))), MARK(nats(z0))) MARK(filter(x0, zprimes, nats(x2))) -> c11(A__FILTER(mark(x0), a__zprimes, a__nats(mark(x2))), MARK(zprimes)) MARK(filter(x0, s(z0), nats(x2))) -> c11(A__FILTER(mark(x0), s(mark(z0)), a__nats(mark(x2))), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, nats(x2))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(sieve(z0), x1, nats(x2))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(nats(z0), x1, nats(x2))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(zprimes, x1, nats(x2))) -> c11(A__FILTER(a__zprimes, mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(cons(z0, z1), x1, nats(x2))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(x0, cons(z0, z1), nats(x2))) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, nats(x2))) -> c11(MARK(x1)) MARK(filter(s(z0), x1, nats(x2))) -> c11(MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(s(s(0))))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), zprimes), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), zprimes)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__zprimes), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), zprimes)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), a__zprimes), MARK(sieve(z0))) MARK(filter(x0, nats(z0), zprimes)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), a__zprimes), MARK(nats(z0))) MARK(filter(x0, zprimes, zprimes)) -> c11(A__FILTER(mark(x0), a__zprimes, a__zprimes), MARK(zprimes)) MARK(filter(x0, cons(z0, z1), zprimes)) -> c11(A__FILTER(mark(x0), cons(mark(z0), z1), a__zprimes), MARK(cons(z0, z1))) MARK(filter(x0, 0, zprimes)) -> c11(A__FILTER(mark(x0), 0, a__zprimes), MARK(0)) MARK(filter(x0, s(z0), zprimes)) -> c11(A__FILTER(mark(x0), s(mark(z0)), a__zprimes), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, zprimes)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__zprimes), MARK(x1)) MARK(filter(sieve(z0), x1, zprimes)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__zprimes), MARK(x1)) MARK(filter(nats(z0), x1, zprimes)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__zprimes), MARK(x1)) MARK(filter(zprimes, x1, zprimes)) -> c11(A__FILTER(a__zprimes, mark(x1), a__zprimes), MARK(x1)) MARK(filter(cons(z0, z1), x1, zprimes)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__zprimes), MARK(x1)) MARK(filter(0, x1, zprimes)) -> c11(A__FILTER(0, mark(x1), a__zprimes), MARK(x1)) MARK(filter(s(z0), x1, zprimes)) -> c11(A__FILTER(s(mark(z0)), mark(x1), a__zprimes), MARK(x1)) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c14_2, c15_1, c16_1, c18_1, c8_1, c10_1, c11_2, c11_1, c12_2, c12_1, c13_2, c13_1, c8_2, c10_2 ---------------------------------------- (165) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(filter(x0, 0, zprimes)) -> c11(A__FILTER(mark(x0), 0, a__zprimes), MARK(0)) ---------------------------------------- (166) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(x0, 0, filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, sieve(x2))) -> c10(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x0)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, sieve(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, sieve(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, sieve(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(0)), MARK(x0)) MARK(filter(x0, x1, sieve(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), sieve(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, sieve(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, sieve(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__sieve(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, sieve(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, sieve(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, sieve(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), a__sieve(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, sieve(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__sieve(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), sieve(x2))) -> c10(MARK(x0)) MARK(filter(x0, 0, sieve(x2))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, sieve(x2))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(mark(x2)), nats(s(mark(x2))))), MARK(x0)) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), nats(mark(x2))), MARK(x0)) MARK(filter(x0, x1, nats(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, nats(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, nats(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, nats(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, nats(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, nats(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(0)), MARK(x0)) MARK(filter(x0, x1, nats(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), nats(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), nats(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), nats(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, nats(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), nats(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, nats(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__nats(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, nats(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, nats(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, nats(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), a__nats(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, 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a__sieve(mark(z0)), a__sieve(mark(x2))), MARK(sieve(z0))) MARK(filter(x0, nats(z0), sieve(x2))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), a__sieve(mark(x2))), MARK(nats(z0))) MARK(filter(x0, zprimes, sieve(x2))) -> c11(A__FILTER(mark(x0), a__zprimes, a__sieve(mark(x2))), MARK(zprimes)) MARK(filter(x0, s(z0), sieve(x2))) -> c11(A__FILTER(mark(x0), s(mark(z0)), a__sieve(mark(x2))), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, sieve(x2))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(sieve(z0), x1, sieve(x2))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(nats(z0), x1, sieve(x2))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(zprimes, x1, sieve(x2))) -> c11(A__FILTER(a__zprimes, mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(cons(z0, z1), x1, sieve(x2))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(x0, cons(z0, z1), sieve(x2))) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, sieve(x2))) -> c11(MARK(x1)) MARK(filter(s(z0), x1, sieve(x2))) -> c11(MARK(x1)) MARK(filter(x0, x1, nats(x2))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(mark(x2)), nats(s(mark(x2))))), MARK(x1)) MARK(filter(x0, x1, nats(x2))) -> c11(A__FILTER(mark(x0), mark(x1), nats(mark(x2))), MARK(x1)) MARK(filter(x0, x1, nats(filter(z0, z1, z2)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x1)) MARK(filter(x0, x1, nats(sieve(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__sieve(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, nats(nats(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__nats(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, nats(zprimes))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__zprimes)), MARK(x1)) MARK(filter(x0, x1, nats(cons(z0, z1)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(cons(mark(z0), z1))), MARK(x1)) MARK(filter(x0, x1, nats(0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(0)), MARK(x1)) MARK(filter(x0, x1, nats(s(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(s(mark(z0)))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), nats(x2))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), nats(x2))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), a__nats(mark(x2))), MARK(sieve(z0))) MARK(filter(x0, nats(z0), nats(x2))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), a__nats(mark(x2))), MARK(nats(z0))) MARK(filter(x0, zprimes, nats(x2))) -> c11(A__FILTER(mark(x0), a__zprimes, a__nats(mark(x2))), MARK(zprimes)) MARK(filter(x0, s(z0), nats(x2))) -> c11(A__FILTER(mark(x0), s(mark(z0)), a__nats(mark(x2))), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, nats(x2))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(sieve(z0), x1, nats(x2))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(nats(z0), x1, nats(x2))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(zprimes, x1, nats(x2))) -> c11(A__FILTER(a__zprimes, mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(cons(z0, z1), x1, nats(x2))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(x0, cons(z0, z1), nats(x2))) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, nats(x2))) -> c11(MARK(x1)) MARK(filter(s(z0), x1, nats(x2))) -> c11(MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(s(s(0))))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), zprimes), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), zprimes)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__zprimes), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), zprimes)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), a__zprimes), MARK(sieve(z0))) MARK(filter(x0, nats(z0), zprimes)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), a__zprimes), MARK(nats(z0))) MARK(filter(x0, zprimes, zprimes)) -> c11(A__FILTER(mark(x0), a__zprimes, a__zprimes), MARK(zprimes)) MARK(filter(x0, cons(z0, z1), zprimes)) -> c11(A__FILTER(mark(x0), cons(mark(z0), z1), a__zprimes), MARK(cons(z0, z1))) MARK(filter(x0, s(z0), zprimes)) -> c11(A__FILTER(mark(x0), s(mark(z0)), a__zprimes), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, zprimes)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__zprimes), MARK(x1)) MARK(filter(sieve(z0), x1, zprimes)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__zprimes), MARK(x1)) MARK(filter(nats(z0), x1, zprimes)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__zprimes), MARK(x1)) MARK(filter(zprimes, x1, zprimes)) -> c11(A__FILTER(a__zprimes, mark(x1), a__zprimes), MARK(x1)) MARK(filter(cons(z0, z1), x1, zprimes)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__zprimes), MARK(x1)) MARK(filter(0, x1, zprimes)) -> c11(A__FILTER(0, mark(x1), a__zprimes), MARK(x1)) MARK(filter(s(z0), x1, zprimes)) -> c11(A__FILTER(s(mark(z0)), mark(x1), a__zprimes), MARK(x1)) S tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, 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c10(A__FILTER(cons(a__sieve(mark(z0)), x1), mark(x2), mark(x3)), MARK(cons(sieve(z0), x1))) MARK(filter(cons(nats(z0), x1), x2, x3)) -> c10(A__FILTER(cons(a__nats(mark(z0)), x1), mark(x2), mark(x3)), MARK(cons(nats(z0), x1))) MARK(filter(cons(zprimes, x1), x2, x3)) -> c10(A__FILTER(cons(a__zprimes, x1), mark(x2), mark(x3)), MARK(cons(zprimes, x1))) MARK(filter(cons(cons(z0, z1), x1), x2, x3)) -> c10(A__FILTER(cons(cons(mark(z0), z1), x1), mark(x2), mark(x3)), MARK(cons(cons(z0, z1), x1))) MARK(filter(cons(0, x1), x2, x3)) -> c10(A__FILTER(cons(0, x1), mark(x2), mark(x3)), MARK(cons(0, x1))) MARK(filter(cons(s(z0), x1), x2, x3)) -> c10(A__FILTER(cons(s(mark(z0)), x1), mark(x2), mark(x3)), MARK(cons(s(z0), x1))) MARK(filter(cons(x0, x1), x2, x3)) -> c10(MARK(cons(x0, x1))) MARK(filter(cons(x0, x1), cons(z0, z1), x3)) -> c10(MARK(cons(x0, x1))) MARK(filter(cons(x0, x1), 0, x3)) -> c10(MARK(cons(x0, x1))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c11(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, filter(x2, x3, x4))) -> c11(MARK(x1)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c11(MARK(x1)) MARK(filter(x0, x1, sieve(x2))) -> c11(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x1)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x1)) MARK(filter(x0, x1, sieve(sieve(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__sieve(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, sieve(nats(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, sieve(zprimes))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__zprimes)), MARK(x1)) MARK(filter(x0, x1, sieve(cons(z0, z1)))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(cons(mark(z0), z1))), MARK(x1)) MARK(filter(x0, x1, sieve(0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(0)), MARK(x1)) MARK(filter(x0, x1, sieve(s(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(s(mark(z0)))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), sieve(x2))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), sieve(x2))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), a__sieve(mark(x2))), MARK(sieve(z0))) MARK(filter(x0, nats(z0), sieve(x2))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), a__sieve(mark(x2))), MARK(nats(z0))) MARK(filter(x0, zprimes, sieve(x2))) -> c11(A__FILTER(mark(x0), a__zprimes, a__sieve(mark(x2))), MARK(zprimes)) MARK(filter(x0, s(z0), sieve(x2))) -> c11(A__FILTER(mark(x0), s(mark(z0)), a__sieve(mark(x2))), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, sieve(x2))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(sieve(z0), x1, sieve(x2))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(nats(z0), x1, sieve(x2))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(zprimes, x1, sieve(x2))) -> c11(A__FILTER(a__zprimes, mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(cons(z0, z1), x1, sieve(x2))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(x0, cons(z0, z1), sieve(x2))) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, sieve(x2))) -> c11(MARK(x1)) MARK(filter(s(z0), x1, sieve(x2))) -> c11(MARK(x1)) MARK(filter(x0, x1, nats(x2))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(mark(x2)), nats(s(mark(x2))))), MARK(x1)) MARK(filter(x0, x1, nats(x2))) -> c11(A__FILTER(mark(x0), mark(x1), nats(mark(x2))), MARK(x1)) MARK(filter(x0, x1, nats(filter(z0, z1, z2)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x1)) MARK(filter(x0, x1, nats(sieve(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__sieve(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, nats(nats(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__nats(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, nats(zprimes))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__zprimes)), MARK(x1)) MARK(filter(x0, x1, nats(cons(z0, z1)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(cons(mark(z0), z1))), MARK(x1)) MARK(filter(x0, x1, nats(0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(0)), MARK(x1)) MARK(filter(x0, x1, nats(s(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(s(mark(z0)))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), nats(x2))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), nats(x2))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), a__nats(mark(x2))), MARK(sieve(z0))) MARK(filter(x0, nats(z0), nats(x2))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), a__nats(mark(x2))), MARK(nats(z0))) MARK(filter(x0, zprimes, nats(x2))) -> c11(A__FILTER(mark(x0), a__zprimes, a__nats(mark(x2))), MARK(zprimes)) MARK(filter(x0, s(z0), nats(x2))) -> c11(A__FILTER(mark(x0), s(mark(z0)), a__nats(mark(x2))), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, nats(x2))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(sieve(z0), x1, nats(x2))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(nats(z0), x1, nats(x2))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(zprimes, x1, nats(x2))) -> c11(A__FILTER(a__zprimes, mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(cons(z0, z1), x1, nats(x2))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(x0, cons(z0, z1), nats(x2))) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, nats(x2))) -> c11(MARK(x1)) MARK(filter(s(z0), x1, nats(x2))) -> c11(MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(s(s(0))))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), zprimes), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), zprimes)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__zprimes), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), zprimes)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), a__zprimes), MARK(sieve(z0))) MARK(filter(x0, nats(z0), zprimes)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), a__zprimes), MARK(nats(z0))) MARK(filter(x0, zprimes, zprimes)) -> c11(A__FILTER(mark(x0), a__zprimes, a__zprimes), MARK(zprimes)) MARK(filter(x0, cons(z0, z1), zprimes)) -> c11(A__FILTER(mark(x0), cons(mark(z0), z1), a__zprimes), MARK(cons(z0, z1))) MARK(filter(x0, s(z0), zprimes)) -> c11(A__FILTER(mark(x0), s(mark(z0)), a__zprimes), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, zprimes)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__zprimes), MARK(x1)) MARK(filter(sieve(z0), x1, zprimes)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__zprimes), MARK(x1)) MARK(filter(nats(z0), x1, zprimes)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__zprimes), MARK(x1)) MARK(filter(zprimes, x1, zprimes)) -> c11(A__FILTER(a__zprimes, mark(x1), a__zprimes), MARK(x1)) MARK(filter(cons(z0, z1), x1, zprimes)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__zprimes), MARK(x1)) MARK(filter(0, x1, zprimes)) -> c11(A__FILTER(0, mark(x1), a__zprimes), MARK(x1)) MARK(filter(s(z0), x1, zprimes)) -> c11(A__FILTER(s(mark(z0)), mark(x1), a__zprimes), MARK(x1)) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c14_2, c15_1, c16_1, c18_1, c8_1, c10_1, c11_2, c11_1, c12_2, c12_1, c13_2, c13_1, c8_2, c10_2 ---------------------------------------- (167) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (168) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(x0, 0, filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, sieve(x2))) -> c10(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x0)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, sieve(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, sieve(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, sieve(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(0)), MARK(x0)) MARK(filter(x0, x1, sieve(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), sieve(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, sieve(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, sieve(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__sieve(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, sieve(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, sieve(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, sieve(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), a__sieve(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, sieve(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__sieve(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), sieve(x2))) -> c10(MARK(x0)) MARK(filter(x0, 0, sieve(x2))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, sieve(x2))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(mark(x2)), nats(s(mark(x2))))), MARK(x0)) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), nats(mark(x2))), MARK(x0)) MARK(filter(x0, x1, nats(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, nats(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, nats(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, nats(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, nats(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, nats(0))) -> 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c10(A__FILTER(a__nats(mark(z0)), a__zprimes, mark(x1)), MARK(nats(z0))) MARK(filter(zprimes, zprimes, x1)) -> c10(A__FILTER(a__zprimes, a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(cons(z0, z1), zprimes, x1)) -> c10(A__FILTER(cons(mark(z0), z1), a__zprimes, mark(x1)), MARK(cons(z0, z1))) MARK(filter(x0, zprimes, x1)) -> c10(MARK(x0)) MARK(filter(s(z0), zprimes, x1)) -> c10(MARK(s(z0))) MARK(filter(x0, s(filter(z0, z1, z2)), x2)) -> c10(A__FILTER(mark(x0), s(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(x0)) MARK(filter(x0, s(sieve(z0)), x2)) -> c10(A__FILTER(mark(x0), s(a__sieve(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, s(nats(z0)), x2)) -> c10(A__FILTER(mark(x0), s(a__nats(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, s(zprimes), x2)) -> c10(A__FILTER(mark(x0), s(a__zprimes), mark(x2)), MARK(x0)) MARK(filter(x0, s(cons(z0, z1)), x2)) -> c10(A__FILTER(mark(x0), s(cons(mark(z0), z1)), mark(x2)), MARK(x0)) MARK(filter(x0, s(0), x2)) -> c10(A__FILTER(mark(x0), 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c10(A__FILTER(a__filter(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, filter(z0, z1, z2), x2))) MARK(filter(filter(x0, sieve(z0), x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), a__sieve(mark(z0)), mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, sieve(z0), x2))) MARK(filter(filter(x0, nats(z0), x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), a__nats(mark(z0)), mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, nats(z0), x2))) MARK(filter(filter(x0, zprimes, x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), a__zprimes, mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, zprimes, x2))) MARK(filter(filter(x0, cons(z0, z1), x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), cons(mark(z0), z1), mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, cons(z0, z1), x2))) MARK(filter(filter(x0, 0, x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), 0, mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, 0, x2))) MARK(filter(filter(x0, s(z0), x2), x3, x4)) 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MARK(filter(0, x1, filter(x2, x3, x4))) -> c11(MARK(x1)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c11(MARK(x1)) MARK(filter(x0, x1, sieve(x2))) -> c11(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x1)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x1)) MARK(filter(x0, x1, sieve(sieve(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__sieve(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, sieve(nats(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, sieve(zprimes))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__zprimes)), MARK(x1)) MARK(filter(x0, x1, sieve(cons(z0, z1)))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(cons(mark(z0), z1))), MARK(x1)) MARK(filter(x0, x1, sieve(0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(0)), MARK(x1)) MARK(filter(x0, x1, sieve(s(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(s(mark(z0)))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), sieve(x2))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), sieve(x2))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), a__sieve(mark(x2))), MARK(sieve(z0))) MARK(filter(x0, nats(z0), sieve(x2))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), a__sieve(mark(x2))), MARK(nats(z0))) MARK(filter(x0, zprimes, sieve(x2))) -> c11(A__FILTER(mark(x0), a__zprimes, a__sieve(mark(x2))), MARK(zprimes)) MARK(filter(x0, s(z0), sieve(x2))) -> c11(A__FILTER(mark(x0), s(mark(z0)), a__sieve(mark(x2))), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, sieve(x2))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(sieve(z0), x1, sieve(x2))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(nats(z0), x1, sieve(x2))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(zprimes, x1, sieve(x2))) -> c11(A__FILTER(a__zprimes, mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(cons(z0, z1), x1, sieve(x2))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(x0, cons(z0, z1), sieve(x2))) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, sieve(x2))) -> c11(MARK(x1)) MARK(filter(s(z0), x1, sieve(x2))) -> c11(MARK(x1)) MARK(filter(x0, x1, nats(x2))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(mark(x2)), nats(s(mark(x2))))), MARK(x1)) MARK(filter(x0, x1, nats(x2))) -> c11(A__FILTER(mark(x0), mark(x1), nats(mark(x2))), MARK(x1)) MARK(filter(x0, x1, nats(filter(z0, z1, z2)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x1)) MARK(filter(x0, x1, nats(sieve(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__sieve(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, nats(nats(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__nats(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, nats(zprimes))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__zprimes)), MARK(x1)) MARK(filter(x0, x1, nats(cons(z0, z1)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(cons(mark(z0), z1))), MARK(x1)) MARK(filter(x0, x1, nats(0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(0)), MARK(x1)) MARK(filter(x0, x1, nats(s(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(s(mark(z0)))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), nats(x2))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), nats(x2))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), a__nats(mark(x2))), MARK(sieve(z0))) MARK(filter(x0, nats(z0), nats(x2))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), a__nats(mark(x2))), MARK(nats(z0))) MARK(filter(x0, zprimes, nats(x2))) -> c11(A__FILTER(mark(x0), a__zprimes, a__nats(mark(x2))), MARK(zprimes)) MARK(filter(x0, s(z0), nats(x2))) -> c11(A__FILTER(mark(x0), s(mark(z0)), a__nats(mark(x2))), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, nats(x2))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(sieve(z0), x1, nats(x2))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(nats(z0), x1, nats(x2))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(zprimes, x1, nats(x2))) -> c11(A__FILTER(a__zprimes, mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(cons(z0, z1), x1, nats(x2))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(x0, cons(z0, z1), nats(x2))) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, nats(x2))) -> c11(MARK(x1)) MARK(filter(s(z0), x1, nats(x2))) -> c11(MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(s(s(0))))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), zprimes), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), zprimes)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__zprimes), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), zprimes)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), a__zprimes), MARK(sieve(z0))) MARK(filter(x0, nats(z0), zprimes)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), a__zprimes), MARK(nats(z0))) MARK(filter(x0, zprimes, zprimes)) -> c11(A__FILTER(mark(x0), a__zprimes, a__zprimes), MARK(zprimes)) MARK(filter(x0, s(z0), zprimes)) -> c11(A__FILTER(mark(x0), s(mark(z0)), a__zprimes), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, zprimes)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__zprimes), MARK(x1)) MARK(filter(sieve(z0), x1, zprimes)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__zprimes), MARK(x1)) MARK(filter(nats(z0), x1, zprimes)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__zprimes), MARK(x1)) MARK(filter(zprimes, x1, zprimes)) -> c11(A__FILTER(a__zprimes, mark(x1), a__zprimes), MARK(x1)) MARK(filter(cons(z0, z1), x1, zprimes)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, cons(z0, z1), zprimes)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, zprimes)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, zprimes)) -> c11(MARK(x1)) S tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> 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c10(A__FILTER(a__nats(a__nats(mark(z0))), mark(x1), mark(x2)), MARK(nats(nats(z0)))) MARK(filter(nats(zprimes), x1, x2)) -> c10(A__FILTER(a__nats(a__zprimes), mark(x1), mark(x2)), MARK(nats(zprimes))) MARK(filter(nats(cons(z0, z1)), x1, x2)) -> c10(A__FILTER(a__nats(cons(mark(z0), z1)), mark(x1), mark(x2)), MARK(nats(cons(z0, z1)))) MARK(filter(nats(0), x1, x2)) -> c10(A__FILTER(a__nats(0), mark(x1), mark(x2)), MARK(nats(0))) MARK(filter(nats(s(z0)), x1, x2)) -> c10(A__FILTER(a__nats(s(mark(z0))), mark(x1), mark(x2)), MARK(nats(s(z0)))) MARK(filter(nats(x0), x1, x2)) -> c10(MARK(nats(x0))) MARK(filter(nats(x0), cons(z0, z1), x2)) -> c10(MARK(nats(x0))) MARK(filter(nats(x0), 0, x2)) -> c10(MARK(nats(x0))) MARK(filter(zprimes, x0, x1)) -> c10(A__FILTER(a__sieve(a__nats(s(s(0)))), mark(x0), mark(x1)), MARK(zprimes)) MARK(filter(zprimes, cons(z0, z1), x1)) -> c10(MARK(zprimes)) MARK(filter(zprimes, 0, x1)) -> c10(MARK(zprimes)) MARK(filter(zprimes, x0, x1)) -> c10(MARK(zprimes)) MARK(filter(cons(filter(z0, z1, z2), x1), x2, x3)) -> c10(A__FILTER(cons(a__filter(mark(z0), mark(z1), mark(z2)), x1), mark(x2), mark(x3)), MARK(cons(filter(z0, z1, z2), x1))) MARK(filter(cons(sieve(z0), x1), x2, x3)) -> c10(A__FILTER(cons(a__sieve(mark(z0)), x1), mark(x2), mark(x3)), MARK(cons(sieve(z0), x1))) MARK(filter(cons(nats(z0), x1), x2, x3)) -> c10(A__FILTER(cons(a__nats(mark(z0)), x1), mark(x2), mark(x3)), MARK(cons(nats(z0), x1))) MARK(filter(cons(zprimes, x1), x2, x3)) -> c10(A__FILTER(cons(a__zprimes, x1), mark(x2), mark(x3)), MARK(cons(zprimes, x1))) MARK(filter(cons(cons(z0, z1), x1), x2, x3)) -> c10(A__FILTER(cons(cons(mark(z0), z1), x1), mark(x2), mark(x3)), MARK(cons(cons(z0, z1), x1))) MARK(filter(cons(0, x1), x2, x3)) -> c10(A__FILTER(cons(0, x1), mark(x2), mark(x3)), MARK(cons(0, x1))) MARK(filter(cons(s(z0), x1), x2, x3)) -> c10(A__FILTER(cons(s(mark(z0)), x1), mark(x2), mark(x3)), MARK(cons(s(z0), x1))) MARK(filter(cons(x0, x1), x2, x3)) -> c10(MARK(cons(x0, x1))) MARK(filter(cons(x0, x1), cons(z0, z1), x3)) -> c10(MARK(cons(x0, x1))) MARK(filter(cons(x0, x1), 0, x3)) -> c10(MARK(cons(x0, x1))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c11(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, filter(x2, x3, x4))) -> c11(MARK(x1)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c11(MARK(x1)) MARK(filter(x0, x1, sieve(x2))) -> c11(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x1)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x1)) MARK(filter(x0, x1, sieve(sieve(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__sieve(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, sieve(nats(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, sieve(zprimes))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__zprimes)), MARK(x1)) MARK(filter(x0, x1, sieve(cons(z0, z1)))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(cons(mark(z0), z1))), MARK(x1)) MARK(filter(x0, x1, sieve(0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(0)), MARK(x1)) MARK(filter(x0, x1, sieve(s(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(s(mark(z0)))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), sieve(x2))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), sieve(x2))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), a__sieve(mark(x2))), MARK(sieve(z0))) MARK(filter(x0, nats(z0), sieve(x2))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), a__sieve(mark(x2))), MARK(nats(z0))) MARK(filter(x0, zprimes, sieve(x2))) -> c11(A__FILTER(mark(x0), a__zprimes, a__sieve(mark(x2))), MARK(zprimes)) MARK(filter(x0, s(z0), sieve(x2))) -> c11(A__FILTER(mark(x0), s(mark(z0)), a__sieve(mark(x2))), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, sieve(x2))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(sieve(z0), x1, sieve(x2))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(nats(z0), x1, sieve(x2))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(zprimes, x1, sieve(x2))) -> c11(A__FILTER(a__zprimes, mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(cons(z0, z1), x1, sieve(x2))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(x0, cons(z0, z1), sieve(x2))) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, sieve(x2))) -> c11(MARK(x1)) MARK(filter(s(z0), x1, sieve(x2))) -> c11(MARK(x1)) MARK(filter(x0, x1, nats(x2))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(mark(x2)), nats(s(mark(x2))))), MARK(x1)) MARK(filter(x0, x1, nats(x2))) -> c11(A__FILTER(mark(x0), mark(x1), nats(mark(x2))), MARK(x1)) MARK(filter(x0, x1, nats(filter(z0, z1, z2)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x1)) MARK(filter(x0, x1, nats(sieve(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__sieve(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, nats(nats(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__nats(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, nats(zprimes))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__zprimes)), MARK(x1)) MARK(filter(x0, x1, nats(cons(z0, z1)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(cons(mark(z0), z1))), MARK(x1)) MARK(filter(x0, x1, nats(0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(0)), MARK(x1)) MARK(filter(x0, x1, nats(s(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(s(mark(z0)))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), nats(x2))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), nats(x2))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), a__nats(mark(x2))), MARK(sieve(z0))) MARK(filter(x0, nats(z0), nats(x2))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), a__nats(mark(x2))), MARK(nats(z0))) MARK(filter(x0, zprimes, nats(x2))) -> c11(A__FILTER(mark(x0), a__zprimes, a__nats(mark(x2))), MARK(zprimes)) MARK(filter(x0, s(z0), nats(x2))) -> c11(A__FILTER(mark(x0), s(mark(z0)), a__nats(mark(x2))), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, nats(x2))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(sieve(z0), x1, nats(x2))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(nats(z0), x1, nats(x2))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(zprimes, x1, nats(x2))) -> c11(A__FILTER(a__zprimes, mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(cons(z0, z1), x1, nats(x2))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(x0, cons(z0, z1), nats(x2))) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, nats(x2))) -> c11(MARK(x1)) MARK(filter(s(z0), x1, nats(x2))) -> c11(MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(s(s(0))))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), zprimes), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), zprimes)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__zprimes), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), zprimes)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), a__zprimes), MARK(sieve(z0))) MARK(filter(x0, nats(z0), zprimes)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), a__zprimes), MARK(nats(z0))) MARK(filter(x0, zprimes, zprimes)) -> c11(A__FILTER(mark(x0), a__zprimes, a__zprimes), MARK(zprimes)) MARK(filter(x0, s(z0), zprimes)) -> c11(A__FILTER(mark(x0), s(mark(z0)), a__zprimes), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, zprimes)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__zprimes), MARK(x1)) MARK(filter(sieve(z0), x1, zprimes)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__zprimes), MARK(x1)) MARK(filter(nats(z0), x1, zprimes)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__zprimes), MARK(x1)) MARK(filter(zprimes, x1, zprimes)) -> c11(A__FILTER(a__zprimes, mark(x1), a__zprimes), MARK(x1)) MARK(filter(cons(z0, z1), x1, zprimes)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, cons(z0, z1), zprimes)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, zprimes)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, zprimes)) -> c11(MARK(x1)) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c14_2, c15_1, c16_1, c18_1, c8_1, c10_1, c11_2, c11_1, c12_2, c12_1, c13_2, c13_1, c8_2, c10_2 ---------------------------------------- (169) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(filter(x0, x1, cons(z0, z1))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(x1)) by MARK(filter(x0, x1, cons(filter(z0, z1, z2), x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(a__filter(mark(z0), mark(z1), mark(z2)), x3)), MARK(x1)) MARK(filter(x0, x1, cons(sieve(z0), x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(a__sieve(mark(z0)), x3)), MARK(x1)) MARK(filter(x0, x1, cons(nats(z0), x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(a__nats(mark(z0)), x3)), MARK(x1)) MARK(filter(x0, x1, cons(zprimes, x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(a__zprimes, x3)), MARK(x1)) MARK(filter(x0, x1, cons(cons(z0, z1), x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(cons(mark(z0), z1), x3)), MARK(x1)) MARK(filter(x0, x1, cons(0, x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(0, x3)), MARK(x1)) MARK(filter(x0, x1, cons(s(z0), x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(s(mark(z0)), x3)), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), cons(x2, x3))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), cons(mark(x2), x3)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), cons(x2, x3))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), cons(mark(x2), x3)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), cons(x2, x3))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), cons(mark(x2), x3)), MARK(nats(z0))) MARK(filter(x0, zprimes, cons(x2, x3))) -> c11(A__FILTER(mark(x0), a__zprimes, cons(mark(x2), x3)), MARK(zprimes)) MARK(filter(x0, cons(z0, z1), cons(x2, x3))) -> c11(A__FILTER(mark(x0), cons(mark(z0), z1), cons(mark(x2), x3)), MARK(cons(z0, z1))) MARK(filter(x0, 0, cons(x2, x3))) -> c11(A__FILTER(mark(x0), 0, cons(mark(x2), x3)), MARK(0)) MARK(filter(x0, s(z0), cons(x2, x3))) -> c11(A__FILTER(mark(x0), s(mark(z0)), cons(mark(x2), x3)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, cons(x2, x3))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(sieve(z0), x1, cons(x2, x3))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(nats(z0), x1, cons(x2, x3))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(zprimes, x1, cons(x2, x3))) -> c11(A__FILTER(a__zprimes, mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(cons(z0, z1), x1, cons(x2, x3))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(0, x1, cons(x2, x3))) -> c11(A__FILTER(0, mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(s(z0), x1, cons(x2, x3))) -> c11(A__FILTER(s(mark(z0)), mark(x1), cons(mark(x2), x3)), MARK(x1)) ---------------------------------------- (170) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(x0, 0, filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, sieve(x2))) -> c10(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x0)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, sieve(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, sieve(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, sieve(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(0)), MARK(x0)) MARK(filter(x0, x1, sieve(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), sieve(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, sieve(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, sieve(x2))) -> c10(A__FILTER(a__filter(mark(z0), 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MARK(filter(s(z0), x1, sieve(x2))) -> c11(MARK(x1)) MARK(filter(x0, x1, nats(x2))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(mark(x2)), nats(s(mark(x2))))), MARK(x1)) MARK(filter(x0, x1, nats(x2))) -> c11(A__FILTER(mark(x0), mark(x1), nats(mark(x2))), MARK(x1)) MARK(filter(x0, x1, nats(filter(z0, z1, z2)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x1)) MARK(filter(x0, x1, nats(sieve(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__sieve(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, nats(nats(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__nats(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, nats(zprimes))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__zprimes)), MARK(x1)) MARK(filter(x0, x1, nats(cons(z0, z1)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(cons(mark(z0), z1))), MARK(x1)) MARK(filter(x0, x1, nats(0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(0)), MARK(x1)) MARK(filter(x0, x1, nats(s(z0)))) -> 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c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(zprimes, x1, nats(x2))) -> c11(A__FILTER(a__zprimes, mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(cons(z0, z1), x1, nats(x2))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(x0, cons(z0, z1), nats(x2))) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, nats(x2))) -> c11(MARK(x1)) MARK(filter(s(z0), x1, nats(x2))) -> c11(MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(s(s(0))))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), zprimes), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), zprimes)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__zprimes), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), zprimes)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), a__zprimes), MARK(sieve(z0))) MARK(filter(x0, nats(z0), zprimes)) -> c11(A__FILTER(mark(x0), 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cons(x2, x3))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), cons(mark(x2), x3)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), cons(x2, x3))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), cons(mark(x2), x3)), MARK(nats(z0))) MARK(filter(x0, zprimes, cons(x2, x3))) -> c11(A__FILTER(mark(x0), a__zprimes, cons(mark(x2), x3)), MARK(zprimes)) MARK(filter(x0, cons(z0, z1), cons(x2, x3))) -> c11(A__FILTER(mark(x0), cons(mark(z0), z1), cons(mark(x2), x3)), MARK(cons(z0, z1))) MARK(filter(x0, 0, cons(x2, x3))) -> c11(A__FILTER(mark(x0), 0, cons(mark(x2), x3)), MARK(0)) MARK(filter(x0, s(z0), cons(x2, x3))) -> c11(A__FILTER(mark(x0), s(mark(z0)), cons(mark(x2), x3)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, cons(x2, x3))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(sieve(z0), x1, cons(x2, x3))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(nats(z0), x1, cons(x2, x3))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(zprimes, x1, cons(x2, x3))) -> c11(A__FILTER(a__zprimes, mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(cons(z0, z1), x1, cons(x2, x3))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(0, x1, cons(x2, x3))) -> c11(A__FILTER(0, mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(s(z0), x1, cons(x2, x3))) -> c11(A__FILTER(s(mark(z0)), mark(x1), cons(mark(x2), x3)), MARK(x1)) S tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(x0, 0, filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, sieve(x2))) -> c10(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x0)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, sieve(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, sieve(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, sieve(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(0)), MARK(x0)) MARK(filter(x0, x1, sieve(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), sieve(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, sieve(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, sieve(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__sieve(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, sieve(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, sieve(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, sieve(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), a__sieve(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, sieve(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__sieve(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), sieve(x2))) -> c10(MARK(x0)) MARK(filter(x0, 0, sieve(x2))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, sieve(x2))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(mark(x2)), nats(s(mark(x2))))), MARK(x0)) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), nats(mark(x2))), MARK(x0)) MARK(filter(x0, x1, nats(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, nats(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, nats(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, nats(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, nats(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, nats(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(0)), MARK(x0)) MARK(filter(x0, x1, nats(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), nats(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), nats(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), nats(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, nats(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), nats(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, nats(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__nats(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, nats(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__nats(mark(x2))), 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mark(x4))), MARK(zprimes)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c11(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, filter(x2, x3, x4))) -> c11(MARK(x1)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c11(MARK(x1)) MARK(filter(x0, x1, sieve(x2))) -> c11(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x1)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x1)) MARK(filter(x0, x1, sieve(sieve(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__sieve(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, sieve(nats(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, sieve(zprimes))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__zprimes)), MARK(x1)) MARK(filter(x0, x1, sieve(cons(z0, z1)))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(cons(mark(z0), z1))), MARK(x1)) MARK(filter(x0, x1, sieve(0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(0)), MARK(x1)) MARK(filter(x0, x1, sieve(s(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(s(mark(z0)))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), sieve(x2))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), sieve(x2))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), a__sieve(mark(x2))), MARK(sieve(z0))) MARK(filter(x0, nats(z0), sieve(x2))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), a__sieve(mark(x2))), MARK(nats(z0))) MARK(filter(x0, zprimes, sieve(x2))) -> c11(A__FILTER(mark(x0), a__zprimes, a__sieve(mark(x2))), MARK(zprimes)) MARK(filter(x0, s(z0), sieve(x2))) -> c11(A__FILTER(mark(x0), s(mark(z0)), a__sieve(mark(x2))), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, sieve(x2))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(sieve(z0), x1, sieve(x2))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(nats(z0), x1, sieve(x2))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(zprimes, x1, sieve(x2))) -> c11(A__FILTER(a__zprimes, mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(cons(z0, z1), x1, sieve(x2))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(x0, cons(z0, z1), sieve(x2))) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, sieve(x2))) -> c11(MARK(x1)) MARK(filter(s(z0), x1, sieve(x2))) -> c11(MARK(x1)) MARK(filter(x0, x1, nats(x2))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(mark(x2)), nats(s(mark(x2))))), MARK(x1)) MARK(filter(x0, x1, nats(x2))) -> c11(A__FILTER(mark(x0), mark(x1), nats(mark(x2))), MARK(x1)) MARK(filter(x0, x1, nats(filter(z0, z1, z2)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x1)) MARK(filter(x0, x1, nats(sieve(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__sieve(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, nats(nats(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__nats(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, nats(zprimes))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__zprimes)), MARK(x1)) MARK(filter(x0, x1, nats(cons(z0, z1)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(cons(mark(z0), z1))), MARK(x1)) MARK(filter(x0, x1, nats(0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(0)), MARK(x1)) MARK(filter(x0, x1, nats(s(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(s(mark(z0)))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), nats(x2))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), nats(x2))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), a__nats(mark(x2))), MARK(sieve(z0))) MARK(filter(x0, nats(z0), nats(x2))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), a__nats(mark(x2))), MARK(nats(z0))) MARK(filter(x0, zprimes, nats(x2))) -> c11(A__FILTER(mark(x0), a__zprimes, a__nats(mark(x2))), MARK(zprimes)) MARK(filter(x0, s(z0), nats(x2))) -> c11(A__FILTER(mark(x0), s(mark(z0)), a__nats(mark(x2))), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, nats(x2))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(sieve(z0), x1, nats(x2))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(nats(z0), x1, nats(x2))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(zprimes, x1, nats(x2))) -> c11(A__FILTER(a__zprimes, mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(cons(z0, z1), x1, nats(x2))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(x0, cons(z0, z1), nats(x2))) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, nats(x2))) -> c11(MARK(x1)) MARK(filter(s(z0), x1, nats(x2))) -> c11(MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(s(s(0))))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), zprimes), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), zprimes)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__zprimes), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), zprimes)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), a__zprimes), MARK(sieve(z0))) MARK(filter(x0, nats(z0), zprimes)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), a__zprimes), MARK(nats(z0))) MARK(filter(x0, zprimes, zprimes)) -> c11(A__FILTER(mark(x0), a__zprimes, a__zprimes), MARK(zprimes)) MARK(filter(x0, s(z0), zprimes)) -> c11(A__FILTER(mark(x0), s(mark(z0)), a__zprimes), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, zprimes)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__zprimes), MARK(x1)) MARK(filter(sieve(z0), x1, zprimes)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__zprimes), MARK(x1)) MARK(filter(nats(z0), x1, zprimes)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__zprimes), MARK(x1)) MARK(filter(zprimes, x1, zprimes)) -> c11(A__FILTER(a__zprimes, mark(x1), a__zprimes), MARK(x1)) MARK(filter(cons(z0, z1), x1, zprimes)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, cons(z0, z1), zprimes)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, zprimes)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, zprimes)) -> c11(MARK(x1)) MARK(filter(x0, x1, cons(filter(z0, z1, z2), x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(a__filter(mark(z0), mark(z1), mark(z2)), x3)), MARK(x1)) MARK(filter(x0, x1, cons(sieve(z0), x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(a__sieve(mark(z0)), x3)), MARK(x1)) MARK(filter(x0, x1, cons(nats(z0), x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(a__nats(mark(z0)), x3)), MARK(x1)) MARK(filter(x0, x1, cons(zprimes, x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(a__zprimes, x3)), MARK(x1)) MARK(filter(x0, x1, cons(cons(z0, z1), x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(cons(mark(z0), z1), x3)), MARK(x1)) MARK(filter(x0, x1, cons(0, x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(0, x3)), MARK(x1)) MARK(filter(x0, x1, cons(s(z0), x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(s(mark(z0)), x3)), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), cons(x2, x3))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), cons(mark(x2), x3)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), cons(x2, x3))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), cons(mark(x2), x3)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), cons(x2, x3))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), cons(mark(x2), x3)), MARK(nats(z0))) MARK(filter(x0, zprimes, cons(x2, x3))) -> c11(A__FILTER(mark(x0), a__zprimes, cons(mark(x2), x3)), MARK(zprimes)) MARK(filter(x0, cons(z0, z1), cons(x2, x3))) -> c11(A__FILTER(mark(x0), cons(mark(z0), z1), cons(mark(x2), x3)), MARK(cons(z0, z1))) MARK(filter(x0, 0, cons(x2, x3))) -> c11(A__FILTER(mark(x0), 0, cons(mark(x2), x3)), MARK(0)) MARK(filter(x0, s(z0), cons(x2, x3))) -> c11(A__FILTER(mark(x0), s(mark(z0)), cons(mark(x2), x3)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, cons(x2, x3))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(sieve(z0), x1, cons(x2, x3))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(nats(z0), x1, cons(x2, x3))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(zprimes, x1, cons(x2, x3))) -> c11(A__FILTER(a__zprimes, mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(cons(z0, z1), x1, cons(x2, x3))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(0, x1, cons(x2, x3))) -> c11(A__FILTER(0, mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(s(z0), x1, cons(x2, x3))) -> c11(A__FILTER(s(mark(z0)), mark(x1), cons(mark(x2), x3)), MARK(x1)) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c14_2, c15_1, c16_1, c18_1, c8_1, c10_1, c11_2, c11_1, c12_2, c12_1, c13_2, c13_1, c8_2, c10_2 ---------------------------------------- (171) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(filter(x0, 0, cons(x2, x3))) -> c11(A__FILTER(mark(x0), 0, cons(mark(x2), x3)), MARK(0)) ---------------------------------------- (172) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), 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c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(nats(z0), x1, nats(x2))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(zprimes, x1, nats(x2))) -> c11(A__FILTER(a__zprimes, mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(cons(z0, z1), x1, nats(x2))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(x0, cons(z0, z1), nats(x2))) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, nats(x2))) -> c11(MARK(x1)) MARK(filter(s(z0), x1, nats(x2))) -> c11(MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(s(s(0))))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), zprimes), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), zprimes)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__zprimes), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), zprimes)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), a__zprimes), MARK(sieve(z0))) MARK(filter(x0, nats(z0), zprimes)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), a__zprimes), MARK(nats(z0))) MARK(filter(x0, zprimes, zprimes)) -> c11(A__FILTER(mark(x0), a__zprimes, a__zprimes), MARK(zprimes)) MARK(filter(x0, s(z0), zprimes)) -> c11(A__FILTER(mark(x0), s(mark(z0)), a__zprimes), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, zprimes)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__zprimes), MARK(x1)) MARK(filter(sieve(z0), x1, zprimes)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__zprimes), MARK(x1)) MARK(filter(nats(z0), x1, zprimes)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__zprimes), MARK(x1)) MARK(filter(zprimes, x1, zprimes)) -> c11(A__FILTER(a__zprimes, mark(x1), a__zprimes), MARK(x1)) MARK(filter(cons(z0, z1), x1, zprimes)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, cons(z0, z1), zprimes)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, zprimes)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, zprimes)) -> c11(MARK(x1)) MARK(filter(x0, x1, cons(filter(z0, z1, z2), x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(a__filter(mark(z0), mark(z1), mark(z2)), x3)), MARK(x1)) MARK(filter(x0, x1, cons(sieve(z0), x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(a__sieve(mark(z0)), x3)), MARK(x1)) MARK(filter(x0, x1, cons(nats(z0), x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(a__nats(mark(z0)), x3)), MARK(x1)) MARK(filter(x0, x1, cons(zprimes, x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(a__zprimes, x3)), MARK(x1)) MARK(filter(x0, x1, cons(cons(z0, z1), x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(cons(mark(z0), z1), x3)), MARK(x1)) MARK(filter(x0, x1, cons(0, x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(0, x3)), MARK(x1)) MARK(filter(x0, x1, cons(s(z0), x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(s(mark(z0)), x3)), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), cons(x2, x3))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), cons(mark(x2), x3)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), cons(x2, x3))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), cons(mark(x2), x3)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), cons(x2, x3))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), cons(mark(x2), x3)), MARK(nats(z0))) MARK(filter(x0, zprimes, cons(x2, x3))) -> c11(A__FILTER(mark(x0), a__zprimes, cons(mark(x2), x3)), MARK(zprimes)) MARK(filter(x0, cons(z0, z1), cons(x2, x3))) -> c11(A__FILTER(mark(x0), cons(mark(z0), z1), cons(mark(x2), x3)), MARK(cons(z0, z1))) MARK(filter(x0, s(z0), cons(x2, x3))) -> c11(A__FILTER(mark(x0), s(mark(z0)), cons(mark(x2), x3)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, cons(x2, x3))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(sieve(z0), x1, cons(x2, x3))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(nats(z0), x1, cons(x2, x3))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(zprimes, x1, cons(x2, x3))) -> c11(A__FILTER(a__zprimes, mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(cons(z0, z1), x1, cons(x2, x3))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(0, x1, cons(x2, x3))) -> c11(A__FILTER(0, mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(s(z0), x1, cons(x2, x3))) -> c11(A__FILTER(s(mark(z0)), mark(x1), cons(mark(x2), x3)), MARK(x1)) S tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), 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MARK(filter(cons(x0, x1), x2, x3)) -> c10(MARK(cons(x0, x1))) MARK(filter(cons(x0, x1), cons(z0, z1), x3)) -> c10(MARK(cons(x0, x1))) MARK(filter(cons(x0, x1), 0, x3)) -> c10(MARK(cons(x0, x1))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c11(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, filter(x2, x3, x4))) -> c11(MARK(x1)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c11(MARK(x1)) MARK(filter(x0, x1, sieve(x2))) -> c11(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x1)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x1)) MARK(filter(x0, x1, sieve(sieve(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__sieve(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, sieve(nats(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, sieve(zprimes))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__zprimes)), MARK(x1)) MARK(filter(x0, x1, sieve(cons(z0, z1)))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(cons(mark(z0), z1))), MARK(x1)) MARK(filter(x0, x1, sieve(0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(0)), MARK(x1)) MARK(filter(x0, x1, sieve(s(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(s(mark(z0)))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), sieve(x2))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), sieve(x2))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), a__sieve(mark(x2))), MARK(sieve(z0))) MARK(filter(x0, nats(z0), sieve(x2))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), a__sieve(mark(x2))), MARK(nats(z0))) MARK(filter(x0, zprimes, sieve(x2))) -> c11(A__FILTER(mark(x0), a__zprimes, a__sieve(mark(x2))), MARK(zprimes)) MARK(filter(x0, s(z0), sieve(x2))) -> c11(A__FILTER(mark(x0), s(mark(z0)), a__sieve(mark(x2))), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, sieve(x2))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(sieve(z0), x1, sieve(x2))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(nats(z0), x1, sieve(x2))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(zprimes, x1, sieve(x2))) -> c11(A__FILTER(a__zprimes, mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(cons(z0, z1), x1, sieve(x2))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(x0, cons(z0, z1), sieve(x2))) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, sieve(x2))) -> c11(MARK(x1)) MARK(filter(s(z0), x1, sieve(x2))) -> c11(MARK(x1)) MARK(filter(x0, x1, nats(x2))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(mark(x2)), nats(s(mark(x2))))), MARK(x1)) MARK(filter(x0, x1, nats(x2))) -> c11(A__FILTER(mark(x0), mark(x1), nats(mark(x2))), MARK(x1)) MARK(filter(x0, x1, nats(filter(z0, z1, z2)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x1)) MARK(filter(x0, x1, nats(sieve(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__sieve(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, nats(nats(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__nats(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, nats(zprimes))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__zprimes)), MARK(x1)) MARK(filter(x0, x1, nats(cons(z0, z1)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(cons(mark(z0), z1))), MARK(x1)) MARK(filter(x0, x1, nats(0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(0)), MARK(x1)) MARK(filter(x0, x1, nats(s(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(s(mark(z0)))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), nats(x2))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), nats(x2))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), a__nats(mark(x2))), MARK(sieve(z0))) MARK(filter(x0, nats(z0), nats(x2))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), a__nats(mark(x2))), MARK(nats(z0))) MARK(filter(x0, zprimes, nats(x2))) -> c11(A__FILTER(mark(x0), a__zprimes, a__nats(mark(x2))), MARK(zprimes)) MARK(filter(x0, s(z0), nats(x2))) -> c11(A__FILTER(mark(x0), s(mark(z0)), a__nats(mark(x2))), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, nats(x2))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(sieve(z0), x1, nats(x2))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(nats(z0), x1, nats(x2))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(zprimes, x1, nats(x2))) -> c11(A__FILTER(a__zprimes, mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(cons(z0, z1), x1, nats(x2))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(x0, cons(z0, z1), nats(x2))) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, nats(x2))) -> c11(MARK(x1)) MARK(filter(s(z0), x1, nats(x2))) -> c11(MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(s(s(0))))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), zprimes), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), zprimes)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__zprimes), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), zprimes)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), a__zprimes), MARK(sieve(z0))) MARK(filter(x0, nats(z0), zprimes)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), a__zprimes), MARK(nats(z0))) MARK(filter(x0, zprimes, zprimes)) -> c11(A__FILTER(mark(x0), a__zprimes, a__zprimes), MARK(zprimes)) MARK(filter(x0, s(z0), zprimes)) -> c11(A__FILTER(mark(x0), s(mark(z0)), a__zprimes), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, zprimes)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__zprimes), MARK(x1)) MARK(filter(sieve(z0), x1, zprimes)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__zprimes), MARK(x1)) MARK(filter(nats(z0), x1, zprimes)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__zprimes), MARK(x1)) MARK(filter(zprimes, x1, zprimes)) -> c11(A__FILTER(a__zprimes, mark(x1), a__zprimes), MARK(x1)) MARK(filter(cons(z0, z1), x1, zprimes)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, cons(z0, z1), zprimes)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, zprimes)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, zprimes)) -> c11(MARK(x1)) MARK(filter(x0, x1, cons(filter(z0, z1, z2), x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(a__filter(mark(z0), mark(z1), mark(z2)), x3)), MARK(x1)) MARK(filter(x0, x1, cons(sieve(z0), x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(a__sieve(mark(z0)), x3)), MARK(x1)) MARK(filter(x0, x1, cons(nats(z0), x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(a__nats(mark(z0)), x3)), MARK(x1)) MARK(filter(x0, x1, cons(zprimes, x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(a__zprimes, x3)), MARK(x1)) MARK(filter(x0, x1, cons(cons(z0, z1), x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(cons(mark(z0), z1), x3)), MARK(x1)) MARK(filter(x0, x1, cons(0, x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(0, x3)), MARK(x1)) MARK(filter(x0, x1, cons(s(z0), x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(s(mark(z0)), x3)), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), cons(x2, x3))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), cons(mark(x2), x3)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), cons(x2, x3))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), cons(mark(x2), x3)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), cons(x2, x3))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), cons(mark(x2), x3)), MARK(nats(z0))) MARK(filter(x0, zprimes, cons(x2, x3))) -> c11(A__FILTER(mark(x0), a__zprimes, cons(mark(x2), x3)), MARK(zprimes)) MARK(filter(x0, cons(z0, z1), cons(x2, x3))) -> c11(A__FILTER(mark(x0), cons(mark(z0), z1), cons(mark(x2), x3)), MARK(cons(z0, z1))) MARK(filter(x0, s(z0), cons(x2, x3))) -> c11(A__FILTER(mark(x0), s(mark(z0)), cons(mark(x2), x3)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, cons(x2, x3))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(sieve(z0), x1, cons(x2, x3))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(nats(z0), x1, cons(x2, x3))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(zprimes, x1, cons(x2, x3))) -> c11(A__FILTER(a__zprimes, mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(cons(z0, z1), x1, cons(x2, x3))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(0, x1, cons(x2, x3))) -> c11(A__FILTER(0, mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(s(z0), x1, cons(x2, x3))) -> c11(A__FILTER(s(mark(z0)), mark(x1), cons(mark(x2), x3)), MARK(x1)) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c14_2, c15_1, c16_1, c18_1, c8_1, c10_1, c11_2, c11_1, c12_2, c12_1, c13_2, c13_1, c8_2, c10_2 ---------------------------------------- (173) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (174) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(x0, 0, filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, sieve(x2))) -> c10(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x0)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, sieve(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, sieve(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, sieve(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(0)), MARK(x0)) MARK(filter(x0, x1, sieve(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), sieve(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, sieve(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, sieve(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), 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a__sieve(mark(z0)), a__sieve(mark(x2))), MARK(sieve(z0))) MARK(filter(x0, nats(z0), sieve(x2))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), a__sieve(mark(x2))), MARK(nats(z0))) MARK(filter(x0, zprimes, sieve(x2))) -> c11(A__FILTER(mark(x0), a__zprimes, a__sieve(mark(x2))), MARK(zprimes)) MARK(filter(x0, s(z0), sieve(x2))) -> c11(A__FILTER(mark(x0), s(mark(z0)), a__sieve(mark(x2))), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, sieve(x2))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(sieve(z0), x1, sieve(x2))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(nats(z0), x1, sieve(x2))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(zprimes, x1, sieve(x2))) -> c11(A__FILTER(a__zprimes, mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(cons(z0, z1), x1, sieve(x2))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(x0, cons(z0, z1), sieve(x2))) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, sieve(x2))) -> c11(MARK(x1)) MARK(filter(s(z0), x1, sieve(x2))) -> c11(MARK(x1)) MARK(filter(x0, x1, nats(x2))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(mark(x2)), nats(s(mark(x2))))), MARK(x1)) MARK(filter(x0, x1, nats(x2))) -> c11(A__FILTER(mark(x0), mark(x1), nats(mark(x2))), MARK(x1)) MARK(filter(x0, x1, nats(filter(z0, z1, z2)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x1)) MARK(filter(x0, x1, nats(sieve(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__sieve(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, nats(nats(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__nats(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, nats(zprimes))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__zprimes)), MARK(x1)) MARK(filter(x0, x1, nats(cons(z0, z1)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(cons(mark(z0), z1))), MARK(x1)) MARK(filter(x0, x1, nats(0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(0)), MARK(x1)) MARK(filter(x0, x1, nats(s(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(s(mark(z0)))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), nats(x2))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), nats(x2))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), a__nats(mark(x2))), MARK(sieve(z0))) MARK(filter(x0, nats(z0), nats(x2))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), a__nats(mark(x2))), MARK(nats(z0))) MARK(filter(x0, zprimes, nats(x2))) -> c11(A__FILTER(mark(x0), a__zprimes, a__nats(mark(x2))), MARK(zprimes)) MARK(filter(x0, s(z0), nats(x2))) -> c11(A__FILTER(mark(x0), s(mark(z0)), a__nats(mark(x2))), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, nats(x2))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(sieve(z0), x1, nats(x2))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(nats(z0), x1, nats(x2))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(zprimes, x1, nats(x2))) -> c11(A__FILTER(a__zprimes, mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(cons(z0, z1), x1, nats(x2))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(x0, cons(z0, z1), nats(x2))) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, nats(x2))) -> c11(MARK(x1)) MARK(filter(s(z0), x1, nats(x2))) -> c11(MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(s(s(0))))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), zprimes), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), zprimes)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__zprimes), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), zprimes)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), a__zprimes), MARK(sieve(z0))) MARK(filter(x0, nats(z0), zprimes)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), a__zprimes), MARK(nats(z0))) MARK(filter(x0, zprimes, zprimes)) -> c11(A__FILTER(mark(x0), a__zprimes, a__zprimes), MARK(zprimes)) MARK(filter(x0, s(z0), zprimes)) -> c11(A__FILTER(mark(x0), s(mark(z0)), a__zprimes), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, zprimes)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__zprimes), MARK(x1)) MARK(filter(sieve(z0), x1, zprimes)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__zprimes), MARK(x1)) MARK(filter(nats(z0), x1, zprimes)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__zprimes), MARK(x1)) MARK(filter(zprimes, x1, zprimes)) -> c11(A__FILTER(a__zprimes, mark(x1), a__zprimes), MARK(x1)) MARK(filter(cons(z0, z1), x1, zprimes)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, cons(z0, z1), zprimes)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, zprimes)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, zprimes)) -> c11(MARK(x1)) MARK(filter(x0, x1, cons(filter(z0, z1, z2), x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(a__filter(mark(z0), mark(z1), mark(z2)), x3)), MARK(x1)) MARK(filter(x0, x1, cons(sieve(z0), x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(a__sieve(mark(z0)), x3)), MARK(x1)) MARK(filter(x0, x1, cons(nats(z0), x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(a__nats(mark(z0)), x3)), MARK(x1)) MARK(filter(x0, x1, cons(zprimes, x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(a__zprimes, x3)), MARK(x1)) MARK(filter(x0, x1, cons(cons(z0, z1), x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(cons(mark(z0), z1), x3)), MARK(x1)) MARK(filter(x0, x1, cons(0, x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(0, x3)), MARK(x1)) MARK(filter(x0, x1, cons(s(z0), x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(s(mark(z0)), x3)), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), cons(x2, x3))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), cons(mark(x2), x3)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), cons(x2, x3))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), cons(mark(x2), x3)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), cons(x2, x3))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), cons(mark(x2), x3)), MARK(nats(z0))) MARK(filter(x0, zprimes, cons(x2, x3))) -> c11(A__FILTER(mark(x0), a__zprimes, cons(mark(x2), x3)), MARK(zprimes)) MARK(filter(x0, s(z0), cons(x2, x3))) -> c11(A__FILTER(mark(x0), s(mark(z0)), cons(mark(x2), x3)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, cons(x2, x3))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(sieve(z0), x1, cons(x2, x3))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(nats(z0), x1, cons(x2, x3))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(zprimes, x1, cons(x2, x3))) -> c11(A__FILTER(a__zprimes, mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(cons(z0, z1), x1, cons(x2, x3))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(x0, cons(z0, z1), cons(x2, x3))) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, cons(x2, x3))) -> c11(MARK(x1)) MARK(filter(s(z0), x1, cons(x2, x3))) -> c11(MARK(x1)) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c14_2, c15_1, c16_1, c18_1, c8_1, c10_1, c11_2, c11_1, c12_2, c12_1, c13_2, c13_1, c8_2, c10_2 ---------------------------------------- (175) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(filter(x0, x1, 0)) -> c11(A__FILTER(mark(x0), mark(x1), 0), MARK(x1)) by MARK(filter(x0, filter(z0, z1, z2), 0)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), 0), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), 0)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), 0), MARK(sieve(z0))) MARK(filter(x0, nats(z0), 0)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), 0), MARK(nats(z0))) MARK(filter(x0, zprimes, 0)) -> c11(A__FILTER(mark(x0), a__zprimes, 0), MARK(zprimes)) MARK(filter(x0, cons(z0, z1), 0)) -> c11(A__FILTER(mark(x0), cons(mark(z0), z1), 0), MARK(cons(z0, z1))) MARK(filter(x0, 0, 0)) -> c11(A__FILTER(mark(x0), 0, 0), MARK(0)) MARK(filter(x0, s(z0), 0)) -> c11(A__FILTER(mark(x0), s(mark(z0)), 0), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, 0)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), 0), MARK(x1)) MARK(filter(sieve(z0), x1, 0)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), 0), MARK(x1)) MARK(filter(nats(z0), x1, 0)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), 0), MARK(x1)) MARK(filter(zprimes, x1, 0)) -> c11(A__FILTER(a__zprimes, mark(x1), 0), MARK(x1)) MARK(filter(cons(z0, z1), x1, 0)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), 0), MARK(x1)) MARK(filter(0, x1, 0)) -> c11(A__FILTER(0, mark(x1), 0), MARK(x1)) MARK(filter(s(z0), x1, 0)) -> c11(A__FILTER(s(mark(z0)), mark(x1), 0), MARK(x1)) ---------------------------------------- (176) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 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a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), 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a__sieve(0)), MARK(x0)) MARK(filter(x0, x1, sieve(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), sieve(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, sieve(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, sieve(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__sieve(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, sieve(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, sieve(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, sieve(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), a__sieve(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, sieve(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__sieve(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), sieve(x2))) -> c10(MARK(x0)) MARK(filter(x0, 0, sieve(x2))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, sieve(x2))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(mark(x2)), nats(s(mark(x2))))), MARK(x0)) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), nats(mark(x2))), MARK(x0)) MARK(filter(x0, x1, nats(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, nats(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, nats(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, nats(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, nats(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, nats(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(0)), MARK(x0)) MARK(filter(x0, x1, nats(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), nats(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), nats(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), nats(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, nats(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), nats(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, nats(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__nats(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, nats(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, nats(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, nats(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), a__nats(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, nats(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__nats(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), nats(x2))) -> c10(MARK(x0)) MARK(filter(x0, 0, nats(x2))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, nats(x2))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, zprimes)) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(s(s(0))))), MARK(x0)) MARK(filter(x0, x1, zprimes)) -> c10(A__FILTER(mark(x0), mark(x1), zprimes), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), zprimes)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__zprimes), MARK(x0)) MARK(filter(x0, sieve(z0), zprimes)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__zprimes), MARK(x0)) MARK(filter(x0, nats(z0), zprimes)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__zprimes), MARK(x0)) MARK(filter(x0, zprimes, zprimes)) -> c10(A__FILTER(mark(x0), a__zprimes, a__zprimes), MARK(x0)) MARK(filter(x0, s(z0), zprimes)) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__zprimes), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, zprimes)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__zprimes), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, zprimes)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__zprimes), MARK(sieve(z0))) MARK(filter(nats(z0), x1, zprimes)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__zprimes), MARK(nats(z0))) MARK(filter(zprimes, x1, zprimes)) -> c10(A__FILTER(a__zprimes, mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, zprimes)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__zprimes), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), zprimes)) -> c10(MARK(x0)) MARK(filter(x0, 0, zprimes)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, zprimes)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, cons(filter(z0, z1, z2), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(a__filter(mark(z0), mark(z1), mark(z2)), x3)), MARK(x0)) MARK(filter(x0, x1, cons(sieve(z0), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(a__sieve(mark(z0)), x3)), MARK(x0)) MARK(filter(x0, x1, cons(nats(z0), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(a__nats(mark(z0)), x3)), MARK(x0)) MARK(filter(x0, x1, cons(zprimes, x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(a__zprimes, x3)), MARK(x0)) MARK(filter(x0, x1, cons(cons(z0, z1), 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mark(z2)), mark(x1), cons(mark(x2), x3)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, cons(x2, x3))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), cons(mark(x2), x3)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, cons(x2, x3))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), cons(mark(x2), x3)), MARK(nats(z0))) MARK(filter(zprimes, x1, cons(x2, x3))) -> c10(A__FILTER(a__zprimes, mark(x1), cons(mark(x2), x3)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, cons(x2, x3))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), cons(mark(x2), x3)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), cons(x2, x3))) -> c10(MARK(x0)) MARK(filter(x0, 0, cons(x2, x3))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, cons(x2, x3))) -> c10(MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), 0)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), 0), MARK(x0)) MARK(filter(x0, sieve(z0), 0)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), 0), MARK(x0)) MARK(filter(x0, nats(z0), 0)) -> 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a__zprimes), MARK(zprimes)) MARK(filter(x0, s(z0), zprimes)) -> c11(A__FILTER(mark(x0), s(mark(z0)), a__zprimes), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, zprimes)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__zprimes), MARK(x1)) MARK(filter(sieve(z0), x1, zprimes)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__zprimes), MARK(x1)) MARK(filter(nats(z0), x1, zprimes)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__zprimes), MARK(x1)) MARK(filter(zprimes, x1, zprimes)) -> c11(A__FILTER(a__zprimes, mark(x1), a__zprimes), MARK(x1)) MARK(filter(cons(z0, z1), x1, zprimes)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, cons(z0, z1), zprimes)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, zprimes)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, zprimes)) -> c11(MARK(x1)) MARK(filter(x0, x1, cons(filter(z0, z1, z2), x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(a__filter(mark(z0), mark(z1), mark(z2)), x3)), MARK(x1)) MARK(filter(x0, x1, cons(sieve(z0), x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(a__sieve(mark(z0)), x3)), MARK(x1)) MARK(filter(x0, x1, cons(nats(z0), x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(a__nats(mark(z0)), x3)), MARK(x1)) MARK(filter(x0, x1, cons(zprimes, x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(a__zprimes, x3)), MARK(x1)) MARK(filter(x0, x1, cons(cons(z0, z1), x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(cons(mark(z0), z1), x3)), MARK(x1)) MARK(filter(x0, x1, cons(0, x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(0, x3)), MARK(x1)) MARK(filter(x0, x1, cons(s(z0), x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(s(mark(z0)), x3)), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), cons(x2, x3))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), cons(mark(x2), x3)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), cons(x2, x3))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), cons(mark(x2), x3)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), cons(x2, x3))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), cons(mark(x2), x3)), MARK(nats(z0))) MARK(filter(x0, zprimes, cons(x2, x3))) -> c11(A__FILTER(mark(x0), a__zprimes, cons(mark(x2), x3)), MARK(zprimes)) MARK(filter(x0, s(z0), cons(x2, x3))) -> c11(A__FILTER(mark(x0), s(mark(z0)), cons(mark(x2), x3)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, cons(x2, x3))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(sieve(z0), x1, cons(x2, x3))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(nats(z0), x1, cons(x2, x3))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(zprimes, x1, cons(x2, x3))) -> c11(A__FILTER(a__zprimes, mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(cons(z0, z1), x1, cons(x2, x3))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(x0, cons(z0, z1), cons(x2, x3))) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, cons(x2, x3))) -> c11(MARK(x1)) MARK(filter(s(z0), x1, cons(x2, x3))) -> c11(MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), 0)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), 0), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), 0)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), 0), MARK(sieve(z0))) MARK(filter(x0, nats(z0), 0)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), 0), MARK(nats(z0))) MARK(filter(x0, zprimes, 0)) -> c11(A__FILTER(mark(x0), a__zprimes, 0), MARK(zprimes)) MARK(filter(x0, cons(z0, z1), 0)) -> c11(A__FILTER(mark(x0), cons(mark(z0), z1), 0), MARK(cons(z0, z1))) MARK(filter(x0, 0, 0)) -> c11(A__FILTER(mark(x0), 0, 0), MARK(0)) MARK(filter(x0, s(z0), 0)) -> c11(A__FILTER(mark(x0), s(mark(z0)), 0), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, 0)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), 0), MARK(x1)) MARK(filter(sieve(z0), x1, 0)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), 0), MARK(x1)) MARK(filter(nats(z0), x1, 0)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), 0), MARK(x1)) MARK(filter(zprimes, x1, 0)) -> c11(A__FILTER(a__zprimes, mark(x1), 0), MARK(x1)) MARK(filter(cons(z0, z1), x1, 0)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), 0), MARK(x1)) MARK(filter(0, x1, 0)) -> c11(A__FILTER(0, mark(x1), 0), MARK(x1)) MARK(filter(s(z0), x1, 0)) -> c11(A__FILTER(s(mark(z0)), mark(x1), 0), MARK(x1)) S tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), 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-> c10(A__FILTER(a__filter(mark(x0), a__sieve(mark(z0)), mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, sieve(z0), x2))) MARK(filter(filter(x0, nats(z0), x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), a__nats(mark(z0)), mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, nats(z0), x2))) MARK(filter(filter(x0, zprimes, x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), a__zprimes, mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, zprimes, x2))) MARK(filter(filter(x0, cons(z0, z1), x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), cons(mark(z0), z1), mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, cons(z0, z1), x2))) MARK(filter(filter(x0, 0, x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), 0, mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, 0, x2))) MARK(filter(filter(x0, s(z0), x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), s(mark(z0)), mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, s(z0), x2))) MARK(filter(filter(filter(z0, z1, z2), x1, x2), x3, x4)) -> 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MARK(filter(x0, filter(z0, z1, z2), cons(x2, x3))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), cons(mark(x2), x3)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), cons(x2, x3))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), cons(mark(x2), x3)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), cons(x2, x3))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), cons(mark(x2), x3)), MARK(nats(z0))) MARK(filter(x0, zprimes, cons(x2, x3))) -> c11(A__FILTER(mark(x0), a__zprimes, cons(mark(x2), x3)), MARK(zprimes)) MARK(filter(x0, s(z0), cons(x2, x3))) -> c11(A__FILTER(mark(x0), s(mark(z0)), cons(mark(x2), x3)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, cons(x2, x3))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(sieve(z0), x1, cons(x2, x3))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(nats(z0), x1, cons(x2, x3))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), 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c11(A__FILTER(mark(x0), 0, 0), MARK(0)) MARK(filter(x0, s(z0), 0)) -> c11(A__FILTER(mark(x0), s(mark(z0)), 0), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, 0)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), 0), MARK(x1)) MARK(filter(sieve(z0), x1, 0)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), 0), MARK(x1)) MARK(filter(nats(z0), x1, 0)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), 0), MARK(x1)) MARK(filter(zprimes, x1, 0)) -> c11(A__FILTER(a__zprimes, mark(x1), 0), MARK(x1)) MARK(filter(cons(z0, z1), x1, 0)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), 0), MARK(x1)) MARK(filter(0, x1, 0)) -> c11(A__FILTER(0, mark(x1), 0), MARK(x1)) MARK(filter(s(z0), x1, 0)) -> c11(A__FILTER(s(mark(z0)), mark(x1), 0), MARK(x1)) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c14_2, c15_1, c16_1, c18_1, c8_1, c10_1, c11_2, c11_1, c12_2, c12_1, c13_2, c13_1, c8_2, c10_2 ---------------------------------------- (177) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(filter(x0, 0, 0)) -> c11(A__FILTER(mark(x0), 0, 0), MARK(0)) ---------------------------------------- (178) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(x0, 0, filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, sieve(x2))) -> c10(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x0)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, sieve(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, sieve(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, sieve(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(0)), MARK(x0)) MARK(filter(x0, x1, sieve(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), sieve(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, sieve(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, sieve(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__sieve(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, sieve(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, sieve(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, sieve(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), a__sieve(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, sieve(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__sieve(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), sieve(x2))) -> c10(MARK(x0)) MARK(filter(x0, 0, sieve(x2))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, sieve(x2))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(mark(x2)), nats(s(mark(x2))))), MARK(x0)) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), nats(mark(x2))), MARK(x0)) MARK(filter(x0, x1, nats(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, nats(sieve(z0)))) -> c10(A__FILTER(mark(x0), 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c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), nats(x2))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), a__nats(mark(x2))), MARK(sieve(z0))) MARK(filter(x0, nats(z0), nats(x2))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), a__nats(mark(x2))), MARK(nats(z0))) MARK(filter(x0, zprimes, nats(x2))) -> c11(A__FILTER(mark(x0), a__zprimes, a__nats(mark(x2))), MARK(zprimes)) MARK(filter(x0, s(z0), nats(x2))) -> c11(A__FILTER(mark(x0), s(mark(z0)), a__nats(mark(x2))), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, nats(x2))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(sieve(z0), x1, nats(x2))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(nats(z0), x1, nats(x2))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(zprimes, x1, nats(x2))) -> 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MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(x0, 0, filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, sieve(x2))) -> c10(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x0)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, sieve(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), 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c10(A__FILTER(mark(x0), s(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, sieve(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__sieve(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, sieve(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, sieve(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, sieve(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), a__sieve(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, sieve(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__sieve(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), sieve(x2))) -> c10(MARK(x0)) MARK(filter(x0, 0, sieve(x2))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, sieve(x2))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(mark(x2)), nats(s(mark(x2))))), MARK(x0)) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), nats(mark(x2))), MARK(x0)) MARK(filter(x0, x1, nats(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, nats(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, nats(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, nats(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, nats(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, nats(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(0)), MARK(x0)) MARK(filter(x0, x1, nats(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), nats(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), 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mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c11(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, filter(x2, x3, x4))) -> c11(MARK(x1)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c11(MARK(x1)) MARK(filter(x0, x1, sieve(x2))) -> c11(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x1)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x1)) MARK(filter(x0, x1, sieve(sieve(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__sieve(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, sieve(nats(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, sieve(zprimes))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__zprimes)), MARK(x1)) MARK(filter(x0, x1, sieve(cons(z0, z1)))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(cons(mark(z0), z1))), MARK(x1)) MARK(filter(x0, x1, sieve(0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(0)), MARK(x1)) MARK(filter(x0, x1, sieve(s(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(s(mark(z0)))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), sieve(x2))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), sieve(x2))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), a__sieve(mark(x2))), MARK(sieve(z0))) MARK(filter(x0, nats(z0), sieve(x2))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), a__sieve(mark(x2))), MARK(nats(z0))) MARK(filter(x0, zprimes, sieve(x2))) -> c11(A__FILTER(mark(x0), a__zprimes, a__sieve(mark(x2))), MARK(zprimes)) MARK(filter(x0, s(z0), sieve(x2))) -> c11(A__FILTER(mark(x0), s(mark(z0)), a__sieve(mark(x2))), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, sieve(x2))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(sieve(z0), x1, sieve(x2))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(nats(z0), x1, sieve(x2))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(zprimes, x1, sieve(x2))) -> c11(A__FILTER(a__zprimes, mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(cons(z0, z1), x1, sieve(x2))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(x0, cons(z0, z1), sieve(x2))) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, sieve(x2))) -> c11(MARK(x1)) MARK(filter(s(z0), x1, sieve(x2))) -> c11(MARK(x1)) MARK(filter(x0, x1, nats(x2))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(mark(x2)), nats(s(mark(x2))))), MARK(x1)) MARK(filter(x0, x1, nats(x2))) -> c11(A__FILTER(mark(x0), mark(x1), nats(mark(x2))), MARK(x1)) MARK(filter(x0, x1, nats(filter(z0, z1, z2)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x1)) MARK(filter(x0, x1, nats(sieve(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__sieve(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, nats(nats(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__nats(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, nats(zprimes))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__zprimes)), MARK(x1)) MARK(filter(x0, x1, nats(cons(z0, z1)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(cons(mark(z0), z1))), MARK(x1)) MARK(filter(x0, x1, nats(0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(0)), MARK(x1)) MARK(filter(x0, x1, nats(s(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(s(mark(z0)))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), nats(x2))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), nats(x2))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), a__nats(mark(x2))), MARK(sieve(z0))) MARK(filter(x0, nats(z0), nats(x2))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), a__nats(mark(x2))), MARK(nats(z0))) MARK(filter(x0, zprimes, nats(x2))) -> c11(A__FILTER(mark(x0), a__zprimes, a__nats(mark(x2))), MARK(zprimes)) MARK(filter(x0, s(z0), nats(x2))) -> c11(A__FILTER(mark(x0), s(mark(z0)), a__nats(mark(x2))), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, nats(x2))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(sieve(z0), x1, nats(x2))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(nats(z0), x1, nats(x2))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(zprimes, x1, nats(x2))) -> c11(A__FILTER(a__zprimes, mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(cons(z0, z1), x1, nats(x2))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(x0, cons(z0, z1), nats(x2))) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, nats(x2))) -> c11(MARK(x1)) MARK(filter(s(z0), x1, nats(x2))) -> c11(MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(s(s(0))))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), zprimes), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), zprimes)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__zprimes), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), zprimes)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), a__zprimes), MARK(sieve(z0))) MARK(filter(x0, nats(z0), zprimes)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), a__zprimes), MARK(nats(z0))) MARK(filter(x0, zprimes, zprimes)) -> c11(A__FILTER(mark(x0), a__zprimes, a__zprimes), MARK(zprimes)) MARK(filter(x0, s(z0), zprimes)) -> c11(A__FILTER(mark(x0), s(mark(z0)), a__zprimes), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, zprimes)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__zprimes), MARK(x1)) MARK(filter(sieve(z0), x1, zprimes)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__zprimes), MARK(x1)) MARK(filter(nats(z0), x1, zprimes)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__zprimes), MARK(x1)) MARK(filter(zprimes, x1, zprimes)) -> c11(A__FILTER(a__zprimes, mark(x1), a__zprimes), MARK(x1)) MARK(filter(cons(z0, z1), x1, zprimes)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, cons(z0, z1), zprimes)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, zprimes)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, zprimes)) -> c11(MARK(x1)) MARK(filter(x0, x1, cons(filter(z0, z1, z2), x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(a__filter(mark(z0), mark(z1), mark(z2)), x3)), MARK(x1)) MARK(filter(x0, x1, cons(sieve(z0), x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(a__sieve(mark(z0)), x3)), MARK(x1)) MARK(filter(x0, x1, cons(nats(z0), x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(a__nats(mark(z0)), x3)), MARK(x1)) MARK(filter(x0, x1, cons(zprimes, x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(a__zprimes, x3)), MARK(x1)) MARK(filter(x0, x1, cons(cons(z0, z1), x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(cons(mark(z0), z1), x3)), MARK(x1)) MARK(filter(x0, x1, cons(0, x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(0, x3)), MARK(x1)) MARK(filter(x0, x1, cons(s(z0), x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(s(mark(z0)), x3)), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), cons(x2, x3))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), cons(mark(x2), x3)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), cons(x2, x3))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), cons(mark(x2), x3)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), cons(x2, x3))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), cons(mark(x2), x3)), MARK(nats(z0))) MARK(filter(x0, zprimes, cons(x2, x3))) -> c11(A__FILTER(mark(x0), a__zprimes, cons(mark(x2), x3)), MARK(zprimes)) MARK(filter(x0, s(z0), cons(x2, x3))) -> c11(A__FILTER(mark(x0), s(mark(z0)), cons(mark(x2), x3)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, cons(x2, x3))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(sieve(z0), x1, cons(x2, x3))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(nats(z0), x1, cons(x2, x3))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(zprimes, x1, cons(x2, x3))) -> c11(A__FILTER(a__zprimes, mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(cons(z0, z1), x1, cons(x2, x3))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(x0, cons(z0, z1), cons(x2, x3))) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, cons(x2, x3))) -> c11(MARK(x1)) MARK(filter(s(z0), x1, cons(x2, x3))) -> c11(MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), 0)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), 0), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), 0)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), 0), MARK(sieve(z0))) MARK(filter(x0, nats(z0), 0)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), 0), MARK(nats(z0))) MARK(filter(x0, zprimes, 0)) -> c11(A__FILTER(mark(x0), a__zprimes, 0), MARK(zprimes)) MARK(filter(x0, cons(z0, z1), 0)) -> c11(A__FILTER(mark(x0), cons(mark(z0), z1), 0), MARK(cons(z0, z1))) MARK(filter(x0, s(z0), 0)) -> c11(A__FILTER(mark(x0), s(mark(z0)), 0), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, 0)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), 0), MARK(x1)) MARK(filter(sieve(z0), x1, 0)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), 0), MARK(x1)) MARK(filter(nats(z0), x1, 0)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), 0), MARK(x1)) MARK(filter(zprimes, x1, 0)) -> c11(A__FILTER(a__zprimes, mark(x1), 0), MARK(x1)) MARK(filter(cons(z0, z1), x1, 0)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), 0), MARK(x1)) MARK(filter(0, x1, 0)) -> c11(A__FILTER(0, mark(x1), 0), MARK(x1)) MARK(filter(s(z0), x1, 0)) -> c11(A__FILTER(s(mark(z0)), mark(x1), 0), MARK(x1)) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c14_2, c15_1, c16_1, c18_1, c8_1, c10_1, c11_2, c11_1, c12_2, c12_1, c13_2, c13_1, c8_2, c10_2 ---------------------------------------- (179) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (180) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, 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a__nats(mark(x2))), MARK(sieve(z0))) MARK(filter(x0, nats(z0), nats(x2))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), a__nats(mark(x2))), MARK(nats(z0))) MARK(filter(x0, zprimes, nats(x2))) -> c11(A__FILTER(mark(x0), a__zprimes, a__nats(mark(x2))), MARK(zprimes)) MARK(filter(x0, s(z0), nats(x2))) -> c11(A__FILTER(mark(x0), s(mark(z0)), a__nats(mark(x2))), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, nats(x2))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(sieve(z0), x1, nats(x2))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(nats(z0), x1, nats(x2))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(zprimes, x1, nats(x2))) -> c11(A__FILTER(a__zprimes, mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(cons(z0, z1), x1, nats(x2))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(x0, cons(z0, z1), nats(x2))) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, nats(x2))) -> c11(MARK(x1)) MARK(filter(s(z0), x1, nats(x2))) -> c11(MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(s(s(0))))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), zprimes), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), zprimes)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__zprimes), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), zprimes)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), a__zprimes), MARK(sieve(z0))) MARK(filter(x0, nats(z0), zprimes)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), a__zprimes), MARK(nats(z0))) MARK(filter(x0, zprimes, zprimes)) -> c11(A__FILTER(mark(x0), a__zprimes, a__zprimes), MARK(zprimes)) MARK(filter(x0, s(z0), zprimes)) -> c11(A__FILTER(mark(x0), s(mark(z0)), a__zprimes), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, zprimes)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__zprimes), MARK(x1)) MARK(filter(sieve(z0), x1, zprimes)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__zprimes), MARK(x1)) MARK(filter(nats(z0), x1, zprimes)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__zprimes), MARK(x1)) MARK(filter(zprimes, x1, zprimes)) -> c11(A__FILTER(a__zprimes, mark(x1), a__zprimes), MARK(x1)) MARK(filter(cons(z0, z1), x1, zprimes)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, cons(z0, z1), zprimes)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, zprimes)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, zprimes)) -> c11(MARK(x1)) MARK(filter(x0, x1, cons(filter(z0, z1, z2), x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(a__filter(mark(z0), mark(z1), mark(z2)), x3)), MARK(x1)) MARK(filter(x0, x1, cons(sieve(z0), x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(a__sieve(mark(z0)), x3)), MARK(x1)) MARK(filter(x0, x1, cons(nats(z0), x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(a__nats(mark(z0)), x3)), MARK(x1)) MARK(filter(x0, x1, cons(zprimes, x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(a__zprimes, x3)), MARK(x1)) MARK(filter(x0, x1, cons(cons(z0, z1), x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(cons(mark(z0), z1), x3)), MARK(x1)) MARK(filter(x0, x1, cons(0, x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(0, x3)), MARK(x1)) MARK(filter(x0, x1, cons(s(z0), x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(s(mark(z0)), x3)), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), cons(x2, x3))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), cons(mark(x2), x3)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), cons(x2, x3))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), cons(mark(x2), x3)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), cons(x2, x3))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), cons(mark(x2), x3)), MARK(nats(z0))) MARK(filter(x0, zprimes, cons(x2, x3))) -> c11(A__FILTER(mark(x0), a__zprimes, cons(mark(x2), x3)), MARK(zprimes)) MARK(filter(x0, s(z0), cons(x2, x3))) -> c11(A__FILTER(mark(x0), s(mark(z0)), cons(mark(x2), x3)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, cons(x2, x3))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(sieve(z0), x1, cons(x2, x3))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(nats(z0), x1, cons(x2, x3))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(zprimes, x1, cons(x2, x3))) -> c11(A__FILTER(a__zprimes, mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(cons(z0, z1), x1, cons(x2, x3))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(x0, cons(z0, z1), cons(x2, x3))) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, cons(x2, x3))) -> c11(MARK(x1)) MARK(filter(s(z0), x1, cons(x2, x3))) -> c11(MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), 0)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), 0), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), 0)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), 0), MARK(sieve(z0))) MARK(filter(x0, nats(z0), 0)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), 0), MARK(nats(z0))) MARK(filter(x0, zprimes, 0)) -> c11(A__FILTER(mark(x0), a__zprimes, 0), MARK(zprimes)) MARK(filter(x0, s(z0), 0)) -> c11(A__FILTER(mark(x0), s(mark(z0)), 0), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, 0)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), 0), MARK(x1)) MARK(filter(sieve(z0), x1, 0)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), 0), MARK(x1)) MARK(filter(nats(z0), x1, 0)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), 0), MARK(x1)) MARK(filter(zprimes, x1, 0)) -> c11(A__FILTER(a__zprimes, mark(x1), 0), MARK(x1)) MARK(filter(cons(z0, z1), x1, 0)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), 0), MARK(x1)) MARK(filter(x0, cons(z0, z1), 0)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, 0)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, 0)) -> c11(MARK(x1)) S tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(x0, 0, filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, sieve(x2))) -> c10(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x0)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, sieve(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, sieve(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, sieve(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(0)), MARK(x0)) MARK(filter(x0, x1, sieve(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), sieve(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, sieve(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, sieve(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__sieve(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, sieve(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), 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mark(x1), a__nats(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, nats(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, nats(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, nats(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, nats(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(0)), MARK(x0)) MARK(filter(x0, x1, nats(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), nats(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), nats(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), nats(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, nats(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), nats(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, nats(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__nats(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, nats(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, nats(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, nats(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), a__nats(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, nats(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__nats(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), nats(x2))) -> c10(MARK(x0)) MARK(filter(x0, 0, nats(x2))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, nats(x2))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, zprimes)) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(s(s(0))))), MARK(x0)) MARK(filter(x0, x1, zprimes)) -> c10(A__FILTER(mark(x0), mark(x1), zprimes), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), zprimes)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__zprimes), MARK(x0)) MARK(filter(x0, sieve(z0), zprimes)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__zprimes), MARK(x0)) MARK(filter(x0, nats(z0), zprimes)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__zprimes), MARK(x0)) MARK(filter(x0, zprimes, zprimes)) -> c10(A__FILTER(mark(x0), a__zprimes, a__zprimes), MARK(x0)) MARK(filter(x0, s(z0), zprimes)) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__zprimes), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, zprimes)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__zprimes), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, zprimes)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__zprimes), MARK(sieve(z0))) MARK(filter(nats(z0), x1, 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mark(z2)), mark(x1), cons(mark(x2), x3)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, cons(x2, x3))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), cons(mark(x2), x3)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, cons(x2, x3))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), cons(mark(x2), x3)), MARK(nats(z0))) MARK(filter(zprimes, x1, cons(x2, x3))) -> c10(A__FILTER(a__zprimes, mark(x1), cons(mark(x2), x3)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, cons(x2, x3))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), cons(mark(x2), x3)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), cons(x2, x3))) -> c10(MARK(x0)) MARK(filter(x0, 0, cons(x2, x3))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, cons(x2, x3))) -> c10(MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), 0)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), 0), MARK(x0)) MARK(filter(x0, sieve(z0), 0)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), 0), MARK(x0)) MARK(filter(x0, nats(z0), 0)) -> 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a__zprimes), MARK(zprimes)) MARK(filter(x0, s(z0), zprimes)) -> c11(A__FILTER(mark(x0), s(mark(z0)), a__zprimes), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, zprimes)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__zprimes), MARK(x1)) MARK(filter(sieve(z0), x1, zprimes)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__zprimes), MARK(x1)) MARK(filter(nats(z0), x1, zprimes)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__zprimes), MARK(x1)) MARK(filter(zprimes, x1, zprimes)) -> c11(A__FILTER(a__zprimes, mark(x1), a__zprimes), MARK(x1)) MARK(filter(cons(z0, z1), x1, zprimes)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, cons(z0, z1), zprimes)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, zprimes)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, zprimes)) -> c11(MARK(x1)) MARK(filter(x0, x1, cons(filter(z0, z1, z2), x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(a__filter(mark(z0), mark(z1), mark(z2)), x3)), MARK(x1)) MARK(filter(x0, x1, cons(sieve(z0), x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(a__sieve(mark(z0)), x3)), MARK(x1)) MARK(filter(x0, x1, cons(nats(z0), x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(a__nats(mark(z0)), x3)), MARK(x1)) MARK(filter(x0, x1, cons(zprimes, x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(a__zprimes, x3)), MARK(x1)) MARK(filter(x0, x1, cons(cons(z0, z1), x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(cons(mark(z0), z1), x3)), MARK(x1)) MARK(filter(x0, x1, cons(0, x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(0, x3)), MARK(x1)) MARK(filter(x0, x1, cons(s(z0), x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(s(mark(z0)), x3)), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), cons(x2, x3))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), cons(mark(x2), x3)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), cons(x2, x3))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), cons(mark(x2), x3)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), cons(x2, x3))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), cons(mark(x2), x3)), MARK(nats(z0))) MARK(filter(x0, zprimes, cons(x2, x3))) -> c11(A__FILTER(mark(x0), a__zprimes, cons(mark(x2), x3)), MARK(zprimes)) MARK(filter(x0, s(z0), cons(x2, x3))) -> c11(A__FILTER(mark(x0), s(mark(z0)), cons(mark(x2), x3)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, cons(x2, x3))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(sieve(z0), x1, cons(x2, x3))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(nats(z0), x1, cons(x2, x3))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(zprimes, x1, cons(x2, x3))) -> c11(A__FILTER(a__zprimes, mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(cons(z0, z1), x1, cons(x2, x3))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(x0, cons(z0, z1), cons(x2, x3))) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, cons(x2, x3))) -> c11(MARK(x1)) MARK(filter(s(z0), x1, cons(x2, x3))) -> c11(MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), 0)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), 0), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), 0)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), 0), MARK(sieve(z0))) MARK(filter(x0, nats(z0), 0)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), 0), MARK(nats(z0))) MARK(filter(x0, zprimes, 0)) -> c11(A__FILTER(mark(x0), a__zprimes, 0), MARK(zprimes)) MARK(filter(x0, s(z0), 0)) -> c11(A__FILTER(mark(x0), s(mark(z0)), 0), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, 0)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), 0), MARK(x1)) MARK(filter(sieve(z0), x1, 0)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), 0), MARK(x1)) MARK(filter(nats(z0), x1, 0)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), 0), MARK(x1)) MARK(filter(zprimes, x1, 0)) -> c11(A__FILTER(a__zprimes, mark(x1), 0), MARK(x1)) MARK(filter(cons(z0, z1), x1, 0)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), 0), MARK(x1)) MARK(filter(x0, cons(z0, z1), 0)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, 0)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, 0)) -> c11(MARK(x1)) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c14_2, c15_1, c16_1, c18_1, c8_1, c10_1, c11_2, c11_1, c12_2, c12_1, c13_2, c13_1, c8_2, c10_2 ---------------------------------------- (181) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(filter(x0, x1, s(z0))) -> c11(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(x1)) by MARK(filter(x0, x1, s(filter(z0, z1, z2)))) -> c11(A__FILTER(mark(x0), mark(x1), s(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x1)) MARK(filter(x0, x1, s(sieve(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), s(a__sieve(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, s(nats(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), s(a__nats(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, s(zprimes))) -> c11(A__FILTER(mark(x0), mark(x1), s(a__zprimes)), MARK(x1)) MARK(filter(x0, x1, s(cons(z0, z1)))) -> c11(A__FILTER(mark(x0), mark(x1), s(cons(mark(z0), z1))), MARK(x1)) MARK(filter(x0, x1, s(0))) -> c11(A__FILTER(mark(x0), mark(x1), s(0)), MARK(x1)) MARK(filter(x0, x1, s(s(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), s(s(mark(z0)))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), s(x2))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), s(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), s(x2))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), s(mark(x2))), MARK(sieve(z0))) MARK(filter(x0, nats(z0), s(x2))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), s(mark(x2))), MARK(nats(z0))) MARK(filter(x0, zprimes, s(x2))) -> c11(A__FILTER(mark(x0), a__zprimes, s(mark(x2))), MARK(zprimes)) MARK(filter(x0, cons(z0, z1), s(x2))) -> c11(A__FILTER(mark(x0), cons(mark(z0), z1), s(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, 0, s(x2))) -> c11(A__FILTER(mark(x0), 0, s(mark(x2))), MARK(0)) MARK(filter(x0, s(z0), s(x2))) -> c11(A__FILTER(mark(x0), s(mark(z0)), s(mark(x2))), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, s(x2))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), s(mark(x2))), MARK(x1)) MARK(filter(sieve(z0), x1, s(x2))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), s(mark(x2))), MARK(x1)) MARK(filter(nats(z0), x1, s(x2))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), s(mark(x2))), MARK(x1)) MARK(filter(zprimes, x1, s(x2))) -> c11(A__FILTER(a__zprimes, mark(x1), s(mark(x2))), MARK(x1)) MARK(filter(cons(z0, z1), x1, s(x2))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), s(mark(x2))), MARK(x1)) MARK(filter(0, x1, s(x2))) -> c11(A__FILTER(0, mark(x1), s(mark(x2))), MARK(x1)) MARK(filter(s(z0), x1, s(x2))) -> c11(A__FILTER(s(mark(z0)), mark(x1), s(mark(x2))), MARK(x1)) ---------------------------------------- (182) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> 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z2), x1, cons(x2, x3))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(sieve(z0), x1, cons(x2, x3))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(nats(z0), x1, cons(x2, x3))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(zprimes, x1, cons(x2, x3))) -> c11(A__FILTER(a__zprimes, mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(cons(z0, z1), x1, cons(x2, x3))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(x0, cons(z0, z1), cons(x2, x3))) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, cons(x2, x3))) -> c11(MARK(x1)) MARK(filter(s(z0), x1, cons(x2, x3))) -> c11(MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), 0)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), 0), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), 0)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), 0), MARK(sieve(z0))) MARK(filter(x0, nats(z0), 0)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), 0), MARK(nats(z0))) MARK(filter(x0, zprimes, 0)) -> c11(A__FILTER(mark(x0), a__zprimes, 0), MARK(zprimes)) MARK(filter(x0, s(z0), 0)) -> c11(A__FILTER(mark(x0), s(mark(z0)), 0), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, 0)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), 0), MARK(x1)) MARK(filter(sieve(z0), x1, 0)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), 0), MARK(x1)) MARK(filter(nats(z0), x1, 0)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), 0), MARK(x1)) MARK(filter(zprimes, x1, 0)) -> c11(A__FILTER(a__zprimes, mark(x1), 0), MARK(x1)) MARK(filter(cons(z0, z1), x1, 0)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), 0), MARK(x1)) MARK(filter(x0, cons(z0, z1), 0)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, 0)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, 0)) -> c11(MARK(x1)) MARK(filter(x0, x1, s(filter(z0, z1, z2)))) -> c11(A__FILTER(mark(x0), mark(x1), s(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x1)) MARK(filter(x0, x1, s(sieve(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), s(a__sieve(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, s(nats(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), s(a__nats(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, s(zprimes))) -> c11(A__FILTER(mark(x0), mark(x1), s(a__zprimes)), MARK(x1)) MARK(filter(x0, x1, s(cons(z0, z1)))) -> c11(A__FILTER(mark(x0), mark(x1), s(cons(mark(z0), z1))), MARK(x1)) MARK(filter(x0, x1, s(0))) -> c11(A__FILTER(mark(x0), mark(x1), s(0)), MARK(x1)) MARK(filter(x0, x1, s(s(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), s(s(mark(z0)))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), s(x2))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), s(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), s(x2))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), s(mark(x2))), MARK(sieve(z0))) MARK(filter(x0, nats(z0), s(x2))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), s(mark(x2))), MARK(nats(z0))) MARK(filter(x0, zprimes, s(x2))) -> c11(A__FILTER(mark(x0), a__zprimes, s(mark(x2))), MARK(zprimes)) MARK(filter(x0, cons(z0, z1), s(x2))) -> c11(A__FILTER(mark(x0), cons(mark(z0), z1), s(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, 0, s(x2))) -> c11(A__FILTER(mark(x0), 0, s(mark(x2))), MARK(0)) MARK(filter(x0, s(z0), s(x2))) -> c11(A__FILTER(mark(x0), s(mark(z0)), s(mark(x2))), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, s(x2))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), s(mark(x2))), MARK(x1)) MARK(filter(sieve(z0), x1, s(x2))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), s(mark(x2))), MARK(x1)) MARK(filter(nats(z0), x1, s(x2))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), s(mark(x2))), MARK(x1)) MARK(filter(zprimes, x1, s(x2))) -> c11(A__FILTER(a__zprimes, mark(x1), s(mark(x2))), MARK(x1)) MARK(filter(cons(z0, z1), x1, s(x2))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), s(mark(x2))), MARK(x1)) MARK(filter(0, x1, s(x2))) -> c11(A__FILTER(0, mark(x1), s(mark(x2))), MARK(x1)) MARK(filter(s(z0), x1, s(x2))) -> c11(A__FILTER(s(mark(z0)), mark(x1), s(mark(x2))), MARK(x1)) S tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, 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cons(x2, x3))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), cons(mark(x2), x3)), MARK(nats(z0))) MARK(filter(x0, zprimes, cons(x2, x3))) -> c11(A__FILTER(mark(x0), a__zprimes, cons(mark(x2), x3)), MARK(zprimes)) MARK(filter(x0, s(z0), cons(x2, x3))) -> c11(A__FILTER(mark(x0), s(mark(z0)), cons(mark(x2), x3)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, cons(x2, x3))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(sieve(z0), x1, cons(x2, x3))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(nats(z0), x1, cons(x2, x3))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(zprimes, x1, cons(x2, x3))) -> c11(A__FILTER(a__zprimes, mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(cons(z0, z1), x1, cons(x2, x3))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(x0, cons(z0, z1), cons(x2, x3))) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, cons(x2, x3))) -> c11(MARK(x1)) MARK(filter(s(z0), x1, cons(x2, x3))) -> c11(MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), 0)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), 0), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), 0)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), 0), MARK(sieve(z0))) MARK(filter(x0, nats(z0), 0)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), 0), MARK(nats(z0))) MARK(filter(x0, zprimes, 0)) -> c11(A__FILTER(mark(x0), a__zprimes, 0), MARK(zprimes)) MARK(filter(x0, s(z0), 0)) -> c11(A__FILTER(mark(x0), s(mark(z0)), 0), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, 0)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), 0), MARK(x1)) MARK(filter(sieve(z0), x1, 0)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), 0), MARK(x1)) MARK(filter(nats(z0), x1, 0)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), 0), MARK(x1)) MARK(filter(zprimes, x1, 0)) -> c11(A__FILTER(a__zprimes, mark(x1), 0), MARK(x1)) MARK(filter(cons(z0, z1), x1, 0)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), 0), MARK(x1)) MARK(filter(x0, cons(z0, z1), 0)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, 0)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, 0)) -> c11(MARK(x1)) MARK(filter(x0, x1, s(filter(z0, z1, z2)))) -> c11(A__FILTER(mark(x0), mark(x1), s(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x1)) MARK(filter(x0, x1, s(sieve(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), s(a__sieve(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, s(nats(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), s(a__nats(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, s(zprimes))) -> c11(A__FILTER(mark(x0), mark(x1), s(a__zprimes)), MARK(x1)) MARK(filter(x0, x1, s(cons(z0, z1)))) -> c11(A__FILTER(mark(x0), mark(x1), s(cons(mark(z0), z1))), MARK(x1)) MARK(filter(x0, x1, s(0))) -> c11(A__FILTER(mark(x0), mark(x1), s(0)), MARK(x1)) MARK(filter(x0, x1, s(s(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), s(s(mark(z0)))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), s(x2))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), s(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), s(x2))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), s(mark(x2))), MARK(sieve(z0))) MARK(filter(x0, nats(z0), s(x2))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), s(mark(x2))), MARK(nats(z0))) MARK(filter(x0, zprimes, s(x2))) -> c11(A__FILTER(mark(x0), a__zprimes, s(mark(x2))), MARK(zprimes)) MARK(filter(x0, cons(z0, z1), s(x2))) -> c11(A__FILTER(mark(x0), cons(mark(z0), z1), s(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, 0, s(x2))) -> c11(A__FILTER(mark(x0), 0, s(mark(x2))), MARK(0)) MARK(filter(x0, s(z0), s(x2))) -> c11(A__FILTER(mark(x0), s(mark(z0)), s(mark(x2))), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, s(x2))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), s(mark(x2))), MARK(x1)) MARK(filter(sieve(z0), x1, s(x2))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), s(mark(x2))), MARK(x1)) MARK(filter(nats(z0), x1, s(x2))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), s(mark(x2))), MARK(x1)) MARK(filter(zprimes, x1, s(x2))) -> c11(A__FILTER(a__zprimes, mark(x1), s(mark(x2))), MARK(x1)) MARK(filter(cons(z0, z1), x1, s(x2))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), s(mark(x2))), MARK(x1)) MARK(filter(0, x1, s(x2))) -> c11(A__FILTER(0, mark(x1), s(mark(x2))), MARK(x1)) MARK(filter(s(z0), x1, s(x2))) -> c11(A__FILTER(s(mark(z0)), mark(x1), s(mark(x2))), MARK(x1)) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c14_2, c15_1, c16_1, c18_1, c8_1, c10_1, c11_2, c11_1, c12_2, c12_1, c13_2, c13_1, c8_2, c10_2 ---------------------------------------- (183) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(filter(x0, 0, s(x2))) -> c11(A__FILTER(mark(x0), 0, s(mark(x2))), MARK(0)) ---------------------------------------- (184) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> 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x1, s(x2))) -> c11(A__FILTER(s(mark(z0)), mark(x1), s(mark(x2))), MARK(x1)) S tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(x0, 0, filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, sieve(x2))) -> c10(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x0)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, sieve(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, sieve(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, sieve(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(0)), MARK(x0)) MARK(filter(x0, x1, sieve(s(z0)))) -> c10(A__FILTER(mark(x0), 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MARK(filter(x0, x1, nats(filter(z0, z1, z2)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x1)) MARK(filter(x0, x1, nats(sieve(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__sieve(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, nats(nats(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__nats(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, nats(zprimes))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__zprimes)), MARK(x1)) MARK(filter(x0, x1, nats(cons(z0, z1)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(cons(mark(z0), z1))), MARK(x1)) MARK(filter(x0, x1, nats(0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(0)), MARK(x1)) MARK(filter(x0, x1, nats(s(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(s(mark(z0)))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), nats(x2))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), nats(x2))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), a__nats(mark(x2))), MARK(sieve(z0))) MARK(filter(x0, nats(z0), nats(x2))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), a__nats(mark(x2))), MARK(nats(z0))) MARK(filter(x0, zprimes, nats(x2))) -> c11(A__FILTER(mark(x0), a__zprimes, a__nats(mark(x2))), MARK(zprimes)) MARK(filter(x0, s(z0), nats(x2))) -> c11(A__FILTER(mark(x0), s(mark(z0)), a__nats(mark(x2))), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, nats(x2))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(sieve(z0), x1, nats(x2))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(nats(z0), x1, nats(x2))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(zprimes, x1, nats(x2))) -> c11(A__FILTER(a__zprimes, mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(cons(z0, z1), x1, nats(x2))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), 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z1))) MARK(filter(0, x1, 0)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, 0)) -> c11(MARK(x1)) MARK(filter(x0, x1, s(filter(z0, z1, z2)))) -> c11(A__FILTER(mark(x0), mark(x1), s(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x1)) MARK(filter(x0, x1, s(sieve(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), s(a__sieve(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, s(nats(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), s(a__nats(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, s(zprimes))) -> c11(A__FILTER(mark(x0), mark(x1), s(a__zprimes)), MARK(x1)) MARK(filter(x0, x1, s(cons(z0, z1)))) -> c11(A__FILTER(mark(x0), mark(x1), s(cons(mark(z0), z1))), MARK(x1)) MARK(filter(x0, x1, s(0))) -> c11(A__FILTER(mark(x0), mark(x1), s(0)), MARK(x1)) MARK(filter(x0, x1, s(s(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), s(s(mark(z0)))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), s(x2))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), s(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), s(x2))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), s(mark(x2))), MARK(sieve(z0))) MARK(filter(x0, nats(z0), s(x2))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), s(mark(x2))), MARK(nats(z0))) MARK(filter(x0, zprimes, s(x2))) -> c11(A__FILTER(mark(x0), a__zprimes, s(mark(x2))), MARK(zprimes)) MARK(filter(x0, cons(z0, z1), s(x2))) -> c11(A__FILTER(mark(x0), cons(mark(z0), z1), s(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, s(z0), s(x2))) -> c11(A__FILTER(mark(x0), s(mark(z0)), s(mark(x2))), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, s(x2))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), s(mark(x2))), MARK(x1)) MARK(filter(sieve(z0), x1, s(x2))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), s(mark(x2))), MARK(x1)) MARK(filter(nats(z0), x1, s(x2))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), s(mark(x2))), MARK(x1)) MARK(filter(zprimes, x1, s(x2))) -> c11(A__FILTER(a__zprimes, mark(x1), s(mark(x2))), MARK(x1)) MARK(filter(cons(z0, z1), x1, s(x2))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), s(mark(x2))), MARK(x1)) MARK(filter(0, x1, s(x2))) -> c11(A__FILTER(0, mark(x1), s(mark(x2))), MARK(x1)) MARK(filter(s(z0), x1, s(x2))) -> c11(A__FILTER(s(mark(z0)), mark(x1), s(mark(x2))), MARK(x1)) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c14_2, c15_1, c16_1, c18_1, c8_1, c10_1, c11_2, c11_1, c12_2, c12_1, c13_2, c13_1, c8_2, c10_2 ---------------------------------------- (185) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (186) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(x0, 0, filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, sieve(x2))) -> c10(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x0)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, sieve(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, sieve(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, sieve(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(0)), MARK(x0)) MARK(filter(x0, x1, sieve(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), sieve(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, sieve(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), sieve(x2))) -> 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c11(A__FILTER(mark(x0), mark(x1), s(cons(mark(z0), z1))), MARK(x1)) MARK(filter(x0, x1, s(0))) -> c11(A__FILTER(mark(x0), mark(x1), s(0)), MARK(x1)) MARK(filter(x0, x1, s(s(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), s(s(mark(z0)))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), s(x2))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), s(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), s(x2))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), s(mark(x2))), MARK(sieve(z0))) MARK(filter(x0, nats(z0), s(x2))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), s(mark(x2))), MARK(nats(z0))) MARK(filter(x0, zprimes, s(x2))) -> c11(A__FILTER(mark(x0), a__zprimes, s(mark(x2))), MARK(zprimes)) MARK(filter(x0, s(z0), s(x2))) -> c11(A__FILTER(mark(x0), s(mark(z0)), s(mark(x2))), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, s(x2))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), s(mark(x2))), MARK(x1)) MARK(filter(sieve(z0), x1, s(x2))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), s(mark(x2))), MARK(x1)) MARK(filter(nats(z0), x1, s(x2))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), s(mark(x2))), MARK(x1)) MARK(filter(zprimes, x1, s(x2))) -> c11(A__FILTER(a__zprimes, mark(x1), s(mark(x2))), MARK(x1)) MARK(filter(cons(z0, z1), x1, s(x2))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), s(mark(x2))), MARK(x1)) MARK(filter(x0, cons(z0, z1), s(x2))) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, s(x2))) -> c11(MARK(x1)) MARK(filter(s(z0), x1, s(x2))) -> c11(MARK(x1)) S tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(x0, 0, filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, sieve(x2))) -> c10(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x0)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, sieve(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, sieve(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, sieve(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(0)), MARK(x0)) MARK(filter(x0, x1, sieve(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), sieve(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, sieve(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, sieve(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__sieve(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, sieve(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, sieve(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, sieve(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), a__sieve(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, sieve(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__sieve(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), sieve(x2))) -> c10(MARK(x0)) MARK(filter(x0, 0, sieve(x2))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, sieve(x2))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(mark(x2)), nats(s(mark(x2))))), MARK(x0)) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), nats(mark(x2))), MARK(x0)) MARK(filter(x0, x1, nats(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, nats(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, nats(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, nats(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, nats(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, nats(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(0)), MARK(x0)) MARK(filter(x0, x1, nats(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), nats(x2))) -> 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c10(A__FILTER(cons(mark(z0), z1), a__zprimes, mark(x1)), MARK(cons(z0, z1))) MARK(filter(x0, zprimes, x1)) -> c10(MARK(x0)) MARK(filter(s(z0), zprimes, x1)) -> c10(MARK(s(z0))) MARK(filter(x0, s(filter(z0, z1, z2)), x2)) -> c10(A__FILTER(mark(x0), s(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(x0)) MARK(filter(x0, s(sieve(z0)), x2)) -> c10(A__FILTER(mark(x0), s(a__sieve(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, s(nats(z0)), x2)) -> c10(A__FILTER(mark(x0), s(a__nats(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(x0, s(zprimes), x2)) -> c10(A__FILTER(mark(x0), s(a__zprimes), mark(x2)), MARK(x0)) MARK(filter(x0, s(cons(z0, z1)), x2)) -> c10(A__FILTER(mark(x0), s(cons(mark(z0), z1)), mark(x2)), MARK(x0)) MARK(filter(x0, s(0), x2)) -> c10(A__FILTER(mark(x0), s(0), mark(x2)), MARK(x0)) MARK(filter(x0, s(s(z0)), x2)) -> c10(A__FILTER(mark(x0), s(s(mark(z0))), mark(x2)), MARK(x0)) MARK(filter(filter(z0, z1, z2), s(x1), x2)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), s(mark(x1)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), s(x1), x2)) -> c10(A__FILTER(a__sieve(mark(z0)), s(mark(x1)), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(z0), s(x1), x2)) -> c10(A__FILTER(a__nats(mark(z0)), s(mark(x1)), mark(x2)), MARK(nats(z0))) MARK(filter(zprimes, s(x1), x2)) -> c10(A__FILTER(a__zprimes, s(mark(x1)), mark(x2)), MARK(zprimes)) MARK(filter(cons(z0, z1), s(x1), x2)) -> c10(A__FILTER(cons(mark(z0), z1), s(mark(x1)), mark(x2)), MARK(cons(z0, z1))) MARK(filter(s(z0), s(x1), x2)) -> c10(MARK(s(z0))) MARK(filter(filter(x0, x1, filter(z0, z1, z2)), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), mark(x3), mark(x4)), MARK(filter(x0, x1, filter(z0, z1, z2)))) MARK(filter(filter(x0, x1, sieve(z0)), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), a__sieve(mark(z0))), mark(x3), mark(x4)), MARK(filter(x0, x1, sieve(z0)))) MARK(filter(filter(x0, x1, nats(z0)), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), a__nats(mark(z0))), mark(x3), mark(x4)), MARK(filter(x0, x1, nats(z0)))) MARK(filter(filter(x0, x1, zprimes), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), a__zprimes), mark(x3), mark(x4)), MARK(filter(x0, x1, zprimes))) MARK(filter(filter(x0, x1, cons(z0, z1)), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), cons(mark(z0), z1)), mark(x3), mark(x4)), MARK(filter(x0, x1, cons(z0, z1)))) MARK(filter(filter(x0, x1, 0), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), 0), mark(x3), mark(x4)), MARK(filter(x0, x1, 0))) MARK(filter(filter(x0, x1, s(z0)), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), mark(x1), s(mark(z0))), mark(x3), mark(x4)), MARK(filter(x0, x1, s(z0)))) MARK(filter(filter(x0, filter(z0, z1, z2), x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, filter(z0, z1, z2), x2))) MARK(filter(filter(x0, sieve(z0), x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), a__sieve(mark(z0)), mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, sieve(z0), x2))) MARK(filter(filter(x0, nats(z0), x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), a__nats(mark(z0)), mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, nats(z0), x2))) MARK(filter(filter(x0, zprimes, x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), a__zprimes, mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, zprimes, x2))) MARK(filter(filter(x0, cons(z0, z1), x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), cons(mark(z0), z1), mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, cons(z0, z1), x2))) MARK(filter(filter(x0, 0, x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), 0, mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, 0, x2))) MARK(filter(filter(x0, s(z0), x2), x3, x4)) -> c10(A__FILTER(a__filter(mark(x0), s(mark(z0)), mark(x2)), mark(x3), mark(x4)), MARK(filter(x0, s(z0), x2))) MARK(filter(filter(filter(z0, z1, z2), x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(filter(z0, z1, z2), x1, x2))) MARK(filter(filter(sieve(z0), x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(a__sieve(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(sieve(z0), x1, x2))) MARK(filter(filter(nats(z0), x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(a__nats(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(nats(z0), x1, x2))) MARK(filter(filter(zprimes, x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(a__zprimes, mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(zprimes, x1, x2))) MARK(filter(filter(cons(z0, z1), x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(cons(mark(z0), z1), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(cons(z0, z1), x1, x2))) MARK(filter(filter(0, x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(0, mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(0, x1, x2))) MARK(filter(filter(s(z0), x1, x2), x3, x4)) -> c10(A__FILTER(a__filter(s(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(filter(s(z0), x1, x2))) MARK(filter(filter(x0, x1, x2), cons(z0, z1), x4)) -> c10(MARK(filter(x0, x1, x2))) MARK(filter(filter(x0, x1, x2), 0, x4)) -> c10(MARK(filter(x0, x1, x2))) MARK(filter(filter(x0, x1, x2), x3, x4)) -> c10(MARK(filter(x0, x1, x2))) MARK(filter(sieve(filter(z0, z1, z2)), x1, x2)) -> c10(A__FILTER(a__sieve(a__filter(mark(z0), mark(z1), mark(z2))), mark(x1), mark(x2)), MARK(sieve(filter(z0, z1, z2)))) MARK(filter(sieve(sieve(z0)), x1, x2)) -> c10(A__FILTER(a__sieve(a__sieve(mark(z0))), mark(x1), mark(x2)), MARK(sieve(sieve(z0)))) MARK(filter(sieve(nats(z0)), x1, x2)) -> c10(A__FILTER(a__sieve(a__nats(mark(z0))), mark(x1), mark(x2)), MARK(sieve(nats(z0)))) MARK(filter(sieve(zprimes), x1, x2)) -> c10(A__FILTER(a__sieve(a__zprimes), mark(x1), mark(x2)), MARK(sieve(zprimes))) MARK(filter(sieve(cons(z0, z1)), x1, x2)) -> c10(A__FILTER(a__sieve(cons(mark(z0), z1)), mark(x1), mark(x2)), MARK(sieve(cons(z0, z1)))) MARK(filter(sieve(s(z0)), x1, x2)) -> c10(A__FILTER(a__sieve(s(mark(z0))), mark(x1), mark(x2)), MARK(sieve(s(z0)))) MARK(filter(sieve(x0), cons(z0, z1), x2)) -> c10(MARK(sieve(x0))) MARK(filter(sieve(x0), 0, x2)) -> c10(MARK(sieve(x0))) MARK(filter(sieve(x0), x1, x2)) -> c10(MARK(sieve(x0))) MARK(filter(sieve(0), x1, x2)) -> c10(A__FILTER(a__sieve(0), mark(x1), mark(x2))) MARK(filter(nats(x0), x1, x2)) -> c10(A__FILTER(cons(mark(mark(x0)), nats(s(mark(x0)))), mark(x1), mark(x2)), MARK(nats(x0))) MARK(filter(nats(filter(z0, z1, z2)), x1, x2)) -> c10(A__FILTER(a__nats(a__filter(mark(z0), mark(z1), mark(z2))), mark(x1), mark(x2)), MARK(nats(filter(z0, z1, z2)))) MARK(filter(nats(sieve(z0)), x1, x2)) -> c10(A__FILTER(a__nats(a__sieve(mark(z0))), mark(x1), mark(x2)), MARK(nats(sieve(z0)))) MARK(filter(nats(nats(z0)), x1, x2)) -> c10(A__FILTER(a__nats(a__nats(mark(z0))), mark(x1), mark(x2)), MARK(nats(nats(z0)))) MARK(filter(nats(zprimes), x1, x2)) -> c10(A__FILTER(a__nats(a__zprimes), mark(x1), mark(x2)), MARK(nats(zprimes))) MARK(filter(nats(cons(z0, z1)), x1, x2)) -> c10(A__FILTER(a__nats(cons(mark(z0), z1)), mark(x1), mark(x2)), MARK(nats(cons(z0, z1)))) MARK(filter(nats(0), x1, x2)) -> c10(A__FILTER(a__nats(0), mark(x1), mark(x2)), MARK(nats(0))) MARK(filter(nats(s(z0)), x1, x2)) -> c10(A__FILTER(a__nats(s(mark(z0))), mark(x1), mark(x2)), MARK(nats(s(z0)))) MARK(filter(nats(x0), x1, x2)) -> c10(MARK(nats(x0))) MARK(filter(nats(x0), cons(z0, z1), x2)) -> c10(MARK(nats(x0))) MARK(filter(nats(x0), 0, x2)) -> c10(MARK(nats(x0))) MARK(filter(zprimes, x0, x1)) -> c10(A__FILTER(a__sieve(a__nats(s(s(0)))), mark(x0), mark(x1)), MARK(zprimes)) MARK(filter(zprimes, cons(z0, z1), x1)) -> c10(MARK(zprimes)) MARK(filter(zprimes, 0, x1)) -> c10(MARK(zprimes)) MARK(filter(zprimes, x0, x1)) -> c10(MARK(zprimes)) MARK(filter(cons(filter(z0, z1, z2), x1), x2, x3)) -> c10(A__FILTER(cons(a__filter(mark(z0), mark(z1), mark(z2)), x1), mark(x2), mark(x3)), MARK(cons(filter(z0, z1, z2), x1))) MARK(filter(cons(sieve(z0), x1), x2, x3)) -> c10(A__FILTER(cons(a__sieve(mark(z0)), x1), mark(x2), mark(x3)), MARK(cons(sieve(z0), x1))) MARK(filter(cons(nats(z0), x1), x2, x3)) -> c10(A__FILTER(cons(a__nats(mark(z0)), x1), mark(x2), mark(x3)), MARK(cons(nats(z0), x1))) MARK(filter(cons(zprimes, x1), x2, x3)) -> c10(A__FILTER(cons(a__zprimes, x1), mark(x2), mark(x3)), MARK(cons(zprimes, x1))) MARK(filter(cons(cons(z0, z1), x1), x2, x3)) -> c10(A__FILTER(cons(cons(mark(z0), z1), x1), mark(x2), mark(x3)), MARK(cons(cons(z0, z1), x1))) MARK(filter(cons(0, x1), x2, x3)) -> c10(A__FILTER(cons(0, x1), mark(x2), mark(x3)), MARK(cons(0, x1))) MARK(filter(cons(s(z0), x1), x2, x3)) -> c10(A__FILTER(cons(s(mark(z0)), x1), mark(x2), mark(x3)), MARK(cons(s(z0), x1))) MARK(filter(cons(x0, x1), x2, x3)) -> c10(MARK(cons(x0, x1))) MARK(filter(cons(x0, x1), cons(z0, z1), x3)) -> c10(MARK(cons(x0, x1))) MARK(filter(cons(x0, x1), 0, x3)) -> c10(MARK(cons(x0, x1))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c11(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, filter(x2, x3, x4))) -> c11(MARK(x1)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c11(MARK(x1)) MARK(filter(x0, x1, sieve(x2))) -> c11(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x1)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x1)) MARK(filter(x0, x1, sieve(sieve(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__sieve(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, sieve(nats(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, sieve(zprimes))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__zprimes)), MARK(x1)) MARK(filter(x0, x1, sieve(cons(z0, z1)))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(cons(mark(z0), z1))), MARK(x1)) MARK(filter(x0, x1, sieve(0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(0)), MARK(x1)) MARK(filter(x0, x1, sieve(s(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(s(mark(z0)))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), sieve(x2))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), sieve(x2))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), a__sieve(mark(x2))), MARK(sieve(z0))) MARK(filter(x0, nats(z0), sieve(x2))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), a__sieve(mark(x2))), MARK(nats(z0))) MARK(filter(x0, zprimes, sieve(x2))) -> c11(A__FILTER(mark(x0), a__zprimes, a__sieve(mark(x2))), MARK(zprimes)) MARK(filter(x0, s(z0), sieve(x2))) -> c11(A__FILTER(mark(x0), s(mark(z0)), a__sieve(mark(x2))), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, sieve(x2))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(sieve(z0), x1, sieve(x2))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(nats(z0), x1, sieve(x2))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(zprimes, x1, sieve(x2))) -> c11(A__FILTER(a__zprimes, mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(cons(z0, z1), x1, sieve(x2))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(x0, cons(z0, z1), sieve(x2))) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, sieve(x2))) -> c11(MARK(x1)) MARK(filter(s(z0), x1, sieve(x2))) -> c11(MARK(x1)) MARK(filter(x0, x1, nats(x2))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(mark(x2)), nats(s(mark(x2))))), MARK(x1)) MARK(filter(x0, x1, nats(x2))) -> c11(A__FILTER(mark(x0), mark(x1), nats(mark(x2))), MARK(x1)) MARK(filter(x0, x1, nats(filter(z0, z1, z2)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x1)) MARK(filter(x0, x1, nats(sieve(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__sieve(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, nats(nats(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__nats(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, nats(zprimes))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__zprimes)), MARK(x1)) MARK(filter(x0, x1, nats(cons(z0, z1)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(cons(mark(z0), z1))), MARK(x1)) MARK(filter(x0, x1, nats(0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(0)), MARK(x1)) MARK(filter(x0, x1, nats(s(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(s(mark(z0)))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), nats(x2))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), nats(x2))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), a__nats(mark(x2))), MARK(sieve(z0))) MARK(filter(x0, nats(z0), nats(x2))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), a__nats(mark(x2))), MARK(nats(z0))) MARK(filter(x0, zprimes, nats(x2))) -> c11(A__FILTER(mark(x0), a__zprimes, a__nats(mark(x2))), MARK(zprimes)) MARK(filter(x0, s(z0), nats(x2))) -> c11(A__FILTER(mark(x0), s(mark(z0)), a__nats(mark(x2))), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, nats(x2))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(sieve(z0), x1, nats(x2))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(nats(z0), x1, nats(x2))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(zprimes, x1, nats(x2))) -> c11(A__FILTER(a__zprimes, mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(cons(z0, z1), x1, nats(x2))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(x0, cons(z0, z1), nats(x2))) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, nats(x2))) -> c11(MARK(x1)) MARK(filter(s(z0), x1, nats(x2))) -> c11(MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(s(s(0))))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), zprimes), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), zprimes)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__zprimes), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), zprimes)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), a__zprimes), MARK(sieve(z0))) MARK(filter(x0, nats(z0), zprimes)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), a__zprimes), MARK(nats(z0))) MARK(filter(x0, zprimes, zprimes)) -> c11(A__FILTER(mark(x0), a__zprimes, a__zprimes), MARK(zprimes)) MARK(filter(x0, s(z0), zprimes)) -> c11(A__FILTER(mark(x0), s(mark(z0)), a__zprimes), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, zprimes)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__zprimes), MARK(x1)) MARK(filter(sieve(z0), x1, zprimes)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__zprimes), MARK(x1)) MARK(filter(nats(z0), x1, zprimes)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__zprimes), MARK(x1)) MARK(filter(zprimes, x1, zprimes)) -> c11(A__FILTER(a__zprimes, mark(x1), a__zprimes), MARK(x1)) MARK(filter(cons(z0, z1), x1, zprimes)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, cons(z0, z1), zprimes)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, zprimes)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, zprimes)) -> c11(MARK(x1)) MARK(filter(x0, x1, cons(filter(z0, z1, z2), x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(a__filter(mark(z0), mark(z1), mark(z2)), x3)), MARK(x1)) MARK(filter(x0, x1, cons(sieve(z0), x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(a__sieve(mark(z0)), x3)), MARK(x1)) MARK(filter(x0, x1, cons(nats(z0), x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(a__nats(mark(z0)), x3)), MARK(x1)) MARK(filter(x0, x1, cons(zprimes, x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(a__zprimes, x3)), MARK(x1)) MARK(filter(x0, x1, cons(cons(z0, z1), x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(cons(mark(z0), z1), x3)), MARK(x1)) MARK(filter(x0, x1, cons(0, x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(0, x3)), MARK(x1)) MARK(filter(x0, x1, cons(s(z0), x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(s(mark(z0)), x3)), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), cons(x2, x3))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), cons(mark(x2), x3)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), cons(x2, x3))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), cons(mark(x2), x3)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), cons(x2, x3))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), cons(mark(x2), x3)), MARK(nats(z0))) MARK(filter(x0, zprimes, cons(x2, x3))) -> c11(A__FILTER(mark(x0), a__zprimes, cons(mark(x2), x3)), MARK(zprimes)) MARK(filter(x0, s(z0), cons(x2, x3))) -> c11(A__FILTER(mark(x0), s(mark(z0)), cons(mark(x2), x3)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, cons(x2, x3))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(sieve(z0), x1, cons(x2, x3))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(nats(z0), x1, cons(x2, x3))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(zprimes, x1, cons(x2, x3))) -> c11(A__FILTER(a__zprimes, mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(cons(z0, z1), x1, cons(x2, x3))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(x0, cons(z0, z1), cons(x2, x3))) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, cons(x2, x3))) -> c11(MARK(x1)) MARK(filter(s(z0), x1, cons(x2, x3))) -> c11(MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), 0)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), 0), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), 0)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), 0), MARK(sieve(z0))) MARK(filter(x0, nats(z0), 0)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), 0), MARK(nats(z0))) MARK(filter(x0, zprimes, 0)) -> c11(A__FILTER(mark(x0), a__zprimes, 0), MARK(zprimes)) MARK(filter(x0, s(z0), 0)) -> c11(A__FILTER(mark(x0), s(mark(z0)), 0), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, 0)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), 0), MARK(x1)) MARK(filter(sieve(z0), x1, 0)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), 0), MARK(x1)) MARK(filter(nats(z0), x1, 0)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), 0), MARK(x1)) MARK(filter(zprimes, x1, 0)) -> c11(A__FILTER(a__zprimes, mark(x1), 0), MARK(x1)) MARK(filter(cons(z0, z1), x1, 0)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), 0), MARK(x1)) MARK(filter(x0, cons(z0, z1), 0)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, 0)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, 0)) -> c11(MARK(x1)) MARK(filter(x0, x1, s(filter(z0, z1, z2)))) -> c11(A__FILTER(mark(x0), mark(x1), s(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x1)) MARK(filter(x0, x1, s(sieve(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), s(a__sieve(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, s(nats(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), s(a__nats(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, s(zprimes))) -> c11(A__FILTER(mark(x0), mark(x1), s(a__zprimes)), MARK(x1)) MARK(filter(x0, x1, s(cons(z0, z1)))) -> c11(A__FILTER(mark(x0), mark(x1), s(cons(mark(z0), z1))), MARK(x1)) MARK(filter(x0, x1, s(0))) -> c11(A__FILTER(mark(x0), mark(x1), s(0)), MARK(x1)) MARK(filter(x0, x1, s(s(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), s(s(mark(z0)))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), s(x2))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), s(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), s(x2))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), s(mark(x2))), MARK(sieve(z0))) MARK(filter(x0, nats(z0), s(x2))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), s(mark(x2))), MARK(nats(z0))) MARK(filter(x0, zprimes, s(x2))) -> c11(A__FILTER(mark(x0), a__zprimes, s(mark(x2))), MARK(zprimes)) MARK(filter(x0, s(z0), s(x2))) -> c11(A__FILTER(mark(x0), s(mark(z0)), s(mark(x2))), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, s(x2))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), s(mark(x2))), MARK(x1)) MARK(filter(sieve(z0), x1, s(x2))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), s(mark(x2))), MARK(x1)) MARK(filter(nats(z0), x1, s(x2))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), s(mark(x2))), MARK(x1)) MARK(filter(zprimes, x1, s(x2))) -> c11(A__FILTER(a__zprimes, mark(x1), s(mark(x2))), MARK(x1)) MARK(filter(cons(z0, z1), x1, s(x2))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), s(mark(x2))), MARK(x1)) MARK(filter(x0, cons(z0, z1), s(x2))) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, s(x2))) -> c11(MARK(x1)) MARK(filter(s(z0), x1, s(x2))) -> c11(MARK(x1)) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c14_2, c15_1, c16_1, c18_1, c8_1, c10_1, c11_2, c11_1, c12_2, c12_1, c13_2, c13_1, c8_2, c10_2 ---------------------------------------- (187) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(filter(x0, filter(z0, z1, z2), x2)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) by MARK(filter(x0, filter(x1, x2, x3), filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), mark(x3)), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(x1, x2, x3))) MARK(filter(x0, filter(x1, x2, x3), sieve(z0))) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), mark(x3)), a__sieve(mark(z0))), MARK(filter(x1, x2, x3))) MARK(filter(x0, filter(x1, x2, x3), nats(z0))) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), mark(x3)), a__nats(mark(z0))), MARK(filter(x1, x2, x3))) MARK(filter(x0, filter(x1, x2, x3), zprimes)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), mark(x3)), a__zprimes), MARK(filter(x1, x2, x3))) MARK(filter(x0, filter(x1, x2, x3), cons(z0, z1))) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), mark(x3)), cons(mark(z0), z1)), MARK(filter(x1, x2, x3))) MARK(filter(x0, filter(x1, x2, x3), 0)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), mark(x3)), 0), MARK(filter(x1, x2, x3))) MARK(filter(x0, filter(x1, x2, x3), s(z0))) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), mark(x3)), s(mark(z0))), MARK(filter(x1, x2, x3))) MARK(filter(x0, filter(x1, x2, x3), x4)) -> c11(A__FILTER(mark(x0), filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(x0, filter(x1, x2, filter(z0, z1, z2)), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__filter(mark(z0), mark(z1), mark(z2))), mark(x4)), MARK(filter(x1, x2, filter(z0, z1, z2)))) MARK(filter(x0, filter(x1, x2, sieve(z0)), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__sieve(mark(z0))), mark(x4)), MARK(filter(x1, x2, sieve(z0)))) MARK(filter(x0, filter(x1, x2, nats(z0)), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__nats(mark(z0))), mark(x4)), MARK(filter(x1, x2, nats(z0)))) MARK(filter(x0, filter(x1, x2, zprimes), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__zprimes), mark(x4)), MARK(filter(x1, x2, zprimes))) MARK(filter(x0, filter(x1, x2, cons(z0, z1)), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), cons(mark(z0), z1)), mark(x4)), MARK(filter(x1, x2, cons(z0, z1)))) MARK(filter(x0, filter(x1, x2, 0), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), 0), mark(x4)), MARK(filter(x1, x2, 0))) MARK(filter(x0, filter(x1, x2, s(z0)), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), s(mark(z0))), mark(x4)), MARK(filter(x1, x2, s(z0)))) MARK(filter(x0, filter(x1, filter(z0, z1, z2), x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), a__filter(mark(z0), mark(z1), mark(z2)), mark(x3)), mark(x4)), MARK(filter(x1, filter(z0, z1, z2), x3))) MARK(filter(x0, filter(x1, sieve(z0), x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), a__sieve(mark(z0)), mark(x3)), mark(x4)), MARK(filter(x1, sieve(z0), x3))) MARK(filter(x0, filter(x1, nats(z0), x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), a__nats(mark(z0)), mark(x3)), mark(x4)), MARK(filter(x1, nats(z0), x3))) MARK(filter(x0, filter(x1, zprimes, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), a__zprimes, mark(x3)), mark(x4)), MARK(filter(x1, zprimes, x3))) MARK(filter(x0, filter(x1, cons(z0, z1), x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), cons(mark(z0), z1), mark(x3)), mark(x4)), MARK(filter(x1, cons(z0, z1), x3))) MARK(filter(x0, filter(x1, 0, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), 0, mark(x3)), mark(x4)), MARK(filter(x1, 0, x3))) MARK(filter(x0, filter(x1, s(z0), x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), s(mark(z0)), mark(x3)), mark(x4)), MARK(filter(x1, s(z0), x3))) MARK(filter(x0, filter(filter(z0, z1, z2), x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x2), mark(x3)), mark(x4)), MARK(filter(filter(z0, z1, z2), x2, x3))) MARK(filter(x0, filter(sieve(z0), x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(a__sieve(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(filter(sieve(z0), x2, x3))) MARK(filter(x0, filter(nats(z0), x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(a__nats(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(filter(nats(z0), x2, x3))) MARK(filter(x0, filter(zprimes, x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(a__zprimes, mark(x2), mark(x3)), mark(x4)), MARK(filter(zprimes, x2, x3))) MARK(filter(x0, filter(cons(z0, z1), x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(cons(mark(z0), z1), mark(x2), mark(x3)), mark(x4)), MARK(filter(cons(z0, z1), x2, x3))) MARK(filter(x0, filter(0, x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(0, mark(x2), mark(x3)), mark(x4)), MARK(filter(0, x2, x3))) MARK(filter(x0, filter(s(z0), x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(s(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(filter(s(z0), x2, x3))) MARK(filter(filter(z0, z1, z2), filter(x1, x2, x3), x4)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(sieve(z0), filter(x1, x2, x3), x4)) -> c11(A__FILTER(a__sieve(mark(z0)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(nats(z0), filter(x1, x2, x3), x4)) -> c11(A__FILTER(a__nats(mark(z0)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(zprimes, filter(x1, x2, x3), x4)) -> c11(A__FILTER(a__zprimes, a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(cons(z0, z1), filter(x1, x2, x3), x4)) -> c11(A__FILTER(cons(mark(z0), z1), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(0, filter(x1, x2, x3), x4)) -> c11(A__FILTER(0, a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(s(z0), filter(x1, x2, x3), x4)) -> c11(A__FILTER(s(mark(z0)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) ---------------------------------------- (188) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(x0, 0, filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, sieve(x2))) -> c10(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x0)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> 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s(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x1)) MARK(filter(x0, x1, s(sieve(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), s(a__sieve(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, s(nats(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), s(a__nats(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, s(zprimes))) -> c11(A__FILTER(mark(x0), mark(x1), s(a__zprimes)), MARK(x1)) MARK(filter(x0, x1, s(cons(z0, z1)))) -> c11(A__FILTER(mark(x0), mark(x1), s(cons(mark(z0), z1))), MARK(x1)) MARK(filter(x0, x1, s(0))) -> c11(A__FILTER(mark(x0), mark(x1), s(0)), MARK(x1)) MARK(filter(x0, x1, s(s(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), s(s(mark(z0)))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), s(x2))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), s(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), s(x2))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), s(mark(x2))), MARK(sieve(z0))) MARK(filter(x0, nats(z0), s(x2))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), s(mark(x2))), MARK(nats(z0))) MARK(filter(x0, zprimes, s(x2))) -> c11(A__FILTER(mark(x0), a__zprimes, s(mark(x2))), MARK(zprimes)) MARK(filter(x0, s(z0), s(x2))) -> c11(A__FILTER(mark(x0), s(mark(z0)), s(mark(x2))), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, s(x2))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), s(mark(x2))), MARK(x1)) MARK(filter(sieve(z0), x1, s(x2))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), s(mark(x2))), MARK(x1)) MARK(filter(nats(z0), x1, s(x2))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), s(mark(x2))), MARK(x1)) MARK(filter(zprimes, x1, s(x2))) -> c11(A__FILTER(a__zprimes, mark(x1), s(mark(x2))), MARK(x1)) MARK(filter(cons(z0, z1), x1, s(x2))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), s(mark(x2))), MARK(x1)) MARK(filter(x0, cons(z0, z1), s(x2))) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, s(x2))) -> c11(MARK(x1)) MARK(filter(s(z0), x1, s(x2))) -> c11(MARK(x1)) MARK(filter(x0, filter(x1, x2, x3), x4)) -> c11(A__FILTER(mark(x0), filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(x0, filter(x1, x2, filter(z0, z1, z2)), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__filter(mark(z0), mark(z1), mark(z2))), mark(x4)), MARK(filter(x1, x2, filter(z0, z1, z2)))) MARK(filter(x0, filter(x1, x2, sieve(z0)), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__sieve(mark(z0))), mark(x4)), MARK(filter(x1, x2, sieve(z0)))) MARK(filter(x0, filter(x1, x2, nats(z0)), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__nats(mark(z0))), mark(x4)), MARK(filter(x1, x2, nats(z0)))) MARK(filter(x0, filter(x1, x2, zprimes), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__zprimes), mark(x4)), MARK(filter(x1, x2, zprimes))) MARK(filter(x0, filter(x1, x2, cons(z0, z1)), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), cons(mark(z0), z1)), mark(x4)), MARK(filter(x1, x2, cons(z0, z1)))) MARK(filter(x0, filter(x1, x2, 0), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), 0), mark(x4)), MARK(filter(x1, x2, 0))) MARK(filter(x0, filter(x1, x2, s(z0)), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), s(mark(z0))), mark(x4)), MARK(filter(x1, x2, s(z0)))) MARK(filter(x0, filter(x1, filter(z0, z1, z2), x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), a__filter(mark(z0), mark(z1), mark(z2)), mark(x3)), mark(x4)), MARK(filter(x1, filter(z0, z1, z2), x3))) MARK(filter(x0, filter(x1, sieve(z0), x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), a__sieve(mark(z0)), mark(x3)), mark(x4)), MARK(filter(x1, sieve(z0), x3))) MARK(filter(x0, filter(x1, nats(z0), x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), a__nats(mark(z0)), mark(x3)), mark(x4)), MARK(filter(x1, nats(z0), x3))) MARK(filter(x0, filter(x1, zprimes, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), a__zprimes, mark(x3)), mark(x4)), MARK(filter(x1, zprimes, x3))) MARK(filter(x0, filter(x1, cons(z0, z1), x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), cons(mark(z0), z1), mark(x3)), mark(x4)), MARK(filter(x1, cons(z0, z1), x3))) MARK(filter(x0, filter(x1, 0, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), 0, mark(x3)), mark(x4)), MARK(filter(x1, 0, x3))) MARK(filter(x0, filter(x1, s(z0), x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), s(mark(z0)), mark(x3)), mark(x4)), MARK(filter(x1, s(z0), x3))) MARK(filter(x0, filter(filter(z0, z1, z2), x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x2), mark(x3)), mark(x4)), MARK(filter(filter(z0, z1, z2), x2, x3))) MARK(filter(x0, filter(sieve(z0), x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(a__sieve(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(filter(sieve(z0), x2, x3))) MARK(filter(x0, filter(nats(z0), x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(a__nats(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(filter(nats(z0), x2, x3))) MARK(filter(x0, filter(zprimes, x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(a__zprimes, mark(x2), mark(x3)), mark(x4)), MARK(filter(zprimes, x2, x3))) MARK(filter(x0, filter(cons(z0, z1), x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(cons(mark(z0), z1), mark(x2), mark(x3)), mark(x4)), MARK(filter(cons(z0, z1), x2, x3))) MARK(filter(x0, filter(0, x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(0, mark(x2), mark(x3)), mark(x4)), MARK(filter(0, x2, x3))) MARK(filter(x0, filter(s(z0), x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(s(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(filter(s(z0), x2, x3))) MARK(filter(filter(z0, z1, z2), filter(x1, x2, x3), x4)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(sieve(z0), filter(x1, x2, x3), x4)) -> c11(A__FILTER(a__sieve(mark(z0)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(nats(z0), filter(x1, x2, x3), x4)) -> c11(A__FILTER(a__nats(mark(z0)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(zprimes, filter(x1, x2, x3), x4)) -> c11(A__FILTER(a__zprimes, a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(cons(z0, z1), filter(x1, x2, x3), x4)) -> c11(A__FILTER(cons(mark(z0), z1), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(0, filter(x1, x2, x3), x4)) -> c11(A__FILTER(0, a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(s(z0), filter(x1, x2, x3), x4)) -> c11(A__FILTER(s(mark(z0)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) S tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), 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mark(x1), a__zprimes), MARK(x1)) MARK(filter(cons(z0, z1), x1, zprimes)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, cons(z0, z1), zprimes)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, zprimes)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, zprimes)) -> c11(MARK(x1)) MARK(filter(x0, x1, cons(filter(z0, z1, z2), x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(a__filter(mark(z0), mark(z1), mark(z2)), x3)), MARK(x1)) MARK(filter(x0, x1, cons(sieve(z0), x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(a__sieve(mark(z0)), x3)), MARK(x1)) MARK(filter(x0, x1, cons(nats(z0), x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(a__nats(mark(z0)), x3)), MARK(x1)) MARK(filter(x0, x1, cons(zprimes, x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(a__zprimes, x3)), MARK(x1)) MARK(filter(x0, x1, cons(cons(z0, z1), x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(cons(mark(z0), z1), x3)), MARK(x1)) MARK(filter(x0, x1, cons(0, x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(0, x3)), MARK(x1)) MARK(filter(x0, x1, cons(s(z0), x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(s(mark(z0)), x3)), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), cons(x2, x3))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), cons(mark(x2), x3)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), cons(x2, x3))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), cons(mark(x2), x3)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), cons(x2, x3))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), cons(mark(x2), x3)), MARK(nats(z0))) MARK(filter(x0, zprimes, cons(x2, x3))) -> c11(A__FILTER(mark(x0), a__zprimes, cons(mark(x2), x3)), MARK(zprimes)) MARK(filter(x0, s(z0), cons(x2, x3))) -> c11(A__FILTER(mark(x0), s(mark(z0)), cons(mark(x2), x3)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, cons(x2, x3))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(sieve(z0), x1, cons(x2, x3))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(nats(z0), x1, cons(x2, x3))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(zprimes, x1, cons(x2, x3))) -> c11(A__FILTER(a__zprimes, mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(cons(z0, z1), x1, cons(x2, x3))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(x0, cons(z0, z1), cons(x2, x3))) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, cons(x2, x3))) -> c11(MARK(x1)) MARK(filter(s(z0), x1, cons(x2, x3))) -> c11(MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), 0)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), 0), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), 0)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), 0), MARK(sieve(z0))) MARK(filter(x0, nats(z0), 0)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), 0), MARK(nats(z0))) MARK(filter(x0, zprimes, 0)) -> c11(A__FILTER(mark(x0), a__zprimes, 0), MARK(zprimes)) MARK(filter(x0, s(z0), 0)) -> c11(A__FILTER(mark(x0), s(mark(z0)), 0), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, 0)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), 0), MARK(x1)) MARK(filter(sieve(z0), x1, 0)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), 0), MARK(x1)) MARK(filter(nats(z0), x1, 0)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), 0), MARK(x1)) MARK(filter(zprimes, x1, 0)) -> c11(A__FILTER(a__zprimes, mark(x1), 0), MARK(x1)) MARK(filter(cons(z0, z1), x1, 0)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), 0), MARK(x1)) MARK(filter(x0, cons(z0, z1), 0)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, 0)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, 0)) -> c11(MARK(x1)) MARK(filter(x0, x1, s(filter(z0, z1, z2)))) -> c11(A__FILTER(mark(x0), mark(x1), s(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x1)) MARK(filter(x0, x1, s(sieve(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), s(a__sieve(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, s(nats(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), s(a__nats(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, s(zprimes))) -> c11(A__FILTER(mark(x0), mark(x1), s(a__zprimes)), MARK(x1)) MARK(filter(x0, x1, s(cons(z0, z1)))) -> c11(A__FILTER(mark(x0), mark(x1), s(cons(mark(z0), z1))), MARK(x1)) MARK(filter(x0, x1, s(0))) -> c11(A__FILTER(mark(x0), mark(x1), s(0)), MARK(x1)) MARK(filter(x0, x1, s(s(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), s(s(mark(z0)))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), s(x2))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), s(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), s(x2))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), s(mark(x2))), MARK(sieve(z0))) MARK(filter(x0, nats(z0), s(x2))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), s(mark(x2))), MARK(nats(z0))) MARK(filter(x0, zprimes, s(x2))) -> c11(A__FILTER(mark(x0), a__zprimes, s(mark(x2))), MARK(zprimes)) MARK(filter(x0, s(z0), s(x2))) -> c11(A__FILTER(mark(x0), s(mark(z0)), s(mark(x2))), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, s(x2))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), s(mark(x2))), MARK(x1)) MARK(filter(sieve(z0), x1, s(x2))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), s(mark(x2))), MARK(x1)) MARK(filter(nats(z0), x1, s(x2))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), s(mark(x2))), MARK(x1)) MARK(filter(zprimes, x1, s(x2))) -> c11(A__FILTER(a__zprimes, mark(x1), s(mark(x2))), MARK(x1)) MARK(filter(cons(z0, z1), x1, s(x2))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), s(mark(x2))), MARK(x1)) MARK(filter(x0, cons(z0, z1), s(x2))) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, s(x2))) -> c11(MARK(x1)) MARK(filter(s(z0), x1, s(x2))) -> c11(MARK(x1)) MARK(filter(x0, filter(x1, x2, x3), x4)) -> c11(A__FILTER(mark(x0), filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(x0, filter(x1, x2, filter(z0, z1, z2)), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__filter(mark(z0), mark(z1), mark(z2))), mark(x4)), MARK(filter(x1, x2, filter(z0, z1, z2)))) MARK(filter(x0, filter(x1, x2, sieve(z0)), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__sieve(mark(z0))), mark(x4)), MARK(filter(x1, x2, sieve(z0)))) MARK(filter(x0, filter(x1, x2, nats(z0)), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__nats(mark(z0))), mark(x4)), MARK(filter(x1, x2, nats(z0)))) MARK(filter(x0, filter(x1, x2, zprimes), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__zprimes), mark(x4)), MARK(filter(x1, x2, zprimes))) MARK(filter(x0, filter(x1, x2, cons(z0, z1)), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), cons(mark(z0), z1)), mark(x4)), MARK(filter(x1, x2, cons(z0, z1)))) MARK(filter(x0, filter(x1, x2, 0), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), 0), mark(x4)), MARK(filter(x1, x2, 0))) MARK(filter(x0, filter(x1, x2, s(z0)), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), s(mark(z0))), mark(x4)), MARK(filter(x1, x2, s(z0)))) MARK(filter(x0, filter(x1, filter(z0, z1, z2), x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), a__filter(mark(z0), mark(z1), mark(z2)), mark(x3)), mark(x4)), MARK(filter(x1, filter(z0, z1, z2), x3))) MARK(filter(x0, filter(x1, sieve(z0), x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), a__sieve(mark(z0)), mark(x3)), mark(x4)), MARK(filter(x1, sieve(z0), x3))) MARK(filter(x0, filter(x1, nats(z0), x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), a__nats(mark(z0)), mark(x3)), mark(x4)), MARK(filter(x1, nats(z0), x3))) MARK(filter(x0, filter(x1, zprimes, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), a__zprimes, mark(x3)), mark(x4)), MARK(filter(x1, zprimes, x3))) MARK(filter(x0, filter(x1, cons(z0, z1), x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), cons(mark(z0), z1), mark(x3)), mark(x4)), MARK(filter(x1, cons(z0, z1), x3))) MARK(filter(x0, filter(x1, 0, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), 0, mark(x3)), mark(x4)), MARK(filter(x1, 0, x3))) MARK(filter(x0, filter(x1, s(z0), x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), s(mark(z0)), mark(x3)), mark(x4)), MARK(filter(x1, s(z0), x3))) MARK(filter(x0, filter(filter(z0, z1, z2), x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x2), mark(x3)), mark(x4)), MARK(filter(filter(z0, z1, z2), x2, x3))) MARK(filter(x0, filter(sieve(z0), x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(a__sieve(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(filter(sieve(z0), x2, x3))) MARK(filter(x0, filter(nats(z0), x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(a__nats(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(filter(nats(z0), x2, x3))) MARK(filter(x0, filter(zprimes, x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(a__zprimes, mark(x2), mark(x3)), mark(x4)), MARK(filter(zprimes, x2, x3))) MARK(filter(x0, filter(cons(z0, z1), x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(cons(mark(z0), z1), mark(x2), mark(x3)), mark(x4)), MARK(filter(cons(z0, z1), x2, x3))) MARK(filter(x0, filter(0, x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(0, mark(x2), mark(x3)), mark(x4)), MARK(filter(0, x2, x3))) MARK(filter(x0, filter(s(z0), x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(s(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(filter(s(z0), x2, x3))) MARK(filter(filter(z0, z1, z2), filter(x1, x2, x3), x4)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(sieve(z0), filter(x1, x2, x3), x4)) -> c11(A__FILTER(a__sieve(mark(z0)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(nats(z0), filter(x1, x2, x3), x4)) -> c11(A__FILTER(a__nats(mark(z0)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(zprimes, filter(x1, x2, x3), x4)) -> c11(A__FILTER(a__zprimes, a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(cons(z0, z1), filter(x1, x2, x3), x4)) -> c11(A__FILTER(cons(mark(z0), z1), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(0, filter(x1, x2, x3), x4)) -> c11(A__FILTER(0, a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(s(z0), filter(x1, x2, x3), x4)) -> c11(A__FILTER(s(mark(z0)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c14_2, c15_1, c16_1, c18_1, c8_1, c10_1, c11_2, c11_1, c12_2, c12_1, c13_2, c13_1, c8_2, c10_2 ---------------------------------------- (189) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (190) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> 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MARK(filter(cons(z0, z1), filter(x1, x2, x3), x4)) -> c11(A__FILTER(cons(mark(z0), z1), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(x0, filter(x1, x2, x3), x4)) -> c11(MARK(filter(x1, x2, x3))) MARK(filter(0, filter(x1, x2, x3), x4)) -> c11(MARK(filter(x1, x2, x3))) MARK(filter(s(z0), filter(x1, x2, x3), x4)) -> c11(MARK(filter(x1, x2, x3))) S tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(x0, 0, filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, sieve(x2))) -> c10(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x0)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, sieve(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, sieve(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, sieve(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(0)), MARK(x0)) MARK(filter(x0, x1, sieve(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), sieve(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, sieve(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, sieve(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__sieve(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, sieve(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, sieve(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, sieve(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), a__sieve(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, sieve(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__sieve(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), sieve(x2))) -> c10(MARK(x0)) MARK(filter(x0, 0, sieve(x2))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, sieve(x2))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(mark(x2)), nats(s(mark(x2))))), MARK(x0)) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), nats(mark(x2))), MARK(x0)) MARK(filter(x0, x1, nats(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), 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MARK(filter(zprimes, x1, s(x2))) -> c11(A__FILTER(a__zprimes, mark(x1), s(mark(x2))), MARK(x1)) MARK(filter(cons(z0, z1), x1, s(x2))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), s(mark(x2))), MARK(x1)) MARK(filter(x0, cons(z0, z1), s(x2))) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, s(x2))) -> c11(MARK(x1)) MARK(filter(s(z0), x1, s(x2))) -> c11(MARK(x1)) MARK(filter(x0, filter(x1, x2, filter(z0, z1, z2)), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__filter(mark(z0), mark(z1), mark(z2))), mark(x4)), MARK(filter(x1, x2, filter(z0, z1, z2)))) MARK(filter(x0, filter(x1, x2, sieve(z0)), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__sieve(mark(z0))), mark(x4)), MARK(filter(x1, x2, sieve(z0)))) MARK(filter(x0, filter(x1, x2, nats(z0)), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__nats(mark(z0))), mark(x4)), MARK(filter(x1, x2, nats(z0)))) MARK(filter(x0, filter(x1, x2, zprimes), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__zprimes), mark(x4)), MARK(filter(x1, x2, zprimes))) MARK(filter(x0, filter(x1, x2, cons(z0, z1)), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), cons(mark(z0), z1)), mark(x4)), MARK(filter(x1, x2, cons(z0, z1)))) MARK(filter(x0, filter(x1, x2, 0), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), 0), mark(x4)), MARK(filter(x1, x2, 0))) MARK(filter(x0, filter(x1, x2, s(z0)), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), s(mark(z0))), mark(x4)), MARK(filter(x1, x2, s(z0)))) MARK(filter(x0, filter(x1, filter(z0, z1, z2), x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), a__filter(mark(z0), mark(z1), mark(z2)), mark(x3)), mark(x4)), MARK(filter(x1, filter(z0, z1, z2), x3))) MARK(filter(x0, filter(x1, sieve(z0), x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), a__sieve(mark(z0)), mark(x3)), mark(x4)), MARK(filter(x1, sieve(z0), x3))) MARK(filter(x0, filter(x1, nats(z0), x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), a__nats(mark(z0)), mark(x3)), mark(x4)), MARK(filter(x1, nats(z0), x3))) MARK(filter(x0, filter(x1, zprimes, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), a__zprimes, mark(x3)), mark(x4)), MARK(filter(x1, zprimes, x3))) MARK(filter(x0, filter(x1, cons(z0, z1), x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), cons(mark(z0), z1), mark(x3)), mark(x4)), MARK(filter(x1, cons(z0, z1), x3))) MARK(filter(x0, filter(x1, 0, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), 0, mark(x3)), mark(x4)), MARK(filter(x1, 0, x3))) MARK(filter(x0, filter(x1, s(z0), x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), s(mark(z0)), mark(x3)), mark(x4)), MARK(filter(x1, s(z0), x3))) MARK(filter(x0, filter(filter(z0, z1, z2), x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x2), mark(x3)), mark(x4)), MARK(filter(filter(z0, z1, z2), x2, x3))) MARK(filter(x0, filter(sieve(z0), x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(a__sieve(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(filter(sieve(z0), x2, x3))) MARK(filter(x0, filter(nats(z0), x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(a__nats(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(filter(nats(z0), x2, x3))) MARK(filter(x0, filter(zprimes, x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(a__zprimes, mark(x2), mark(x3)), mark(x4)), MARK(filter(zprimes, x2, x3))) MARK(filter(x0, filter(cons(z0, z1), x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(cons(mark(z0), z1), mark(x2), mark(x3)), mark(x4)), MARK(filter(cons(z0, z1), x2, x3))) MARK(filter(x0, filter(0, x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(0, mark(x2), mark(x3)), mark(x4)), MARK(filter(0, x2, x3))) MARK(filter(x0, filter(s(z0), x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(s(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(filter(s(z0), x2, x3))) MARK(filter(filter(z0, z1, z2), filter(x1, x2, x3), x4)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(sieve(z0), filter(x1, x2, x3), x4)) -> c11(A__FILTER(a__sieve(mark(z0)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(nats(z0), filter(x1, x2, x3), x4)) -> c11(A__FILTER(a__nats(mark(z0)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(zprimes, filter(x1, x2, x3), x4)) -> c11(A__FILTER(a__zprimes, a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(cons(z0, z1), filter(x1, x2, x3), x4)) -> c11(A__FILTER(cons(mark(z0), z1), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(x0, filter(x1, x2, x3), x4)) -> c11(MARK(filter(x1, x2, x3))) MARK(filter(0, filter(x1, x2, x3), x4)) -> c11(MARK(filter(x1, x2, x3))) MARK(filter(s(z0), filter(x1, x2, x3), x4)) -> c11(MARK(filter(x1, x2, x3))) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c14_2, c15_1, c16_1, c18_1, c8_1, c10_1, c11_2, c11_1, c12_2, c12_1, c13_2, c13_1, c8_2, c10_2 ---------------------------------------- (191) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(filter(x0, sieve(z0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) by MARK(filter(x0, sieve(x1), filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), a__sieve(mark(x1)), a__filter(mark(z0), mark(z1), mark(z2))), MARK(sieve(x1))) MARK(filter(x0, sieve(x1), sieve(z0))) -> c11(A__FILTER(mark(x0), a__sieve(mark(x1)), a__sieve(mark(z0))), MARK(sieve(x1))) MARK(filter(x0, sieve(x1), nats(z0))) -> c11(A__FILTER(mark(x0), a__sieve(mark(x1)), a__nats(mark(z0))), MARK(sieve(x1))) MARK(filter(x0, sieve(x1), zprimes)) -> c11(A__FILTER(mark(x0), a__sieve(mark(x1)), a__zprimes), MARK(sieve(x1))) MARK(filter(x0, sieve(x1), cons(z0, z1))) -> c11(A__FILTER(mark(x0), a__sieve(mark(x1)), cons(mark(z0), z1)), MARK(sieve(x1))) MARK(filter(x0, sieve(x1), 0)) -> c11(A__FILTER(mark(x0), a__sieve(mark(x1)), 0), MARK(sieve(x1))) MARK(filter(x0, sieve(x1), s(z0))) -> c11(A__FILTER(mark(x0), a__sieve(mark(x1)), s(mark(z0))), MARK(sieve(x1))) MARK(filter(x0, sieve(x1), x2)) -> c11(A__FILTER(mark(x0), sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(x0, sieve(filter(z0, z1, z2)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(sieve(filter(z0, z1, z2)))) MARK(filter(x0, sieve(sieve(z0)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(a__sieve(mark(z0))), mark(x2)), MARK(sieve(sieve(z0)))) MARK(filter(x0, sieve(nats(z0)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(a__nats(mark(z0))), mark(x2)), MARK(sieve(nats(z0)))) MARK(filter(x0, sieve(zprimes), x2)) -> c11(A__FILTER(mark(x0), a__sieve(a__zprimes), mark(x2)), MARK(sieve(zprimes))) MARK(filter(x0, sieve(cons(z0, z1)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(cons(mark(z0), z1)), mark(x2)), MARK(sieve(cons(z0, z1)))) MARK(filter(x0, sieve(0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(0), mark(x2)), MARK(sieve(0))) MARK(filter(x0, sieve(s(z0)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(s(mark(z0))), mark(x2)), MARK(sieve(s(z0)))) MARK(filter(filter(z0, z1, z2), sieve(x1), x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(sieve(z0), sieve(x1), x2)) -> c11(A__FILTER(a__sieve(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(nats(z0), sieve(x1), x2)) -> c11(A__FILTER(a__nats(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(zprimes, sieve(x1), x2)) -> c11(A__FILTER(a__zprimes, a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(cons(z0, z1), sieve(x1), x2)) -> c11(A__FILTER(cons(mark(z0), z1), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(0, sieve(x1), x2)) -> c11(A__FILTER(0, a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(s(z0), sieve(x1), x2)) -> c11(A__FILTER(s(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) ---------------------------------------- (192) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, 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MARK(filter(x0, x1, cons(cons(z0, z1), x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(cons(mark(z0), z1), x3)), MARK(x1)) MARK(filter(x0, x1, cons(0, x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(0, x3)), MARK(x1)) MARK(filter(x0, x1, cons(s(z0), x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(s(mark(z0)), x3)), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), cons(x2, x3))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), cons(mark(x2), x3)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), cons(x2, x3))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), cons(mark(x2), x3)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), cons(x2, x3))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), cons(mark(x2), x3)), MARK(nats(z0))) MARK(filter(x0, zprimes, cons(x2, x3))) -> c11(A__FILTER(mark(x0), a__zprimes, cons(mark(x2), x3)), MARK(zprimes)) MARK(filter(x0, s(z0), cons(x2, x3))) -> c11(A__FILTER(mark(x0), s(mark(z0)), cons(mark(x2), x3)), MARK(s(z0))) MARK(filter(filter(z0, z1, 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MARK(sieve(z0))) MARK(filter(x0, nats(z0), 0)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), 0), MARK(nats(z0))) MARK(filter(x0, zprimes, 0)) -> c11(A__FILTER(mark(x0), a__zprimes, 0), MARK(zprimes)) MARK(filter(x0, s(z0), 0)) -> c11(A__FILTER(mark(x0), s(mark(z0)), 0), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, 0)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), 0), MARK(x1)) MARK(filter(sieve(z0), x1, 0)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), 0), MARK(x1)) MARK(filter(nats(z0), x1, 0)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), 0), MARK(x1)) MARK(filter(zprimes, x1, 0)) -> c11(A__FILTER(a__zprimes, mark(x1), 0), MARK(x1)) MARK(filter(cons(z0, z1), x1, 0)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), 0), MARK(x1)) MARK(filter(x0, cons(z0, z1), 0)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, 0)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, 0)) -> c11(MARK(x1)) MARK(filter(x0, x1, s(filter(z0, z1, z2)))) -> c11(A__FILTER(mark(x0), mark(x1), s(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x1)) MARK(filter(x0, x1, s(sieve(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), s(a__sieve(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, s(nats(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), s(a__nats(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, s(zprimes))) -> c11(A__FILTER(mark(x0), mark(x1), s(a__zprimes)), MARK(x1)) MARK(filter(x0, x1, s(cons(z0, z1)))) -> c11(A__FILTER(mark(x0), mark(x1), s(cons(mark(z0), z1))), MARK(x1)) MARK(filter(x0, x1, s(0))) -> c11(A__FILTER(mark(x0), mark(x1), s(0)), MARK(x1)) MARK(filter(x0, x1, s(s(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), s(s(mark(z0)))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), s(x2))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), s(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), s(x2))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), s(mark(x2))), MARK(sieve(z0))) MARK(filter(x0, nats(z0), s(x2))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), s(mark(x2))), MARK(nats(z0))) MARK(filter(x0, zprimes, s(x2))) -> c11(A__FILTER(mark(x0), a__zprimes, s(mark(x2))), MARK(zprimes)) MARK(filter(x0, s(z0), s(x2))) -> c11(A__FILTER(mark(x0), s(mark(z0)), s(mark(x2))), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, s(x2))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), s(mark(x2))), MARK(x1)) MARK(filter(sieve(z0), x1, s(x2))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), s(mark(x2))), MARK(x1)) MARK(filter(nats(z0), x1, s(x2))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), s(mark(x2))), MARK(x1)) MARK(filter(zprimes, x1, s(x2))) -> c11(A__FILTER(a__zprimes, mark(x1), s(mark(x2))), MARK(x1)) MARK(filter(cons(z0, z1), x1, s(x2))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), s(mark(x2))), MARK(x1)) MARK(filter(x0, cons(z0, z1), s(x2))) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, s(x2))) -> c11(MARK(x1)) MARK(filter(s(z0), x1, s(x2))) -> c11(MARK(x1)) MARK(filter(x0, filter(x1, x2, filter(z0, z1, z2)), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__filter(mark(z0), mark(z1), mark(z2))), mark(x4)), MARK(filter(x1, x2, filter(z0, z1, z2)))) MARK(filter(x0, filter(x1, x2, sieve(z0)), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__sieve(mark(z0))), mark(x4)), MARK(filter(x1, x2, sieve(z0)))) MARK(filter(x0, filter(x1, x2, nats(z0)), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__nats(mark(z0))), mark(x4)), MARK(filter(x1, x2, nats(z0)))) MARK(filter(x0, filter(x1, x2, zprimes), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__zprimes), mark(x4)), MARK(filter(x1, x2, zprimes))) MARK(filter(x0, filter(x1, x2, cons(z0, z1)), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), cons(mark(z0), z1)), mark(x4)), MARK(filter(x1, x2, cons(z0, z1)))) MARK(filter(x0, filter(x1, x2, 0), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), 0), mark(x4)), MARK(filter(x1, x2, 0))) MARK(filter(x0, filter(x1, x2, s(z0)), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), s(mark(z0))), mark(x4)), MARK(filter(x1, x2, s(z0)))) MARK(filter(x0, filter(x1, filter(z0, z1, z2), x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), a__filter(mark(z0), mark(z1), mark(z2)), mark(x3)), mark(x4)), MARK(filter(x1, filter(z0, z1, z2), x3))) MARK(filter(x0, filter(x1, sieve(z0), x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), a__sieve(mark(z0)), mark(x3)), mark(x4)), MARK(filter(x1, sieve(z0), x3))) MARK(filter(x0, filter(x1, nats(z0), x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), a__nats(mark(z0)), mark(x3)), mark(x4)), MARK(filter(x1, nats(z0), x3))) MARK(filter(x0, filter(x1, zprimes, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), a__zprimes, mark(x3)), mark(x4)), MARK(filter(x1, zprimes, x3))) MARK(filter(x0, filter(x1, cons(z0, z1), x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), cons(mark(z0), z1), mark(x3)), mark(x4)), MARK(filter(x1, cons(z0, z1), x3))) MARK(filter(x0, filter(x1, 0, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), 0, mark(x3)), mark(x4)), MARK(filter(x1, 0, x3))) MARK(filter(x0, filter(x1, s(z0), x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), s(mark(z0)), mark(x3)), mark(x4)), MARK(filter(x1, s(z0), x3))) MARK(filter(x0, filter(filter(z0, z1, z2), x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x2), mark(x3)), mark(x4)), MARK(filter(filter(z0, z1, z2), x2, x3))) MARK(filter(x0, filter(sieve(z0), x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(a__sieve(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(filter(sieve(z0), x2, x3))) MARK(filter(x0, filter(nats(z0), x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(a__nats(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(filter(nats(z0), x2, x3))) MARK(filter(x0, filter(zprimes, x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(a__zprimes, mark(x2), mark(x3)), mark(x4)), MARK(filter(zprimes, x2, x3))) MARK(filter(x0, filter(cons(z0, z1), x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(cons(mark(z0), z1), mark(x2), mark(x3)), mark(x4)), MARK(filter(cons(z0, z1), x2, x3))) MARK(filter(x0, filter(0, x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(0, mark(x2), mark(x3)), mark(x4)), MARK(filter(0, x2, x3))) MARK(filter(x0, filter(s(z0), x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(s(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(filter(s(z0), x2, x3))) MARK(filter(filter(z0, z1, z2), filter(x1, x2, x3), x4)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(sieve(z0), filter(x1, x2, x3), x4)) -> c11(A__FILTER(a__sieve(mark(z0)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(nats(z0), filter(x1, x2, x3), x4)) -> c11(A__FILTER(a__nats(mark(z0)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(zprimes, filter(x1, x2, x3), x4)) -> c11(A__FILTER(a__zprimes, a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(cons(z0, z1), filter(x1, x2, x3), x4)) -> c11(A__FILTER(cons(mark(z0), z1), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(x0, filter(x1, x2, x3), x4)) -> c11(MARK(filter(x1, x2, x3))) MARK(filter(0, filter(x1, x2, x3), x4)) -> c11(MARK(filter(x1, x2, x3))) MARK(filter(s(z0), filter(x1, x2, x3), x4)) -> c11(MARK(filter(x1, x2, x3))) MARK(filter(x0, sieve(x1), x2)) -> c11(A__FILTER(mark(x0), sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(x0, sieve(filter(z0, z1, z2)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(sieve(filter(z0, z1, z2)))) MARK(filter(x0, sieve(sieve(z0)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(a__sieve(mark(z0))), mark(x2)), MARK(sieve(sieve(z0)))) MARK(filter(x0, sieve(nats(z0)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(a__nats(mark(z0))), mark(x2)), MARK(sieve(nats(z0)))) MARK(filter(x0, sieve(zprimes), x2)) -> c11(A__FILTER(mark(x0), a__sieve(a__zprimes), mark(x2)), MARK(sieve(zprimes))) MARK(filter(x0, sieve(cons(z0, z1)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(cons(mark(z0), z1)), mark(x2)), MARK(sieve(cons(z0, z1)))) MARK(filter(x0, sieve(0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(0), mark(x2)), MARK(sieve(0))) MARK(filter(x0, sieve(s(z0)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(s(mark(z0))), mark(x2)), MARK(sieve(s(z0)))) MARK(filter(filter(z0, z1, z2), sieve(x1), x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(sieve(z0), sieve(x1), x2)) -> c11(A__FILTER(a__sieve(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(nats(z0), sieve(x1), x2)) -> c11(A__FILTER(a__nats(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(zprimes, sieve(x1), x2)) -> c11(A__FILTER(a__zprimes, a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(cons(z0, z1), sieve(x1), x2)) -> c11(A__FILTER(cons(mark(z0), z1), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(0, sieve(x1), x2)) -> c11(A__FILTER(0, a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(s(z0), sieve(x1), x2)) -> c11(A__FILTER(s(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) S tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, 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MARK(filter(sieve(z0), x2, x3))) MARK(filter(x0, filter(nats(z0), x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(a__nats(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(filter(nats(z0), x2, x3))) MARK(filter(x0, filter(zprimes, x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(a__zprimes, mark(x2), mark(x3)), mark(x4)), MARK(filter(zprimes, x2, x3))) MARK(filter(x0, filter(cons(z0, z1), x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(cons(mark(z0), z1), mark(x2), mark(x3)), mark(x4)), MARK(filter(cons(z0, z1), x2, x3))) MARK(filter(x0, filter(0, x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(0, mark(x2), mark(x3)), mark(x4)), MARK(filter(0, x2, x3))) MARK(filter(x0, filter(s(z0), x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(s(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(filter(s(z0), x2, x3))) MARK(filter(filter(z0, z1, z2), filter(x1, x2, x3), x4)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(sieve(z0), filter(x1, x2, x3), x4)) -> c11(A__FILTER(a__sieve(mark(z0)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(nats(z0), filter(x1, x2, x3), x4)) -> c11(A__FILTER(a__nats(mark(z0)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(zprimes, filter(x1, x2, x3), x4)) -> c11(A__FILTER(a__zprimes, a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(cons(z0, z1), filter(x1, x2, x3), x4)) -> c11(A__FILTER(cons(mark(z0), z1), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(x0, filter(x1, x2, x3), x4)) -> c11(MARK(filter(x1, x2, x3))) MARK(filter(0, filter(x1, x2, x3), x4)) -> c11(MARK(filter(x1, x2, x3))) MARK(filter(s(z0), filter(x1, x2, x3), x4)) -> c11(MARK(filter(x1, x2, x3))) MARK(filter(x0, sieve(x1), x2)) -> c11(A__FILTER(mark(x0), sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(x0, sieve(filter(z0, z1, z2)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(sieve(filter(z0, z1, z2)))) MARK(filter(x0, sieve(sieve(z0)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(a__sieve(mark(z0))), mark(x2)), MARK(sieve(sieve(z0)))) MARK(filter(x0, sieve(nats(z0)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(a__nats(mark(z0))), mark(x2)), MARK(sieve(nats(z0)))) MARK(filter(x0, sieve(zprimes), x2)) -> c11(A__FILTER(mark(x0), a__sieve(a__zprimes), mark(x2)), MARK(sieve(zprimes))) MARK(filter(x0, sieve(cons(z0, z1)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(cons(mark(z0), z1)), mark(x2)), MARK(sieve(cons(z0, z1)))) MARK(filter(x0, sieve(0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(0), mark(x2)), MARK(sieve(0))) MARK(filter(x0, sieve(s(z0)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(s(mark(z0))), mark(x2)), MARK(sieve(s(z0)))) MARK(filter(filter(z0, z1, z2), sieve(x1), x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(sieve(z0), sieve(x1), x2)) -> c11(A__FILTER(a__sieve(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(nats(z0), sieve(x1), x2)) -> c11(A__FILTER(a__nats(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(zprimes, sieve(x1), x2)) -> c11(A__FILTER(a__zprimes, a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(cons(z0, z1), sieve(x1), x2)) -> c11(A__FILTER(cons(mark(z0), z1), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(0, sieve(x1), x2)) -> c11(A__FILTER(0, a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(s(z0), sieve(x1), x2)) -> c11(A__FILTER(s(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c14_2, c15_1, c16_1, c18_1, c8_1, c10_1, c11_2, c11_1, c12_2, c12_1, c13_2, c13_1, c8_2, c10_2 ---------------------------------------- (193) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing tuple parts ---------------------------------------- (194) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(x0, 0, filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, sieve(x2))) -> c10(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x0)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> 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a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, sieve(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, sieve(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__sieve(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, sieve(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, sieve(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, sieve(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), a__sieve(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, sieve(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__sieve(mark(x2))), 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s(mark(x2))), MARK(nats(z0))) MARK(filter(x0, zprimes, s(x2))) -> c11(A__FILTER(mark(x0), a__zprimes, s(mark(x2))), MARK(zprimes)) MARK(filter(x0, s(z0), s(x2))) -> c11(A__FILTER(mark(x0), s(mark(z0)), s(mark(x2))), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, s(x2))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), s(mark(x2))), MARK(x1)) MARK(filter(sieve(z0), x1, s(x2))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), s(mark(x2))), MARK(x1)) MARK(filter(nats(z0), x1, s(x2))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), s(mark(x2))), MARK(x1)) MARK(filter(zprimes, x1, s(x2))) -> c11(A__FILTER(a__zprimes, mark(x1), s(mark(x2))), MARK(x1)) MARK(filter(cons(z0, z1), x1, s(x2))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), s(mark(x2))), MARK(x1)) MARK(filter(x0, cons(z0, z1), s(x2))) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, s(x2))) -> c11(MARK(x1)) MARK(filter(s(z0), x1, s(x2))) -> c11(MARK(x1)) MARK(filter(x0, filter(x1, x2, filter(z0, z1, z2)), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__filter(mark(z0), mark(z1), mark(z2))), mark(x4)), MARK(filter(x1, x2, filter(z0, z1, z2)))) MARK(filter(x0, filter(x1, x2, sieve(z0)), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__sieve(mark(z0))), mark(x4)), MARK(filter(x1, x2, sieve(z0)))) MARK(filter(x0, filter(x1, x2, nats(z0)), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__nats(mark(z0))), mark(x4)), MARK(filter(x1, x2, nats(z0)))) MARK(filter(x0, filter(x1, x2, zprimes), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__zprimes), mark(x4)), MARK(filter(x1, x2, zprimes))) MARK(filter(x0, filter(x1, x2, cons(z0, z1)), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), cons(mark(z0), z1)), mark(x4)), MARK(filter(x1, x2, cons(z0, z1)))) MARK(filter(x0, filter(x1, x2, 0), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), 0), mark(x4)), MARK(filter(x1, x2, 0))) MARK(filter(x0, filter(x1, x2, s(z0)), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), s(mark(z0))), mark(x4)), MARK(filter(x1, x2, s(z0)))) MARK(filter(x0, filter(x1, filter(z0, z1, z2), x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), a__filter(mark(z0), mark(z1), mark(z2)), mark(x3)), mark(x4)), MARK(filter(x1, filter(z0, z1, z2), x3))) MARK(filter(x0, filter(x1, sieve(z0), x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), a__sieve(mark(z0)), mark(x3)), mark(x4)), MARK(filter(x1, sieve(z0), x3))) MARK(filter(x0, filter(x1, nats(z0), x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), a__nats(mark(z0)), mark(x3)), mark(x4)), MARK(filter(x1, nats(z0), x3))) MARK(filter(x0, filter(x1, zprimes, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), a__zprimes, mark(x3)), mark(x4)), MARK(filter(x1, zprimes, x3))) MARK(filter(x0, filter(x1, cons(z0, z1), x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), cons(mark(z0), z1), mark(x3)), mark(x4)), MARK(filter(x1, cons(z0, z1), x3))) MARK(filter(x0, filter(x1, 0, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), 0, mark(x3)), mark(x4)), MARK(filter(x1, 0, x3))) MARK(filter(x0, filter(x1, s(z0), x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), s(mark(z0)), mark(x3)), mark(x4)), MARK(filter(x1, s(z0), x3))) MARK(filter(x0, filter(filter(z0, z1, z2), x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x2), mark(x3)), mark(x4)), MARK(filter(filter(z0, z1, z2), x2, x3))) MARK(filter(x0, filter(sieve(z0), x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(a__sieve(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(filter(sieve(z0), x2, x3))) MARK(filter(x0, filter(nats(z0), x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(a__nats(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(filter(nats(z0), x2, x3))) MARK(filter(x0, filter(zprimes, x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(a__zprimes, mark(x2), mark(x3)), mark(x4)), MARK(filter(zprimes, x2, x3))) MARK(filter(x0, filter(cons(z0, z1), x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(cons(mark(z0), z1), mark(x2), mark(x3)), mark(x4)), MARK(filter(cons(z0, z1), x2, x3))) MARK(filter(x0, filter(0, x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(0, mark(x2), mark(x3)), mark(x4)), MARK(filter(0, x2, x3))) MARK(filter(x0, filter(s(z0), x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(s(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(filter(s(z0), x2, x3))) MARK(filter(filter(z0, z1, z2), filter(x1, x2, x3), x4)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(sieve(z0), filter(x1, x2, x3), x4)) -> c11(A__FILTER(a__sieve(mark(z0)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(nats(z0), filter(x1, x2, x3), x4)) -> c11(A__FILTER(a__nats(mark(z0)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(zprimes, filter(x1, x2, x3), x4)) -> c11(A__FILTER(a__zprimes, a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(cons(z0, z1), filter(x1, x2, x3), x4)) -> c11(A__FILTER(cons(mark(z0), z1), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(x0, filter(x1, x2, x3), x4)) -> c11(MARK(filter(x1, x2, x3))) MARK(filter(0, filter(x1, x2, x3), x4)) -> c11(MARK(filter(x1, x2, x3))) MARK(filter(s(z0), filter(x1, x2, x3), x4)) -> c11(MARK(filter(x1, x2, x3))) MARK(filter(x0, sieve(filter(z0, z1, z2)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(sieve(filter(z0, z1, z2)))) MARK(filter(x0, sieve(sieve(z0)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(a__sieve(mark(z0))), mark(x2)), MARK(sieve(sieve(z0)))) MARK(filter(x0, sieve(nats(z0)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(a__nats(mark(z0))), mark(x2)), MARK(sieve(nats(z0)))) MARK(filter(x0, sieve(zprimes), x2)) -> c11(A__FILTER(mark(x0), a__sieve(a__zprimes), mark(x2)), MARK(sieve(zprimes))) MARK(filter(x0, sieve(cons(z0, z1)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(cons(mark(z0), z1)), mark(x2)), MARK(sieve(cons(z0, z1)))) MARK(filter(x0, sieve(s(z0)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(s(mark(z0))), mark(x2)), MARK(sieve(s(z0)))) MARK(filter(filter(z0, z1, z2), sieve(x1), x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(sieve(z0), sieve(x1), x2)) -> c11(A__FILTER(a__sieve(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(nats(z0), sieve(x1), x2)) -> c11(A__FILTER(a__nats(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(zprimes, sieve(x1), x2)) -> c11(A__FILTER(a__zprimes, a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(cons(z0, z1), sieve(x1), x2)) -> c11(A__FILTER(cons(mark(z0), z1), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(x0, sieve(x1), x2)) -> c11(MARK(sieve(x1))) MARK(filter(x0, sieve(0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(0), mark(x2))) MARK(filter(0, sieve(x1), x2)) -> c11(MARK(sieve(x1))) MARK(filter(s(z0), sieve(x1), x2)) -> c11(MARK(sieve(x1))) S tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 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c11(A__FILTER(mark(x0), mark(x1), cons(a__nats(mark(z0)), x3)), MARK(x1)) MARK(filter(x0, x1, cons(zprimes, x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(a__zprimes, x3)), MARK(x1)) MARK(filter(x0, x1, cons(cons(z0, z1), x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(cons(mark(z0), z1), x3)), MARK(x1)) MARK(filter(x0, x1, cons(0, x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(0, x3)), MARK(x1)) MARK(filter(x0, x1, cons(s(z0), x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(s(mark(z0)), x3)), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), cons(x2, x3))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), cons(mark(x2), x3)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), cons(x2, x3))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), cons(mark(x2), x3)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), cons(x2, x3))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), cons(mark(x2), x3)), MARK(nats(z0))) MARK(filter(x0, zprimes, cons(x2, x3))) -> c11(A__FILTER(mark(x0), a__zprimes, cons(mark(x2), x3)), MARK(zprimes)) MARK(filter(x0, s(z0), cons(x2, x3))) -> c11(A__FILTER(mark(x0), s(mark(z0)), cons(mark(x2), x3)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, cons(x2, x3))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(sieve(z0), x1, cons(x2, x3))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(nats(z0), x1, cons(x2, x3))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(zprimes, x1, cons(x2, x3))) -> c11(A__FILTER(a__zprimes, mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(cons(z0, z1), x1, cons(x2, x3))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(x0, cons(z0, z1), cons(x2, x3))) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, cons(x2, x3))) -> c11(MARK(x1)) MARK(filter(s(z0), x1, cons(x2, x3))) -> c11(MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), 0)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), 0), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), 0)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), 0), MARK(sieve(z0))) MARK(filter(x0, nats(z0), 0)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), 0), MARK(nats(z0))) MARK(filter(x0, zprimes, 0)) -> c11(A__FILTER(mark(x0), a__zprimes, 0), MARK(zprimes)) MARK(filter(x0, s(z0), 0)) -> c11(A__FILTER(mark(x0), s(mark(z0)), 0), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, 0)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), 0), MARK(x1)) MARK(filter(sieve(z0), x1, 0)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), 0), MARK(x1)) MARK(filter(nats(z0), x1, 0)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), 0), MARK(x1)) MARK(filter(zprimes, x1, 0)) -> c11(A__FILTER(a__zprimes, mark(x1), 0), MARK(x1)) MARK(filter(cons(z0, z1), x1, 0)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), 0), MARK(x1)) MARK(filter(x0, cons(z0, z1), 0)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, 0)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, 0)) -> c11(MARK(x1)) MARK(filter(x0, x1, s(filter(z0, z1, z2)))) -> c11(A__FILTER(mark(x0), mark(x1), s(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x1)) MARK(filter(x0, x1, s(sieve(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), s(a__sieve(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, s(nats(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), s(a__nats(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, s(zprimes))) -> c11(A__FILTER(mark(x0), mark(x1), s(a__zprimes)), MARK(x1)) MARK(filter(x0, x1, s(cons(z0, z1)))) -> c11(A__FILTER(mark(x0), mark(x1), s(cons(mark(z0), z1))), MARK(x1)) MARK(filter(x0, x1, s(0))) -> c11(A__FILTER(mark(x0), mark(x1), s(0)), MARK(x1)) MARK(filter(x0, x1, s(s(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), s(s(mark(z0)))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), s(x2))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), s(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), s(x2))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), s(mark(x2))), MARK(sieve(z0))) MARK(filter(x0, nats(z0), s(x2))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), s(mark(x2))), MARK(nats(z0))) MARK(filter(x0, zprimes, s(x2))) -> c11(A__FILTER(mark(x0), a__zprimes, s(mark(x2))), MARK(zprimes)) MARK(filter(x0, s(z0), s(x2))) -> c11(A__FILTER(mark(x0), s(mark(z0)), s(mark(x2))), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, s(x2))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), s(mark(x2))), MARK(x1)) MARK(filter(sieve(z0), x1, s(x2))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), s(mark(x2))), MARK(x1)) MARK(filter(nats(z0), x1, s(x2))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), s(mark(x2))), MARK(x1)) MARK(filter(zprimes, x1, s(x2))) -> c11(A__FILTER(a__zprimes, mark(x1), s(mark(x2))), MARK(x1)) MARK(filter(cons(z0, z1), x1, s(x2))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), s(mark(x2))), MARK(x1)) MARK(filter(x0, cons(z0, z1), s(x2))) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, s(x2))) -> c11(MARK(x1)) MARK(filter(s(z0), x1, s(x2))) -> c11(MARK(x1)) MARK(filter(x0, filter(x1, x2, filter(z0, z1, z2)), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__filter(mark(z0), mark(z1), mark(z2))), mark(x4)), MARK(filter(x1, x2, filter(z0, z1, z2)))) MARK(filter(x0, filter(x1, x2, sieve(z0)), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__sieve(mark(z0))), mark(x4)), MARK(filter(x1, x2, sieve(z0)))) MARK(filter(x0, filter(x1, x2, nats(z0)), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__nats(mark(z0))), mark(x4)), MARK(filter(x1, x2, nats(z0)))) MARK(filter(x0, filter(x1, x2, zprimes), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__zprimes), mark(x4)), MARK(filter(x1, x2, zprimes))) MARK(filter(x0, filter(x1, x2, cons(z0, z1)), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), cons(mark(z0), z1)), mark(x4)), MARK(filter(x1, x2, cons(z0, z1)))) MARK(filter(x0, filter(x1, x2, 0), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), 0), mark(x4)), MARK(filter(x1, x2, 0))) MARK(filter(x0, filter(x1, x2, s(z0)), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), s(mark(z0))), mark(x4)), MARK(filter(x1, x2, s(z0)))) MARK(filter(x0, filter(x1, filter(z0, z1, z2), x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), a__filter(mark(z0), mark(z1), mark(z2)), mark(x3)), mark(x4)), MARK(filter(x1, filter(z0, z1, z2), x3))) MARK(filter(x0, filter(x1, sieve(z0), x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), a__sieve(mark(z0)), mark(x3)), mark(x4)), MARK(filter(x1, sieve(z0), x3))) MARK(filter(x0, filter(x1, nats(z0), x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), a__nats(mark(z0)), mark(x3)), mark(x4)), MARK(filter(x1, nats(z0), x3))) MARK(filter(x0, filter(x1, zprimes, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), a__zprimes, mark(x3)), mark(x4)), MARK(filter(x1, zprimes, x3))) MARK(filter(x0, filter(x1, cons(z0, z1), x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), cons(mark(z0), z1), mark(x3)), mark(x4)), MARK(filter(x1, cons(z0, z1), x3))) MARK(filter(x0, filter(x1, 0, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), 0, mark(x3)), mark(x4)), MARK(filter(x1, 0, x3))) MARK(filter(x0, filter(x1, s(z0), x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), s(mark(z0)), mark(x3)), mark(x4)), MARK(filter(x1, s(z0), x3))) MARK(filter(x0, filter(filter(z0, z1, z2), x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x2), mark(x3)), mark(x4)), MARK(filter(filter(z0, z1, z2), x2, x3))) MARK(filter(x0, filter(sieve(z0), x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(a__sieve(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(filter(sieve(z0), x2, x3))) MARK(filter(x0, filter(nats(z0), x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(a__nats(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(filter(nats(z0), x2, x3))) MARK(filter(x0, filter(zprimes, x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(a__zprimes, mark(x2), mark(x3)), mark(x4)), MARK(filter(zprimes, x2, x3))) MARK(filter(x0, filter(cons(z0, z1), x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(cons(mark(z0), z1), mark(x2), mark(x3)), mark(x4)), MARK(filter(cons(z0, z1), x2, x3))) MARK(filter(x0, filter(0, x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(0, mark(x2), mark(x3)), mark(x4)), MARK(filter(0, x2, x3))) MARK(filter(x0, filter(s(z0), x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(s(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(filter(s(z0), x2, x3))) MARK(filter(filter(z0, z1, z2), filter(x1, x2, x3), x4)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(sieve(z0), filter(x1, x2, x3), x4)) -> c11(A__FILTER(a__sieve(mark(z0)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(nats(z0), filter(x1, x2, x3), x4)) -> c11(A__FILTER(a__nats(mark(z0)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(zprimes, filter(x1, x2, x3), x4)) -> c11(A__FILTER(a__zprimes, a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(cons(z0, z1), filter(x1, x2, x3), x4)) -> c11(A__FILTER(cons(mark(z0), z1), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(x0, filter(x1, x2, x3), x4)) -> c11(MARK(filter(x1, x2, x3))) MARK(filter(0, filter(x1, x2, x3), x4)) -> c11(MARK(filter(x1, x2, x3))) MARK(filter(s(z0), filter(x1, x2, x3), x4)) -> c11(MARK(filter(x1, x2, x3))) MARK(filter(x0, sieve(filter(z0, z1, z2)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(sieve(filter(z0, z1, z2)))) MARK(filter(x0, sieve(sieve(z0)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(a__sieve(mark(z0))), mark(x2)), MARK(sieve(sieve(z0)))) MARK(filter(x0, sieve(nats(z0)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(a__nats(mark(z0))), mark(x2)), MARK(sieve(nats(z0)))) MARK(filter(x0, sieve(zprimes), x2)) -> c11(A__FILTER(mark(x0), a__sieve(a__zprimes), mark(x2)), MARK(sieve(zprimes))) MARK(filter(x0, sieve(cons(z0, z1)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(cons(mark(z0), z1)), mark(x2)), MARK(sieve(cons(z0, z1)))) MARK(filter(x0, sieve(s(z0)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(s(mark(z0))), mark(x2)), MARK(sieve(s(z0)))) MARK(filter(filter(z0, z1, z2), sieve(x1), x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(sieve(z0), sieve(x1), x2)) -> c11(A__FILTER(a__sieve(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(nats(z0), sieve(x1), x2)) -> c11(A__FILTER(a__nats(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(zprimes, sieve(x1), x2)) -> c11(A__FILTER(a__zprimes, a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(cons(z0, z1), sieve(x1), x2)) -> c11(A__FILTER(cons(mark(z0), z1), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(x0, sieve(x1), x2)) -> c11(MARK(sieve(x1))) MARK(filter(x0, sieve(0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(0), mark(x2))) MARK(filter(0, sieve(x1), x2)) -> c11(MARK(sieve(x1))) MARK(filter(s(z0), sieve(x1), x2)) -> c11(MARK(sieve(x1))) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c14_2, c15_1, c16_1, c18_1, c8_1, c10_1, c11_2, c11_1, c12_2, c12_1, c13_2, c13_1, c8_2, c10_2 ---------------------------------------- (195) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(filter(x0, nats(z0), x2)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) by MARK(filter(x0, nats(x1), filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), a__nats(mark(x1)), a__filter(mark(z0), mark(z1), mark(z2))), MARK(nats(x1))) MARK(filter(x0, nats(x1), sieve(z0))) -> c11(A__FILTER(mark(x0), a__nats(mark(x1)), a__sieve(mark(z0))), MARK(nats(x1))) MARK(filter(x0, nats(x1), nats(z0))) -> c11(A__FILTER(mark(x0), a__nats(mark(x1)), a__nats(mark(z0))), MARK(nats(x1))) MARK(filter(x0, nats(x1), zprimes)) -> c11(A__FILTER(mark(x0), a__nats(mark(x1)), a__zprimes), MARK(nats(x1))) MARK(filter(x0, nats(x1), cons(z0, z1))) -> c11(A__FILTER(mark(x0), a__nats(mark(x1)), cons(mark(z0), z1)), MARK(nats(x1))) MARK(filter(x0, nats(x1), 0)) -> c11(A__FILTER(mark(x0), a__nats(mark(x1)), 0), MARK(nats(x1))) MARK(filter(x0, nats(x1), s(z0))) -> c11(A__FILTER(mark(x0), a__nats(mark(x1)), s(mark(z0))), MARK(nats(x1))) MARK(filter(x0, nats(x1), x2)) -> c11(A__FILTER(mark(x0), cons(mark(mark(x1)), nats(s(mark(x1)))), mark(x2)), MARK(nats(x1))) MARK(filter(x0, nats(x1), x2)) -> c11(A__FILTER(mark(x0), nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(x0, nats(filter(z0, z1, z2)), x2)) -> c11(A__FILTER(mark(x0), a__nats(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(nats(filter(z0, z1, z2)))) MARK(filter(x0, nats(sieve(z0)), x2)) -> c11(A__FILTER(mark(x0), a__nats(a__sieve(mark(z0))), mark(x2)), MARK(nats(sieve(z0)))) MARK(filter(x0, nats(nats(z0)), x2)) -> c11(A__FILTER(mark(x0), a__nats(a__nats(mark(z0))), mark(x2)), MARK(nats(nats(z0)))) MARK(filter(x0, nats(zprimes), x2)) -> c11(A__FILTER(mark(x0), a__nats(a__zprimes), mark(x2)), MARK(nats(zprimes))) MARK(filter(x0, nats(cons(z0, z1)), x2)) -> c11(A__FILTER(mark(x0), a__nats(cons(mark(z0), z1)), mark(x2)), MARK(nats(cons(z0, z1)))) MARK(filter(x0, nats(0), x2)) -> c11(A__FILTER(mark(x0), a__nats(0), mark(x2)), MARK(nats(0))) MARK(filter(x0, nats(s(z0)), x2)) -> c11(A__FILTER(mark(x0), a__nats(s(mark(z0))), mark(x2)), MARK(nats(s(z0)))) MARK(filter(filter(z0, z1, z2), nats(x1), x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(sieve(z0), nats(x1), x2)) -> c11(A__FILTER(a__sieve(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(nats(z0), nats(x1), x2)) -> c11(A__FILTER(a__nats(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(zprimes, nats(x1), x2)) -> c11(A__FILTER(a__zprimes, a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(cons(z0, z1), nats(x1), x2)) -> c11(A__FILTER(cons(mark(z0), z1), a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(0, nats(x1), x2)) -> c11(A__FILTER(0, a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(s(z0), nats(x1), x2)) -> c11(A__FILTER(s(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(x0, nats(x1), x2)) -> c11(MARK(nats(x1))) ---------------------------------------- (196) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(x0, 0, filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, sieve(x2))) -> c10(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x0)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, sieve(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, sieve(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, sieve(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(0)), MARK(x0)) MARK(filter(x0, x1, sieve(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), sieve(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, sieve(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, sieve(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), 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mark(x2), a__zprimes), mark(x4)), MARK(filter(x1, x2, zprimes))) MARK(filter(x0, filter(x1, x2, cons(z0, z1)), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), cons(mark(z0), z1)), mark(x4)), MARK(filter(x1, x2, cons(z0, z1)))) MARK(filter(x0, filter(x1, x2, 0), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), 0), mark(x4)), MARK(filter(x1, x2, 0))) MARK(filter(x0, filter(x1, x2, s(z0)), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), s(mark(z0))), mark(x4)), MARK(filter(x1, x2, s(z0)))) MARK(filter(x0, filter(x1, filter(z0, z1, z2), x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), a__filter(mark(z0), mark(z1), mark(z2)), mark(x3)), mark(x4)), MARK(filter(x1, filter(z0, z1, z2), x3))) MARK(filter(x0, filter(x1, sieve(z0), x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), a__sieve(mark(z0)), mark(x3)), mark(x4)), MARK(filter(x1, sieve(z0), x3))) MARK(filter(x0, filter(x1, nats(z0), x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), a__nats(mark(z0)), mark(x3)), mark(x4)), MARK(filter(x1, nats(z0), x3))) MARK(filter(x0, filter(x1, zprimes, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), a__zprimes, mark(x3)), mark(x4)), MARK(filter(x1, zprimes, x3))) MARK(filter(x0, filter(x1, cons(z0, z1), x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), cons(mark(z0), z1), mark(x3)), mark(x4)), MARK(filter(x1, cons(z0, z1), x3))) MARK(filter(x0, filter(x1, 0, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), 0, mark(x3)), mark(x4)), MARK(filter(x1, 0, x3))) MARK(filter(x0, filter(x1, s(z0), x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), s(mark(z0)), mark(x3)), mark(x4)), MARK(filter(x1, s(z0), x3))) MARK(filter(x0, filter(filter(z0, z1, z2), x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x2), mark(x3)), mark(x4)), MARK(filter(filter(z0, z1, z2), x2, x3))) MARK(filter(x0, filter(sieve(z0), x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(a__sieve(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(filter(sieve(z0), x2, x3))) MARK(filter(x0, filter(nats(z0), x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(a__nats(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(filter(nats(z0), x2, x3))) MARK(filter(x0, filter(zprimes, x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(a__zprimes, mark(x2), mark(x3)), mark(x4)), MARK(filter(zprimes, x2, x3))) MARK(filter(x0, filter(cons(z0, z1), x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(cons(mark(z0), z1), mark(x2), mark(x3)), mark(x4)), MARK(filter(cons(z0, z1), x2, x3))) MARK(filter(x0, filter(0, x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(0, mark(x2), mark(x3)), mark(x4)), MARK(filter(0, x2, x3))) MARK(filter(x0, filter(s(z0), x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(s(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(filter(s(z0), x2, x3))) MARK(filter(filter(z0, z1, z2), filter(x1, x2, x3), x4)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(sieve(z0), filter(x1, x2, x3), x4)) -> c11(A__FILTER(a__sieve(mark(z0)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(nats(z0), filter(x1, x2, x3), x4)) -> c11(A__FILTER(a__nats(mark(z0)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(zprimes, filter(x1, x2, x3), x4)) -> c11(A__FILTER(a__zprimes, a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(cons(z0, z1), filter(x1, x2, x3), x4)) -> c11(A__FILTER(cons(mark(z0), z1), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(x0, filter(x1, x2, x3), x4)) -> c11(MARK(filter(x1, x2, x3))) MARK(filter(0, filter(x1, x2, x3), x4)) -> c11(MARK(filter(x1, x2, x3))) MARK(filter(s(z0), filter(x1, x2, x3), x4)) -> c11(MARK(filter(x1, x2, x3))) MARK(filter(x0, sieve(filter(z0, z1, z2)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(sieve(filter(z0, z1, z2)))) MARK(filter(x0, sieve(sieve(z0)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(a__sieve(mark(z0))), mark(x2)), MARK(sieve(sieve(z0)))) MARK(filter(x0, sieve(nats(z0)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(a__nats(mark(z0))), mark(x2)), MARK(sieve(nats(z0)))) MARK(filter(x0, sieve(zprimes), x2)) -> c11(A__FILTER(mark(x0), a__sieve(a__zprimes), mark(x2)), MARK(sieve(zprimes))) MARK(filter(x0, sieve(cons(z0, z1)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(cons(mark(z0), z1)), mark(x2)), MARK(sieve(cons(z0, z1)))) MARK(filter(x0, sieve(s(z0)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(s(mark(z0))), mark(x2)), MARK(sieve(s(z0)))) MARK(filter(filter(z0, z1, z2), sieve(x1), x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(sieve(z0), sieve(x1), x2)) -> c11(A__FILTER(a__sieve(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(nats(z0), sieve(x1), x2)) -> c11(A__FILTER(a__nats(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(zprimes, sieve(x1), x2)) -> c11(A__FILTER(a__zprimes, a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(cons(z0, z1), sieve(x1), x2)) -> c11(A__FILTER(cons(mark(z0), z1), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(x0, sieve(x1), x2)) -> c11(MARK(sieve(x1))) MARK(filter(x0, sieve(0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(0), mark(x2))) MARK(filter(0, sieve(x1), x2)) -> c11(MARK(sieve(x1))) MARK(filter(s(z0), sieve(x1), x2)) -> c11(MARK(sieve(x1))) MARK(filter(x0, nats(x1), x2)) -> c11(A__FILTER(mark(x0), cons(mark(mark(x1)), nats(s(mark(x1)))), mark(x2)), MARK(nats(x1))) MARK(filter(x0, nats(x1), x2)) -> c11(A__FILTER(mark(x0), nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(x0, nats(filter(z0, z1, z2)), x2)) -> c11(A__FILTER(mark(x0), a__nats(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(nats(filter(z0, z1, z2)))) MARK(filter(x0, nats(sieve(z0)), x2)) -> c11(A__FILTER(mark(x0), a__nats(a__sieve(mark(z0))), mark(x2)), MARK(nats(sieve(z0)))) MARK(filter(x0, nats(nats(z0)), x2)) -> c11(A__FILTER(mark(x0), a__nats(a__nats(mark(z0))), mark(x2)), MARK(nats(nats(z0)))) MARK(filter(x0, nats(zprimes), x2)) -> c11(A__FILTER(mark(x0), a__nats(a__zprimes), mark(x2)), MARK(nats(zprimes))) MARK(filter(x0, nats(cons(z0, z1)), x2)) -> c11(A__FILTER(mark(x0), a__nats(cons(mark(z0), z1)), mark(x2)), MARK(nats(cons(z0, z1)))) MARK(filter(x0, nats(0), x2)) -> c11(A__FILTER(mark(x0), a__nats(0), mark(x2)), MARK(nats(0))) MARK(filter(x0, nats(s(z0)), x2)) -> c11(A__FILTER(mark(x0), a__nats(s(mark(z0))), mark(x2)), MARK(nats(s(z0)))) MARK(filter(filter(z0, z1, z2), nats(x1), x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(sieve(z0), nats(x1), x2)) -> c11(A__FILTER(a__sieve(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(nats(z0), nats(x1), x2)) -> c11(A__FILTER(a__nats(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(zprimes, nats(x1), x2)) -> c11(A__FILTER(a__zprimes, a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(cons(z0, z1), nats(x1), x2)) -> c11(A__FILTER(cons(mark(z0), z1), a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(0, nats(x1), x2)) -> c11(A__FILTER(0, a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(s(z0), nats(x1), x2)) -> c11(A__FILTER(s(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(x0, nats(x1), x2)) -> c11(MARK(nats(x1))) S tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(x0, 0, filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, sieve(x2))) -> c10(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x0)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, sieve(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, sieve(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, sieve(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(0)), MARK(x0)) MARK(filter(x0, x1, sieve(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), sieve(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, sieve(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, sieve(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__sieve(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, sieve(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, sieve(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, sieve(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), a__sieve(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, sieve(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__sieve(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), sieve(x2))) -> c10(MARK(x0)) MARK(filter(x0, 0, sieve(x2))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, sieve(x2))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(mark(x2)), nats(s(mark(x2))))), MARK(x0)) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), nats(mark(x2))), MARK(x0)) MARK(filter(x0, x1, nats(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, nats(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, nats(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, nats(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, nats(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, nats(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(0)), MARK(x0)) MARK(filter(x0, x1, nats(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), nats(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), nats(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), nats(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, nats(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), nats(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, nats(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__nats(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, nats(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, nats(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, nats(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), a__nats(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, nats(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__nats(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), nats(x2))) -> c10(MARK(x0)) MARK(filter(x0, 0, nats(x2))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, nats(x2))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, zprimes)) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(s(s(0))))), MARK(x0)) MARK(filter(x0, x1, zprimes)) -> c10(A__FILTER(mark(x0), mark(x1), zprimes), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), zprimes)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__zprimes), MARK(x0)) MARK(filter(x0, sieve(z0), zprimes)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__zprimes), MARK(x0)) MARK(filter(x0, nats(z0), zprimes)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__zprimes), MARK(x0)) MARK(filter(x0, zprimes, zprimes)) -> c10(A__FILTER(mark(x0), a__zprimes, a__zprimes), MARK(x0)) MARK(filter(x0, s(z0), zprimes)) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__zprimes), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, zprimes)) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__zprimes), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, zprimes)) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__zprimes), MARK(sieve(z0))) MARK(filter(nats(z0), x1, zprimes)) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__zprimes), MARK(nats(z0))) MARK(filter(zprimes, x1, zprimes)) -> c10(A__FILTER(a__zprimes, mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, zprimes)) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__zprimes), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), zprimes)) -> c10(MARK(x0)) MARK(filter(x0, 0, zprimes)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, zprimes)) -> c10(MARK(s(z0))) MARK(filter(x0, x1, cons(filter(z0, z1, z2), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(a__filter(mark(z0), mark(z1), mark(z2)), x3)), MARK(x0)) MARK(filter(x0, x1, cons(sieve(z0), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(a__sieve(mark(z0)), x3)), MARK(x0)) MARK(filter(x0, x1, cons(nats(z0), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(a__nats(mark(z0)), x3)), MARK(x0)) MARK(filter(x0, x1, cons(zprimes, x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(a__zprimes, x3)), MARK(x0)) MARK(filter(x0, x1, cons(cons(z0, z1), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(cons(mark(z0), z1), x3)), MARK(x0)) MARK(filter(x0, x1, cons(0, x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(0, x3)), MARK(x0)) MARK(filter(x0, x1, cons(s(z0), x3))) -> c10(A__FILTER(mark(x0), mark(x1), cons(s(mark(z0)), x3)), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), cons(x2, x3))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), cons(mark(x2), x3)), MARK(x0)) MARK(filter(x0, sieve(z0), cons(x2, x3))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), cons(mark(x2), x3)), MARK(x0)) MARK(filter(x0, nats(z0), cons(x2, x3))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), cons(mark(x2), x3)), MARK(x0)) MARK(filter(x0, zprimes, cons(x2, x3))) -> c10(A__FILTER(mark(x0), a__zprimes, cons(mark(x2), x3)), MARK(x0)) MARK(filter(x0, s(z0), cons(x2, x3))) -> c10(A__FILTER(mark(x0), s(mark(z0)), cons(mark(x2), x3)), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, cons(x2, x3))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), cons(mark(x2), x3)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, cons(x2, x3))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), cons(mark(x2), x3)), MARK(sieve(z0))) MARK(filter(nats(z0), x1, cons(x2, x3))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), cons(mark(x2), x3)), MARK(nats(z0))) MARK(filter(zprimes, x1, cons(x2, x3))) -> c10(A__FILTER(a__zprimes, mark(x1), cons(mark(x2), x3)), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, cons(x2, x3))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), cons(mark(x2), x3)), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), cons(x2, x3))) -> c10(MARK(x0)) MARK(filter(x0, 0, cons(x2, x3))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, cons(x2, x3))) -> c10(MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), 0)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), 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cons(filter(z0, z1, z2), x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(a__filter(mark(z0), mark(z1), mark(z2)), x3)), MARK(x1)) MARK(filter(x0, x1, cons(sieve(z0), x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(a__sieve(mark(z0)), x3)), MARK(x1)) MARK(filter(x0, x1, cons(nats(z0), x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(a__nats(mark(z0)), x3)), MARK(x1)) MARK(filter(x0, x1, cons(zprimes, x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(a__zprimes, x3)), MARK(x1)) MARK(filter(x0, x1, cons(cons(z0, z1), x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(cons(mark(z0), z1), x3)), MARK(x1)) MARK(filter(x0, x1, cons(0, x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(0, x3)), MARK(x1)) MARK(filter(x0, x1, cons(s(z0), x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(s(mark(z0)), x3)), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), cons(x2, x3))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), cons(mark(x2), x3)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), cons(x2, x3))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), cons(mark(x2), x3)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), cons(x2, x3))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), cons(mark(x2), x3)), MARK(nats(z0))) MARK(filter(x0, zprimes, cons(x2, x3))) -> c11(A__FILTER(mark(x0), a__zprimes, cons(mark(x2), x3)), MARK(zprimes)) MARK(filter(x0, s(z0), cons(x2, x3))) -> c11(A__FILTER(mark(x0), s(mark(z0)), cons(mark(x2), x3)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, cons(x2, x3))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(sieve(z0), x1, cons(x2, x3))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(nats(z0), x1, cons(x2, x3))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(zprimes, x1, cons(x2, x3))) -> c11(A__FILTER(a__zprimes, mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(cons(z0, z1), x1, cons(x2, x3))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(x0, cons(z0, z1), cons(x2, x3))) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, cons(x2, x3))) -> c11(MARK(x1)) MARK(filter(s(z0), x1, cons(x2, x3))) -> c11(MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), 0)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), 0), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), 0)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), 0), MARK(sieve(z0))) MARK(filter(x0, nats(z0), 0)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), 0), MARK(nats(z0))) MARK(filter(x0, zprimes, 0)) -> c11(A__FILTER(mark(x0), a__zprimes, 0), MARK(zprimes)) MARK(filter(x0, s(z0), 0)) -> c11(A__FILTER(mark(x0), s(mark(z0)), 0), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, 0)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), 0), MARK(x1)) MARK(filter(sieve(z0), x1, 0)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), 0), MARK(x1)) MARK(filter(nats(z0), x1, 0)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), 0), MARK(x1)) MARK(filter(zprimes, x1, 0)) -> c11(A__FILTER(a__zprimes, mark(x1), 0), MARK(x1)) MARK(filter(cons(z0, z1), x1, 0)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), 0), MARK(x1)) MARK(filter(x0, cons(z0, z1), 0)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, 0)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, 0)) -> c11(MARK(x1)) MARK(filter(x0, x1, s(filter(z0, z1, z2)))) -> c11(A__FILTER(mark(x0), mark(x1), s(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x1)) MARK(filter(x0, x1, s(sieve(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), s(a__sieve(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, s(nats(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), s(a__nats(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, s(zprimes))) -> c11(A__FILTER(mark(x0), mark(x1), s(a__zprimes)), MARK(x1)) MARK(filter(x0, x1, s(cons(z0, z1)))) -> c11(A__FILTER(mark(x0), mark(x1), s(cons(mark(z0), z1))), MARK(x1)) MARK(filter(x0, x1, s(0))) -> c11(A__FILTER(mark(x0), mark(x1), s(0)), MARK(x1)) MARK(filter(x0, x1, s(s(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), s(s(mark(z0)))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), s(x2))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), s(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), s(x2))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), s(mark(x2))), MARK(sieve(z0))) MARK(filter(x0, nats(z0), s(x2))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), s(mark(x2))), MARK(nats(z0))) MARK(filter(x0, zprimes, s(x2))) -> c11(A__FILTER(mark(x0), a__zprimes, s(mark(x2))), MARK(zprimes)) MARK(filter(x0, s(z0), s(x2))) -> c11(A__FILTER(mark(x0), s(mark(z0)), s(mark(x2))), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, s(x2))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), s(mark(x2))), MARK(x1)) MARK(filter(sieve(z0), x1, s(x2))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), s(mark(x2))), MARK(x1)) MARK(filter(nats(z0), x1, s(x2))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), s(mark(x2))), MARK(x1)) MARK(filter(zprimes, x1, s(x2))) -> c11(A__FILTER(a__zprimes, mark(x1), s(mark(x2))), MARK(x1)) MARK(filter(cons(z0, z1), x1, s(x2))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), s(mark(x2))), MARK(x1)) MARK(filter(x0, cons(z0, z1), s(x2))) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, s(x2))) -> c11(MARK(x1)) MARK(filter(s(z0), x1, s(x2))) -> c11(MARK(x1)) MARK(filter(x0, filter(x1, x2, filter(z0, z1, z2)), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__filter(mark(z0), mark(z1), mark(z2))), mark(x4)), MARK(filter(x1, x2, filter(z0, z1, z2)))) MARK(filter(x0, filter(x1, x2, sieve(z0)), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__sieve(mark(z0))), mark(x4)), MARK(filter(x1, x2, sieve(z0)))) MARK(filter(x0, filter(x1, x2, nats(z0)), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__nats(mark(z0))), mark(x4)), MARK(filter(x1, x2, nats(z0)))) MARK(filter(x0, filter(x1, x2, zprimes), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__zprimes), mark(x4)), MARK(filter(x1, x2, zprimes))) MARK(filter(x0, filter(x1, x2, cons(z0, z1)), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), cons(mark(z0), z1)), mark(x4)), MARK(filter(x1, x2, cons(z0, z1)))) MARK(filter(x0, filter(x1, x2, 0), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), 0), mark(x4)), MARK(filter(x1, x2, 0))) MARK(filter(x0, filter(x1, x2, s(z0)), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), s(mark(z0))), mark(x4)), MARK(filter(x1, x2, s(z0)))) MARK(filter(x0, filter(x1, filter(z0, z1, z2), x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), a__filter(mark(z0), mark(z1), mark(z2)), mark(x3)), mark(x4)), MARK(filter(x1, filter(z0, z1, z2), x3))) MARK(filter(x0, filter(x1, sieve(z0), x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), a__sieve(mark(z0)), mark(x3)), mark(x4)), MARK(filter(x1, sieve(z0), x3))) MARK(filter(x0, filter(x1, nats(z0), x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), a__nats(mark(z0)), mark(x3)), mark(x4)), MARK(filter(x1, nats(z0), x3))) MARK(filter(x0, filter(x1, zprimes, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), a__zprimes, mark(x3)), mark(x4)), MARK(filter(x1, zprimes, x3))) MARK(filter(x0, filter(x1, cons(z0, z1), x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), cons(mark(z0), z1), mark(x3)), mark(x4)), MARK(filter(x1, cons(z0, z1), x3))) MARK(filter(x0, filter(x1, 0, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), 0, mark(x3)), mark(x4)), MARK(filter(x1, 0, x3))) MARK(filter(x0, filter(x1, s(z0), x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), s(mark(z0)), mark(x3)), mark(x4)), MARK(filter(x1, s(z0), x3))) MARK(filter(x0, filter(filter(z0, z1, z2), x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x2), mark(x3)), mark(x4)), MARK(filter(filter(z0, z1, z2), x2, x3))) MARK(filter(x0, filter(sieve(z0), x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(a__sieve(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(filter(sieve(z0), x2, x3))) MARK(filter(x0, filter(nats(z0), x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(a__nats(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(filter(nats(z0), x2, x3))) MARK(filter(x0, filter(zprimes, x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(a__zprimes, mark(x2), mark(x3)), mark(x4)), MARK(filter(zprimes, x2, x3))) MARK(filter(x0, filter(cons(z0, z1), x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(cons(mark(z0), z1), mark(x2), mark(x3)), mark(x4)), MARK(filter(cons(z0, z1), x2, x3))) MARK(filter(x0, filter(0, x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(0, mark(x2), mark(x3)), mark(x4)), MARK(filter(0, x2, x3))) MARK(filter(x0, filter(s(z0), x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(s(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(filter(s(z0), x2, x3))) MARK(filter(filter(z0, z1, z2), filter(x1, x2, x3), x4)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(sieve(z0), filter(x1, x2, x3), x4)) -> c11(A__FILTER(a__sieve(mark(z0)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(nats(z0), filter(x1, x2, x3), x4)) -> c11(A__FILTER(a__nats(mark(z0)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(zprimes, filter(x1, x2, x3), x4)) -> c11(A__FILTER(a__zprimes, a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(cons(z0, z1), filter(x1, x2, x3), x4)) -> c11(A__FILTER(cons(mark(z0), z1), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(x0, filter(x1, x2, x3), x4)) -> c11(MARK(filter(x1, x2, x3))) MARK(filter(0, filter(x1, x2, x3), x4)) -> c11(MARK(filter(x1, x2, x3))) MARK(filter(s(z0), filter(x1, x2, x3), x4)) -> c11(MARK(filter(x1, x2, x3))) MARK(filter(x0, sieve(filter(z0, z1, z2)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(sieve(filter(z0, z1, z2)))) MARK(filter(x0, sieve(sieve(z0)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(a__sieve(mark(z0))), mark(x2)), MARK(sieve(sieve(z0)))) MARK(filter(x0, sieve(nats(z0)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(a__nats(mark(z0))), mark(x2)), MARK(sieve(nats(z0)))) MARK(filter(x0, sieve(zprimes), x2)) -> c11(A__FILTER(mark(x0), a__sieve(a__zprimes), mark(x2)), MARK(sieve(zprimes))) MARK(filter(x0, sieve(cons(z0, z1)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(cons(mark(z0), z1)), mark(x2)), MARK(sieve(cons(z0, z1)))) MARK(filter(x0, sieve(s(z0)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(s(mark(z0))), mark(x2)), MARK(sieve(s(z0)))) MARK(filter(filter(z0, z1, z2), sieve(x1), x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(sieve(z0), sieve(x1), x2)) -> c11(A__FILTER(a__sieve(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(nats(z0), sieve(x1), x2)) -> c11(A__FILTER(a__nats(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(zprimes, sieve(x1), x2)) -> c11(A__FILTER(a__zprimes, a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(cons(z0, z1), sieve(x1), x2)) -> c11(A__FILTER(cons(mark(z0), z1), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(x0, sieve(x1), x2)) -> c11(MARK(sieve(x1))) MARK(filter(x0, sieve(0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(0), mark(x2))) MARK(filter(0, sieve(x1), x2)) -> c11(MARK(sieve(x1))) MARK(filter(s(z0), sieve(x1), x2)) -> c11(MARK(sieve(x1))) MARK(filter(x0, nats(x1), x2)) -> c11(A__FILTER(mark(x0), cons(mark(mark(x1)), nats(s(mark(x1)))), mark(x2)), MARK(nats(x1))) MARK(filter(x0, nats(x1), x2)) -> c11(A__FILTER(mark(x0), nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(x0, nats(filter(z0, z1, z2)), x2)) -> c11(A__FILTER(mark(x0), a__nats(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(nats(filter(z0, z1, z2)))) MARK(filter(x0, nats(sieve(z0)), x2)) -> c11(A__FILTER(mark(x0), a__nats(a__sieve(mark(z0))), mark(x2)), MARK(nats(sieve(z0)))) MARK(filter(x0, nats(nats(z0)), x2)) -> c11(A__FILTER(mark(x0), a__nats(a__nats(mark(z0))), mark(x2)), MARK(nats(nats(z0)))) MARK(filter(x0, nats(zprimes), x2)) -> c11(A__FILTER(mark(x0), a__nats(a__zprimes), mark(x2)), MARK(nats(zprimes))) MARK(filter(x0, nats(cons(z0, z1)), x2)) -> c11(A__FILTER(mark(x0), a__nats(cons(mark(z0), z1)), mark(x2)), MARK(nats(cons(z0, z1)))) MARK(filter(x0, nats(0), x2)) -> c11(A__FILTER(mark(x0), a__nats(0), mark(x2)), MARK(nats(0))) MARK(filter(x0, nats(s(z0)), x2)) -> c11(A__FILTER(mark(x0), a__nats(s(mark(z0))), mark(x2)), MARK(nats(s(z0)))) MARK(filter(filter(z0, z1, z2), nats(x1), x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(sieve(z0), nats(x1), x2)) -> c11(A__FILTER(a__sieve(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(nats(z0), nats(x1), x2)) -> c11(A__FILTER(a__nats(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(zprimes, nats(x1), x2)) -> c11(A__FILTER(a__zprimes, a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(cons(z0, z1), nats(x1), x2)) -> c11(A__FILTER(cons(mark(z0), z1), a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(0, nats(x1), x2)) -> c11(A__FILTER(0, a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(s(z0), nats(x1), x2)) -> c11(A__FILTER(s(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(x0, nats(x1), x2)) -> c11(MARK(nats(x1))) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c14_2, c15_1, c16_1, c18_1, c8_1, c10_1, c11_2, c11_1, c12_2, c12_1, c13_2, c13_1, c8_2, c10_2 ---------------------------------------- (197) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing tuple parts ---------------------------------------- (198) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(x0, 0, filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, sieve(x2))) -> c10(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x0)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, sieve(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, sieve(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, sieve(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(0)), MARK(x0)) MARK(filter(x0, x1, sieve(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), sieve(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, sieve(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, sieve(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__sieve(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, sieve(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, sieve(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, sieve(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), a__sieve(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, sieve(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__sieve(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), sieve(x2))) -> c10(MARK(x0)) MARK(filter(x0, 0, sieve(x2))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, sieve(x2))) -> c10(MARK(s(z0))) MARK(filter(x0, 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a__filter(mark(z0), mark(z1), mark(z2)), mark(x3)), mark(x4)), MARK(filter(x1, filter(z0, z1, z2), x3))) MARK(filter(x0, filter(x1, sieve(z0), x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), a__sieve(mark(z0)), mark(x3)), mark(x4)), MARK(filter(x1, sieve(z0), x3))) MARK(filter(x0, filter(x1, nats(z0), x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), a__nats(mark(z0)), mark(x3)), mark(x4)), MARK(filter(x1, nats(z0), x3))) MARK(filter(x0, filter(x1, zprimes, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), a__zprimes, mark(x3)), mark(x4)), MARK(filter(x1, zprimes, x3))) MARK(filter(x0, filter(x1, cons(z0, z1), x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), cons(mark(z0), z1), mark(x3)), mark(x4)), MARK(filter(x1, cons(z0, z1), x3))) MARK(filter(x0, filter(x1, 0, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), 0, mark(x3)), mark(x4)), MARK(filter(x1, 0, x3))) MARK(filter(x0, filter(x1, s(z0), x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), s(mark(z0)), mark(x3)), mark(x4)), MARK(filter(x1, s(z0), x3))) MARK(filter(x0, filter(filter(z0, z1, z2), x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x2), mark(x3)), mark(x4)), MARK(filter(filter(z0, z1, z2), x2, x3))) MARK(filter(x0, filter(sieve(z0), x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(a__sieve(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(filter(sieve(z0), x2, x3))) MARK(filter(x0, filter(nats(z0), x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(a__nats(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(filter(nats(z0), x2, x3))) MARK(filter(x0, filter(zprimes, x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(a__zprimes, mark(x2), mark(x3)), mark(x4)), MARK(filter(zprimes, x2, x3))) MARK(filter(x0, filter(cons(z0, z1), x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(cons(mark(z0), z1), mark(x2), mark(x3)), mark(x4)), MARK(filter(cons(z0, z1), x2, x3))) MARK(filter(x0, filter(0, x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(0, mark(x2), mark(x3)), mark(x4)), MARK(filter(0, x2, x3))) MARK(filter(x0, filter(s(z0), x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(s(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(filter(s(z0), x2, x3))) MARK(filter(filter(z0, z1, z2), filter(x1, x2, x3), x4)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(sieve(z0), filter(x1, x2, x3), x4)) -> c11(A__FILTER(a__sieve(mark(z0)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(nats(z0), filter(x1, x2, x3), x4)) -> c11(A__FILTER(a__nats(mark(z0)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(zprimes, filter(x1, x2, x3), x4)) -> c11(A__FILTER(a__zprimes, a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(cons(z0, z1), filter(x1, x2, x3), x4)) -> c11(A__FILTER(cons(mark(z0), z1), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(x0, filter(x1, x2, x3), x4)) -> c11(MARK(filter(x1, x2, x3))) MARK(filter(0, filter(x1, x2, x3), x4)) -> c11(MARK(filter(x1, x2, x3))) MARK(filter(s(z0), filter(x1, x2, x3), x4)) -> c11(MARK(filter(x1, x2, x3))) MARK(filter(x0, sieve(filter(z0, z1, z2)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(sieve(filter(z0, z1, z2)))) MARK(filter(x0, sieve(sieve(z0)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(a__sieve(mark(z0))), mark(x2)), MARK(sieve(sieve(z0)))) MARK(filter(x0, sieve(nats(z0)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(a__nats(mark(z0))), mark(x2)), MARK(sieve(nats(z0)))) MARK(filter(x0, sieve(zprimes), x2)) -> c11(A__FILTER(mark(x0), a__sieve(a__zprimes), mark(x2)), MARK(sieve(zprimes))) MARK(filter(x0, sieve(cons(z0, z1)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(cons(mark(z0), z1)), mark(x2)), MARK(sieve(cons(z0, z1)))) MARK(filter(x0, sieve(s(z0)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(s(mark(z0))), mark(x2)), MARK(sieve(s(z0)))) MARK(filter(filter(z0, z1, z2), sieve(x1), x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(sieve(z0), sieve(x1), x2)) -> c11(A__FILTER(a__sieve(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(nats(z0), sieve(x1), x2)) -> c11(A__FILTER(a__nats(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(zprimes, sieve(x1), x2)) -> c11(A__FILTER(a__zprimes, a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(cons(z0, z1), sieve(x1), x2)) -> c11(A__FILTER(cons(mark(z0), z1), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(x0, sieve(x1), x2)) -> c11(MARK(sieve(x1))) MARK(filter(x0, sieve(0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(0), mark(x2))) MARK(filter(0, sieve(x1), x2)) -> c11(MARK(sieve(x1))) MARK(filter(s(z0), sieve(x1), x2)) -> c11(MARK(sieve(x1))) MARK(filter(x0, nats(filter(z0, z1, z2)), x2)) -> c11(A__FILTER(mark(x0), a__nats(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(nats(filter(z0, z1, z2)))) MARK(filter(x0, nats(sieve(z0)), x2)) -> c11(A__FILTER(mark(x0), a__nats(a__sieve(mark(z0))), mark(x2)), MARK(nats(sieve(z0)))) MARK(filter(x0, nats(nats(z0)), x2)) -> c11(A__FILTER(mark(x0), a__nats(a__nats(mark(z0))), mark(x2)), MARK(nats(nats(z0)))) MARK(filter(x0, nats(zprimes), x2)) -> c11(A__FILTER(mark(x0), a__nats(a__zprimes), mark(x2)), MARK(nats(zprimes))) MARK(filter(x0, nats(cons(z0, z1)), x2)) -> c11(A__FILTER(mark(x0), a__nats(cons(mark(z0), z1)), mark(x2)), MARK(nats(cons(z0, z1)))) MARK(filter(x0, nats(0), x2)) -> c11(A__FILTER(mark(x0), a__nats(0), mark(x2)), MARK(nats(0))) MARK(filter(x0, nats(s(z0)), x2)) -> c11(A__FILTER(mark(x0), a__nats(s(mark(z0))), mark(x2)), MARK(nats(s(z0)))) MARK(filter(filter(z0, z1, z2), nats(x1), x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(sieve(z0), nats(x1), x2)) -> c11(A__FILTER(a__sieve(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(nats(z0), nats(x1), x2)) -> c11(A__FILTER(a__nats(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(zprimes, nats(x1), x2)) -> c11(A__FILTER(a__zprimes, a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(cons(z0, z1), nats(x1), x2)) -> c11(A__FILTER(cons(mark(z0), z1), a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(x0, nats(x1), x2)) -> c11(MARK(nats(x1))) MARK(filter(0, nats(x1), x2)) -> c11(MARK(nats(x1))) MARK(filter(s(z0), nats(x1), x2)) -> c11(MARK(nats(x1))) S tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 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a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> 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a__filter(mark(z0), mark(z1), mark(z2)), cons(mark(x2), x3)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), cons(x2, x3))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), cons(mark(x2), x3)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), cons(x2, x3))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), cons(mark(x2), x3)), MARK(nats(z0))) MARK(filter(x0, zprimes, cons(x2, x3))) -> c11(A__FILTER(mark(x0), a__zprimes, cons(mark(x2), x3)), MARK(zprimes)) MARK(filter(x0, s(z0), cons(x2, x3))) -> c11(A__FILTER(mark(x0), s(mark(z0)), cons(mark(x2), x3)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, cons(x2, x3))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(sieve(z0), x1, cons(x2, x3))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(nats(z0), x1, cons(x2, x3))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(zprimes, x1, cons(x2, x3))) -> c11(A__FILTER(a__zprimes, mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(cons(z0, z1), x1, cons(x2, x3))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(x0, cons(z0, z1), cons(x2, x3))) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, cons(x2, x3))) -> c11(MARK(x1)) MARK(filter(s(z0), x1, cons(x2, x3))) -> c11(MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), 0)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), 0), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), 0)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), 0), MARK(sieve(z0))) MARK(filter(x0, nats(z0), 0)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), 0), MARK(nats(z0))) MARK(filter(x0, zprimes, 0)) -> c11(A__FILTER(mark(x0), a__zprimes, 0), MARK(zprimes)) MARK(filter(x0, s(z0), 0)) -> c11(A__FILTER(mark(x0), s(mark(z0)), 0), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, 0)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), 0), MARK(x1)) MARK(filter(sieve(z0), x1, 0)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), 0), MARK(x1)) MARK(filter(nats(z0), x1, 0)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), 0), MARK(x1)) MARK(filter(zprimes, x1, 0)) -> c11(A__FILTER(a__zprimes, mark(x1), 0), MARK(x1)) MARK(filter(cons(z0, z1), x1, 0)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), 0), MARK(x1)) MARK(filter(x0, cons(z0, z1), 0)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, 0)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, 0)) -> c11(MARK(x1)) MARK(filter(x0, x1, s(filter(z0, z1, z2)))) -> c11(A__FILTER(mark(x0), mark(x1), s(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x1)) MARK(filter(x0, x1, s(sieve(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), s(a__sieve(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, s(nats(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), s(a__nats(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, s(zprimes))) -> c11(A__FILTER(mark(x0), mark(x1), s(a__zprimes)), MARK(x1)) MARK(filter(x0, x1, s(cons(z0, z1)))) -> c11(A__FILTER(mark(x0), mark(x1), s(cons(mark(z0), z1))), MARK(x1)) MARK(filter(x0, x1, s(0))) -> c11(A__FILTER(mark(x0), mark(x1), s(0)), MARK(x1)) MARK(filter(x0, x1, s(s(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), s(s(mark(z0)))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), s(x2))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), s(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), s(x2))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), s(mark(x2))), MARK(sieve(z0))) MARK(filter(x0, nats(z0), s(x2))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), s(mark(x2))), MARK(nats(z0))) MARK(filter(x0, zprimes, s(x2))) -> c11(A__FILTER(mark(x0), a__zprimes, s(mark(x2))), MARK(zprimes)) MARK(filter(x0, s(z0), s(x2))) -> c11(A__FILTER(mark(x0), s(mark(z0)), s(mark(x2))), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, s(x2))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), s(mark(x2))), MARK(x1)) MARK(filter(sieve(z0), x1, s(x2))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), s(mark(x2))), MARK(x1)) MARK(filter(nats(z0), x1, s(x2))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), s(mark(x2))), MARK(x1)) MARK(filter(zprimes, x1, s(x2))) -> c11(A__FILTER(a__zprimes, mark(x1), s(mark(x2))), MARK(x1)) MARK(filter(cons(z0, z1), x1, s(x2))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), s(mark(x2))), MARK(x1)) MARK(filter(x0, cons(z0, z1), s(x2))) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, s(x2))) -> c11(MARK(x1)) MARK(filter(s(z0), x1, s(x2))) -> c11(MARK(x1)) MARK(filter(x0, filter(x1, x2, filter(z0, z1, z2)), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__filter(mark(z0), mark(z1), mark(z2))), mark(x4)), MARK(filter(x1, x2, filter(z0, z1, z2)))) MARK(filter(x0, filter(x1, x2, sieve(z0)), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__sieve(mark(z0))), mark(x4)), MARK(filter(x1, x2, sieve(z0)))) MARK(filter(x0, filter(x1, x2, nats(z0)), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__nats(mark(z0))), mark(x4)), MARK(filter(x1, x2, nats(z0)))) MARK(filter(x0, filter(x1, x2, zprimes), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__zprimes), mark(x4)), MARK(filter(x1, x2, zprimes))) MARK(filter(x0, filter(x1, x2, cons(z0, z1)), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), cons(mark(z0), z1)), mark(x4)), MARK(filter(x1, x2, cons(z0, z1)))) MARK(filter(x0, filter(x1, x2, 0), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), 0), mark(x4)), MARK(filter(x1, x2, 0))) MARK(filter(x0, filter(x1, x2, s(z0)), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), s(mark(z0))), mark(x4)), MARK(filter(x1, x2, s(z0)))) MARK(filter(x0, filter(x1, filter(z0, z1, z2), x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), a__filter(mark(z0), mark(z1), mark(z2)), mark(x3)), mark(x4)), MARK(filter(x1, filter(z0, z1, z2), x3))) MARK(filter(x0, filter(x1, sieve(z0), x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), a__sieve(mark(z0)), mark(x3)), mark(x4)), MARK(filter(x1, sieve(z0), x3))) MARK(filter(x0, filter(x1, nats(z0), x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), a__nats(mark(z0)), mark(x3)), mark(x4)), MARK(filter(x1, nats(z0), x3))) MARK(filter(x0, filter(x1, zprimes, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), a__zprimes, mark(x3)), mark(x4)), MARK(filter(x1, zprimes, x3))) MARK(filter(x0, filter(x1, cons(z0, z1), x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), cons(mark(z0), z1), mark(x3)), mark(x4)), MARK(filter(x1, cons(z0, z1), x3))) MARK(filter(x0, filter(x1, 0, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), 0, mark(x3)), mark(x4)), MARK(filter(x1, 0, x3))) MARK(filter(x0, filter(x1, s(z0), x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), s(mark(z0)), mark(x3)), mark(x4)), MARK(filter(x1, s(z0), x3))) MARK(filter(x0, filter(filter(z0, z1, z2), x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x2), mark(x3)), mark(x4)), MARK(filter(filter(z0, z1, z2), x2, x3))) MARK(filter(x0, filter(sieve(z0), x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(a__sieve(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(filter(sieve(z0), x2, x3))) MARK(filter(x0, filter(nats(z0), x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(a__nats(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(filter(nats(z0), x2, x3))) MARK(filter(x0, filter(zprimes, x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(a__zprimes, mark(x2), mark(x3)), mark(x4)), MARK(filter(zprimes, x2, x3))) MARK(filter(x0, filter(cons(z0, z1), x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(cons(mark(z0), z1), mark(x2), mark(x3)), mark(x4)), MARK(filter(cons(z0, z1), x2, x3))) MARK(filter(x0, filter(0, x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(0, mark(x2), mark(x3)), mark(x4)), MARK(filter(0, x2, x3))) MARK(filter(x0, filter(s(z0), x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(s(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(filter(s(z0), x2, x3))) MARK(filter(filter(z0, z1, z2), filter(x1, x2, x3), x4)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(sieve(z0), filter(x1, x2, x3), x4)) -> c11(A__FILTER(a__sieve(mark(z0)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(nats(z0), filter(x1, x2, x3), x4)) -> c11(A__FILTER(a__nats(mark(z0)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(zprimes, filter(x1, x2, x3), x4)) -> c11(A__FILTER(a__zprimes, a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(cons(z0, z1), filter(x1, x2, x3), x4)) -> c11(A__FILTER(cons(mark(z0), z1), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(x0, filter(x1, x2, x3), x4)) -> c11(MARK(filter(x1, x2, x3))) MARK(filter(0, filter(x1, x2, x3), x4)) -> c11(MARK(filter(x1, x2, x3))) MARK(filter(s(z0), filter(x1, x2, x3), x4)) -> c11(MARK(filter(x1, x2, x3))) MARK(filter(x0, sieve(filter(z0, z1, z2)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(sieve(filter(z0, z1, z2)))) MARK(filter(x0, sieve(sieve(z0)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(a__sieve(mark(z0))), mark(x2)), MARK(sieve(sieve(z0)))) MARK(filter(x0, sieve(nats(z0)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(a__nats(mark(z0))), mark(x2)), MARK(sieve(nats(z0)))) MARK(filter(x0, sieve(zprimes), x2)) -> c11(A__FILTER(mark(x0), a__sieve(a__zprimes), mark(x2)), MARK(sieve(zprimes))) MARK(filter(x0, sieve(cons(z0, z1)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(cons(mark(z0), z1)), mark(x2)), MARK(sieve(cons(z0, z1)))) MARK(filter(x0, sieve(s(z0)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(s(mark(z0))), mark(x2)), MARK(sieve(s(z0)))) MARK(filter(filter(z0, z1, z2), sieve(x1), x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(sieve(z0), sieve(x1), x2)) -> c11(A__FILTER(a__sieve(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(nats(z0), sieve(x1), x2)) -> c11(A__FILTER(a__nats(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(zprimes, sieve(x1), x2)) -> c11(A__FILTER(a__zprimes, a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(cons(z0, z1), sieve(x1), x2)) -> c11(A__FILTER(cons(mark(z0), z1), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(x0, sieve(x1), x2)) -> c11(MARK(sieve(x1))) MARK(filter(x0, sieve(0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(0), mark(x2))) MARK(filter(0, sieve(x1), x2)) -> c11(MARK(sieve(x1))) MARK(filter(s(z0), sieve(x1), x2)) -> c11(MARK(sieve(x1))) MARK(filter(x0, nats(filter(z0, z1, z2)), x2)) -> c11(A__FILTER(mark(x0), a__nats(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(nats(filter(z0, z1, z2)))) MARK(filter(x0, nats(sieve(z0)), x2)) -> c11(A__FILTER(mark(x0), a__nats(a__sieve(mark(z0))), mark(x2)), MARK(nats(sieve(z0)))) MARK(filter(x0, nats(nats(z0)), x2)) -> c11(A__FILTER(mark(x0), a__nats(a__nats(mark(z0))), mark(x2)), MARK(nats(nats(z0)))) MARK(filter(x0, nats(zprimes), x2)) -> c11(A__FILTER(mark(x0), a__nats(a__zprimes), mark(x2)), MARK(nats(zprimes))) MARK(filter(x0, nats(cons(z0, z1)), x2)) -> c11(A__FILTER(mark(x0), a__nats(cons(mark(z0), z1)), mark(x2)), MARK(nats(cons(z0, z1)))) MARK(filter(x0, nats(0), x2)) -> c11(A__FILTER(mark(x0), a__nats(0), mark(x2)), MARK(nats(0))) MARK(filter(x0, nats(s(z0)), x2)) -> c11(A__FILTER(mark(x0), a__nats(s(mark(z0))), mark(x2)), MARK(nats(s(z0)))) MARK(filter(filter(z0, z1, z2), nats(x1), x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(sieve(z0), nats(x1), x2)) -> c11(A__FILTER(a__sieve(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(nats(z0), nats(x1), x2)) -> c11(A__FILTER(a__nats(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(zprimes, nats(x1), x2)) -> c11(A__FILTER(a__zprimes, a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(cons(z0, z1), nats(x1), x2)) -> c11(A__FILTER(cons(mark(z0), z1), a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(x0, nats(x1), x2)) -> c11(MARK(nats(x1))) MARK(filter(0, nats(x1), x2)) -> c11(MARK(nats(x1))) MARK(filter(s(z0), nats(x1), x2)) -> c11(MARK(nats(x1))) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c14_2, c15_1, c16_1, c18_1, c8_1, c10_1, c11_2, c11_1, c12_2, c12_1, c13_2, c13_1, c8_2, c10_2 ---------------------------------------- (199) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(filter(x0, zprimes, x2)) -> c11(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(zprimes)) by MARK(filter(x0, zprimes, filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), a__zprimes, a__filter(mark(z0), mark(z1), mark(z2))), MARK(zprimes)) MARK(filter(x0, zprimes, sieve(z0))) -> c11(A__FILTER(mark(x0), a__zprimes, a__sieve(mark(z0))), MARK(zprimes)) MARK(filter(x0, zprimes, nats(z0))) -> c11(A__FILTER(mark(x0), a__zprimes, a__nats(mark(z0))), MARK(zprimes)) MARK(filter(x0, zprimes, zprimes)) -> c11(A__FILTER(mark(x0), a__zprimes, a__zprimes), MARK(zprimes)) MARK(filter(x0, zprimes, cons(z0, z1))) -> c11(A__FILTER(mark(x0), a__zprimes, cons(mark(z0), z1)), MARK(zprimes)) MARK(filter(x0, zprimes, 0)) -> c11(A__FILTER(mark(x0), a__zprimes, 0), MARK(zprimes)) MARK(filter(x0, zprimes, s(z0))) -> c11(A__FILTER(mark(x0), a__zprimes, s(mark(z0))), MARK(zprimes)) MARK(filter(x0, zprimes, x1)) -> c11(A__FILTER(mark(x0), a__sieve(a__nats(s(s(0)))), mark(x1)), MARK(zprimes)) MARK(filter(x0, zprimes, x1)) -> c11(A__FILTER(mark(x0), zprimes, mark(x1)), MARK(zprimes)) MARK(filter(filter(z0, z1, z2), zprimes, x1)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(sieve(z0), zprimes, x1)) -> c11(A__FILTER(a__sieve(mark(z0)), a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(nats(z0), zprimes, x1)) -> c11(A__FILTER(a__nats(mark(z0)), a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(zprimes, zprimes, x1)) -> c11(A__FILTER(a__zprimes, a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(cons(z0, z1), zprimes, x1)) -> c11(A__FILTER(cons(mark(z0), z1), a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(0, zprimes, x1)) -> c11(A__FILTER(0, a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(s(z0), zprimes, x1)) -> c11(A__FILTER(s(mark(z0)), a__zprimes, mark(x1)), MARK(zprimes)) ---------------------------------------- (200) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(x0, 0, filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, sieve(x2))) -> c10(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x0)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, sieve(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, sieve(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, sieve(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(0)), MARK(x0)) MARK(filter(x0, x1, sieve(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), sieve(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, sieve(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, sieve(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__sieve(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, sieve(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, sieve(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, sieve(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), a__sieve(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, sieve(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__sieve(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), sieve(x2))) -> c10(MARK(x0)) MARK(filter(x0, 0, sieve(x2))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, sieve(x2))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(mark(x2)), nats(s(mark(x2))))), MARK(x0)) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), nats(mark(x2))), MARK(x0)) MARK(filter(x0, x1, nats(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, nats(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, nats(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, nats(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, nats(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, nats(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(0)), MARK(x0)) MARK(filter(x0, x1, nats(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), nats(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), nats(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), nats(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__nats(mark(x2))), MARK(x0)) 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MARK(filter(x0, filter(x1, x2, cons(z0, z1)), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), cons(mark(z0), z1)), mark(x4)), MARK(filter(x1, x2, cons(z0, z1)))) MARK(filter(x0, filter(x1, x2, 0), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), 0), mark(x4)), MARK(filter(x1, x2, 0))) MARK(filter(x0, filter(x1, x2, s(z0)), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), s(mark(z0))), mark(x4)), MARK(filter(x1, x2, s(z0)))) MARK(filter(x0, filter(x1, filter(z0, z1, z2), x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), a__filter(mark(z0), mark(z1), mark(z2)), mark(x3)), mark(x4)), MARK(filter(x1, filter(z0, z1, z2), x3))) MARK(filter(x0, filter(x1, sieve(z0), x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), a__sieve(mark(z0)), mark(x3)), mark(x4)), MARK(filter(x1, sieve(z0), x3))) MARK(filter(x0, filter(x1, nats(z0), x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), a__nats(mark(z0)), mark(x3)), mark(x4)), MARK(filter(x1, 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MARK(filter(sieve(z0), x2, x3))) MARK(filter(x0, filter(nats(z0), x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(a__nats(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(filter(nats(z0), x2, x3))) MARK(filter(x0, filter(zprimes, x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(a__zprimes, mark(x2), mark(x3)), mark(x4)), MARK(filter(zprimes, x2, x3))) MARK(filter(x0, filter(cons(z0, z1), x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(cons(mark(z0), z1), mark(x2), mark(x3)), mark(x4)), MARK(filter(cons(z0, z1), x2, x3))) MARK(filter(x0, filter(0, x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(0, mark(x2), mark(x3)), mark(x4)), MARK(filter(0, x2, x3))) MARK(filter(x0, filter(s(z0), x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(s(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(filter(s(z0), x2, x3))) MARK(filter(filter(z0, z1, z2), filter(x1, x2, x3), x4)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(sieve(z0), filter(x1, x2, x3), x4)) -> c11(A__FILTER(a__sieve(mark(z0)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(nats(z0), filter(x1, x2, x3), x4)) -> c11(A__FILTER(a__nats(mark(z0)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(zprimes, filter(x1, x2, x3), x4)) -> c11(A__FILTER(a__zprimes, a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(cons(z0, z1), filter(x1, x2, x3), x4)) -> c11(A__FILTER(cons(mark(z0), z1), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(x0, filter(x1, x2, x3), x4)) -> c11(MARK(filter(x1, x2, x3))) MARK(filter(0, filter(x1, x2, x3), x4)) -> c11(MARK(filter(x1, x2, x3))) MARK(filter(s(z0), filter(x1, x2, x3), x4)) -> c11(MARK(filter(x1, x2, x3))) MARK(filter(x0, sieve(filter(z0, z1, z2)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(a__filter(mark(z0), 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sieve(x1), x2)) -> c11(A__FILTER(a__nats(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(zprimes, sieve(x1), x2)) -> c11(A__FILTER(a__zprimes, a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(cons(z0, z1), sieve(x1), x2)) -> c11(A__FILTER(cons(mark(z0), z1), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(x0, sieve(x1), x2)) -> c11(MARK(sieve(x1))) MARK(filter(x0, sieve(0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(0), mark(x2))) MARK(filter(0, sieve(x1), x2)) -> c11(MARK(sieve(x1))) MARK(filter(s(z0), sieve(x1), x2)) -> c11(MARK(sieve(x1))) MARK(filter(x0, nats(filter(z0, z1, z2)), x2)) -> c11(A__FILTER(mark(x0), a__nats(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(nats(filter(z0, z1, z2)))) MARK(filter(x0, nats(sieve(z0)), x2)) -> c11(A__FILTER(mark(x0), a__nats(a__sieve(mark(z0))), mark(x2)), MARK(nats(sieve(z0)))) MARK(filter(x0, nats(nats(z0)), x2)) -> c11(A__FILTER(mark(x0), a__nats(a__nats(mark(z0))), mark(x2)), MARK(nats(nats(z0)))) MARK(filter(x0, nats(zprimes), x2)) -> c11(A__FILTER(mark(x0), a__nats(a__zprimes), mark(x2)), MARK(nats(zprimes))) MARK(filter(x0, nats(cons(z0, z1)), x2)) -> c11(A__FILTER(mark(x0), a__nats(cons(mark(z0), z1)), mark(x2)), MARK(nats(cons(z0, z1)))) MARK(filter(x0, nats(0), x2)) -> c11(A__FILTER(mark(x0), a__nats(0), mark(x2)), MARK(nats(0))) MARK(filter(x0, nats(s(z0)), x2)) -> c11(A__FILTER(mark(x0), a__nats(s(mark(z0))), mark(x2)), MARK(nats(s(z0)))) MARK(filter(filter(z0, z1, z2), nats(x1), x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(sieve(z0), nats(x1), x2)) -> c11(A__FILTER(a__sieve(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(nats(z0), nats(x1), x2)) -> c11(A__FILTER(a__nats(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(zprimes, nats(x1), x2)) -> c11(A__FILTER(a__zprimes, a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(cons(z0, z1), nats(x1), x2)) -> c11(A__FILTER(cons(mark(z0), z1), a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(x0, nats(x1), x2)) -> c11(MARK(nats(x1))) MARK(filter(0, nats(x1), x2)) -> c11(MARK(nats(x1))) MARK(filter(s(z0), nats(x1), x2)) -> c11(MARK(nats(x1))) MARK(filter(x0, zprimes, x1)) -> c11(A__FILTER(mark(x0), a__sieve(a__nats(s(s(0)))), mark(x1)), MARK(zprimes)) MARK(filter(x0, zprimes, x1)) -> c11(A__FILTER(mark(x0), zprimes, mark(x1)), MARK(zprimes)) MARK(filter(filter(z0, z1, z2), zprimes, x1)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(sieve(z0), zprimes, x1)) -> c11(A__FILTER(a__sieve(mark(z0)), a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(nats(z0), zprimes, x1)) -> c11(A__FILTER(a__nats(mark(z0)), a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(zprimes, zprimes, x1)) -> c11(A__FILTER(a__zprimes, a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(cons(z0, z1), zprimes, x1)) -> c11(A__FILTER(cons(mark(z0), z1), a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(0, zprimes, x1)) -> c11(A__FILTER(0, a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(s(z0), zprimes, x1)) -> c11(A__FILTER(s(mark(z0)), a__zprimes, mark(x1)), MARK(zprimes)) S tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, 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MARK(filter(x0, sieve(x1), x2)) -> c11(MARK(sieve(x1))) MARK(filter(x0, sieve(0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(0), mark(x2))) MARK(filter(0, sieve(x1), x2)) -> c11(MARK(sieve(x1))) MARK(filter(s(z0), sieve(x1), x2)) -> c11(MARK(sieve(x1))) MARK(filter(x0, nats(filter(z0, z1, z2)), x2)) -> c11(A__FILTER(mark(x0), a__nats(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(nats(filter(z0, z1, z2)))) MARK(filter(x0, nats(sieve(z0)), x2)) -> c11(A__FILTER(mark(x0), a__nats(a__sieve(mark(z0))), mark(x2)), MARK(nats(sieve(z0)))) MARK(filter(x0, nats(nats(z0)), x2)) -> c11(A__FILTER(mark(x0), a__nats(a__nats(mark(z0))), mark(x2)), MARK(nats(nats(z0)))) MARK(filter(x0, nats(zprimes), x2)) -> c11(A__FILTER(mark(x0), a__nats(a__zprimes), mark(x2)), MARK(nats(zprimes))) MARK(filter(x0, nats(cons(z0, z1)), x2)) -> c11(A__FILTER(mark(x0), a__nats(cons(mark(z0), z1)), mark(x2)), MARK(nats(cons(z0, z1)))) MARK(filter(x0, nats(0), x2)) -> c11(A__FILTER(mark(x0), a__nats(0), mark(x2)), MARK(nats(0))) MARK(filter(x0, nats(s(z0)), x2)) -> c11(A__FILTER(mark(x0), a__nats(s(mark(z0))), mark(x2)), MARK(nats(s(z0)))) MARK(filter(filter(z0, z1, z2), nats(x1), x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(sieve(z0), nats(x1), x2)) -> c11(A__FILTER(a__sieve(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(nats(z0), nats(x1), x2)) -> c11(A__FILTER(a__nats(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(zprimes, nats(x1), x2)) -> c11(A__FILTER(a__zprimes, a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(cons(z0, z1), nats(x1), x2)) -> c11(A__FILTER(cons(mark(z0), z1), a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(x0, nats(x1), x2)) -> c11(MARK(nats(x1))) MARK(filter(0, nats(x1), x2)) -> c11(MARK(nats(x1))) MARK(filter(s(z0), nats(x1), x2)) -> c11(MARK(nats(x1))) MARK(filter(x0, zprimes, x1)) -> c11(A__FILTER(mark(x0), a__sieve(a__nats(s(s(0)))), mark(x1)), MARK(zprimes)) MARK(filter(x0, zprimes, x1)) -> c11(A__FILTER(mark(x0), zprimes, mark(x1)), MARK(zprimes)) MARK(filter(filter(z0, z1, z2), zprimes, x1)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(sieve(z0), zprimes, x1)) -> c11(A__FILTER(a__sieve(mark(z0)), a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(nats(z0), zprimes, x1)) -> c11(A__FILTER(a__nats(mark(z0)), a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(zprimes, zprimes, x1)) -> c11(A__FILTER(a__zprimes, a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(cons(z0, z1), zprimes, x1)) -> c11(A__FILTER(cons(mark(z0), z1), a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(0, zprimes, x1)) -> c11(A__FILTER(0, a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(s(z0), zprimes, x1)) -> c11(A__FILTER(s(mark(z0)), a__zprimes, mark(x1)), MARK(zprimes)) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c14_2, c15_1, c16_1, c18_1, c8_1, c10_1, c11_2, c11_1, c12_2, c12_1, c13_2, c13_1, c8_2, c10_2 ---------------------------------------- (201) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (202) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(x0, 0, filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, sieve(x2))) -> c10(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x0)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, sieve(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, sieve(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, sieve(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(0)), MARK(x0)) MARK(filter(x0, x1, sieve(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), sieve(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__sieve(mark(x2))), MARK(x0)) 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MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c10(A__FILTER(cons(mark(z0), z1), 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c10(A__FILTER(mark(x0), mark(x1), a__sieve(0)), MARK(x0)) MARK(filter(x0, x1, sieve(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), sieve(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, sieve(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, sieve(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__sieve(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, sieve(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), 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mark(x1), a__nats(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, nats(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, nats(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, nats(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, nats(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(0)), MARK(x0)) MARK(filter(x0, x1, nats(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), nats(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), nats(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), nats(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, nats(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), nats(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, nats(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__nats(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, nats(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, nats(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, nats(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), a__nats(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, nats(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__nats(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), nats(x2))) -> c10(MARK(x0)) MARK(filter(x0, 0, nats(x2))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, nats(x2))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, zprimes)) 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sieve(x1), x2)) -> c11(A__FILTER(a__nats(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(zprimes, sieve(x1), x2)) -> c11(A__FILTER(a__zprimes, a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(cons(z0, z1), sieve(x1), x2)) -> c11(A__FILTER(cons(mark(z0), z1), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(x0, sieve(x1), x2)) -> c11(MARK(sieve(x1))) MARK(filter(x0, sieve(0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(0), mark(x2))) MARK(filter(0, sieve(x1), x2)) -> c11(MARK(sieve(x1))) MARK(filter(s(z0), sieve(x1), x2)) -> c11(MARK(sieve(x1))) MARK(filter(x0, nats(filter(z0, z1, z2)), x2)) -> c11(A__FILTER(mark(x0), a__nats(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(nats(filter(z0, z1, z2)))) MARK(filter(x0, nats(sieve(z0)), x2)) -> c11(A__FILTER(mark(x0), a__nats(a__sieve(mark(z0))), mark(x2)), MARK(nats(sieve(z0)))) MARK(filter(x0, nats(nats(z0)), x2)) -> c11(A__FILTER(mark(x0), a__nats(a__nats(mark(z0))), mark(x2)), MARK(nats(nats(z0)))) MARK(filter(x0, nats(zprimes), x2)) -> c11(A__FILTER(mark(x0), a__nats(a__zprimes), mark(x2)), MARK(nats(zprimes))) MARK(filter(x0, nats(cons(z0, z1)), x2)) -> c11(A__FILTER(mark(x0), a__nats(cons(mark(z0), z1)), mark(x2)), MARK(nats(cons(z0, z1)))) MARK(filter(x0, nats(0), x2)) -> c11(A__FILTER(mark(x0), a__nats(0), mark(x2)), MARK(nats(0))) MARK(filter(x0, nats(s(z0)), x2)) -> c11(A__FILTER(mark(x0), a__nats(s(mark(z0))), mark(x2)), MARK(nats(s(z0)))) MARK(filter(filter(z0, z1, z2), nats(x1), x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(sieve(z0), nats(x1), x2)) -> c11(A__FILTER(a__sieve(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(nats(z0), nats(x1), x2)) -> c11(A__FILTER(a__nats(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(zprimes, nats(x1), x2)) -> c11(A__FILTER(a__zprimes, a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(cons(z0, z1), nats(x1), x2)) -> c11(A__FILTER(cons(mark(z0), z1), a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(x0, nats(x1), x2)) -> c11(MARK(nats(x1))) MARK(filter(0, nats(x1), x2)) -> c11(MARK(nats(x1))) MARK(filter(s(z0), nats(x1), x2)) -> c11(MARK(nats(x1))) MARK(filter(x0, zprimes, x1)) -> c11(A__FILTER(mark(x0), a__sieve(a__nats(s(s(0)))), mark(x1)), MARK(zprimes)) MARK(filter(filter(z0, z1, z2), zprimes, x1)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(sieve(z0), zprimes, x1)) -> c11(A__FILTER(a__sieve(mark(z0)), a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(nats(z0), zprimes, x1)) -> c11(A__FILTER(a__nats(mark(z0)), a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(zprimes, zprimes, x1)) -> c11(A__FILTER(a__zprimes, a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(cons(z0, z1), zprimes, x1)) -> c11(A__FILTER(cons(mark(z0), z1), a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(x0, zprimes, x1)) -> c11(MARK(zprimes)) MARK(filter(0, zprimes, x1)) -> c11(MARK(zprimes)) MARK(filter(s(z0), zprimes, x1)) -> c11(MARK(zprimes)) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c14_2, c15_1, c16_1, c18_1, c8_1, c10_1, c11_2, c11_1, c12_2, c12_1, c13_2, c13_1, c8_2, c10_2 ---------------------------------------- (203) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(filter(x0, s(z0), x2)) -> c11(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(s(z0))) by MARK(filter(x0, s(x1), filter(z0, z1, z2))) -> c11(A__FILTER(mark(x0), s(mark(x1)), a__filter(mark(z0), mark(z1), mark(z2))), MARK(s(x1))) MARK(filter(x0, s(x1), sieve(z0))) -> c11(A__FILTER(mark(x0), s(mark(x1)), a__sieve(mark(z0))), MARK(s(x1))) MARK(filter(x0, s(x1), nats(z0))) -> c11(A__FILTER(mark(x0), s(mark(x1)), a__nats(mark(z0))), MARK(s(x1))) MARK(filter(x0, s(x1), zprimes)) -> c11(A__FILTER(mark(x0), s(mark(x1)), a__zprimes), MARK(s(x1))) MARK(filter(x0, s(x1), cons(z0, z1))) -> c11(A__FILTER(mark(x0), s(mark(x1)), cons(mark(z0), z1)), MARK(s(x1))) MARK(filter(x0, s(x1), 0)) -> c11(A__FILTER(mark(x0), s(mark(x1)), 0), MARK(s(x1))) MARK(filter(x0, s(x1), s(z0))) -> c11(A__FILTER(mark(x0), s(mark(x1)), s(mark(z0))), MARK(s(x1))) MARK(filter(x0, s(filter(z0, z1, z2)), x2)) -> c11(A__FILTER(mark(x0), s(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(s(filter(z0, z1, z2)))) MARK(filter(x0, s(sieve(z0)), x2)) -> c11(A__FILTER(mark(x0), s(a__sieve(mark(z0))), mark(x2)), MARK(s(sieve(z0)))) MARK(filter(x0, s(nats(z0)), x2)) -> c11(A__FILTER(mark(x0), s(a__nats(mark(z0))), mark(x2)), MARK(s(nats(z0)))) MARK(filter(x0, s(zprimes), x2)) -> c11(A__FILTER(mark(x0), s(a__zprimes), mark(x2)), MARK(s(zprimes))) MARK(filter(x0, s(cons(z0, z1)), x2)) -> c11(A__FILTER(mark(x0), s(cons(mark(z0), z1)), mark(x2)), MARK(s(cons(z0, z1)))) MARK(filter(x0, s(0), x2)) -> c11(A__FILTER(mark(x0), s(0), mark(x2)), MARK(s(0))) MARK(filter(x0, s(s(z0)), x2)) -> c11(A__FILTER(mark(x0), s(s(mark(z0))), mark(x2)), MARK(s(s(z0)))) MARK(filter(filter(z0, z1, z2), s(x1), x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), s(mark(x1)), mark(x2)), MARK(s(x1))) MARK(filter(sieve(z0), s(x1), x2)) -> c11(A__FILTER(a__sieve(mark(z0)), s(mark(x1)), mark(x2)), MARK(s(x1))) MARK(filter(nats(z0), s(x1), x2)) -> c11(A__FILTER(a__nats(mark(z0)), s(mark(x1)), mark(x2)), MARK(s(x1))) MARK(filter(zprimes, s(x1), x2)) -> c11(A__FILTER(a__zprimes, s(mark(x1)), mark(x2)), MARK(s(x1))) MARK(filter(cons(z0, z1), s(x1), x2)) -> c11(A__FILTER(cons(mark(z0), z1), s(mark(x1)), mark(x2)), MARK(s(x1))) MARK(filter(0, s(x1), x2)) -> c11(A__FILTER(0, s(mark(x1)), mark(x2)), MARK(s(x1))) MARK(filter(s(z0), s(x1), x2)) -> c11(A__FILTER(s(mark(z0)), s(mark(x1)), mark(x2)), MARK(s(x1))) MARK(filter(x0, s(x1), x2)) -> c11(MARK(s(x1))) ---------------------------------------- (204) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(x0, 0, filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, sieve(x2))) -> c10(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x0)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, sieve(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, sieve(cons(z0, z1)))) 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MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(x0, 0, filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, sieve(x2))) -> c10(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x0)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, sieve(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, sieve(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, sieve(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(0)), MARK(x0)) MARK(filter(x0, x1, sieve(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), sieve(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, sieve(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, sieve(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__sieve(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, sieve(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, sieve(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, sieve(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), a__sieve(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, sieve(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__sieve(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), sieve(x2))) -> c10(MARK(x0)) MARK(filter(x0, 0, sieve(x2))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, sieve(x2))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(mark(x2)), nats(s(mark(x2))))), MARK(x0)) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), nats(mark(x2))), MARK(x0)) MARK(filter(x0, x1, nats(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, nats(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, nats(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, nats(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, nats(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, nats(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(0)), MARK(x0)) MARK(filter(x0, x1, nats(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), nats(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), nats(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), nats(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, nats(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), nats(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, nats(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__nats(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, nats(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, nats(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, nats(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), a__nats(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, nats(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__nats(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), nats(x2))) -> c10(MARK(x0)) 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c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(sieve(z0), filter(x1, x2, x3), x4)) -> c11(A__FILTER(a__sieve(mark(z0)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(nats(z0), filter(x1, x2, x3), x4)) -> c11(A__FILTER(a__nats(mark(z0)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(zprimes, filter(x1, x2, x3), x4)) -> c11(A__FILTER(a__zprimes, a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(cons(z0, z1), filter(x1, x2, x3), x4)) -> c11(A__FILTER(cons(mark(z0), z1), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(x0, filter(x1, x2, x3), x4)) -> c11(MARK(filter(x1, x2, x3))) MARK(filter(0, filter(x1, x2, x3), x4)) -> c11(MARK(filter(x1, x2, x3))) MARK(filter(s(z0), filter(x1, x2, x3), x4)) -> c11(MARK(filter(x1, x2, x3))) MARK(filter(x0, sieve(filter(z0, z1, z2)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(sieve(filter(z0, z1, z2)))) MARK(filter(x0, sieve(sieve(z0)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(a__sieve(mark(z0))), mark(x2)), MARK(sieve(sieve(z0)))) MARK(filter(x0, sieve(nats(z0)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(a__nats(mark(z0))), mark(x2)), MARK(sieve(nats(z0)))) MARK(filter(x0, sieve(zprimes), x2)) -> c11(A__FILTER(mark(x0), a__sieve(a__zprimes), mark(x2)), MARK(sieve(zprimes))) MARK(filter(x0, sieve(cons(z0, z1)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(cons(mark(z0), z1)), mark(x2)), MARK(sieve(cons(z0, z1)))) MARK(filter(x0, sieve(s(z0)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(s(mark(z0))), mark(x2)), MARK(sieve(s(z0)))) MARK(filter(filter(z0, z1, z2), sieve(x1), x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(sieve(z0), sieve(x1), x2)) -> c11(A__FILTER(a__sieve(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(nats(z0), sieve(x1), x2)) -> c11(A__FILTER(a__nats(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(zprimes, sieve(x1), x2)) -> c11(A__FILTER(a__zprimes, a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(cons(z0, z1), sieve(x1), x2)) -> c11(A__FILTER(cons(mark(z0), z1), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(x0, sieve(x1), x2)) -> c11(MARK(sieve(x1))) MARK(filter(x0, sieve(0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(0), mark(x2))) MARK(filter(0, sieve(x1), x2)) -> c11(MARK(sieve(x1))) MARK(filter(s(z0), sieve(x1), x2)) -> c11(MARK(sieve(x1))) MARK(filter(x0, nats(filter(z0, z1, z2)), x2)) -> c11(A__FILTER(mark(x0), a__nats(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(nats(filter(z0, z1, z2)))) MARK(filter(x0, nats(sieve(z0)), x2)) -> c11(A__FILTER(mark(x0), a__nats(a__sieve(mark(z0))), mark(x2)), MARK(nats(sieve(z0)))) MARK(filter(x0, nats(nats(z0)), x2)) -> c11(A__FILTER(mark(x0), a__nats(a__nats(mark(z0))), mark(x2)), MARK(nats(nats(z0)))) MARK(filter(x0, nats(zprimes), x2)) -> c11(A__FILTER(mark(x0), a__nats(a__zprimes), mark(x2)), MARK(nats(zprimes))) MARK(filter(x0, nats(cons(z0, z1)), x2)) -> c11(A__FILTER(mark(x0), a__nats(cons(mark(z0), z1)), mark(x2)), MARK(nats(cons(z0, z1)))) MARK(filter(x0, nats(0), x2)) -> c11(A__FILTER(mark(x0), a__nats(0), mark(x2)), MARK(nats(0))) MARK(filter(x0, nats(s(z0)), x2)) -> c11(A__FILTER(mark(x0), a__nats(s(mark(z0))), mark(x2)), MARK(nats(s(z0)))) MARK(filter(filter(z0, z1, z2), nats(x1), x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(sieve(z0), nats(x1), x2)) -> c11(A__FILTER(a__sieve(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(nats(z0), nats(x1), x2)) -> c11(A__FILTER(a__nats(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(zprimes, nats(x1), x2)) -> c11(A__FILTER(a__zprimes, a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(cons(z0, z1), nats(x1), x2)) -> c11(A__FILTER(cons(mark(z0), z1), a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(x0, nats(x1), x2)) -> c11(MARK(nats(x1))) MARK(filter(0, nats(x1), x2)) -> c11(MARK(nats(x1))) MARK(filter(s(z0), nats(x1), x2)) -> c11(MARK(nats(x1))) MARK(filter(x0, zprimes, x1)) -> c11(A__FILTER(mark(x0), a__sieve(a__nats(s(s(0)))), mark(x1)), MARK(zprimes)) MARK(filter(filter(z0, z1, z2), zprimes, x1)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(sieve(z0), zprimes, x1)) -> c11(A__FILTER(a__sieve(mark(z0)), a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(nats(z0), zprimes, x1)) -> c11(A__FILTER(a__nats(mark(z0)), a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(zprimes, zprimes, x1)) -> c11(A__FILTER(a__zprimes, a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(cons(z0, z1), zprimes, x1)) -> c11(A__FILTER(cons(mark(z0), z1), a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(x0, zprimes, x1)) -> c11(MARK(zprimes)) MARK(filter(0, zprimes, x1)) -> c11(MARK(zprimes)) MARK(filter(s(z0), zprimes, x1)) -> c11(MARK(zprimes)) MARK(filter(x0, s(filter(z0, z1, z2)), x2)) -> c11(A__FILTER(mark(x0), s(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(s(filter(z0, z1, z2)))) MARK(filter(x0, s(sieve(z0)), x2)) -> c11(A__FILTER(mark(x0), s(a__sieve(mark(z0))), mark(x2)), MARK(s(sieve(z0)))) MARK(filter(x0, s(nats(z0)), x2)) -> c11(A__FILTER(mark(x0), s(a__nats(mark(z0))), mark(x2)), MARK(s(nats(z0)))) MARK(filter(x0, s(zprimes), x2)) -> c11(A__FILTER(mark(x0), s(a__zprimes), mark(x2)), MARK(s(zprimes))) MARK(filter(x0, s(cons(z0, z1)), x2)) -> c11(A__FILTER(mark(x0), s(cons(mark(z0), z1)), mark(x2)), MARK(s(cons(z0, z1)))) MARK(filter(x0, s(0), x2)) -> c11(A__FILTER(mark(x0), s(0), mark(x2)), MARK(s(0))) MARK(filter(x0, s(s(z0)), x2)) -> c11(A__FILTER(mark(x0), s(s(mark(z0))), mark(x2)), MARK(s(s(z0)))) MARK(filter(filter(z0, z1, z2), s(x1), x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), s(mark(x1)), mark(x2)), MARK(s(x1))) MARK(filter(sieve(z0), s(x1), x2)) -> c11(A__FILTER(a__sieve(mark(z0)), s(mark(x1)), mark(x2)), MARK(s(x1))) MARK(filter(nats(z0), s(x1), x2)) -> c11(A__FILTER(a__nats(mark(z0)), s(mark(x1)), mark(x2)), MARK(s(x1))) MARK(filter(zprimes, s(x1), x2)) -> c11(A__FILTER(a__zprimes, s(mark(x1)), mark(x2)), MARK(s(x1))) MARK(filter(cons(z0, z1), s(x1), x2)) -> c11(A__FILTER(cons(mark(z0), z1), s(mark(x1)), mark(x2)), MARK(s(x1))) MARK(filter(0, s(x1), x2)) -> c11(A__FILTER(0, s(mark(x1)), mark(x2)), MARK(s(x1))) MARK(filter(s(z0), s(x1), x2)) -> c11(A__FILTER(s(mark(z0)), s(mark(x1)), mark(x2)), MARK(s(x1))) MARK(filter(x0, s(x1), x2)) -> c11(MARK(s(x1))) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c14_2, c15_1, c16_1, c18_1, c8_1, c10_1, c11_2, c11_1, c12_2, c12_1, c13_2, c13_1, c8_2, c10_2 ---------------------------------------- (205) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (206) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, 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-> c11(A__FILTER(a__nats(mark(z0)), a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(zprimes, zprimes, x1)) -> c11(A__FILTER(a__zprimes, a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(cons(z0, z1), zprimes, x1)) -> c11(A__FILTER(cons(mark(z0), z1), a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(x0, zprimes, x1)) -> c11(MARK(zprimes)) MARK(filter(0, zprimes, x1)) -> c11(MARK(zprimes)) MARK(filter(s(z0), zprimes, x1)) -> c11(MARK(zprimes)) MARK(filter(x0, s(filter(z0, z1, z2)), x2)) -> c11(A__FILTER(mark(x0), s(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(s(filter(z0, z1, z2)))) MARK(filter(x0, s(sieve(z0)), x2)) -> c11(A__FILTER(mark(x0), s(a__sieve(mark(z0))), mark(x2)), MARK(s(sieve(z0)))) MARK(filter(x0, s(nats(z0)), x2)) -> c11(A__FILTER(mark(x0), s(a__nats(mark(z0))), mark(x2)), MARK(s(nats(z0)))) MARK(filter(x0, s(zprimes), x2)) -> c11(A__FILTER(mark(x0), s(a__zprimes), mark(x2)), MARK(s(zprimes))) MARK(filter(x0, s(cons(z0, z1)), x2)) -> c11(A__FILTER(mark(x0), s(cons(mark(z0), z1)), mark(x2)), MARK(s(cons(z0, z1)))) MARK(filter(x0, s(0), x2)) -> c11(A__FILTER(mark(x0), s(0), mark(x2)), MARK(s(0))) MARK(filter(x0, s(s(z0)), x2)) -> c11(A__FILTER(mark(x0), s(s(mark(z0))), mark(x2)), MARK(s(s(z0)))) MARK(filter(filter(z0, z1, z2), s(x1), x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), s(mark(x1)), mark(x2)), MARK(s(x1))) MARK(filter(sieve(z0), s(x1), x2)) -> c11(A__FILTER(a__sieve(mark(z0)), s(mark(x1)), mark(x2)), MARK(s(x1))) MARK(filter(nats(z0), s(x1), x2)) -> c11(A__FILTER(a__nats(mark(z0)), s(mark(x1)), mark(x2)), MARK(s(x1))) MARK(filter(zprimes, s(x1), x2)) -> c11(A__FILTER(a__zprimes, s(mark(x1)), mark(x2)), MARK(s(x1))) MARK(filter(cons(z0, z1), s(x1), x2)) -> c11(A__FILTER(cons(mark(z0), z1), s(mark(x1)), mark(x2)), MARK(s(x1))) MARK(filter(x0, s(x1), x2)) -> c11(MARK(s(x1))) MARK(filter(0, s(x1), x2)) -> c11(MARK(s(x1))) MARK(filter(s(z0), s(x1), x2)) -> c11(MARK(s(x1))) S tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(x0, 0, filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, sieve(x2))) -> c10(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x0)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, sieve(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, sieve(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, sieve(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(0)), MARK(x0)) MARK(filter(x0, x1, sieve(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), sieve(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, sieve(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), sieve(x2))) -> 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c11(A__FILTER(mark(x0), mark(x1), s(cons(mark(z0), z1))), MARK(x1)) MARK(filter(x0, x1, s(0))) -> c11(A__FILTER(mark(x0), mark(x1), s(0)), MARK(x1)) MARK(filter(x0, x1, s(s(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), s(s(mark(z0)))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), s(x2))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), s(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), s(x2))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), s(mark(x2))), MARK(sieve(z0))) MARK(filter(x0, nats(z0), s(x2))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), s(mark(x2))), MARK(nats(z0))) MARK(filter(x0, zprimes, s(x2))) -> c11(A__FILTER(mark(x0), a__zprimes, s(mark(x2))), MARK(zprimes)) MARK(filter(x0, s(z0), s(x2))) -> c11(A__FILTER(mark(x0), s(mark(z0)), s(mark(x2))), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, s(x2))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), s(mark(x2))), MARK(x1)) MARK(filter(sieve(z0), x1, s(x2))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), s(mark(x2))), MARK(x1)) MARK(filter(nats(z0), x1, s(x2))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), s(mark(x2))), MARK(x1)) MARK(filter(zprimes, x1, s(x2))) -> c11(A__FILTER(a__zprimes, mark(x1), s(mark(x2))), MARK(x1)) MARK(filter(cons(z0, z1), x1, s(x2))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), s(mark(x2))), MARK(x1)) MARK(filter(x0, cons(z0, z1), s(x2))) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, s(x2))) -> c11(MARK(x1)) MARK(filter(s(z0), x1, s(x2))) -> c11(MARK(x1)) MARK(filter(x0, filter(x1, x2, filter(z0, z1, z2)), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__filter(mark(z0), mark(z1), mark(z2))), mark(x4)), MARK(filter(x1, x2, filter(z0, z1, z2)))) MARK(filter(x0, filter(x1, x2, sieve(z0)), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__sieve(mark(z0))), mark(x4)), MARK(filter(x1, x2, sieve(z0)))) MARK(filter(x0, filter(x1, x2, nats(z0)), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__nats(mark(z0))), mark(x4)), MARK(filter(x1, x2, nats(z0)))) MARK(filter(x0, filter(x1, x2, zprimes), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__zprimes), mark(x4)), MARK(filter(x1, x2, zprimes))) MARK(filter(x0, filter(x1, x2, cons(z0, z1)), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), cons(mark(z0), z1)), mark(x4)), MARK(filter(x1, x2, cons(z0, z1)))) MARK(filter(x0, filter(x1, x2, 0), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), 0), mark(x4)), MARK(filter(x1, x2, 0))) MARK(filter(x0, filter(x1, x2, s(z0)), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), s(mark(z0))), mark(x4)), MARK(filter(x1, x2, s(z0)))) MARK(filter(x0, filter(x1, filter(z0, z1, z2), x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), a__filter(mark(z0), mark(z1), mark(z2)), mark(x3)), mark(x4)), MARK(filter(x1, filter(z0, z1, z2), x3))) MARK(filter(x0, filter(x1, sieve(z0), x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), a__sieve(mark(z0)), mark(x3)), mark(x4)), MARK(filter(x1, sieve(z0), x3))) MARK(filter(x0, filter(x1, nats(z0), x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), a__nats(mark(z0)), mark(x3)), mark(x4)), MARK(filter(x1, nats(z0), x3))) MARK(filter(x0, filter(x1, zprimes, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), a__zprimes, mark(x3)), mark(x4)), MARK(filter(x1, zprimes, x3))) MARK(filter(x0, filter(x1, cons(z0, z1), x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), cons(mark(z0), z1), mark(x3)), mark(x4)), MARK(filter(x1, cons(z0, z1), x3))) MARK(filter(x0, filter(x1, 0, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), 0, mark(x3)), mark(x4)), MARK(filter(x1, 0, x3))) MARK(filter(x0, filter(x1, s(z0), x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), s(mark(z0)), mark(x3)), mark(x4)), MARK(filter(x1, s(z0), x3))) MARK(filter(x0, filter(filter(z0, z1, z2), x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x2), mark(x3)), mark(x4)), MARK(filter(filter(z0, z1, z2), x2, x3))) MARK(filter(x0, filter(sieve(z0), x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(a__sieve(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(filter(sieve(z0), x2, x3))) MARK(filter(x0, filter(nats(z0), x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(a__nats(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(filter(nats(z0), x2, x3))) MARK(filter(x0, filter(zprimes, x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(a__zprimes, mark(x2), mark(x3)), mark(x4)), MARK(filter(zprimes, x2, x3))) MARK(filter(x0, filter(cons(z0, z1), x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(cons(mark(z0), z1), mark(x2), mark(x3)), mark(x4)), MARK(filter(cons(z0, z1), x2, x3))) MARK(filter(x0, filter(0, x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(0, mark(x2), mark(x3)), mark(x4)), MARK(filter(0, x2, x3))) MARK(filter(x0, filter(s(z0), x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(s(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(filter(s(z0), x2, x3))) MARK(filter(filter(z0, z1, z2), filter(x1, x2, x3), x4)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(sieve(z0), filter(x1, x2, x3), x4)) -> c11(A__FILTER(a__sieve(mark(z0)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(nats(z0), filter(x1, x2, x3), x4)) -> c11(A__FILTER(a__nats(mark(z0)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(zprimes, filter(x1, x2, x3), x4)) -> c11(A__FILTER(a__zprimes, a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(cons(z0, z1), filter(x1, x2, x3), x4)) -> c11(A__FILTER(cons(mark(z0), z1), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(x0, filter(x1, x2, x3), x4)) -> c11(MARK(filter(x1, x2, x3))) MARK(filter(0, filter(x1, x2, x3), x4)) -> c11(MARK(filter(x1, x2, x3))) MARK(filter(s(z0), filter(x1, x2, x3), x4)) -> c11(MARK(filter(x1, x2, x3))) MARK(filter(x0, sieve(filter(z0, z1, z2)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(sieve(filter(z0, z1, z2)))) MARK(filter(x0, sieve(sieve(z0)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(a__sieve(mark(z0))), mark(x2)), MARK(sieve(sieve(z0)))) MARK(filter(x0, sieve(nats(z0)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(a__nats(mark(z0))), mark(x2)), MARK(sieve(nats(z0)))) MARK(filter(x0, sieve(zprimes), x2)) -> c11(A__FILTER(mark(x0), a__sieve(a__zprimes), mark(x2)), MARK(sieve(zprimes))) MARK(filter(x0, sieve(cons(z0, z1)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(cons(mark(z0), z1)), mark(x2)), MARK(sieve(cons(z0, z1)))) MARK(filter(x0, sieve(s(z0)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(s(mark(z0))), mark(x2)), MARK(sieve(s(z0)))) MARK(filter(filter(z0, z1, z2), sieve(x1), x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(sieve(z0), sieve(x1), x2)) -> c11(A__FILTER(a__sieve(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(nats(z0), sieve(x1), x2)) -> c11(A__FILTER(a__nats(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(zprimes, sieve(x1), x2)) -> c11(A__FILTER(a__zprimes, a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(cons(z0, z1), sieve(x1), x2)) -> c11(A__FILTER(cons(mark(z0), z1), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(x0, sieve(x1), x2)) -> c11(MARK(sieve(x1))) MARK(filter(x0, sieve(0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(0), mark(x2))) MARK(filter(0, sieve(x1), x2)) -> c11(MARK(sieve(x1))) MARK(filter(s(z0), sieve(x1), x2)) -> c11(MARK(sieve(x1))) MARK(filter(x0, nats(filter(z0, z1, z2)), x2)) -> c11(A__FILTER(mark(x0), a__nats(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(nats(filter(z0, z1, z2)))) MARK(filter(x0, nats(sieve(z0)), x2)) -> c11(A__FILTER(mark(x0), a__nats(a__sieve(mark(z0))), mark(x2)), MARK(nats(sieve(z0)))) MARK(filter(x0, nats(nats(z0)), x2)) -> c11(A__FILTER(mark(x0), a__nats(a__nats(mark(z0))), mark(x2)), MARK(nats(nats(z0)))) MARK(filter(x0, nats(zprimes), x2)) -> c11(A__FILTER(mark(x0), a__nats(a__zprimes), mark(x2)), MARK(nats(zprimes))) MARK(filter(x0, nats(cons(z0, z1)), x2)) -> c11(A__FILTER(mark(x0), a__nats(cons(mark(z0), z1)), mark(x2)), MARK(nats(cons(z0, z1)))) MARK(filter(x0, nats(0), x2)) -> c11(A__FILTER(mark(x0), a__nats(0), mark(x2)), MARK(nats(0))) MARK(filter(x0, nats(s(z0)), x2)) -> c11(A__FILTER(mark(x0), a__nats(s(mark(z0))), mark(x2)), MARK(nats(s(z0)))) MARK(filter(filter(z0, z1, z2), nats(x1), x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(sieve(z0), nats(x1), x2)) -> c11(A__FILTER(a__sieve(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(nats(z0), nats(x1), x2)) -> c11(A__FILTER(a__nats(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(zprimes, nats(x1), x2)) -> c11(A__FILTER(a__zprimes, a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(cons(z0, z1), nats(x1), x2)) -> c11(A__FILTER(cons(mark(z0), z1), a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(x0, nats(x1), x2)) -> c11(MARK(nats(x1))) MARK(filter(0, nats(x1), x2)) -> c11(MARK(nats(x1))) MARK(filter(s(z0), nats(x1), x2)) -> c11(MARK(nats(x1))) MARK(filter(x0, zprimes, x1)) -> c11(A__FILTER(mark(x0), a__sieve(a__nats(s(s(0)))), mark(x1)), MARK(zprimes)) MARK(filter(filter(z0, z1, z2), zprimes, x1)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(sieve(z0), zprimes, x1)) -> c11(A__FILTER(a__sieve(mark(z0)), a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(nats(z0), zprimes, x1)) -> c11(A__FILTER(a__nats(mark(z0)), a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(zprimes, zprimes, x1)) -> c11(A__FILTER(a__zprimes, a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(cons(z0, z1), zprimes, x1)) -> c11(A__FILTER(cons(mark(z0), z1), a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(x0, zprimes, x1)) -> c11(MARK(zprimes)) MARK(filter(0, zprimes, x1)) -> c11(MARK(zprimes)) MARK(filter(s(z0), zprimes, x1)) -> c11(MARK(zprimes)) MARK(filter(x0, s(filter(z0, z1, z2)), x2)) -> c11(A__FILTER(mark(x0), s(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(s(filter(z0, z1, z2)))) MARK(filter(x0, s(sieve(z0)), x2)) -> c11(A__FILTER(mark(x0), s(a__sieve(mark(z0))), mark(x2)), MARK(s(sieve(z0)))) MARK(filter(x0, s(nats(z0)), x2)) -> c11(A__FILTER(mark(x0), s(a__nats(mark(z0))), mark(x2)), MARK(s(nats(z0)))) MARK(filter(x0, s(zprimes), x2)) -> c11(A__FILTER(mark(x0), s(a__zprimes), mark(x2)), MARK(s(zprimes))) MARK(filter(x0, s(cons(z0, z1)), x2)) -> c11(A__FILTER(mark(x0), s(cons(mark(z0), z1)), mark(x2)), MARK(s(cons(z0, z1)))) MARK(filter(x0, s(0), x2)) -> c11(A__FILTER(mark(x0), s(0), mark(x2)), MARK(s(0))) MARK(filter(x0, s(s(z0)), x2)) -> c11(A__FILTER(mark(x0), s(s(mark(z0))), mark(x2)), MARK(s(s(z0)))) MARK(filter(filter(z0, z1, z2), s(x1), x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), s(mark(x1)), mark(x2)), MARK(s(x1))) MARK(filter(sieve(z0), s(x1), x2)) -> c11(A__FILTER(a__sieve(mark(z0)), s(mark(x1)), mark(x2)), MARK(s(x1))) MARK(filter(nats(z0), s(x1), x2)) -> c11(A__FILTER(a__nats(mark(z0)), s(mark(x1)), mark(x2)), MARK(s(x1))) MARK(filter(zprimes, s(x1), x2)) -> c11(A__FILTER(a__zprimes, s(mark(x1)), mark(x2)), MARK(s(x1))) MARK(filter(cons(z0, z1), s(x1), x2)) -> c11(A__FILTER(cons(mark(z0), z1), s(mark(x1)), mark(x2)), MARK(s(x1))) MARK(filter(x0, s(x1), x2)) -> c11(MARK(s(x1))) MARK(filter(0, s(x1), x2)) -> c11(MARK(s(x1))) MARK(filter(s(z0), s(x1), x2)) -> c11(MARK(s(x1))) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c14_2, c15_1, c16_1, c18_1, c8_1, c10_1, c11_2, c11_1, c12_2, c12_1, c13_2, c13_1, c8_2, c10_2 ---------------------------------------- (207) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(filter(filter(z0, z1, z2), x1, x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x1)) by MARK(filter(filter(x0, x1, x2), x3, filter(z0, z1, z2))) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), mark(x2)), mark(x3), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x3)) MARK(filter(filter(x0, x1, x2), x3, sieve(z0))) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), mark(x2)), mark(x3), a__sieve(mark(z0))), MARK(x3)) MARK(filter(filter(x0, x1, x2), x3, nats(z0))) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), mark(x2)), mark(x3), a__nats(mark(z0))), MARK(x3)) MARK(filter(filter(x0, x1, x2), x3, zprimes)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), mark(x2)), mark(x3), a__zprimes), MARK(x3)) MARK(filter(filter(x0, x1, x2), x3, cons(z0, z1))) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), mark(x2)), mark(x3), cons(mark(z0), z1)), MARK(x3)) MARK(filter(filter(x0, x1, x2), x3, 0)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), mark(x2)), mark(x3), 0), MARK(x3)) MARK(filter(filter(x0, x1, x2), x3, s(z0))) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), mark(x2)), mark(x3), s(mark(z0))), MARK(x3)) MARK(filter(filter(x0, x1, x2), filter(z0, z1, z2), x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), mark(x2)), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4)), MARK(filter(z0, z1, z2))) MARK(filter(filter(x0, x1, x2), sieve(z0), x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), mark(x2)), a__sieve(mark(z0)), mark(x4)), MARK(sieve(z0))) MARK(filter(filter(x0, x1, x2), nats(z0), x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), mark(x2)), a__nats(mark(z0)), mark(x4)), MARK(nats(z0))) MARK(filter(filter(x0, x1, x2), zprimes, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), mark(x2)), a__zprimes, mark(x4)), MARK(zprimes)) MARK(filter(filter(x0, x1, x2), cons(z0, z1), x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), mark(x2)), cons(mark(z0), z1), mark(x4)), MARK(cons(z0, z1))) MARK(filter(filter(x0, x1, x2), 0, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), mark(x2)), 0, mark(x4)), MARK(0)) MARK(filter(filter(x0, x1, x2), s(z0), x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), mark(x2)), s(mark(z0)), mark(x4)), MARK(s(z0))) MARK(filter(filter(x0, x1, x2), x3, x4)) -> c11(A__FILTER(filter(mark(x0), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, filter(z0, z1, z2)), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, sieve(z0)), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), a__sieve(mark(z0))), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, nats(z0)), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), a__nats(mark(z0))), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, zprimes), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), a__zprimes), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, cons(z0, z1)), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), cons(mark(z0), z1)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, 0), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), 0), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, s(z0)), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), s(mark(z0))), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, filter(z0, z1, z2), x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, sieve(z0), x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), a__sieve(mark(z0)), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, nats(z0), x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), a__nats(mark(z0)), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, zprimes, x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), a__zprimes, mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, cons(z0, z1), x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), cons(mark(z0), z1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, 0, x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), 0, mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, s(z0), x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), s(mark(z0)), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(filter(z0, z1, z2), x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(sieve(z0), x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(a__sieve(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(nats(z0), x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(a__nats(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(zprimes, x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(a__zprimes, mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(cons(z0, z1), x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(cons(mark(z0), z1), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(0, x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(0, mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(s(z0), x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(s(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) ---------------------------------------- (208) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(x0, 0, filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, sieve(x2))) -> c10(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x0)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, sieve(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, sieve(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, sieve(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(0)), MARK(x0)) MARK(filter(x0, x1, sieve(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), sieve(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), 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s(x2))) -> c11(MARK(x1)) MARK(filter(s(z0), x1, s(x2))) -> c11(MARK(x1)) MARK(filter(x0, filter(x1, x2, filter(z0, z1, z2)), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__filter(mark(z0), mark(z1), mark(z2))), mark(x4)), MARK(filter(x1, x2, filter(z0, z1, z2)))) MARK(filter(x0, filter(x1, x2, sieve(z0)), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__sieve(mark(z0))), mark(x4)), MARK(filter(x1, x2, sieve(z0)))) MARK(filter(x0, filter(x1, x2, nats(z0)), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__nats(mark(z0))), mark(x4)), MARK(filter(x1, x2, nats(z0)))) MARK(filter(x0, filter(x1, x2, zprimes), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__zprimes), mark(x4)), MARK(filter(x1, x2, zprimes))) MARK(filter(x0, filter(x1, x2, cons(z0, z1)), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), cons(mark(z0), z1)), mark(x4)), MARK(filter(x1, x2, cons(z0, z1)))) MARK(filter(x0, filter(x1, x2, 0), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), 0), mark(x4)), MARK(filter(x1, x2, 0))) MARK(filter(x0, filter(x1, x2, s(z0)), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), s(mark(z0))), mark(x4)), MARK(filter(x1, x2, s(z0)))) MARK(filter(x0, filter(x1, filter(z0, z1, z2), x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), a__filter(mark(z0), mark(z1), mark(z2)), mark(x3)), mark(x4)), MARK(filter(x1, filter(z0, z1, z2), x3))) MARK(filter(x0, filter(x1, sieve(z0), x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), a__sieve(mark(z0)), mark(x3)), mark(x4)), MARK(filter(x1, sieve(z0), x3))) MARK(filter(x0, filter(x1, nats(z0), x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), a__nats(mark(z0)), mark(x3)), mark(x4)), MARK(filter(x1, nats(z0), x3))) MARK(filter(x0, filter(x1, zprimes, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), a__zprimes, mark(x3)), mark(x4)), MARK(filter(x1, zprimes, x3))) MARK(filter(x0, filter(x1, cons(z0, z1), x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), cons(mark(z0), z1), mark(x3)), mark(x4)), MARK(filter(x1, cons(z0, z1), x3))) MARK(filter(x0, filter(x1, 0, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), 0, mark(x3)), mark(x4)), MARK(filter(x1, 0, x3))) MARK(filter(x0, filter(x1, s(z0), x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), s(mark(z0)), mark(x3)), mark(x4)), MARK(filter(x1, s(z0), x3))) MARK(filter(x0, filter(filter(z0, z1, z2), x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x2), mark(x3)), mark(x4)), MARK(filter(filter(z0, z1, z2), x2, x3))) MARK(filter(x0, filter(sieve(z0), x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(a__sieve(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(filter(sieve(z0), x2, x3))) MARK(filter(x0, filter(nats(z0), x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(a__nats(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(filter(nats(z0), x2, x3))) MARK(filter(x0, filter(zprimes, x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(a__zprimes, mark(x2), mark(x3)), mark(x4)), MARK(filter(zprimes, x2, x3))) MARK(filter(x0, filter(cons(z0, z1), x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(cons(mark(z0), z1), mark(x2), mark(x3)), mark(x4)), MARK(filter(cons(z0, z1), x2, x3))) MARK(filter(x0, filter(0, x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(0, mark(x2), mark(x3)), mark(x4)), MARK(filter(0, x2, x3))) MARK(filter(x0, filter(s(z0), x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(s(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(filter(s(z0), x2, x3))) MARK(filter(filter(z0, z1, z2), filter(x1, x2, x3), x4)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(sieve(z0), filter(x1, x2, x3), x4)) -> c11(A__FILTER(a__sieve(mark(z0)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(nats(z0), filter(x1, x2, x3), x4)) -> c11(A__FILTER(a__nats(mark(z0)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(zprimes, filter(x1, x2, x3), x4)) -> c11(A__FILTER(a__zprimes, a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(cons(z0, z1), filter(x1, x2, x3), x4)) -> c11(A__FILTER(cons(mark(z0), z1), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(x0, filter(x1, x2, x3), x4)) -> c11(MARK(filter(x1, x2, x3))) MARK(filter(0, filter(x1, x2, x3), x4)) -> c11(MARK(filter(x1, x2, x3))) MARK(filter(s(z0), filter(x1, x2, x3), x4)) -> c11(MARK(filter(x1, x2, x3))) MARK(filter(x0, sieve(filter(z0, z1, z2)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(sieve(filter(z0, z1, z2)))) MARK(filter(x0, sieve(sieve(z0)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(a__sieve(mark(z0))), mark(x2)), MARK(sieve(sieve(z0)))) MARK(filter(x0, sieve(nats(z0)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(a__nats(mark(z0))), mark(x2)), MARK(sieve(nats(z0)))) MARK(filter(x0, sieve(zprimes), x2)) -> c11(A__FILTER(mark(x0), a__sieve(a__zprimes), mark(x2)), MARK(sieve(zprimes))) MARK(filter(x0, sieve(cons(z0, z1)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(cons(mark(z0), z1)), mark(x2)), MARK(sieve(cons(z0, z1)))) MARK(filter(x0, sieve(s(z0)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(s(mark(z0))), mark(x2)), MARK(sieve(s(z0)))) MARK(filter(filter(z0, z1, z2), sieve(x1), x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(sieve(z0), sieve(x1), x2)) -> c11(A__FILTER(a__sieve(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(nats(z0), sieve(x1), x2)) -> c11(A__FILTER(a__nats(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(zprimes, sieve(x1), x2)) -> c11(A__FILTER(a__zprimes, a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(cons(z0, z1), sieve(x1), x2)) -> c11(A__FILTER(cons(mark(z0), z1), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(x0, sieve(x1), x2)) -> c11(MARK(sieve(x1))) MARK(filter(x0, sieve(0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(0), mark(x2))) MARK(filter(0, sieve(x1), x2)) -> c11(MARK(sieve(x1))) MARK(filter(s(z0), sieve(x1), x2)) -> c11(MARK(sieve(x1))) MARK(filter(x0, nats(filter(z0, z1, z2)), x2)) -> c11(A__FILTER(mark(x0), a__nats(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(nats(filter(z0, z1, z2)))) MARK(filter(x0, nats(sieve(z0)), x2)) -> c11(A__FILTER(mark(x0), a__nats(a__sieve(mark(z0))), mark(x2)), MARK(nats(sieve(z0)))) MARK(filter(x0, nats(nats(z0)), x2)) -> c11(A__FILTER(mark(x0), a__nats(a__nats(mark(z0))), mark(x2)), MARK(nats(nats(z0)))) MARK(filter(x0, nats(zprimes), x2)) -> c11(A__FILTER(mark(x0), a__nats(a__zprimes), mark(x2)), MARK(nats(zprimes))) MARK(filter(x0, nats(cons(z0, z1)), x2)) -> c11(A__FILTER(mark(x0), a__nats(cons(mark(z0), z1)), mark(x2)), MARK(nats(cons(z0, z1)))) MARK(filter(x0, nats(0), x2)) -> c11(A__FILTER(mark(x0), a__nats(0), mark(x2)), MARK(nats(0))) MARK(filter(x0, nats(s(z0)), x2)) -> c11(A__FILTER(mark(x0), a__nats(s(mark(z0))), mark(x2)), MARK(nats(s(z0)))) MARK(filter(filter(z0, z1, z2), nats(x1), x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(sieve(z0), nats(x1), x2)) -> c11(A__FILTER(a__sieve(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(nats(z0), nats(x1), x2)) -> c11(A__FILTER(a__nats(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(zprimes, nats(x1), x2)) -> c11(A__FILTER(a__zprimes, a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(cons(z0, z1), nats(x1), x2)) -> c11(A__FILTER(cons(mark(z0), z1), a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(x0, nats(x1), x2)) -> c11(MARK(nats(x1))) MARK(filter(0, nats(x1), x2)) -> c11(MARK(nats(x1))) MARK(filter(s(z0), nats(x1), x2)) -> c11(MARK(nats(x1))) MARK(filter(x0, zprimes, x1)) -> c11(A__FILTER(mark(x0), a__sieve(a__nats(s(s(0)))), mark(x1)), MARK(zprimes)) MARK(filter(filter(z0, z1, z2), zprimes, x1)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(sieve(z0), zprimes, x1)) -> c11(A__FILTER(a__sieve(mark(z0)), a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(nats(z0), zprimes, x1)) -> c11(A__FILTER(a__nats(mark(z0)), a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(zprimes, zprimes, x1)) -> c11(A__FILTER(a__zprimes, a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(cons(z0, z1), zprimes, x1)) -> c11(A__FILTER(cons(mark(z0), z1), a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(x0, zprimes, x1)) -> c11(MARK(zprimes)) MARK(filter(0, zprimes, x1)) -> c11(MARK(zprimes)) MARK(filter(s(z0), zprimes, x1)) -> c11(MARK(zprimes)) MARK(filter(x0, s(filter(z0, z1, z2)), x2)) -> c11(A__FILTER(mark(x0), s(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), 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mark(x2)), MARK(s(x1))) MARK(filter(zprimes, s(x1), x2)) -> c11(A__FILTER(a__zprimes, s(mark(x1)), mark(x2)), MARK(s(x1))) MARK(filter(cons(z0, z1), s(x1), x2)) -> c11(A__FILTER(cons(mark(z0), z1), s(mark(x1)), mark(x2)), MARK(s(x1))) MARK(filter(x0, s(x1), x2)) -> c11(MARK(s(x1))) MARK(filter(0, s(x1), x2)) -> c11(MARK(s(x1))) MARK(filter(s(z0), s(x1), x2)) -> c11(MARK(s(x1))) MARK(filter(filter(x0, x1, x2), cons(z0, z1), x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), mark(x2)), cons(mark(z0), z1), mark(x4)), MARK(cons(z0, z1))) MARK(filter(filter(x0, x1, x2), 0, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), mark(x2)), 0, mark(x4)), MARK(0)) MARK(filter(filter(x0, x1, x2), x3, x4)) -> c11(A__FILTER(filter(mark(x0), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, filter(z0, z1, z2)), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, sieve(z0)), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), a__sieve(mark(z0))), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, nats(z0)), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), a__nats(mark(z0))), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, zprimes), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), a__zprimes), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, cons(z0, z1)), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), cons(mark(z0), z1)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, 0), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), 0), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, s(z0)), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), s(mark(z0))), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, filter(z0, z1, z2), x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, sieve(z0), x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), a__sieve(mark(z0)), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, nats(z0), x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), a__nats(mark(z0)), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, zprimes, x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), a__zprimes, mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, cons(z0, z1), x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), cons(mark(z0), z1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, 0, x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), 0, mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, s(z0), x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), s(mark(z0)), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(filter(z0, z1, z2), x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(sieve(z0), x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(a__sieve(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(nats(z0), x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(a__nats(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(zprimes, x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(a__zprimes, mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(cons(z0, z1), x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(cons(mark(z0), z1), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(0, x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(0, mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(s(z0), x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(s(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) S tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(x0, 0, filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, sieve(x2))) -> c10(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x0)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, sieve(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, sieve(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, sieve(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(0)), MARK(x0)) MARK(filter(x0, x1, sieve(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), sieve(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, sieve(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, sieve(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__sieve(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, sieve(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, sieve(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, sieve(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), a__sieve(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, sieve(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__sieve(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), sieve(x2))) -> c10(MARK(x0)) MARK(filter(x0, 0, sieve(x2))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, sieve(x2))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(mark(x2)), nats(s(mark(x2))))), MARK(x0)) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), nats(mark(x2))), MARK(x0)) MARK(filter(x0, x1, nats(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, nats(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, nats(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, nats(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, nats(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, nats(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(0)), MARK(x0)) MARK(filter(x0, x1, nats(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), nats(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), nats(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), nats(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, nats(x2))) -> 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MARK(filter(sieve(z0), x2, x3))) MARK(filter(x0, filter(nats(z0), x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(a__nats(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(filter(nats(z0), x2, x3))) MARK(filter(x0, filter(zprimes, x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(a__zprimes, mark(x2), mark(x3)), mark(x4)), MARK(filter(zprimes, x2, x3))) MARK(filter(x0, filter(cons(z0, z1), x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(cons(mark(z0), z1), mark(x2), mark(x3)), mark(x4)), MARK(filter(cons(z0, z1), x2, x3))) MARK(filter(x0, filter(0, x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(0, mark(x2), mark(x3)), mark(x4)), MARK(filter(0, x2, x3))) MARK(filter(x0, filter(s(z0), x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(s(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(filter(s(z0), x2, x3))) MARK(filter(filter(z0, z1, z2), filter(x1, x2, x3), x4)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(sieve(z0), filter(x1, x2, x3), x4)) -> c11(A__FILTER(a__sieve(mark(z0)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(nats(z0), filter(x1, x2, x3), x4)) -> c11(A__FILTER(a__nats(mark(z0)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(zprimes, filter(x1, x2, x3), x4)) -> c11(A__FILTER(a__zprimes, a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(cons(z0, z1), filter(x1, x2, x3), x4)) -> c11(A__FILTER(cons(mark(z0), z1), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(x0, filter(x1, x2, x3), x4)) -> c11(MARK(filter(x1, x2, x3))) MARK(filter(0, filter(x1, x2, x3), x4)) -> c11(MARK(filter(x1, x2, x3))) MARK(filter(s(z0), filter(x1, x2, x3), x4)) -> c11(MARK(filter(x1, x2, x3))) MARK(filter(x0, sieve(filter(z0, z1, z2)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(sieve(filter(z0, z1, z2)))) MARK(filter(x0, sieve(sieve(z0)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(a__sieve(mark(z0))), mark(x2)), MARK(sieve(sieve(z0)))) MARK(filter(x0, sieve(nats(z0)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(a__nats(mark(z0))), mark(x2)), MARK(sieve(nats(z0)))) MARK(filter(x0, sieve(zprimes), x2)) -> c11(A__FILTER(mark(x0), a__sieve(a__zprimes), mark(x2)), MARK(sieve(zprimes))) MARK(filter(x0, sieve(cons(z0, z1)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(cons(mark(z0), z1)), mark(x2)), MARK(sieve(cons(z0, z1)))) MARK(filter(x0, sieve(s(z0)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(s(mark(z0))), mark(x2)), MARK(sieve(s(z0)))) MARK(filter(filter(z0, z1, z2), sieve(x1), x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(sieve(z0), sieve(x1), x2)) -> c11(A__FILTER(a__sieve(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(nats(z0), sieve(x1), x2)) -> c11(A__FILTER(a__nats(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(zprimes, sieve(x1), x2)) -> c11(A__FILTER(a__zprimes, a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(cons(z0, z1), sieve(x1), x2)) -> c11(A__FILTER(cons(mark(z0), z1), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(x0, sieve(x1), x2)) -> c11(MARK(sieve(x1))) MARK(filter(x0, sieve(0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(0), mark(x2))) MARK(filter(0, sieve(x1), x2)) -> c11(MARK(sieve(x1))) MARK(filter(s(z0), sieve(x1), x2)) -> c11(MARK(sieve(x1))) MARK(filter(x0, nats(filter(z0, z1, z2)), x2)) -> c11(A__FILTER(mark(x0), a__nats(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(nats(filter(z0, z1, z2)))) MARK(filter(x0, nats(sieve(z0)), x2)) -> c11(A__FILTER(mark(x0), a__nats(a__sieve(mark(z0))), mark(x2)), MARK(nats(sieve(z0)))) MARK(filter(x0, nats(nats(z0)), x2)) -> c11(A__FILTER(mark(x0), a__nats(a__nats(mark(z0))), mark(x2)), MARK(nats(nats(z0)))) MARK(filter(x0, nats(zprimes), x2)) -> c11(A__FILTER(mark(x0), a__nats(a__zprimes), mark(x2)), MARK(nats(zprimes))) MARK(filter(x0, nats(cons(z0, z1)), x2)) -> c11(A__FILTER(mark(x0), a__nats(cons(mark(z0), z1)), mark(x2)), MARK(nats(cons(z0, z1)))) MARK(filter(x0, nats(0), x2)) -> c11(A__FILTER(mark(x0), a__nats(0), mark(x2)), MARK(nats(0))) MARK(filter(x0, nats(s(z0)), x2)) -> c11(A__FILTER(mark(x0), a__nats(s(mark(z0))), mark(x2)), MARK(nats(s(z0)))) MARK(filter(filter(z0, z1, z2), nats(x1), x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(sieve(z0), nats(x1), x2)) -> c11(A__FILTER(a__sieve(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(nats(z0), nats(x1), x2)) -> c11(A__FILTER(a__nats(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(zprimes, nats(x1), x2)) -> c11(A__FILTER(a__zprimes, a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(cons(z0, z1), nats(x1), x2)) -> c11(A__FILTER(cons(mark(z0), z1), a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(x0, nats(x1), x2)) -> c11(MARK(nats(x1))) MARK(filter(0, nats(x1), x2)) -> c11(MARK(nats(x1))) MARK(filter(s(z0), nats(x1), x2)) -> c11(MARK(nats(x1))) MARK(filter(x0, zprimes, x1)) -> c11(A__FILTER(mark(x0), a__sieve(a__nats(s(s(0)))), mark(x1)), MARK(zprimes)) MARK(filter(filter(z0, z1, z2), zprimes, x1)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(sieve(z0), zprimes, x1)) -> c11(A__FILTER(a__sieve(mark(z0)), a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(nats(z0), zprimes, x1)) -> c11(A__FILTER(a__nats(mark(z0)), a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(zprimes, zprimes, x1)) -> c11(A__FILTER(a__zprimes, a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(cons(z0, z1), zprimes, x1)) -> c11(A__FILTER(cons(mark(z0), z1), a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(x0, zprimes, x1)) -> c11(MARK(zprimes)) MARK(filter(0, zprimes, x1)) -> c11(MARK(zprimes)) MARK(filter(s(z0), zprimes, x1)) -> c11(MARK(zprimes)) MARK(filter(x0, s(filter(z0, z1, z2)), x2)) -> c11(A__FILTER(mark(x0), s(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(s(filter(z0, z1, z2)))) MARK(filter(x0, s(sieve(z0)), x2)) -> c11(A__FILTER(mark(x0), s(a__sieve(mark(z0))), mark(x2)), MARK(s(sieve(z0)))) MARK(filter(x0, s(nats(z0)), x2)) -> c11(A__FILTER(mark(x0), s(a__nats(mark(z0))), mark(x2)), MARK(s(nats(z0)))) MARK(filter(x0, s(zprimes), x2)) -> c11(A__FILTER(mark(x0), s(a__zprimes), mark(x2)), MARK(s(zprimes))) MARK(filter(x0, s(cons(z0, z1)), x2)) -> c11(A__FILTER(mark(x0), s(cons(mark(z0), z1)), mark(x2)), MARK(s(cons(z0, z1)))) MARK(filter(x0, s(0), x2)) -> c11(A__FILTER(mark(x0), s(0), mark(x2)), MARK(s(0))) MARK(filter(x0, s(s(z0)), x2)) -> c11(A__FILTER(mark(x0), s(s(mark(z0))), mark(x2)), MARK(s(s(z0)))) MARK(filter(filter(z0, z1, z2), s(x1), x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), s(mark(x1)), mark(x2)), MARK(s(x1))) MARK(filter(sieve(z0), s(x1), x2)) -> c11(A__FILTER(a__sieve(mark(z0)), s(mark(x1)), mark(x2)), MARK(s(x1))) MARK(filter(nats(z0), s(x1), x2)) -> c11(A__FILTER(a__nats(mark(z0)), s(mark(x1)), mark(x2)), MARK(s(x1))) MARK(filter(zprimes, s(x1), x2)) -> c11(A__FILTER(a__zprimes, s(mark(x1)), mark(x2)), MARK(s(x1))) MARK(filter(cons(z0, z1), s(x1), x2)) -> c11(A__FILTER(cons(mark(z0), z1), s(mark(x1)), mark(x2)), MARK(s(x1))) MARK(filter(x0, s(x1), x2)) -> c11(MARK(s(x1))) MARK(filter(0, s(x1), x2)) -> c11(MARK(s(x1))) MARK(filter(s(z0), s(x1), x2)) -> c11(MARK(s(x1))) MARK(filter(filter(x0, x1, x2), cons(z0, z1), x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), mark(x2)), cons(mark(z0), z1), mark(x4)), MARK(cons(z0, z1))) MARK(filter(filter(x0, x1, x2), 0, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), mark(x2)), 0, mark(x4)), MARK(0)) MARK(filter(filter(x0, x1, x2), x3, x4)) -> c11(A__FILTER(filter(mark(x0), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, filter(z0, z1, z2)), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, sieve(z0)), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), a__sieve(mark(z0))), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, nats(z0)), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), a__nats(mark(z0))), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, zprimes), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), a__zprimes), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, cons(z0, z1)), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), cons(mark(z0), z1)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, 0), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), 0), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, s(z0)), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), s(mark(z0))), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, filter(z0, z1, z2), x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, sieve(z0), x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), a__sieve(mark(z0)), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, nats(z0), x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), a__nats(mark(z0)), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, zprimes, x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), a__zprimes, mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, cons(z0, z1), x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), cons(mark(z0), z1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, 0, x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), 0, mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, s(z0), x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), s(mark(z0)), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(filter(z0, z1, z2), x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(sieve(z0), x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(a__sieve(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(nats(z0), x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(a__nats(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(zprimes, x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(a__zprimes, mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(cons(z0, z1), x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(cons(mark(z0), z1), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(0, x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(0, mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(s(z0), x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(s(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c14_2, c15_1, c16_1, c18_1, c8_1, c10_1, c11_2, c11_1, c12_2, c12_1, c13_2, c13_1, c8_2, c10_2 ---------------------------------------- (209) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(filter(filter(x0, x1, x2), 0, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), mark(x2)), 0, mark(x4)), MARK(0)) ---------------------------------------- (210) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(x0, 0, filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, sieve(x2))) -> c10(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x0)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, sieve(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, sieve(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, sieve(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(0)), MARK(x0)) MARK(filter(x0, x1, sieve(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), sieve(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, sieve(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, sieve(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__sieve(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, sieve(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, sieve(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, sieve(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), a__sieve(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, sieve(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__sieve(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), sieve(x2))) -> c10(MARK(x0)) MARK(filter(x0, 0, sieve(x2))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, sieve(x2))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(mark(x2)), nats(s(mark(x2))))), MARK(x0)) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), nats(mark(x2))), MARK(x0)) MARK(filter(x0, x1, nats(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, nats(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, nats(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, nats(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, nats(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, nats(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(0)), MARK(x0)) MARK(filter(x0, x1, nats(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), nats(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), nats(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), nats(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, nats(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), nats(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, nats(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__nats(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, nats(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, nats(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, nats(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), a__nats(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, nats(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__nats(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), nats(x2))) -> c10(MARK(x0)) MARK(filter(x0, 0, nats(x2))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, nats(x2))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, zprimes)) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(s(s(0))))), MARK(x0)) MARK(filter(x0, x1, zprimes)) -> c10(A__FILTER(mark(x0), mark(x1), zprimes), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), zprimes)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__zprimes), MARK(x0)) MARK(filter(x0, 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c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(nats(z0), x1, sieve(x2))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(zprimes, x1, sieve(x2))) -> c11(A__FILTER(a__zprimes, mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(cons(z0, z1), x1, sieve(x2))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(x0, cons(z0, z1), sieve(x2))) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, sieve(x2))) -> c11(MARK(x1)) MARK(filter(s(z0), x1, sieve(x2))) -> c11(MARK(x1)) MARK(filter(x0, x1, nats(x2))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(mark(x2)), nats(s(mark(x2))))), MARK(x1)) MARK(filter(x0, x1, nats(x2))) -> c11(A__FILTER(mark(x0), mark(x1), nats(mark(x2))), MARK(x1)) MARK(filter(x0, x1, nats(filter(z0, z1, z2)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x1)) MARK(filter(x0, x1, nats(sieve(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__sieve(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, nats(nats(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__nats(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, nats(zprimes))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__zprimes)), MARK(x1)) MARK(filter(x0, x1, nats(cons(z0, z1)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(cons(mark(z0), z1))), MARK(x1)) MARK(filter(x0, x1, nats(0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(0)), MARK(x1)) MARK(filter(x0, x1, nats(s(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(s(mark(z0)))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), nats(x2))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), nats(x2))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), a__nats(mark(x2))), MARK(sieve(z0))) MARK(filter(x0, nats(z0), nats(x2))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), a__nats(mark(x2))), MARK(nats(z0))) MARK(filter(x0, zprimes, nats(x2))) -> c11(A__FILTER(mark(x0), a__zprimes, a__nats(mark(x2))), MARK(zprimes)) MARK(filter(x0, s(z0), nats(x2))) -> c11(A__FILTER(mark(x0), s(mark(z0)), a__nats(mark(x2))), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, nats(x2))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(sieve(z0), x1, nats(x2))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(nats(z0), x1, nats(x2))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(zprimes, x1, nats(x2))) -> c11(A__FILTER(a__zprimes, mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(cons(z0, z1), x1, nats(x2))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(x0, cons(z0, z1), nats(x2))) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, nats(x2))) -> c11(MARK(x1)) MARK(filter(s(z0), x1, nats(x2))) -> c11(MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(s(s(0))))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), zprimes), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), zprimes)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__zprimes), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), zprimes)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), a__zprimes), MARK(sieve(z0))) MARK(filter(x0, nats(z0), zprimes)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), a__zprimes), MARK(nats(z0))) MARK(filter(x0, zprimes, zprimes)) -> c11(A__FILTER(mark(x0), a__zprimes, a__zprimes), MARK(zprimes)) MARK(filter(x0, s(z0), zprimes)) -> c11(A__FILTER(mark(x0), s(mark(z0)), a__zprimes), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, zprimes)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__zprimes), MARK(x1)) MARK(filter(sieve(z0), x1, zprimes)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), 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mark(x2)), MARK(sieve(sieve(z0)))) MARK(filter(x0, sieve(nats(z0)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(a__nats(mark(z0))), mark(x2)), MARK(sieve(nats(z0)))) MARK(filter(x0, sieve(zprimes), x2)) -> c11(A__FILTER(mark(x0), a__sieve(a__zprimes), mark(x2)), MARK(sieve(zprimes))) MARK(filter(x0, sieve(cons(z0, z1)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(cons(mark(z0), z1)), mark(x2)), MARK(sieve(cons(z0, z1)))) MARK(filter(x0, sieve(s(z0)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(s(mark(z0))), mark(x2)), MARK(sieve(s(z0)))) MARK(filter(filter(z0, z1, z2), sieve(x1), x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(sieve(z0), sieve(x1), x2)) -> c11(A__FILTER(a__sieve(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(nats(z0), sieve(x1), x2)) -> c11(A__FILTER(a__nats(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(zprimes, sieve(x1), x2)) -> c11(A__FILTER(a__zprimes, a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(cons(z0, z1), sieve(x1), x2)) -> c11(A__FILTER(cons(mark(z0), z1), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(x0, sieve(x1), x2)) -> c11(MARK(sieve(x1))) MARK(filter(x0, sieve(0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(0), mark(x2))) MARK(filter(0, sieve(x1), x2)) -> c11(MARK(sieve(x1))) MARK(filter(s(z0), sieve(x1), x2)) -> c11(MARK(sieve(x1))) MARK(filter(x0, nats(filter(z0, z1, z2)), x2)) -> c11(A__FILTER(mark(x0), a__nats(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(nats(filter(z0, z1, z2)))) MARK(filter(x0, nats(sieve(z0)), x2)) -> c11(A__FILTER(mark(x0), a__nats(a__sieve(mark(z0))), mark(x2)), MARK(nats(sieve(z0)))) MARK(filter(x0, nats(nats(z0)), x2)) -> c11(A__FILTER(mark(x0), a__nats(a__nats(mark(z0))), mark(x2)), MARK(nats(nats(z0)))) MARK(filter(x0, nats(zprimes), x2)) -> c11(A__FILTER(mark(x0), a__nats(a__zprimes), mark(x2)), MARK(nats(zprimes))) MARK(filter(x0, nats(cons(z0, z1)), x2)) -> c11(A__FILTER(mark(x0), a__nats(cons(mark(z0), z1)), mark(x2)), MARK(nats(cons(z0, z1)))) MARK(filter(x0, nats(0), x2)) -> c11(A__FILTER(mark(x0), a__nats(0), mark(x2)), MARK(nats(0))) MARK(filter(x0, nats(s(z0)), x2)) -> c11(A__FILTER(mark(x0), a__nats(s(mark(z0))), mark(x2)), MARK(nats(s(z0)))) MARK(filter(filter(z0, z1, z2), nats(x1), x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(sieve(z0), nats(x1), x2)) -> c11(A__FILTER(a__sieve(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(nats(z0), nats(x1), x2)) -> c11(A__FILTER(a__nats(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(zprimes, nats(x1), x2)) -> c11(A__FILTER(a__zprimes, a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(cons(z0, z1), nats(x1), x2)) -> c11(A__FILTER(cons(mark(z0), z1), a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(x0, nats(x1), x2)) -> c11(MARK(nats(x1))) MARK(filter(0, nats(x1), x2)) -> c11(MARK(nats(x1))) MARK(filter(s(z0), nats(x1), x2)) -> c11(MARK(nats(x1))) MARK(filter(x0, zprimes, x1)) -> c11(A__FILTER(mark(x0), a__sieve(a__nats(s(s(0)))), mark(x1)), MARK(zprimes)) MARK(filter(filter(z0, z1, z2), zprimes, x1)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(sieve(z0), zprimes, x1)) -> c11(A__FILTER(a__sieve(mark(z0)), a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(nats(z0), zprimes, x1)) -> c11(A__FILTER(a__nats(mark(z0)), a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(zprimes, zprimes, x1)) -> c11(A__FILTER(a__zprimes, a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(cons(z0, z1), zprimes, x1)) -> c11(A__FILTER(cons(mark(z0), z1), a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(x0, zprimes, x1)) -> c11(MARK(zprimes)) MARK(filter(0, zprimes, x1)) -> c11(MARK(zprimes)) MARK(filter(s(z0), zprimes, x1)) -> c11(MARK(zprimes)) MARK(filter(x0, s(filter(z0, z1, z2)), x2)) -> c11(A__FILTER(mark(x0), s(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(s(filter(z0, z1, z2)))) MARK(filter(x0, s(sieve(z0)), x2)) -> c11(A__FILTER(mark(x0), s(a__sieve(mark(z0))), mark(x2)), MARK(s(sieve(z0)))) MARK(filter(x0, s(nats(z0)), x2)) -> c11(A__FILTER(mark(x0), s(a__nats(mark(z0))), mark(x2)), MARK(s(nats(z0)))) MARK(filter(x0, s(zprimes), x2)) -> c11(A__FILTER(mark(x0), s(a__zprimes), mark(x2)), MARK(s(zprimes))) MARK(filter(x0, s(cons(z0, z1)), x2)) -> c11(A__FILTER(mark(x0), s(cons(mark(z0), z1)), mark(x2)), MARK(s(cons(z0, z1)))) MARK(filter(x0, s(0), x2)) -> c11(A__FILTER(mark(x0), s(0), mark(x2)), MARK(s(0))) MARK(filter(x0, s(s(z0)), x2)) -> c11(A__FILTER(mark(x0), s(s(mark(z0))), mark(x2)), MARK(s(s(z0)))) MARK(filter(filter(z0, z1, z2), s(x1), x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), s(mark(x1)), mark(x2)), MARK(s(x1))) MARK(filter(sieve(z0), s(x1), x2)) -> c11(A__FILTER(a__sieve(mark(z0)), s(mark(x1)), mark(x2)), MARK(s(x1))) MARK(filter(nats(z0), s(x1), x2)) -> c11(A__FILTER(a__nats(mark(z0)), s(mark(x1)), mark(x2)), MARK(s(x1))) MARK(filter(zprimes, s(x1), x2)) -> c11(A__FILTER(a__zprimes, s(mark(x1)), mark(x2)), MARK(s(x1))) MARK(filter(cons(z0, z1), s(x1), x2)) -> c11(A__FILTER(cons(mark(z0), z1), s(mark(x1)), mark(x2)), MARK(s(x1))) MARK(filter(x0, s(x1), x2)) -> c11(MARK(s(x1))) MARK(filter(0, s(x1), x2)) -> c11(MARK(s(x1))) MARK(filter(s(z0), s(x1), x2)) -> c11(MARK(s(x1))) MARK(filter(filter(x0, x1, x2), cons(z0, z1), x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), mark(x2)), cons(mark(z0), z1), mark(x4)), MARK(cons(z0, z1))) MARK(filter(filter(x0, x1, x2), x3, x4)) -> c11(A__FILTER(filter(mark(x0), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, filter(z0, z1, z2)), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, sieve(z0)), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), a__sieve(mark(z0))), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, nats(z0)), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), a__nats(mark(z0))), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, zprimes), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), a__zprimes), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, cons(z0, z1)), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), cons(mark(z0), z1)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, 0), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), 0), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, s(z0)), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), s(mark(z0))), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, filter(z0, z1, z2), x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, sieve(z0), x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), a__sieve(mark(z0)), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, nats(z0), x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), a__nats(mark(z0)), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, zprimes, x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), a__zprimes, mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, cons(z0, z1), x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), cons(mark(z0), z1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, 0, x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), 0, mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, s(z0), x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), s(mark(z0)), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(filter(z0, z1, z2), x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(sieve(z0), x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(a__sieve(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(nats(z0), x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(a__nats(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(zprimes, x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(a__zprimes, mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(cons(z0, z1), x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(cons(mark(z0), z1), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(0, x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(0, mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(s(z0), x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(s(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c14_2, c15_1, c16_1, c18_1, c8_1, c10_1, c11_2, c11_1, c12_2, c12_1, c13_2, c13_1, c8_2, c10_2 ---------------------------------------- (211) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (212) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 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-> c11(A__FILTER(a__nats(mark(z0)), a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(zprimes, zprimes, x1)) -> c11(A__FILTER(a__zprimes, a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(cons(z0, z1), zprimes, x1)) -> c11(A__FILTER(cons(mark(z0), z1), a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(x0, zprimes, x1)) -> c11(MARK(zprimes)) MARK(filter(0, zprimes, x1)) -> c11(MARK(zprimes)) MARK(filter(s(z0), zprimes, x1)) -> c11(MARK(zprimes)) MARK(filter(x0, s(filter(z0, z1, z2)), x2)) -> c11(A__FILTER(mark(x0), s(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(s(filter(z0, z1, z2)))) MARK(filter(x0, s(sieve(z0)), x2)) -> c11(A__FILTER(mark(x0), s(a__sieve(mark(z0))), mark(x2)), MARK(s(sieve(z0)))) MARK(filter(x0, s(nats(z0)), x2)) -> c11(A__FILTER(mark(x0), s(a__nats(mark(z0))), mark(x2)), MARK(s(nats(z0)))) MARK(filter(x0, s(zprimes), x2)) -> c11(A__FILTER(mark(x0), s(a__zprimes), mark(x2)), MARK(s(zprimes))) MARK(filter(x0, s(cons(z0, z1)), x2)) -> c11(A__FILTER(mark(x0), s(cons(mark(z0), z1)), mark(x2)), MARK(s(cons(z0, z1)))) MARK(filter(x0, s(0), x2)) -> c11(A__FILTER(mark(x0), s(0), mark(x2)), MARK(s(0))) MARK(filter(x0, s(s(z0)), x2)) -> c11(A__FILTER(mark(x0), s(s(mark(z0))), mark(x2)), MARK(s(s(z0)))) MARK(filter(filter(z0, z1, z2), s(x1), x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), s(mark(x1)), mark(x2)), MARK(s(x1))) MARK(filter(sieve(z0), s(x1), x2)) -> c11(A__FILTER(a__sieve(mark(z0)), s(mark(x1)), mark(x2)), MARK(s(x1))) MARK(filter(nats(z0), s(x1), x2)) -> c11(A__FILTER(a__nats(mark(z0)), s(mark(x1)), mark(x2)), MARK(s(x1))) MARK(filter(zprimes, s(x1), x2)) -> c11(A__FILTER(a__zprimes, s(mark(x1)), mark(x2)), MARK(s(x1))) MARK(filter(cons(z0, z1), s(x1), x2)) -> c11(A__FILTER(cons(mark(z0), z1), s(mark(x1)), mark(x2)), MARK(s(x1))) MARK(filter(x0, s(x1), x2)) -> c11(MARK(s(x1))) MARK(filter(0, s(x1), x2)) -> c11(MARK(s(x1))) MARK(filter(s(z0), s(x1), x2)) -> c11(MARK(s(x1))) MARK(filter(filter(x0, x1, filter(z0, z1, z2)), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, sieve(z0)), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), a__sieve(mark(z0))), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, nats(z0)), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), a__nats(mark(z0))), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, zprimes), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), a__zprimes), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, cons(z0, z1)), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), cons(mark(z0), z1)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, 0), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), 0), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, s(z0)), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), s(mark(z0))), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, filter(z0, z1, z2), x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, sieve(z0), x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), a__sieve(mark(z0)), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, nats(z0), x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), a__nats(mark(z0)), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, zprimes, x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), a__zprimes, mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, cons(z0, z1), x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), cons(mark(z0), z1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, 0, x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), 0, mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, s(z0), x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), s(mark(z0)), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(filter(z0, z1, z2), x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(sieve(z0), x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(a__sieve(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(nats(z0), x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(a__nats(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(zprimes, x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(a__zprimes, mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(cons(z0, z1), x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(cons(mark(z0), z1), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(0, x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(0, mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(s(z0), x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(s(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, x2), cons(z0, z1), x4)) -> c11(MARK(cons(z0, z1))) MARK(filter(filter(x0, x1, x2), x3, x4)) -> c11(MARK(x3)) S tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) 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c11(A__FILTER(mark(x0), mark(x1), s(cons(mark(z0), z1))), MARK(x1)) MARK(filter(x0, x1, s(0))) -> c11(A__FILTER(mark(x0), mark(x1), s(0)), MARK(x1)) MARK(filter(x0, x1, s(s(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), s(s(mark(z0)))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), s(x2))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), s(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), s(x2))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), s(mark(x2))), MARK(sieve(z0))) MARK(filter(x0, nats(z0), s(x2))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), s(mark(x2))), MARK(nats(z0))) MARK(filter(x0, zprimes, s(x2))) -> c11(A__FILTER(mark(x0), a__zprimes, s(mark(x2))), MARK(zprimes)) MARK(filter(x0, s(z0), s(x2))) -> c11(A__FILTER(mark(x0), s(mark(z0)), s(mark(x2))), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, s(x2))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), s(mark(x2))), MARK(x1)) MARK(filter(sieve(z0), x1, s(x2))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), s(mark(x2))), MARK(x1)) MARK(filter(nats(z0), x1, s(x2))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), s(mark(x2))), MARK(x1)) MARK(filter(zprimes, x1, s(x2))) -> c11(A__FILTER(a__zprimes, mark(x1), s(mark(x2))), MARK(x1)) MARK(filter(cons(z0, z1), x1, s(x2))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), s(mark(x2))), MARK(x1)) MARK(filter(x0, cons(z0, z1), s(x2))) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, s(x2))) -> c11(MARK(x1)) MARK(filter(s(z0), x1, s(x2))) -> c11(MARK(x1)) MARK(filter(x0, filter(x1, x2, filter(z0, z1, z2)), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__filter(mark(z0), mark(z1), mark(z2))), mark(x4)), MARK(filter(x1, x2, filter(z0, z1, z2)))) MARK(filter(x0, filter(x1, x2, sieve(z0)), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__sieve(mark(z0))), mark(x4)), MARK(filter(x1, x2, sieve(z0)))) MARK(filter(x0, filter(x1, x2, nats(z0)), x4)) -> c11(A__FILTER(mark(x0), 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c11(A__FILTER(mark(x0), a__filter(mark(x1), a__sieve(mark(z0)), mark(x3)), mark(x4)), MARK(filter(x1, sieve(z0), x3))) MARK(filter(x0, filter(x1, nats(z0), x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), a__nats(mark(z0)), mark(x3)), mark(x4)), MARK(filter(x1, nats(z0), x3))) MARK(filter(x0, filter(x1, zprimes, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), a__zprimes, mark(x3)), mark(x4)), MARK(filter(x1, zprimes, x3))) MARK(filter(x0, filter(x1, cons(z0, z1), x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), cons(mark(z0), z1), mark(x3)), mark(x4)), MARK(filter(x1, cons(z0, z1), x3))) MARK(filter(x0, filter(x1, 0, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), 0, mark(x3)), mark(x4)), MARK(filter(x1, 0, x3))) MARK(filter(x0, filter(x1, s(z0), x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), s(mark(z0)), mark(x3)), mark(x4)), MARK(filter(x1, s(z0), x3))) MARK(filter(x0, filter(filter(z0, z1, z2), x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x2), mark(x3)), mark(x4)), MARK(filter(filter(z0, z1, z2), x2, x3))) MARK(filter(x0, filter(sieve(z0), x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(a__sieve(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(filter(sieve(z0), x2, x3))) MARK(filter(x0, filter(nats(z0), x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(a__nats(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(filter(nats(z0), x2, x3))) MARK(filter(x0, filter(zprimes, x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(a__zprimes, mark(x2), mark(x3)), mark(x4)), MARK(filter(zprimes, x2, x3))) MARK(filter(x0, filter(cons(z0, z1), x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(cons(mark(z0), z1), mark(x2), mark(x3)), mark(x4)), MARK(filter(cons(z0, z1), x2, x3))) MARK(filter(x0, filter(0, x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(0, mark(x2), mark(x3)), mark(x4)), MARK(filter(0, x2, x3))) MARK(filter(x0, filter(s(z0), x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(s(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(filter(s(z0), x2, x3))) MARK(filter(filter(z0, z1, z2), filter(x1, x2, x3), x4)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(sieve(z0), filter(x1, x2, x3), x4)) -> c11(A__FILTER(a__sieve(mark(z0)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(nats(z0), filter(x1, x2, x3), x4)) -> c11(A__FILTER(a__nats(mark(z0)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(zprimes, filter(x1, x2, x3), x4)) -> c11(A__FILTER(a__zprimes, a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(cons(z0, z1), filter(x1, x2, x3), x4)) -> c11(A__FILTER(cons(mark(z0), z1), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(x0, filter(x1, x2, x3), x4)) -> c11(MARK(filter(x1, x2, x3))) MARK(filter(0, filter(x1, x2, x3), x4)) -> c11(MARK(filter(x1, x2, x3))) MARK(filter(s(z0), filter(x1, x2, x3), x4)) -> c11(MARK(filter(x1, x2, x3))) MARK(filter(x0, sieve(filter(z0, z1, z2)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(sieve(filter(z0, z1, z2)))) MARK(filter(x0, sieve(sieve(z0)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(a__sieve(mark(z0))), mark(x2)), MARK(sieve(sieve(z0)))) MARK(filter(x0, sieve(nats(z0)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(a__nats(mark(z0))), mark(x2)), MARK(sieve(nats(z0)))) MARK(filter(x0, sieve(zprimes), x2)) -> c11(A__FILTER(mark(x0), a__sieve(a__zprimes), mark(x2)), MARK(sieve(zprimes))) MARK(filter(x0, sieve(cons(z0, z1)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(cons(mark(z0), z1)), mark(x2)), MARK(sieve(cons(z0, z1)))) MARK(filter(x0, sieve(s(z0)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(s(mark(z0))), mark(x2)), MARK(sieve(s(z0)))) MARK(filter(filter(z0, z1, z2), sieve(x1), x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(sieve(z0), sieve(x1), x2)) -> c11(A__FILTER(a__sieve(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(nats(z0), sieve(x1), x2)) -> c11(A__FILTER(a__nats(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(zprimes, sieve(x1), x2)) -> c11(A__FILTER(a__zprimes, a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(cons(z0, z1), sieve(x1), x2)) -> c11(A__FILTER(cons(mark(z0), z1), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(x0, sieve(x1), x2)) -> c11(MARK(sieve(x1))) MARK(filter(x0, sieve(0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(0), mark(x2))) MARK(filter(0, sieve(x1), x2)) -> c11(MARK(sieve(x1))) MARK(filter(s(z0), sieve(x1), x2)) -> c11(MARK(sieve(x1))) MARK(filter(x0, nats(filter(z0, z1, z2)), x2)) -> c11(A__FILTER(mark(x0), a__nats(a__filter(mark(z0), 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c11(A__FILTER(a__sieve(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(nats(z0), nats(x1), x2)) -> c11(A__FILTER(a__nats(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(zprimes, nats(x1), x2)) -> c11(A__FILTER(a__zprimes, a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(cons(z0, z1), nats(x1), x2)) -> c11(A__FILTER(cons(mark(z0), z1), a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(x0, nats(x1), x2)) -> c11(MARK(nats(x1))) MARK(filter(0, nats(x1), x2)) -> c11(MARK(nats(x1))) MARK(filter(s(z0), nats(x1), x2)) -> c11(MARK(nats(x1))) MARK(filter(x0, zprimes, x1)) -> c11(A__FILTER(mark(x0), a__sieve(a__nats(s(s(0)))), mark(x1)), MARK(zprimes)) MARK(filter(filter(z0, z1, z2), zprimes, x1)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(sieve(z0), zprimes, x1)) -> c11(A__FILTER(a__sieve(mark(z0)), a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(nats(z0), zprimes, x1)) -> c11(A__FILTER(a__nats(mark(z0)), a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(zprimes, zprimes, x1)) -> c11(A__FILTER(a__zprimes, a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(cons(z0, z1), zprimes, x1)) -> c11(A__FILTER(cons(mark(z0), z1), a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(x0, zprimes, x1)) -> c11(MARK(zprimes)) MARK(filter(0, zprimes, x1)) -> c11(MARK(zprimes)) MARK(filter(s(z0), zprimes, x1)) -> c11(MARK(zprimes)) MARK(filter(x0, s(filter(z0, z1, z2)), x2)) -> c11(A__FILTER(mark(x0), s(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(s(filter(z0, z1, z2)))) MARK(filter(x0, s(sieve(z0)), x2)) -> c11(A__FILTER(mark(x0), s(a__sieve(mark(z0))), mark(x2)), MARK(s(sieve(z0)))) MARK(filter(x0, s(nats(z0)), x2)) -> c11(A__FILTER(mark(x0), s(a__nats(mark(z0))), mark(x2)), MARK(s(nats(z0)))) MARK(filter(x0, s(zprimes), x2)) -> c11(A__FILTER(mark(x0), s(a__zprimes), mark(x2)), MARK(s(zprimes))) MARK(filter(x0, s(cons(z0, z1)), x2)) -> c11(A__FILTER(mark(x0), s(cons(mark(z0), z1)), mark(x2)), MARK(s(cons(z0, z1)))) MARK(filter(x0, s(0), x2)) -> c11(A__FILTER(mark(x0), s(0), mark(x2)), MARK(s(0))) MARK(filter(x0, s(s(z0)), x2)) -> c11(A__FILTER(mark(x0), s(s(mark(z0))), mark(x2)), MARK(s(s(z0)))) MARK(filter(filter(z0, z1, z2), s(x1), x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), s(mark(x1)), mark(x2)), MARK(s(x1))) MARK(filter(sieve(z0), s(x1), x2)) -> c11(A__FILTER(a__sieve(mark(z0)), s(mark(x1)), mark(x2)), MARK(s(x1))) MARK(filter(nats(z0), s(x1), x2)) -> c11(A__FILTER(a__nats(mark(z0)), s(mark(x1)), mark(x2)), MARK(s(x1))) MARK(filter(zprimes, s(x1), x2)) -> c11(A__FILTER(a__zprimes, s(mark(x1)), mark(x2)), MARK(s(x1))) MARK(filter(cons(z0, z1), s(x1), x2)) -> c11(A__FILTER(cons(mark(z0), z1), s(mark(x1)), mark(x2)), MARK(s(x1))) MARK(filter(x0, s(x1), x2)) -> c11(MARK(s(x1))) MARK(filter(0, s(x1), x2)) -> c11(MARK(s(x1))) MARK(filter(s(z0), s(x1), x2)) -> c11(MARK(s(x1))) MARK(filter(filter(x0, x1, filter(z0, z1, z2)), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, sieve(z0)), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), a__sieve(mark(z0))), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, nats(z0)), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), a__nats(mark(z0))), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, zprimes), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), a__zprimes), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, cons(z0, z1)), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), cons(mark(z0), z1)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, 0), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), 0), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, s(z0)), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), s(mark(z0))), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, filter(z0, z1, z2), x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, sieve(z0), x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), a__sieve(mark(z0)), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, nats(z0), x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), a__nats(mark(z0)), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, zprimes, x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), a__zprimes, mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, cons(z0, z1), x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), cons(mark(z0), z1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, 0, x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), 0, mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, s(z0), x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), s(mark(z0)), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(filter(z0, z1, z2), x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(sieve(z0), x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(a__sieve(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(nats(z0), x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(a__nats(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(zprimes, x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(a__zprimes, mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(cons(z0, z1), x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(cons(mark(z0), z1), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(0, x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(0, mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(s(z0), x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(s(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, x2), cons(z0, z1), x4)) -> c11(MARK(cons(z0, z1))) MARK(filter(filter(x0, x1, x2), x3, x4)) -> c11(MARK(x3)) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c14_2, c15_1, c16_1, c18_1, c8_1, c10_1, c11_2, c11_1, c12_2, c12_1, c13_2, c13_1, c8_2, c10_2 ---------------------------------------- (213) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(filter(sieve(z0), x1, x2)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x1)) by MARK(filter(sieve(x0), x1, filter(z0, z1, z2))) -> c11(A__FILTER(a__sieve(mark(x0)), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x1)) MARK(filter(sieve(x0), x1, sieve(z0))) -> c11(A__FILTER(a__sieve(mark(x0)), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(sieve(x0), x1, nats(z0))) -> c11(A__FILTER(a__sieve(mark(x0)), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(sieve(x0), x1, zprimes)) -> c11(A__FILTER(a__sieve(mark(x0)), mark(x1), a__zprimes), MARK(x1)) MARK(filter(sieve(x0), x1, cons(z0, z1))) -> c11(A__FILTER(a__sieve(mark(x0)), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(sieve(x0), x1, 0)) -> c11(A__FILTER(a__sieve(mark(x0)), mark(x1), 0), MARK(x1)) MARK(filter(sieve(x0), x1, s(z0))) -> c11(A__FILTER(a__sieve(mark(x0)), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(sieve(x0), filter(z0, z1, z2), x2)) -> c11(A__FILTER(a__sieve(mark(x0)), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(sieve(x0), sieve(z0), x2)) -> c11(A__FILTER(a__sieve(mark(x0)), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(sieve(x0), nats(z0), x2)) -> c11(A__FILTER(a__sieve(mark(x0)), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(sieve(x0), zprimes, x2)) -> c11(A__FILTER(a__sieve(mark(x0)), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(sieve(x0), cons(z0, z1), x2)) -> c11(A__FILTER(a__sieve(mark(x0)), cons(mark(z0), z1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(sieve(x0), 0, x2)) -> c11(A__FILTER(a__sieve(mark(x0)), 0, mark(x2)), MARK(0)) MARK(filter(sieve(x0), s(z0), x2)) -> c11(A__FILTER(a__sieve(mark(x0)), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(sieve(x0), x1, x2)) -> c11(A__FILTER(sieve(mark(x0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(filter(z0, z1, z2)), x1, x2)) -> c11(A__FILTER(a__sieve(a__filter(mark(z0), mark(z1), mark(z2))), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(sieve(z0)), x1, x2)) -> c11(A__FILTER(a__sieve(a__sieve(mark(z0))), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(nats(z0)), x1, x2)) -> c11(A__FILTER(a__sieve(a__nats(mark(z0))), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(zprimes), x1, x2)) -> c11(A__FILTER(a__sieve(a__zprimes), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(cons(z0, z1)), x1, x2)) -> c11(A__FILTER(a__sieve(cons(mark(z0), z1)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(0), x1, x2)) -> c11(A__FILTER(a__sieve(0), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(s(z0)), x1, x2)) -> c11(A__FILTER(a__sieve(s(mark(z0))), mark(x1), mark(x2)), MARK(x1)) ---------------------------------------- (214) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__zprimes, mark(x1), 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-> c10(MARK(sieve(x0))) MARK(filter(sieve(x0), 0, x2)) -> c10(MARK(sieve(x0))) MARK(filter(sieve(x0), x1, x2)) -> c10(MARK(sieve(x0))) MARK(filter(sieve(0), x1, x2)) -> c10(A__FILTER(a__sieve(0), mark(x1), mark(x2))) MARK(filter(nats(x0), x1, x2)) -> c10(A__FILTER(cons(mark(mark(x0)), nats(s(mark(x0)))), mark(x1), mark(x2)), MARK(nats(x0))) MARK(filter(nats(filter(z0, z1, z2)), x1, x2)) -> c10(A__FILTER(a__nats(a__filter(mark(z0), mark(z1), mark(z2))), mark(x1), mark(x2)), MARK(nats(filter(z0, z1, z2)))) MARK(filter(nats(sieve(z0)), x1, x2)) -> c10(A__FILTER(a__nats(a__sieve(mark(z0))), mark(x1), mark(x2)), MARK(nats(sieve(z0)))) MARK(filter(nats(nats(z0)), x1, x2)) -> c10(A__FILTER(a__nats(a__nats(mark(z0))), mark(x1), mark(x2)), MARK(nats(nats(z0)))) MARK(filter(nats(zprimes), x1, x2)) -> c10(A__FILTER(a__nats(a__zprimes), mark(x1), mark(x2)), MARK(nats(zprimes))) MARK(filter(nats(cons(z0, z1)), x1, x2)) -> c10(A__FILTER(a__nats(cons(mark(z0), z1)), mark(x1), mark(x2)), MARK(nats(cons(z0, z1)))) MARK(filter(nats(0), x1, x2)) -> c10(A__FILTER(a__nats(0), mark(x1), mark(x2)), MARK(nats(0))) MARK(filter(nats(s(z0)), x1, x2)) -> c10(A__FILTER(a__nats(s(mark(z0))), mark(x1), mark(x2)), MARK(nats(s(z0)))) MARK(filter(nats(x0), x1, x2)) -> c10(MARK(nats(x0))) MARK(filter(nats(x0), cons(z0, z1), x2)) -> c10(MARK(nats(x0))) MARK(filter(nats(x0), 0, x2)) -> c10(MARK(nats(x0))) MARK(filter(zprimes, x0, x1)) -> c10(A__FILTER(a__sieve(a__nats(s(s(0)))), mark(x0), mark(x1)), MARK(zprimes)) MARK(filter(zprimes, cons(z0, z1), x1)) -> c10(MARK(zprimes)) MARK(filter(zprimes, 0, x1)) -> c10(MARK(zprimes)) MARK(filter(zprimes, x0, x1)) -> c10(MARK(zprimes)) MARK(filter(cons(filter(z0, z1, z2), x1), x2, x3)) -> c10(A__FILTER(cons(a__filter(mark(z0), mark(z1), mark(z2)), x1), mark(x2), mark(x3)), MARK(cons(filter(z0, z1, z2), x1))) MARK(filter(cons(sieve(z0), x1), x2, x3)) -> c10(A__FILTER(cons(a__sieve(mark(z0)), x1), mark(x2), mark(x3)), MARK(cons(sieve(z0), x1))) MARK(filter(cons(nats(z0), x1), x2, x3)) -> c10(A__FILTER(cons(a__nats(mark(z0)), x1), mark(x2), mark(x3)), MARK(cons(nats(z0), x1))) MARK(filter(cons(zprimes, x1), x2, x3)) -> c10(A__FILTER(cons(a__zprimes, x1), mark(x2), mark(x3)), MARK(cons(zprimes, x1))) MARK(filter(cons(cons(z0, z1), x1), x2, x3)) -> c10(A__FILTER(cons(cons(mark(z0), z1), x1), mark(x2), mark(x3)), MARK(cons(cons(z0, z1), x1))) MARK(filter(cons(0, x1), x2, x3)) -> c10(A__FILTER(cons(0, x1), mark(x2), mark(x3)), MARK(cons(0, x1))) MARK(filter(cons(s(z0), x1), x2, x3)) -> c10(A__FILTER(cons(s(mark(z0)), x1), mark(x2), mark(x3)), MARK(cons(s(z0), x1))) MARK(filter(cons(x0, x1), x2, x3)) -> c10(MARK(cons(x0, x1))) MARK(filter(cons(x0, x1), cons(z0, z1), x3)) -> c10(MARK(cons(x0, x1))) MARK(filter(cons(x0, x1), 0, x3)) -> c10(MARK(cons(x0, x1))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x1)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c11(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c11(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c11(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x1)) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, filter(x2, x3, x4))) -> c11(MARK(x1)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c11(MARK(x1)) MARK(filter(x0, x1, sieve(x2))) -> c11(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x1)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x1)) MARK(filter(x0, x1, sieve(sieve(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__sieve(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, sieve(nats(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, sieve(zprimes))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__zprimes)), MARK(x1)) MARK(filter(x0, x1, sieve(cons(z0, z1)))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(cons(mark(z0), z1))), MARK(x1)) MARK(filter(x0, x1, sieve(0))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(0)), MARK(x1)) MARK(filter(x0, x1, sieve(s(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(s(mark(z0)))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), sieve(x2))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), sieve(x2))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), a__sieve(mark(x2))), MARK(sieve(z0))) MARK(filter(x0, nats(z0), sieve(x2))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), a__sieve(mark(x2))), MARK(nats(z0))) MARK(filter(x0, zprimes, sieve(x2))) -> c11(A__FILTER(mark(x0), a__zprimes, a__sieve(mark(x2))), MARK(zprimes)) MARK(filter(x0, s(z0), sieve(x2))) -> c11(A__FILTER(mark(x0), s(mark(z0)), a__sieve(mark(x2))), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, sieve(x2))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(sieve(z0), x1, sieve(x2))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(nats(z0), x1, sieve(x2))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(zprimes, x1, sieve(x2))) -> c11(A__FILTER(a__zprimes, mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(cons(z0, z1), x1, sieve(x2))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(x0, cons(z0, z1), sieve(x2))) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, sieve(x2))) -> c11(MARK(x1)) MARK(filter(s(z0), x1, sieve(x2))) -> c11(MARK(x1)) MARK(filter(x0, x1, nats(x2))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(mark(x2)), nats(s(mark(x2))))), MARK(x1)) MARK(filter(x0, x1, nats(x2))) -> c11(A__FILTER(mark(x0), mark(x1), nats(mark(x2))), MARK(x1)) MARK(filter(x0, x1, nats(filter(z0, z1, z2)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x1)) MARK(filter(x0, x1, nats(sieve(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__sieve(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, nats(nats(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__nats(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, nats(zprimes))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__zprimes)), MARK(x1)) MARK(filter(x0, x1, nats(cons(z0, z1)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(cons(mark(z0), z1))), MARK(x1)) MARK(filter(x0, x1, nats(0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(0)), MARK(x1)) MARK(filter(x0, x1, nats(s(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(s(mark(z0)))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), nats(x2))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), nats(x2))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), a__nats(mark(x2))), MARK(sieve(z0))) MARK(filter(x0, nats(z0), nats(x2))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), a__nats(mark(x2))), MARK(nats(z0))) MARK(filter(x0, zprimes, nats(x2))) -> c11(A__FILTER(mark(x0), a__zprimes, a__nats(mark(x2))), MARK(zprimes)) MARK(filter(x0, s(z0), nats(x2))) -> c11(A__FILTER(mark(x0), s(mark(z0)), a__nats(mark(x2))), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, nats(x2))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(sieve(z0), x1, nats(x2))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(nats(z0), x1, nats(x2))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(zprimes, x1, nats(x2))) -> c11(A__FILTER(a__zprimes, mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(cons(z0, z1), x1, nats(x2))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(x0, cons(z0, z1), nats(x2))) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, nats(x2))) -> c11(MARK(x1)) MARK(filter(s(z0), x1, nats(x2))) -> c11(MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(s(s(0))))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), zprimes), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), zprimes)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__zprimes), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), zprimes)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), a__zprimes), MARK(sieve(z0))) MARK(filter(x0, nats(z0), zprimes)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), a__zprimes), MARK(nats(z0))) MARK(filter(x0, zprimes, zprimes)) -> c11(A__FILTER(mark(x0), a__zprimes, a__zprimes), MARK(zprimes)) MARK(filter(x0, s(z0), zprimes)) -> c11(A__FILTER(mark(x0), s(mark(z0)), a__zprimes), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, zprimes)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__zprimes), MARK(x1)) MARK(filter(sieve(z0), x1, zprimes)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__zprimes), MARK(x1)) MARK(filter(nats(z0), x1, zprimes)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__zprimes), MARK(x1)) MARK(filter(zprimes, x1, zprimes)) -> c11(A__FILTER(a__zprimes, mark(x1), a__zprimes), MARK(x1)) MARK(filter(cons(z0, z1), x1, zprimes)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, cons(z0, z1), zprimes)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, zprimes)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, zprimes)) -> c11(MARK(x1)) MARK(filter(x0, x1, cons(filter(z0, z1, z2), x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(a__filter(mark(z0), mark(z1), mark(z2)), x3)), MARK(x1)) MARK(filter(x0, x1, cons(sieve(z0), x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(a__sieve(mark(z0)), x3)), MARK(x1)) MARK(filter(x0, x1, cons(nats(z0), x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(a__nats(mark(z0)), x3)), MARK(x1)) MARK(filter(x0, x1, cons(zprimes, x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(a__zprimes, x3)), MARK(x1)) MARK(filter(x0, x1, cons(cons(z0, z1), x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(cons(mark(z0), z1), x3)), MARK(x1)) MARK(filter(x0, x1, cons(0, x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(0, x3)), MARK(x1)) MARK(filter(x0, x1, cons(s(z0), x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(s(mark(z0)), x3)), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), cons(x2, x3))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), cons(mark(x2), x3)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), cons(x2, x3))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), cons(mark(x2), x3)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), cons(x2, x3))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), cons(mark(x2), x3)), MARK(nats(z0))) MARK(filter(x0, zprimes, cons(x2, x3))) -> c11(A__FILTER(mark(x0), a__zprimes, cons(mark(x2), x3)), MARK(zprimes)) MARK(filter(x0, s(z0), cons(x2, x3))) -> c11(A__FILTER(mark(x0), s(mark(z0)), cons(mark(x2), x3)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, cons(x2, x3))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(sieve(z0), x1, cons(x2, x3))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(nats(z0), x1, cons(x2, x3))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(zprimes, x1, cons(x2, x3))) -> c11(A__FILTER(a__zprimes, mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(cons(z0, z1), x1, cons(x2, x3))) -> 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mark(x1), a__sieve(mark(z0))), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, nats(z0)), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), a__nats(mark(z0))), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, zprimes), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), a__zprimes), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, cons(z0, z1)), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), cons(mark(z0), z1)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, 0), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), 0), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, s(z0)), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), s(mark(z0))), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, filter(z0, z1, z2), x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, sieve(z0), x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), a__sieve(mark(z0)), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, nats(z0), x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), a__nats(mark(z0)), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, zprimes, x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), a__zprimes, mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, cons(z0, z1), x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), cons(mark(z0), z1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, 0, x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), 0, mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, s(z0), x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), s(mark(z0)), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(filter(z0, z1, z2), x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(sieve(z0), x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(a__sieve(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(nats(z0), x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(a__nats(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(zprimes, x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(a__zprimes, mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(cons(z0, z1), x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(cons(mark(z0), z1), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(0, x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(0, mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(s(z0), x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(s(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, x2), cons(z0, z1), x4)) -> c11(MARK(cons(z0, z1))) MARK(filter(filter(x0, x1, x2), x3, x4)) -> c11(MARK(x3)) MARK(filter(sieve(x0), cons(z0, z1), x2)) -> c11(A__FILTER(a__sieve(mark(x0)), cons(mark(z0), z1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(sieve(x0), 0, x2)) -> c11(A__FILTER(a__sieve(mark(x0)), 0, mark(x2)), MARK(0)) MARK(filter(sieve(x0), x1, x2)) -> c11(A__FILTER(sieve(mark(x0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(filter(z0, z1, z2)), x1, x2)) -> c11(A__FILTER(a__sieve(a__filter(mark(z0), mark(z1), mark(z2))), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(sieve(z0)), x1, x2)) -> c11(A__FILTER(a__sieve(a__sieve(mark(z0))), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(nats(z0)), x1, x2)) -> c11(A__FILTER(a__sieve(a__nats(mark(z0))), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(zprimes), x1, x2)) -> c11(A__FILTER(a__sieve(a__zprimes), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(cons(z0, z1)), x1, x2)) -> c11(A__FILTER(a__sieve(cons(mark(z0), z1)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(0), x1, x2)) -> c11(A__FILTER(a__sieve(0), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(s(z0)), x1, x2)) -> c11(A__FILTER(a__sieve(s(mark(z0))), mark(x1), mark(x2)), MARK(x1)) S tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__zprimes, mark(x1), 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MARK(filter(s(z0), x1, sieve(x2))) -> c11(MARK(x1)) MARK(filter(x0, x1, nats(x2))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(mark(x2)), nats(s(mark(x2))))), MARK(x1)) MARK(filter(x0, x1, nats(x2))) -> c11(A__FILTER(mark(x0), mark(x1), nats(mark(x2))), MARK(x1)) MARK(filter(x0, x1, nats(filter(z0, z1, z2)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x1)) MARK(filter(x0, x1, nats(sieve(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__sieve(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, nats(nats(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__nats(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, nats(zprimes))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__zprimes)), MARK(x1)) MARK(filter(x0, x1, nats(cons(z0, z1)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(cons(mark(z0), z1))), MARK(x1)) MARK(filter(x0, x1, nats(0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(0)), MARK(x1)) MARK(filter(x0, x1, nats(s(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(s(mark(z0)))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), nats(x2))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), nats(x2))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), a__nats(mark(x2))), MARK(sieve(z0))) MARK(filter(x0, nats(z0), nats(x2))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), a__nats(mark(x2))), MARK(nats(z0))) MARK(filter(x0, zprimes, nats(x2))) -> c11(A__FILTER(mark(x0), a__zprimes, a__nats(mark(x2))), MARK(zprimes)) MARK(filter(x0, s(z0), nats(x2))) -> c11(A__FILTER(mark(x0), s(mark(z0)), a__nats(mark(x2))), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, nats(x2))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(sieve(z0), x1, nats(x2))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(nats(z0), x1, nats(x2))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(zprimes, x1, nats(x2))) -> c11(A__FILTER(a__zprimes, mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(cons(z0, z1), x1, nats(x2))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(x0, cons(z0, z1), nats(x2))) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, nats(x2))) -> c11(MARK(x1)) MARK(filter(s(z0), x1, nats(x2))) -> c11(MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(s(s(0))))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), zprimes), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), zprimes)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__zprimes), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), zprimes)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), a__zprimes), MARK(sieve(z0))) MARK(filter(x0, nats(z0), zprimes)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), a__zprimes), MARK(nats(z0))) MARK(filter(x0, zprimes, zprimes)) -> c11(A__FILTER(mark(x0), a__zprimes, a__zprimes), MARK(zprimes)) MARK(filter(x0, s(z0), zprimes)) -> c11(A__FILTER(mark(x0), s(mark(z0)), a__zprimes), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, zprimes)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__zprimes), MARK(x1)) MARK(filter(sieve(z0), x1, zprimes)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__zprimes), MARK(x1)) MARK(filter(nats(z0), x1, zprimes)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__zprimes), MARK(x1)) MARK(filter(zprimes, x1, zprimes)) -> c11(A__FILTER(a__zprimes, mark(x1), a__zprimes), MARK(x1)) MARK(filter(cons(z0, z1), x1, zprimes)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, cons(z0, z1), zprimes)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, zprimes)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, zprimes)) -> c11(MARK(x1)) MARK(filter(x0, x1, 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mark(x1), a__sieve(mark(z0))), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, nats(z0)), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), a__nats(mark(z0))), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, zprimes), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), a__zprimes), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, cons(z0, z1)), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), cons(mark(z0), z1)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, 0), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), 0), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, s(z0)), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), s(mark(z0))), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, filter(z0, z1, z2), x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, sieve(z0), x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), a__sieve(mark(z0)), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, nats(z0), x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), a__nats(mark(z0)), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, zprimes, x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), a__zprimes, mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, cons(z0, z1), x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), cons(mark(z0), z1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, 0, x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), 0, mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, s(z0), x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), s(mark(z0)), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(filter(z0, z1, z2), x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(sieve(z0), x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(a__sieve(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(nats(z0), x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(a__nats(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(zprimes, x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(a__zprimes, mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(cons(z0, z1), x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(cons(mark(z0), z1), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(0, x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(0, mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(s(z0), x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(s(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, x2), cons(z0, z1), x4)) -> c11(MARK(cons(z0, z1))) MARK(filter(filter(x0, x1, x2), x3, x4)) -> c11(MARK(x3)) MARK(filter(sieve(x0), cons(z0, z1), x2)) -> c11(A__FILTER(a__sieve(mark(x0)), cons(mark(z0), z1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(sieve(x0), 0, x2)) -> c11(A__FILTER(a__sieve(mark(x0)), 0, mark(x2)), MARK(0)) MARK(filter(sieve(x0), x1, x2)) -> c11(A__FILTER(sieve(mark(x0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(filter(z0, z1, z2)), x1, x2)) -> c11(A__FILTER(a__sieve(a__filter(mark(z0), mark(z1), mark(z2))), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(sieve(z0)), x1, x2)) -> c11(A__FILTER(a__sieve(a__sieve(mark(z0))), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(nats(z0)), x1, x2)) -> c11(A__FILTER(a__sieve(a__nats(mark(z0))), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(zprimes), x1, x2)) -> c11(A__FILTER(a__sieve(a__zprimes), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(cons(z0, z1)), x1, x2)) -> c11(A__FILTER(a__sieve(cons(mark(z0), z1)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(0), x1, x2)) -> c11(A__FILTER(a__sieve(0), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(s(z0)), x1, x2)) -> c11(A__FILTER(a__sieve(s(mark(z0))), mark(x1), mark(x2)), MARK(x1)) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c14_2, c15_1, c16_1, c18_1, c8_1, c10_1, c11_2, c11_1, c12_2, c12_1, c13_2, c13_1, c8_2, c10_2 ---------------------------------------- (215) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(filter(sieve(x0), 0, x2)) -> c11(A__FILTER(a__sieve(mark(x0)), 0, mark(x2)), MARK(0)) ---------------------------------------- (216) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(x0, 0, filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, sieve(x2))) -> c10(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x0)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, sieve(sieve(z0)))) -> 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mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(s(z0), x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(s(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, x2), cons(z0, z1), x4)) -> c11(MARK(cons(z0, z1))) MARK(filter(filter(x0, x1, x2), x3, x4)) -> c11(MARK(x3)) MARK(filter(sieve(x0), cons(z0, z1), x2)) -> c11(A__FILTER(a__sieve(mark(x0)), cons(mark(z0), z1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(sieve(x0), x1, x2)) -> c11(A__FILTER(sieve(mark(x0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(filter(z0, z1, z2)), x1, x2)) -> c11(A__FILTER(a__sieve(a__filter(mark(z0), mark(z1), mark(z2))), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(sieve(z0)), x1, x2)) -> c11(A__FILTER(a__sieve(a__sieve(mark(z0))), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(nats(z0)), x1, x2)) -> c11(A__FILTER(a__sieve(a__nats(mark(z0))), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(zprimes), x1, x2)) -> c11(A__FILTER(a__sieve(a__zprimes), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(cons(z0, z1)), x1, x2)) -> c11(A__FILTER(a__sieve(cons(mark(z0), z1)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(0), x1, x2)) -> c11(A__FILTER(a__sieve(0), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(s(z0)), x1, x2)) -> c11(A__FILTER(a__sieve(s(mark(z0))), mark(x1), mark(x2)), MARK(x1)) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c14_2, c15_1, c16_1, c18_1, c8_1, c10_1, c11_2, c11_1, c12_2, c12_1, c13_2, c13_1, c8_2, c10_2 ---------------------------------------- (217) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (218) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(x0, 0, filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, sieve(x2))) -> c10(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x0)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, sieve(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, sieve(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, sieve(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(0)), MARK(x0)) MARK(filter(x0, x1, sieve(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), sieve(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, sieve(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, sieve(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__sieve(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, sieve(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, sieve(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, sieve(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), a__sieve(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, sieve(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__sieve(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), sieve(x2))) -> c10(MARK(x0)) MARK(filter(x0, 0, sieve(x2))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, sieve(x2))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(mark(x2)), nats(s(mark(x2))))), MARK(x0)) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), nats(mark(x2))), MARK(x0)) MARK(filter(x0, x1, nats(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, nats(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, nats(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, nats(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, nats(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(cons(mark(z0), 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MARK(nats(z0))) MARK(filter(x0, zprimes, nats(x2))) -> c11(A__FILTER(mark(x0), a__zprimes, a__nats(mark(x2))), MARK(zprimes)) MARK(filter(x0, s(z0), nats(x2))) -> c11(A__FILTER(mark(x0), s(mark(z0)), a__nats(mark(x2))), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, nats(x2))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(sieve(z0), x1, nats(x2))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(nats(z0), x1, nats(x2))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(zprimes, x1, nats(x2))) -> c11(A__FILTER(a__zprimes, mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(cons(z0, z1), x1, nats(x2))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(x0, cons(z0, z1), nats(x2))) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, nats(x2))) -> c11(MARK(x1)) MARK(filter(s(z0), x1, nats(x2))) -> c11(MARK(x1)) 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c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__filter(mark(z0), mark(z1), mark(z2))), mark(x4)), MARK(filter(x1, x2, filter(z0, z1, z2)))) MARK(filter(x0, filter(x1, x2, sieve(z0)), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__sieve(mark(z0))), mark(x4)), MARK(filter(x1, x2, sieve(z0)))) MARK(filter(x0, filter(x1, x2, nats(z0)), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__nats(mark(z0))), mark(x4)), MARK(filter(x1, x2, nats(z0)))) MARK(filter(x0, filter(x1, x2, zprimes), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__zprimes), mark(x4)), MARK(filter(x1, x2, zprimes))) MARK(filter(x0, filter(x1, x2, cons(z0, z1)), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), cons(mark(z0), z1)), mark(x4)), MARK(filter(x1, x2, cons(z0, z1)))) MARK(filter(x0, filter(x1, x2, 0), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), 0), mark(x4)), MARK(filter(x1, x2, 0))) MARK(filter(x0, filter(x1, x2, s(z0)), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), s(mark(z0))), mark(x4)), MARK(filter(x1, x2, s(z0)))) MARK(filter(x0, filter(x1, filter(z0, z1, z2), x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), a__filter(mark(z0), mark(z1), mark(z2)), mark(x3)), mark(x4)), MARK(filter(x1, filter(z0, z1, z2), x3))) MARK(filter(x0, filter(x1, sieve(z0), x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), a__sieve(mark(z0)), mark(x3)), mark(x4)), MARK(filter(x1, sieve(z0), x3))) MARK(filter(x0, filter(x1, nats(z0), x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), a__nats(mark(z0)), mark(x3)), mark(x4)), MARK(filter(x1, nats(z0), x3))) MARK(filter(x0, filter(x1, zprimes, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), a__zprimes, mark(x3)), mark(x4)), MARK(filter(x1, zprimes, x3))) MARK(filter(x0, filter(x1, cons(z0, z1), x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), cons(mark(z0), z1), mark(x3)), mark(x4)), MARK(filter(x1, cons(z0, z1), x3))) MARK(filter(x0, filter(x1, 0, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), 0, mark(x3)), mark(x4)), MARK(filter(x1, 0, x3))) MARK(filter(x0, filter(x1, s(z0), x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), s(mark(z0)), mark(x3)), mark(x4)), MARK(filter(x1, s(z0), x3))) MARK(filter(x0, filter(filter(z0, z1, z2), x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x2), mark(x3)), mark(x4)), MARK(filter(filter(z0, z1, z2), x2, x3))) MARK(filter(x0, filter(sieve(z0), x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(a__sieve(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(filter(sieve(z0), x2, x3))) MARK(filter(x0, filter(nats(z0), x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(a__nats(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(filter(nats(z0), x2, x3))) MARK(filter(x0, filter(zprimes, x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(a__zprimes, mark(x2), mark(x3)), mark(x4)), MARK(filter(zprimes, x2, x3))) MARK(filter(x0, filter(cons(z0, z1), x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(cons(mark(z0), z1), mark(x2), mark(x3)), mark(x4)), MARK(filter(cons(z0, z1), x2, x3))) MARK(filter(x0, filter(0, x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(0, mark(x2), mark(x3)), mark(x4)), MARK(filter(0, x2, x3))) MARK(filter(x0, filter(s(z0), x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(s(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(filter(s(z0), x2, x3))) MARK(filter(filter(z0, z1, z2), filter(x1, x2, x3), x4)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(sieve(z0), filter(x1, x2, x3), x4)) -> c11(A__FILTER(a__sieve(mark(z0)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(nats(z0), filter(x1, x2, x3), x4)) -> c11(A__FILTER(a__nats(mark(z0)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(zprimes, filter(x1, x2, x3), x4)) -> c11(A__FILTER(a__zprimes, a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(cons(z0, z1), filter(x1, x2, x3), x4)) -> c11(A__FILTER(cons(mark(z0), z1), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(x0, filter(x1, x2, x3), x4)) -> c11(MARK(filter(x1, x2, x3))) MARK(filter(0, filter(x1, x2, x3), x4)) -> c11(MARK(filter(x1, x2, x3))) MARK(filter(s(z0), filter(x1, x2, x3), x4)) -> c11(MARK(filter(x1, x2, x3))) MARK(filter(x0, sieve(filter(z0, z1, z2)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(sieve(filter(z0, z1, z2)))) MARK(filter(x0, sieve(sieve(z0)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(a__sieve(mark(z0))), mark(x2)), MARK(sieve(sieve(z0)))) MARK(filter(x0, sieve(nats(z0)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(a__nats(mark(z0))), mark(x2)), MARK(sieve(nats(z0)))) MARK(filter(x0, sieve(zprimes), x2)) -> c11(A__FILTER(mark(x0), a__sieve(a__zprimes), mark(x2)), MARK(sieve(zprimes))) MARK(filter(x0, sieve(cons(z0, z1)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(cons(mark(z0), z1)), mark(x2)), MARK(sieve(cons(z0, z1)))) MARK(filter(x0, sieve(s(z0)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(s(mark(z0))), mark(x2)), MARK(sieve(s(z0)))) MARK(filter(filter(z0, z1, z2), sieve(x1), x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(sieve(z0), sieve(x1), x2)) -> c11(A__FILTER(a__sieve(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(nats(z0), sieve(x1), x2)) -> c11(A__FILTER(a__nats(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(zprimes, sieve(x1), x2)) -> c11(A__FILTER(a__zprimes, a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(cons(z0, z1), sieve(x1), x2)) -> c11(A__FILTER(cons(mark(z0), z1), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(x0, sieve(x1), x2)) -> c11(MARK(sieve(x1))) MARK(filter(x0, sieve(0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(0), mark(x2))) MARK(filter(0, sieve(x1), x2)) -> c11(MARK(sieve(x1))) MARK(filter(s(z0), sieve(x1), x2)) -> c11(MARK(sieve(x1))) MARK(filter(x0, nats(filter(z0, z1, z2)), x2)) -> c11(A__FILTER(mark(x0), a__nats(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(nats(filter(z0, z1, z2)))) MARK(filter(x0, nats(sieve(z0)), x2)) -> c11(A__FILTER(mark(x0), a__nats(a__sieve(mark(z0))), mark(x2)), MARK(nats(sieve(z0)))) MARK(filter(x0, nats(nats(z0)), x2)) -> c11(A__FILTER(mark(x0), a__nats(a__nats(mark(z0))), mark(x2)), MARK(nats(nats(z0)))) MARK(filter(x0, nats(zprimes), x2)) -> c11(A__FILTER(mark(x0), a__nats(a__zprimes), mark(x2)), MARK(nats(zprimes))) MARK(filter(x0, nats(cons(z0, z1)), x2)) -> c11(A__FILTER(mark(x0), a__nats(cons(mark(z0), z1)), mark(x2)), MARK(nats(cons(z0, z1)))) MARK(filter(x0, nats(0), x2)) -> c11(A__FILTER(mark(x0), a__nats(0), mark(x2)), MARK(nats(0))) MARK(filter(x0, nats(s(z0)), x2)) -> c11(A__FILTER(mark(x0), a__nats(s(mark(z0))), mark(x2)), MARK(nats(s(z0)))) MARK(filter(filter(z0, z1, z2), nats(x1), x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(sieve(z0), nats(x1), x2)) -> c11(A__FILTER(a__sieve(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(nats(z0), nats(x1), x2)) -> c11(A__FILTER(a__nats(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(zprimes, nats(x1), x2)) -> c11(A__FILTER(a__zprimes, a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(cons(z0, z1), nats(x1), x2)) -> c11(A__FILTER(cons(mark(z0), z1), a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(x0, nats(x1), x2)) -> c11(MARK(nats(x1))) MARK(filter(0, nats(x1), x2)) -> c11(MARK(nats(x1))) MARK(filter(s(z0), nats(x1), x2)) -> c11(MARK(nats(x1))) MARK(filter(x0, zprimes, x1)) -> c11(A__FILTER(mark(x0), a__sieve(a__nats(s(s(0)))), mark(x1)), MARK(zprimes)) MARK(filter(filter(z0, z1, z2), zprimes, x1)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(sieve(z0), zprimes, x1)) -> c11(A__FILTER(a__sieve(mark(z0)), a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(nats(z0), zprimes, x1)) -> c11(A__FILTER(a__nats(mark(z0)), a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(zprimes, zprimes, x1)) -> c11(A__FILTER(a__zprimes, a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(cons(z0, z1), zprimes, x1)) -> c11(A__FILTER(cons(mark(z0), z1), a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(x0, zprimes, x1)) -> c11(MARK(zprimes)) MARK(filter(0, zprimes, x1)) -> c11(MARK(zprimes)) MARK(filter(s(z0), zprimes, x1)) -> c11(MARK(zprimes)) MARK(filter(x0, s(filter(z0, z1, z2)), x2)) -> c11(A__FILTER(mark(x0), s(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(s(filter(z0, z1, z2)))) MARK(filter(x0, s(sieve(z0)), x2)) -> c11(A__FILTER(mark(x0), s(a__sieve(mark(z0))), mark(x2)), MARK(s(sieve(z0)))) MARK(filter(x0, s(nats(z0)), x2)) -> c11(A__FILTER(mark(x0), s(a__nats(mark(z0))), mark(x2)), MARK(s(nats(z0)))) MARK(filter(x0, s(zprimes), x2)) -> c11(A__FILTER(mark(x0), s(a__zprimes), mark(x2)), MARK(s(zprimes))) MARK(filter(x0, s(cons(z0, z1)), x2)) -> c11(A__FILTER(mark(x0), s(cons(mark(z0), z1)), mark(x2)), MARK(s(cons(z0, z1)))) MARK(filter(x0, s(0), x2)) -> c11(A__FILTER(mark(x0), s(0), mark(x2)), MARK(s(0))) MARK(filter(x0, s(s(z0)), x2)) -> c11(A__FILTER(mark(x0), s(s(mark(z0))), mark(x2)), MARK(s(s(z0)))) MARK(filter(filter(z0, z1, z2), s(x1), x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), s(mark(x1)), mark(x2)), MARK(s(x1))) MARK(filter(sieve(z0), s(x1), x2)) -> c11(A__FILTER(a__sieve(mark(z0)), s(mark(x1)), mark(x2)), MARK(s(x1))) MARK(filter(nats(z0), s(x1), x2)) -> c11(A__FILTER(a__nats(mark(z0)), s(mark(x1)), mark(x2)), MARK(s(x1))) MARK(filter(zprimes, s(x1), x2)) -> c11(A__FILTER(a__zprimes, s(mark(x1)), mark(x2)), MARK(s(x1))) MARK(filter(cons(z0, z1), s(x1), x2)) -> c11(A__FILTER(cons(mark(z0), z1), s(mark(x1)), mark(x2)), MARK(s(x1))) MARK(filter(x0, s(x1), x2)) -> c11(MARK(s(x1))) MARK(filter(0, s(x1), x2)) -> c11(MARK(s(x1))) MARK(filter(s(z0), s(x1), x2)) -> c11(MARK(s(x1))) MARK(filter(filter(x0, x1, filter(z0, z1, z2)), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, sieve(z0)), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), a__sieve(mark(z0))), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, nats(z0)), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), a__nats(mark(z0))), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, zprimes), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), a__zprimes), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, cons(z0, z1)), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), cons(mark(z0), z1)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, 0), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), 0), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, s(z0)), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), s(mark(z0))), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, filter(z0, z1, z2), x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, sieve(z0), x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), a__sieve(mark(z0)), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, nats(z0), x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), a__nats(mark(z0)), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, zprimes, x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), a__zprimes, mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, cons(z0, z1), 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MARK(filter(filter(cons(z0, z1), x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(cons(mark(z0), z1), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(0, x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(0, mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(s(z0), x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(s(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, x2), cons(z0, z1), x4)) -> c11(MARK(cons(z0, z1))) MARK(filter(filter(x0, x1, x2), x3, x4)) -> c11(MARK(x3)) MARK(filter(sieve(filter(z0, z1, z2)), x1, x2)) -> c11(A__FILTER(a__sieve(a__filter(mark(z0), mark(z1), mark(z2))), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(sieve(z0)), x1, x2)) -> c11(A__FILTER(a__sieve(a__sieve(mark(z0))), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(nats(z0)), x1, x2)) -> c11(A__FILTER(a__sieve(a__nats(mark(z0))), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(zprimes), x1, x2)) -> c11(A__FILTER(a__sieve(a__zprimes), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(cons(z0, z1)), x1, x2)) -> c11(A__FILTER(a__sieve(cons(mark(z0), z1)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(0), x1, x2)) -> c11(A__FILTER(a__sieve(0), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(s(z0)), x1, x2)) -> c11(A__FILTER(a__sieve(s(mark(z0))), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(x0), cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(sieve(x0), x1, x2)) -> c11(MARK(x1)) S tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(x0, 0, filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, sieve(x2))) -> c10(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x0)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, sieve(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, sieve(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, sieve(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(0)), MARK(x0)) MARK(filter(x0, x1, sieve(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), sieve(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, sieve(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, sieve(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__sieve(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, sieve(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, sieve(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, sieve(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), a__sieve(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, sieve(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__sieve(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), sieve(x2))) -> c10(MARK(x0)) MARK(filter(x0, 0, sieve(x2))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, sieve(x2))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(mark(x2)), nats(s(mark(x2))))), MARK(x0)) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), nats(mark(x2))), MARK(x0)) MARK(filter(x0, x1, nats(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, nats(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, nats(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, nats(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, nats(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, nats(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(0)), MARK(x0)) MARK(filter(x0, x1, nats(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), nats(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), nats(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), nats(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, nats(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), nats(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, nats(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__nats(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, nats(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, nats(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, nats(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), a__nats(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, nats(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__nats(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), nats(x2))) -> c10(MARK(x0)) MARK(filter(x0, 0, nats(x2))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, nats(x2))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, zprimes)) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(s(s(0))))), MARK(x0)) MARK(filter(x0, x1, zprimes)) -> c10(A__FILTER(mark(x0), mark(x1), zprimes), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), zprimes)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__zprimes), MARK(x0)) MARK(filter(x0, sieve(z0), zprimes)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__zprimes), MARK(x0)) MARK(filter(x0, nats(z0), 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MARK(filter(cons(z0, z1), filter(x1, x2, x3), x4)) -> c11(A__FILTER(cons(mark(z0), z1), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(x0, filter(x1, x2, x3), x4)) -> c11(MARK(filter(x1, x2, x3))) MARK(filter(0, filter(x1, x2, x3), x4)) -> c11(MARK(filter(x1, x2, x3))) MARK(filter(s(z0), filter(x1, x2, x3), x4)) -> c11(MARK(filter(x1, x2, x3))) MARK(filter(x0, sieve(filter(z0, z1, z2)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(sieve(filter(z0, z1, z2)))) MARK(filter(x0, sieve(sieve(z0)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(a__sieve(mark(z0))), mark(x2)), MARK(sieve(sieve(z0)))) MARK(filter(x0, sieve(nats(z0)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(a__nats(mark(z0))), mark(x2)), MARK(sieve(nats(z0)))) MARK(filter(x0, sieve(zprimes), x2)) -> c11(A__FILTER(mark(x0), a__sieve(a__zprimes), mark(x2)), MARK(sieve(zprimes))) MARK(filter(x0, sieve(cons(z0, z1)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(cons(mark(z0), z1)), mark(x2)), MARK(sieve(cons(z0, z1)))) MARK(filter(x0, sieve(s(z0)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(s(mark(z0))), mark(x2)), MARK(sieve(s(z0)))) MARK(filter(filter(z0, z1, z2), sieve(x1), x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(sieve(z0), sieve(x1), x2)) -> c11(A__FILTER(a__sieve(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(nats(z0), sieve(x1), x2)) -> c11(A__FILTER(a__nats(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(zprimes, sieve(x1), x2)) -> c11(A__FILTER(a__zprimes, a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(cons(z0, z1), sieve(x1), x2)) -> c11(A__FILTER(cons(mark(z0), z1), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(x0, sieve(x1), x2)) -> c11(MARK(sieve(x1))) MARK(filter(x0, sieve(0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(0), mark(x2))) MARK(filter(0, sieve(x1), x2)) -> c11(MARK(sieve(x1))) MARK(filter(s(z0), sieve(x1), x2)) -> c11(MARK(sieve(x1))) MARK(filter(x0, nats(filter(z0, z1, z2)), x2)) -> c11(A__FILTER(mark(x0), a__nats(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(nats(filter(z0, z1, z2)))) MARK(filter(x0, nats(sieve(z0)), x2)) -> c11(A__FILTER(mark(x0), a__nats(a__sieve(mark(z0))), mark(x2)), MARK(nats(sieve(z0)))) MARK(filter(x0, nats(nats(z0)), x2)) -> c11(A__FILTER(mark(x0), a__nats(a__nats(mark(z0))), mark(x2)), MARK(nats(nats(z0)))) MARK(filter(x0, nats(zprimes), x2)) -> c11(A__FILTER(mark(x0), a__nats(a__zprimes), mark(x2)), MARK(nats(zprimes))) MARK(filter(x0, nats(cons(z0, z1)), x2)) -> c11(A__FILTER(mark(x0), a__nats(cons(mark(z0), z1)), mark(x2)), MARK(nats(cons(z0, z1)))) MARK(filter(x0, nats(0), x2)) -> c11(A__FILTER(mark(x0), a__nats(0), mark(x2)), MARK(nats(0))) MARK(filter(x0, nats(s(z0)), x2)) -> c11(A__FILTER(mark(x0), a__nats(s(mark(z0))), mark(x2)), MARK(nats(s(z0)))) MARK(filter(filter(z0, z1, z2), nats(x1), x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(sieve(z0), nats(x1), x2)) -> c11(A__FILTER(a__sieve(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(nats(z0), nats(x1), x2)) -> c11(A__FILTER(a__nats(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(zprimes, nats(x1), x2)) -> c11(A__FILTER(a__zprimes, a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(cons(z0, z1), nats(x1), x2)) -> c11(A__FILTER(cons(mark(z0), z1), a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(x0, nats(x1), x2)) -> c11(MARK(nats(x1))) MARK(filter(0, nats(x1), x2)) -> c11(MARK(nats(x1))) MARK(filter(s(z0), nats(x1), x2)) -> c11(MARK(nats(x1))) MARK(filter(x0, zprimes, x1)) -> c11(A__FILTER(mark(x0), a__sieve(a__nats(s(s(0)))), mark(x1)), MARK(zprimes)) MARK(filter(filter(z0, z1, z2), zprimes, x1)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(sieve(z0), zprimes, x1)) -> c11(A__FILTER(a__sieve(mark(z0)), a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(nats(z0), zprimes, x1)) -> c11(A__FILTER(a__nats(mark(z0)), a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(zprimes, zprimes, x1)) -> c11(A__FILTER(a__zprimes, a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(cons(z0, z1), zprimes, x1)) -> c11(A__FILTER(cons(mark(z0), z1), a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(x0, zprimes, x1)) -> c11(MARK(zprimes)) MARK(filter(0, zprimes, x1)) -> c11(MARK(zprimes)) MARK(filter(s(z0), zprimes, x1)) -> c11(MARK(zprimes)) MARK(filter(x0, s(filter(z0, z1, z2)), x2)) -> c11(A__FILTER(mark(x0), s(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(s(filter(z0, z1, z2)))) MARK(filter(x0, s(sieve(z0)), x2)) -> c11(A__FILTER(mark(x0), s(a__sieve(mark(z0))), mark(x2)), MARK(s(sieve(z0)))) MARK(filter(x0, s(nats(z0)), x2)) -> c11(A__FILTER(mark(x0), s(a__nats(mark(z0))), mark(x2)), MARK(s(nats(z0)))) MARK(filter(x0, s(zprimes), x2)) -> c11(A__FILTER(mark(x0), s(a__zprimes), mark(x2)), MARK(s(zprimes))) MARK(filter(x0, s(cons(z0, z1)), x2)) -> c11(A__FILTER(mark(x0), s(cons(mark(z0), z1)), mark(x2)), MARK(s(cons(z0, z1)))) MARK(filter(x0, s(0), x2)) -> c11(A__FILTER(mark(x0), s(0), mark(x2)), MARK(s(0))) MARK(filter(x0, s(s(z0)), x2)) -> c11(A__FILTER(mark(x0), s(s(mark(z0))), mark(x2)), MARK(s(s(z0)))) MARK(filter(filter(z0, z1, z2), s(x1), x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), s(mark(x1)), mark(x2)), MARK(s(x1))) MARK(filter(sieve(z0), s(x1), x2)) -> c11(A__FILTER(a__sieve(mark(z0)), s(mark(x1)), mark(x2)), MARK(s(x1))) MARK(filter(nats(z0), s(x1), x2)) -> c11(A__FILTER(a__nats(mark(z0)), s(mark(x1)), mark(x2)), MARK(s(x1))) MARK(filter(zprimes, s(x1), x2)) -> c11(A__FILTER(a__zprimes, s(mark(x1)), mark(x2)), MARK(s(x1))) MARK(filter(cons(z0, z1), s(x1), x2)) -> c11(A__FILTER(cons(mark(z0), z1), s(mark(x1)), mark(x2)), MARK(s(x1))) MARK(filter(x0, s(x1), x2)) -> c11(MARK(s(x1))) MARK(filter(0, s(x1), x2)) -> c11(MARK(s(x1))) MARK(filter(s(z0), s(x1), x2)) -> c11(MARK(s(x1))) MARK(filter(filter(x0, x1, filter(z0, z1, z2)), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, sieve(z0)), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), a__sieve(mark(z0))), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, nats(z0)), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), a__nats(mark(z0))), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, zprimes), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), a__zprimes), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, cons(z0, z1)), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), cons(mark(z0), z1)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, 0), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), 0), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, s(z0)), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), s(mark(z0))), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, filter(z0, z1, z2), x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, sieve(z0), x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), a__sieve(mark(z0)), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, nats(z0), x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), a__nats(mark(z0)), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, zprimes, x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), a__zprimes, mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, cons(z0, z1), x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), cons(mark(z0), z1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, 0, x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), 0, mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, s(z0), x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), s(mark(z0)), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(filter(z0, z1, z2), x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(sieve(z0), x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(a__sieve(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(nats(z0), x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(a__nats(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(zprimes, x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(a__zprimes, mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(cons(z0, z1), x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(cons(mark(z0), z1), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(0, x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(0, mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(s(z0), x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(s(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, x2), cons(z0, z1), x4)) -> c11(MARK(cons(z0, z1))) MARK(filter(filter(x0, x1, x2), x3, x4)) -> c11(MARK(x3)) MARK(filter(sieve(filter(z0, z1, z2)), x1, x2)) -> c11(A__FILTER(a__sieve(a__filter(mark(z0), mark(z1), mark(z2))), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(sieve(z0)), x1, x2)) -> c11(A__FILTER(a__sieve(a__sieve(mark(z0))), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(nats(z0)), x1, x2)) -> c11(A__FILTER(a__sieve(a__nats(mark(z0))), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(zprimes), x1, x2)) -> c11(A__FILTER(a__sieve(a__zprimes), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(cons(z0, z1)), x1, x2)) -> c11(A__FILTER(a__sieve(cons(mark(z0), z1)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(0), x1, x2)) -> c11(A__FILTER(a__sieve(0), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(s(z0)), x1, x2)) -> c11(A__FILTER(a__sieve(s(mark(z0))), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(x0), cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(sieve(x0), x1, x2)) -> c11(MARK(x1)) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c14_2, c15_1, c16_1, c18_1, c8_1, c10_1, c11_2, c11_1, c12_2, c12_1, c13_2, c13_1, c8_2, c10_2 ---------------------------------------- (219) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(filter(nats(z0), x1, x2)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x1)) by MARK(filter(nats(x0), x1, filter(z0, z1, z2))) -> c11(A__FILTER(a__nats(mark(x0)), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x1)) MARK(filter(nats(x0), x1, sieve(z0))) -> c11(A__FILTER(a__nats(mark(x0)), mark(x1), a__sieve(mark(z0))), MARK(x1)) MARK(filter(nats(x0), x1, nats(z0))) -> c11(A__FILTER(a__nats(mark(x0)), mark(x1), a__nats(mark(z0))), MARK(x1)) MARK(filter(nats(x0), x1, zprimes)) -> c11(A__FILTER(a__nats(mark(x0)), mark(x1), a__zprimes), MARK(x1)) MARK(filter(nats(x0), x1, cons(z0, z1))) -> c11(A__FILTER(a__nats(mark(x0)), mark(x1), cons(mark(z0), z1)), MARK(x1)) MARK(filter(nats(x0), x1, 0)) -> c11(A__FILTER(a__nats(mark(x0)), mark(x1), 0), MARK(x1)) MARK(filter(nats(x0), x1, s(z0))) -> c11(A__FILTER(a__nats(mark(x0)), mark(x1), s(mark(z0))), MARK(x1)) MARK(filter(nats(x0), filter(z0, z1, z2), x2)) -> c11(A__FILTER(a__nats(mark(x0)), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(filter(z0, z1, z2))) MARK(filter(nats(x0), sieve(z0), x2)) -> c11(A__FILTER(a__nats(mark(x0)), a__sieve(mark(z0)), mark(x2)), MARK(sieve(z0))) MARK(filter(nats(x0), nats(z0), x2)) -> c11(A__FILTER(a__nats(mark(x0)), a__nats(mark(z0)), mark(x2)), MARK(nats(z0))) MARK(filter(nats(x0), zprimes, x2)) -> c11(A__FILTER(a__nats(mark(x0)), a__zprimes, mark(x2)), MARK(zprimes)) MARK(filter(nats(x0), cons(z0, z1), x2)) -> c11(A__FILTER(a__nats(mark(x0)), cons(mark(z0), z1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(nats(x0), 0, x2)) -> c11(A__FILTER(a__nats(mark(x0)), 0, mark(x2)), MARK(0)) MARK(filter(nats(x0), s(z0), x2)) -> c11(A__FILTER(a__nats(mark(x0)), s(mark(z0)), mark(x2)), MARK(s(z0))) MARK(filter(nats(x0), x1, x2)) -> c11(A__FILTER(cons(mark(mark(x0)), nats(s(mark(x0)))), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(x0), x1, x2)) -> c11(A__FILTER(nats(mark(x0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(filter(z0, z1, z2)), x1, x2)) -> c11(A__FILTER(a__nats(a__filter(mark(z0), mark(z1), mark(z2))), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(sieve(z0)), x1, x2)) -> c11(A__FILTER(a__nats(a__sieve(mark(z0))), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(nats(z0)), x1, x2)) -> c11(A__FILTER(a__nats(a__nats(mark(z0))), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(zprimes), x1, x2)) -> c11(A__FILTER(a__nats(a__zprimes), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(cons(z0, z1)), x1, x2)) -> c11(A__FILTER(a__nats(cons(mark(z0), z1)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(0), x1, x2)) -> c11(A__FILTER(a__nats(0), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(s(z0)), x1, x2)) -> c11(A__FILTER(a__nats(s(mark(z0))), mark(x1), mark(x2)), MARK(x1)) ---------------------------------------- (220) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(x0, 0, filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, sieve(x2))) -> c10(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x0)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, sieve(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, sieve(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, sieve(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(0)), MARK(x0)) MARK(filter(x0, x1, sieve(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), sieve(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, sieve(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, sieve(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__sieve(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, sieve(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, sieve(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, sieve(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), a__sieve(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, sieve(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__sieve(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), sieve(x2))) -> c10(MARK(x0)) MARK(filter(x0, 0, sieve(x2))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, sieve(x2))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(mark(x2)), nats(s(mark(x2))))), MARK(x0)) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), nats(mark(x2))), MARK(x0)) MARK(filter(x0, x1, nats(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, nats(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, nats(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, nats(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, nats(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, nats(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(0)), MARK(x0)) MARK(filter(x0, x1, nats(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), nats(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x2))), MARK(x0)) 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c11(A__FILTER(mark(x0), a__filter(s(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(filter(s(z0), x2, x3))) MARK(filter(filter(z0, z1, z2), filter(x1, x2, x3), x4)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(sieve(z0), filter(x1, x2, x3), x4)) -> c11(A__FILTER(a__sieve(mark(z0)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(nats(z0), filter(x1, x2, x3), x4)) -> c11(A__FILTER(a__nats(mark(z0)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(zprimes, filter(x1, x2, x3), x4)) -> c11(A__FILTER(a__zprimes, a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(cons(z0, z1), filter(x1, x2, x3), x4)) -> c11(A__FILTER(cons(mark(z0), z1), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(x0, filter(x1, x2, x3), x4)) -> c11(MARK(filter(x1, x2, x3))) MARK(filter(0, filter(x1, x2, x3), x4)) -> c11(MARK(filter(x1, x2, x3))) MARK(filter(s(z0), filter(x1, x2, x3), x4)) -> c11(MARK(filter(x1, x2, x3))) MARK(filter(x0, sieve(filter(z0, z1, z2)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(sieve(filter(z0, z1, z2)))) MARK(filter(x0, sieve(sieve(z0)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(a__sieve(mark(z0))), mark(x2)), MARK(sieve(sieve(z0)))) MARK(filter(x0, sieve(nats(z0)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(a__nats(mark(z0))), mark(x2)), MARK(sieve(nats(z0)))) MARK(filter(x0, sieve(zprimes), x2)) -> c11(A__FILTER(mark(x0), a__sieve(a__zprimes), mark(x2)), MARK(sieve(zprimes))) MARK(filter(x0, sieve(cons(z0, z1)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(cons(mark(z0), z1)), mark(x2)), MARK(sieve(cons(z0, z1)))) MARK(filter(x0, sieve(s(z0)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(s(mark(z0))), mark(x2)), MARK(sieve(s(z0)))) MARK(filter(filter(z0, z1, z2), sieve(x1), x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(sieve(z0), sieve(x1), x2)) -> c11(A__FILTER(a__sieve(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(nats(z0), sieve(x1), x2)) -> c11(A__FILTER(a__nats(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(zprimes, sieve(x1), x2)) -> c11(A__FILTER(a__zprimes, a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(cons(z0, z1), sieve(x1), x2)) -> c11(A__FILTER(cons(mark(z0), z1), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(x0, sieve(x1), x2)) -> c11(MARK(sieve(x1))) MARK(filter(x0, sieve(0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(0), mark(x2))) MARK(filter(0, sieve(x1), x2)) -> c11(MARK(sieve(x1))) MARK(filter(s(z0), sieve(x1), x2)) -> c11(MARK(sieve(x1))) MARK(filter(x0, nats(filter(z0, z1, z2)), x2)) -> c11(A__FILTER(mark(x0), a__nats(a__filter(mark(z0), 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c11(A__FILTER(a__sieve(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(nats(z0), nats(x1), x2)) -> c11(A__FILTER(a__nats(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(zprimes, nats(x1), x2)) -> c11(A__FILTER(a__zprimes, a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(cons(z0, z1), nats(x1), x2)) -> c11(A__FILTER(cons(mark(z0), z1), a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(x0, nats(x1), x2)) -> c11(MARK(nats(x1))) MARK(filter(0, nats(x1), x2)) -> c11(MARK(nats(x1))) MARK(filter(s(z0), nats(x1), x2)) -> c11(MARK(nats(x1))) MARK(filter(x0, zprimes, x1)) -> c11(A__FILTER(mark(x0), a__sieve(a__nats(s(s(0)))), mark(x1)), MARK(zprimes)) MARK(filter(filter(z0, z1, z2), zprimes, x1)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(sieve(z0), zprimes, x1)) -> c11(A__FILTER(a__sieve(mark(z0)), a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(nats(z0), zprimes, x1)) -> c11(A__FILTER(a__nats(mark(z0)), a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(zprimes, zprimes, x1)) -> c11(A__FILTER(a__zprimes, a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(cons(z0, z1), zprimes, x1)) -> c11(A__FILTER(cons(mark(z0), z1), a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(x0, zprimes, x1)) -> c11(MARK(zprimes)) MARK(filter(0, zprimes, x1)) -> c11(MARK(zprimes)) MARK(filter(s(z0), zprimes, x1)) -> c11(MARK(zprimes)) MARK(filter(x0, s(filter(z0, z1, z2)), x2)) -> c11(A__FILTER(mark(x0), s(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(s(filter(z0, z1, z2)))) MARK(filter(x0, s(sieve(z0)), x2)) -> c11(A__FILTER(mark(x0), s(a__sieve(mark(z0))), mark(x2)), MARK(s(sieve(z0)))) MARK(filter(x0, s(nats(z0)), x2)) -> c11(A__FILTER(mark(x0), s(a__nats(mark(z0))), mark(x2)), MARK(s(nats(z0)))) MARK(filter(x0, s(zprimes), x2)) -> c11(A__FILTER(mark(x0), s(a__zprimes), mark(x2)), MARK(s(zprimes))) MARK(filter(x0, s(cons(z0, z1)), x2)) -> c11(A__FILTER(mark(x0), s(cons(mark(z0), z1)), mark(x2)), MARK(s(cons(z0, z1)))) MARK(filter(x0, s(0), x2)) -> c11(A__FILTER(mark(x0), s(0), mark(x2)), MARK(s(0))) MARK(filter(x0, s(s(z0)), x2)) -> c11(A__FILTER(mark(x0), s(s(mark(z0))), mark(x2)), MARK(s(s(z0)))) MARK(filter(filter(z0, z1, z2), s(x1), x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), s(mark(x1)), mark(x2)), MARK(s(x1))) MARK(filter(sieve(z0), s(x1), x2)) -> c11(A__FILTER(a__sieve(mark(z0)), s(mark(x1)), mark(x2)), MARK(s(x1))) MARK(filter(nats(z0), s(x1), x2)) -> c11(A__FILTER(a__nats(mark(z0)), s(mark(x1)), mark(x2)), MARK(s(x1))) MARK(filter(zprimes, s(x1), x2)) -> c11(A__FILTER(a__zprimes, s(mark(x1)), mark(x2)), MARK(s(x1))) MARK(filter(cons(z0, z1), s(x1), x2)) -> c11(A__FILTER(cons(mark(z0), z1), s(mark(x1)), mark(x2)), MARK(s(x1))) MARK(filter(x0, s(x1), x2)) -> c11(MARK(s(x1))) MARK(filter(0, s(x1), x2)) -> c11(MARK(s(x1))) MARK(filter(s(z0), s(x1), x2)) -> c11(MARK(s(x1))) MARK(filter(filter(x0, 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MARK(filter(filter(x0, filter(z0, z1, z2), x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, sieve(z0), x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), a__sieve(mark(z0)), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, nats(z0), x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), a__nats(mark(z0)), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, zprimes, x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), a__zprimes, mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, cons(z0, z1), x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), cons(mark(z0), z1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, 0, x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), 0, mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, s(z0), x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), s(mark(z0)), mark(x2)), mark(x3), 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MARK(filter(sieve(x0), cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(sieve(x0), x1, x2)) -> c11(MARK(x1)) MARK(filter(nats(x0), cons(z0, z1), x2)) -> c11(A__FILTER(a__nats(mark(x0)), cons(mark(z0), z1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(nats(x0), 0, x2)) -> c11(A__FILTER(a__nats(mark(x0)), 0, mark(x2)), MARK(0)) MARK(filter(nats(x0), x1, x2)) -> c11(A__FILTER(cons(mark(mark(x0)), nats(s(mark(x0)))), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(x0), x1, x2)) -> c11(A__FILTER(nats(mark(x0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(filter(z0, z1, z2)), x1, x2)) -> c11(A__FILTER(a__nats(a__filter(mark(z0), mark(z1), mark(z2))), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(sieve(z0)), x1, x2)) -> c11(A__FILTER(a__nats(a__sieve(mark(z0))), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(nats(z0)), x1, x2)) -> c11(A__FILTER(a__nats(a__nats(mark(z0))), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(zprimes), x1, x2)) -> c11(A__FILTER(a__nats(a__zprimes), 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MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> 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x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(x0, 0, filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, sieve(x2))) -> c10(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x0)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, sieve(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, sieve(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, sieve(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(0)), MARK(x0)) MARK(filter(x0, x1, sieve(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), sieve(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, sieve(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, sieve(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__sieve(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, sieve(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, sieve(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, sieve(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), a__sieve(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, sieve(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__sieve(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), sieve(x2))) -> c10(MARK(x0)) MARK(filter(x0, 0, sieve(x2))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, sieve(x2))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(mark(x2)), nats(s(mark(x2))))), MARK(x0)) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), nats(mark(x2))), MARK(x0)) MARK(filter(x0, x1, nats(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, nats(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, nats(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, nats(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, nats(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, nats(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(0)), MARK(x0)) MARK(filter(x0, x1, nats(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), nats(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), 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c11(A__FILTER(mark(x0), a__filter(s(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(filter(s(z0), x2, x3))) MARK(filter(filter(z0, z1, z2), filter(x1, x2, x3), x4)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(sieve(z0), filter(x1, x2, x3), x4)) -> c11(A__FILTER(a__sieve(mark(z0)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(nats(z0), filter(x1, x2, x3), x4)) -> c11(A__FILTER(a__nats(mark(z0)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(zprimes, filter(x1, x2, x3), x4)) -> c11(A__FILTER(a__zprimes, a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(cons(z0, z1), filter(x1, x2, x3), x4)) -> c11(A__FILTER(cons(mark(z0), z1), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(x0, filter(x1, x2, x3), x4)) -> c11(MARK(filter(x1, x2, x3))) MARK(filter(0, filter(x1, x2, x3), x4)) -> c11(MARK(filter(x1, x2, x3))) MARK(filter(s(z0), filter(x1, x2, x3), x4)) -> c11(MARK(filter(x1, x2, x3))) MARK(filter(x0, sieve(filter(z0, z1, z2)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(sieve(filter(z0, z1, z2)))) MARK(filter(x0, sieve(sieve(z0)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(a__sieve(mark(z0))), mark(x2)), MARK(sieve(sieve(z0)))) MARK(filter(x0, sieve(nats(z0)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(a__nats(mark(z0))), mark(x2)), MARK(sieve(nats(z0)))) MARK(filter(x0, sieve(zprimes), x2)) -> c11(A__FILTER(mark(x0), a__sieve(a__zprimes), mark(x2)), MARK(sieve(zprimes))) MARK(filter(x0, sieve(cons(z0, z1)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(cons(mark(z0), z1)), mark(x2)), MARK(sieve(cons(z0, z1)))) MARK(filter(x0, sieve(s(z0)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(s(mark(z0))), mark(x2)), MARK(sieve(s(z0)))) MARK(filter(filter(z0, z1, z2), sieve(x1), x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(sieve(z0), sieve(x1), x2)) -> c11(A__FILTER(a__sieve(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(nats(z0), sieve(x1), x2)) -> c11(A__FILTER(a__nats(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(zprimes, sieve(x1), x2)) -> c11(A__FILTER(a__zprimes, a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(cons(z0, z1), sieve(x1), x2)) -> c11(A__FILTER(cons(mark(z0), z1), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(x0, sieve(x1), x2)) -> c11(MARK(sieve(x1))) MARK(filter(x0, sieve(0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(0), mark(x2))) MARK(filter(0, sieve(x1), x2)) -> c11(MARK(sieve(x1))) MARK(filter(s(z0), sieve(x1), x2)) -> c11(MARK(sieve(x1))) MARK(filter(x0, nats(filter(z0, z1, z2)), x2)) -> c11(A__FILTER(mark(x0), a__nats(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(nats(filter(z0, z1, z2)))) MARK(filter(x0, nats(sieve(z0)), x2)) -> c11(A__FILTER(mark(x0), a__nats(a__sieve(mark(z0))), mark(x2)), MARK(nats(sieve(z0)))) MARK(filter(x0, nats(nats(z0)), x2)) -> c11(A__FILTER(mark(x0), a__nats(a__nats(mark(z0))), mark(x2)), MARK(nats(nats(z0)))) MARK(filter(x0, nats(zprimes), x2)) -> c11(A__FILTER(mark(x0), a__nats(a__zprimes), mark(x2)), MARK(nats(zprimes))) MARK(filter(x0, nats(cons(z0, z1)), x2)) -> c11(A__FILTER(mark(x0), a__nats(cons(mark(z0), z1)), mark(x2)), MARK(nats(cons(z0, z1)))) MARK(filter(x0, nats(0), x2)) -> c11(A__FILTER(mark(x0), a__nats(0), mark(x2)), MARK(nats(0))) MARK(filter(x0, nats(s(z0)), x2)) -> c11(A__FILTER(mark(x0), a__nats(s(mark(z0))), mark(x2)), MARK(nats(s(z0)))) MARK(filter(filter(z0, z1, z2), nats(x1), x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(sieve(z0), nats(x1), x2)) -> c11(A__FILTER(a__sieve(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(nats(z0), nats(x1), x2)) -> c11(A__FILTER(a__nats(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(zprimes, nats(x1), x2)) -> c11(A__FILTER(a__zprimes, a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(cons(z0, z1), nats(x1), x2)) -> c11(A__FILTER(cons(mark(z0), z1), a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(x0, nats(x1), x2)) -> c11(MARK(nats(x1))) MARK(filter(0, nats(x1), x2)) -> c11(MARK(nats(x1))) MARK(filter(s(z0), nats(x1), x2)) -> c11(MARK(nats(x1))) MARK(filter(x0, zprimes, x1)) -> c11(A__FILTER(mark(x0), a__sieve(a__nats(s(s(0)))), mark(x1)), MARK(zprimes)) MARK(filter(filter(z0, z1, z2), zprimes, x1)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(sieve(z0), zprimes, x1)) -> c11(A__FILTER(a__sieve(mark(z0)), a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(nats(z0), zprimes, x1)) -> c11(A__FILTER(a__nats(mark(z0)), a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(zprimes, zprimes, x1)) -> c11(A__FILTER(a__zprimes, a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(cons(z0, z1), zprimes, x1)) -> c11(A__FILTER(cons(mark(z0), z1), a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(x0, zprimes, x1)) -> c11(MARK(zprimes)) MARK(filter(0, zprimes, x1)) -> c11(MARK(zprimes)) MARK(filter(s(z0), zprimes, x1)) -> c11(MARK(zprimes)) MARK(filter(x0, s(filter(z0, z1, z2)), x2)) -> c11(A__FILTER(mark(x0), s(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(s(filter(z0, z1, z2)))) MARK(filter(x0, s(sieve(z0)), x2)) -> c11(A__FILTER(mark(x0), s(a__sieve(mark(z0))), mark(x2)), MARK(s(sieve(z0)))) MARK(filter(x0, s(nats(z0)), x2)) -> c11(A__FILTER(mark(x0), s(a__nats(mark(z0))), mark(x2)), MARK(s(nats(z0)))) MARK(filter(x0, s(zprimes), x2)) -> c11(A__FILTER(mark(x0), s(a__zprimes), mark(x2)), MARK(s(zprimes))) MARK(filter(x0, s(cons(z0, z1)), x2)) -> c11(A__FILTER(mark(x0), s(cons(mark(z0), z1)), mark(x2)), MARK(s(cons(z0, z1)))) MARK(filter(x0, s(0), x2)) -> c11(A__FILTER(mark(x0), s(0), mark(x2)), MARK(s(0))) MARK(filter(x0, s(s(z0)), x2)) -> c11(A__FILTER(mark(x0), s(s(mark(z0))), mark(x2)), MARK(s(s(z0)))) MARK(filter(filter(z0, z1, z2), s(x1), x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), s(mark(x1)), mark(x2)), MARK(s(x1))) MARK(filter(sieve(z0), s(x1), x2)) -> c11(A__FILTER(a__sieve(mark(z0)), s(mark(x1)), mark(x2)), MARK(s(x1))) MARK(filter(nats(z0), s(x1), x2)) -> c11(A__FILTER(a__nats(mark(z0)), s(mark(x1)), mark(x2)), MARK(s(x1))) MARK(filter(zprimes, s(x1), x2)) -> c11(A__FILTER(a__zprimes, s(mark(x1)), mark(x2)), MARK(s(x1))) MARK(filter(cons(z0, z1), s(x1), x2)) -> c11(A__FILTER(cons(mark(z0), z1), s(mark(x1)), mark(x2)), MARK(s(x1))) MARK(filter(x0, s(x1), x2)) -> c11(MARK(s(x1))) MARK(filter(0, s(x1), x2)) -> c11(MARK(s(x1))) MARK(filter(s(z0), s(x1), x2)) -> c11(MARK(s(x1))) MARK(filter(filter(x0, x1, filter(z0, z1, z2)), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, sieve(z0)), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), a__sieve(mark(z0))), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, nats(z0)), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), a__nats(mark(z0))), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, zprimes), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), a__zprimes), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, cons(z0, z1)), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), cons(mark(z0), z1)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, 0), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), 0), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, s(z0)), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), s(mark(z0))), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, filter(z0, z1, z2), x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, sieve(z0), x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), a__sieve(mark(z0)), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, nats(z0), x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), a__nats(mark(z0)), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, zprimes, x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), a__zprimes, mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, cons(z0, z1), x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), cons(mark(z0), z1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, 0, x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), 0, mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, s(z0), x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), s(mark(z0)), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(filter(z0, z1, z2), x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(sieve(z0), x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(a__sieve(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(nats(z0), x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(a__nats(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(zprimes, x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(a__zprimes, mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(cons(z0, z1), x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(cons(mark(z0), z1), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(0, x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(0, mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(s(z0), x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(s(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, x2), cons(z0, z1), x4)) -> c11(MARK(cons(z0, z1))) MARK(filter(filter(x0, x1, x2), x3, x4)) -> c11(MARK(x3)) MARK(filter(sieve(filter(z0, z1, z2)), x1, x2)) -> c11(A__FILTER(a__sieve(a__filter(mark(z0), mark(z1), mark(z2))), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(sieve(z0)), x1, x2)) -> c11(A__FILTER(a__sieve(a__sieve(mark(z0))), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(nats(z0)), x1, x2)) -> c11(A__FILTER(a__sieve(a__nats(mark(z0))), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(zprimes), x1, x2)) -> c11(A__FILTER(a__sieve(a__zprimes), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(cons(z0, z1)), x1, x2)) -> c11(A__FILTER(a__sieve(cons(mark(z0), z1)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(0), x1, x2)) -> c11(A__FILTER(a__sieve(0), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(s(z0)), x1, x2)) -> c11(A__FILTER(a__sieve(s(mark(z0))), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(x0), cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(sieve(x0), x1, x2)) -> c11(MARK(x1)) MARK(filter(nats(x0), cons(z0, z1), x2)) -> c11(A__FILTER(a__nats(mark(x0)), cons(mark(z0), z1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(nats(x0), 0, x2)) -> c11(A__FILTER(a__nats(mark(x0)), 0, mark(x2)), MARK(0)) MARK(filter(nats(x0), x1, x2)) -> c11(A__FILTER(cons(mark(mark(x0)), nats(s(mark(x0)))), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(x0), x1, x2)) -> c11(A__FILTER(nats(mark(x0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(filter(z0, z1, z2)), x1, x2)) -> c11(A__FILTER(a__nats(a__filter(mark(z0), mark(z1), mark(z2))), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(sieve(z0)), x1, x2)) -> c11(A__FILTER(a__nats(a__sieve(mark(z0))), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(nats(z0)), x1, x2)) -> c11(A__FILTER(a__nats(a__nats(mark(z0))), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(zprimes), x1, x2)) -> c11(A__FILTER(a__nats(a__zprimes), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(cons(z0, z1)), x1, x2)) -> c11(A__FILTER(a__nats(cons(mark(z0), z1)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(0), x1, x2)) -> c11(A__FILTER(a__nats(0), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(s(z0)), x1, x2)) -> c11(A__FILTER(a__nats(s(mark(z0))), mark(x1), mark(x2)), MARK(x1)) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c14_2, c15_1, c16_1, c18_1, c8_1, c10_1, c11_2, c11_1, c12_2, c12_1, c13_2, c13_1, c8_2, c10_2 ---------------------------------------- (221) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(filter(nats(x0), 0, x2)) -> c11(A__FILTER(a__nats(mark(x0)), 0, mark(x2)), MARK(0)) ---------------------------------------- (222) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(x0, 0, filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, sieve(x2))) -> c10(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x0)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, sieve(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, sieve(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, sieve(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(0)), MARK(x0)) MARK(filter(x0, x1, sieve(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), sieve(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, sieve(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, sieve(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__sieve(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, sieve(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, sieve(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, sieve(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), a__sieve(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, sieve(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__sieve(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), sieve(x2))) -> c10(MARK(x0)) MARK(filter(x0, 0, sieve(x2))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, sieve(x2))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(mark(x2)), nats(s(mark(x2))))), MARK(x0)) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), nats(mark(x2))), MARK(x0)) MARK(filter(x0, x1, nats(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, nats(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, nats(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, nats(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, nats(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, nats(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(0)), MARK(x0)) MARK(filter(x0, x1, nats(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), nats(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), nats(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), nats(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, nats(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), nats(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, nats(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__nats(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, nats(x2))) -> 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c11(A__FILTER(mark(x0), mark(x1), a__sieve(0)), MARK(x1)) MARK(filter(x0, x1, sieve(s(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(s(mark(z0)))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), sieve(x2))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), sieve(x2))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), a__sieve(mark(x2))), MARK(sieve(z0))) MARK(filter(x0, nats(z0), sieve(x2))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), a__sieve(mark(x2))), MARK(nats(z0))) MARK(filter(x0, zprimes, sieve(x2))) -> c11(A__FILTER(mark(x0), a__zprimes, a__sieve(mark(x2))), MARK(zprimes)) MARK(filter(x0, s(z0), sieve(x2))) -> c11(A__FILTER(mark(x0), s(mark(z0)), a__sieve(mark(x2))), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, sieve(x2))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(sieve(z0), x1, sieve(x2))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(nats(z0), x1, sieve(x2))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(zprimes, x1, sieve(x2))) -> c11(A__FILTER(a__zprimes, mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(cons(z0, z1), x1, sieve(x2))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(x0, cons(z0, z1), sieve(x2))) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, sieve(x2))) -> c11(MARK(x1)) MARK(filter(s(z0), x1, sieve(x2))) -> c11(MARK(x1)) MARK(filter(x0, x1, nats(x2))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(mark(x2)), nats(s(mark(x2))))), MARK(x1)) MARK(filter(x0, x1, nats(x2))) -> c11(A__FILTER(mark(x0), mark(x1), nats(mark(x2))), MARK(x1)) MARK(filter(x0, x1, nats(filter(z0, z1, z2)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x1)) MARK(filter(x0, x1, nats(sieve(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__sieve(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, nats(nats(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__nats(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, nats(zprimes))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__zprimes)), MARK(x1)) MARK(filter(x0, x1, nats(cons(z0, z1)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(cons(mark(z0), z1))), MARK(x1)) MARK(filter(x0, x1, nats(0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(0)), MARK(x1)) MARK(filter(x0, x1, nats(s(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(s(mark(z0)))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), nats(x2))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), nats(x2))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), a__nats(mark(x2))), MARK(sieve(z0))) MARK(filter(x0, nats(z0), nats(x2))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), a__nats(mark(x2))), MARK(nats(z0))) MARK(filter(x0, zprimes, nats(x2))) -> c11(A__FILTER(mark(x0), a__zprimes, a__nats(mark(x2))), MARK(zprimes)) MARK(filter(x0, s(z0), nats(x2))) -> c11(A__FILTER(mark(x0), s(mark(z0)), a__nats(mark(x2))), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, nats(x2))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(sieve(z0), x1, nats(x2))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(nats(z0), x1, nats(x2))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(zprimes, x1, nats(x2))) -> c11(A__FILTER(a__zprimes, mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(cons(z0, z1), x1, nats(x2))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(x0, cons(z0, z1), nats(x2))) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, nats(x2))) -> c11(MARK(x1)) MARK(filter(s(z0), x1, nats(x2))) -> c11(MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(s(s(0))))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), zprimes), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), zprimes)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__zprimes), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), zprimes)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), a__zprimes), MARK(sieve(z0))) MARK(filter(x0, nats(z0), zprimes)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), a__zprimes), MARK(nats(z0))) MARK(filter(x0, zprimes, zprimes)) -> c11(A__FILTER(mark(x0), a__zprimes, a__zprimes), MARK(zprimes)) MARK(filter(x0, s(z0), zprimes)) -> c11(A__FILTER(mark(x0), s(mark(z0)), a__zprimes), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, zprimes)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__zprimes), MARK(x1)) MARK(filter(sieve(z0), x1, zprimes)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), 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MARK(filter(x0, x1, cons(cons(z0, z1), x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(cons(mark(z0), z1), x3)), MARK(x1)) MARK(filter(x0, x1, cons(0, x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(0, x3)), MARK(x1)) MARK(filter(x0, x1, cons(s(z0), x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(s(mark(z0)), x3)), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), cons(x2, x3))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), cons(mark(x2), x3)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), cons(x2, x3))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), cons(mark(x2), x3)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), cons(x2, x3))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), cons(mark(x2), x3)), MARK(nats(z0))) MARK(filter(x0, zprimes, cons(x2, x3))) -> c11(A__FILTER(mark(x0), a__zprimes, cons(mark(x2), x3)), MARK(zprimes)) MARK(filter(x0, s(z0), cons(x2, x3))) -> c11(A__FILTER(mark(x0), s(mark(z0)), cons(mark(x2), x3)), MARK(s(z0))) MARK(filter(filter(z0, z1, 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MARK(sieve(z0))) MARK(filter(x0, nats(z0), 0)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), 0), MARK(nats(z0))) MARK(filter(x0, zprimes, 0)) -> c11(A__FILTER(mark(x0), a__zprimes, 0), MARK(zprimes)) MARK(filter(x0, s(z0), 0)) -> c11(A__FILTER(mark(x0), s(mark(z0)), 0), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, 0)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), 0), MARK(x1)) MARK(filter(sieve(z0), x1, 0)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), 0), MARK(x1)) MARK(filter(nats(z0), x1, 0)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), 0), MARK(x1)) MARK(filter(zprimes, x1, 0)) -> c11(A__FILTER(a__zprimes, mark(x1), 0), MARK(x1)) MARK(filter(cons(z0, z1), x1, 0)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), 0), MARK(x1)) MARK(filter(x0, cons(z0, z1), 0)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, 0)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, 0)) -> c11(MARK(x1)) MARK(filter(x0, x1, s(filter(z0, z1, z2)))) -> c11(A__FILTER(mark(x0), mark(x1), 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s(mark(x2))), MARK(nats(z0))) MARK(filter(x0, zprimes, s(x2))) -> c11(A__FILTER(mark(x0), a__zprimes, s(mark(x2))), MARK(zprimes)) MARK(filter(x0, s(z0), s(x2))) -> c11(A__FILTER(mark(x0), s(mark(z0)), s(mark(x2))), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, s(x2))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), s(mark(x2))), MARK(x1)) MARK(filter(sieve(z0), x1, s(x2))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), s(mark(x2))), MARK(x1)) MARK(filter(nats(z0), x1, s(x2))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), s(mark(x2))), MARK(x1)) MARK(filter(zprimes, x1, s(x2))) -> c11(A__FILTER(a__zprimes, mark(x1), s(mark(x2))), MARK(x1)) MARK(filter(cons(z0, z1), x1, s(x2))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), s(mark(x2))), MARK(x1)) MARK(filter(x0, cons(z0, z1), s(x2))) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, s(x2))) -> c11(MARK(x1)) MARK(filter(s(z0), x1, s(x2))) -> c11(MARK(x1)) MARK(filter(x0, filter(x1, x2, filter(z0, z1, z2)), x4)) -> 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-> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), s(mark(z0))), mark(x4)), MARK(filter(x1, x2, s(z0)))) MARK(filter(x0, filter(x1, filter(z0, z1, z2), x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), a__filter(mark(z0), mark(z1), mark(z2)), mark(x3)), mark(x4)), MARK(filter(x1, filter(z0, z1, z2), x3))) MARK(filter(x0, filter(x1, sieve(z0), x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), a__sieve(mark(z0)), mark(x3)), mark(x4)), MARK(filter(x1, sieve(z0), x3))) MARK(filter(x0, filter(x1, nats(z0), x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), a__nats(mark(z0)), mark(x3)), mark(x4)), MARK(filter(x1, nats(z0), x3))) MARK(filter(x0, filter(x1, zprimes, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), a__zprimes, mark(x3)), mark(x4)), MARK(filter(x1, zprimes, x3))) MARK(filter(x0, filter(x1, cons(z0, z1), x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), cons(mark(z0), z1), mark(x3)), mark(x4)), MARK(filter(x1, cons(z0, z1), x3))) MARK(filter(x0, filter(x1, 0, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), 0, mark(x3)), mark(x4)), MARK(filter(x1, 0, x3))) MARK(filter(x0, filter(x1, s(z0), x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), s(mark(z0)), mark(x3)), mark(x4)), MARK(filter(x1, s(z0), x3))) MARK(filter(x0, filter(filter(z0, z1, z2), x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x2), mark(x3)), mark(x4)), MARK(filter(filter(z0, z1, z2), x2, x3))) MARK(filter(x0, filter(sieve(z0), x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(a__sieve(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(filter(sieve(z0), x2, x3))) MARK(filter(x0, filter(nats(z0), x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(a__nats(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(filter(nats(z0), x2, x3))) MARK(filter(x0, filter(zprimes, x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(a__zprimes, mark(x2), mark(x3)), mark(x4)), MARK(filter(zprimes, x2, x3))) MARK(filter(x0, filter(cons(z0, z1), x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(cons(mark(z0), z1), mark(x2), mark(x3)), mark(x4)), MARK(filter(cons(z0, z1), x2, x3))) MARK(filter(x0, filter(0, x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(0, mark(x2), mark(x3)), mark(x4)), MARK(filter(0, x2, x3))) MARK(filter(x0, filter(s(z0), x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(s(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(filter(s(z0), x2, x3))) MARK(filter(filter(z0, z1, z2), filter(x1, x2, x3), x4)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(sieve(z0), filter(x1, x2, x3), x4)) -> c11(A__FILTER(a__sieve(mark(z0)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(nats(z0), filter(x1, x2, x3), x4)) -> c11(A__FILTER(a__nats(mark(z0)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(zprimes, filter(x1, x2, x3), x4)) -> c11(A__FILTER(a__zprimes, a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(cons(z0, z1), filter(x1, x2, x3), x4)) -> c11(A__FILTER(cons(mark(z0), z1), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(x0, filter(x1, x2, x3), x4)) -> c11(MARK(filter(x1, x2, x3))) MARK(filter(0, filter(x1, x2, x3), x4)) -> c11(MARK(filter(x1, x2, x3))) MARK(filter(s(z0), filter(x1, x2, x3), x4)) -> c11(MARK(filter(x1, x2, x3))) MARK(filter(x0, sieve(filter(z0, z1, z2)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(sieve(filter(z0, z1, z2)))) MARK(filter(x0, sieve(sieve(z0)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(a__sieve(mark(z0))), mark(x2)), MARK(sieve(sieve(z0)))) MARK(filter(x0, sieve(nats(z0)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(a__nats(mark(z0))), mark(x2)), MARK(sieve(nats(z0)))) MARK(filter(x0, sieve(zprimes), x2)) -> c11(A__FILTER(mark(x0), a__sieve(a__zprimes), mark(x2)), MARK(sieve(zprimes))) MARK(filter(x0, sieve(cons(z0, z1)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(cons(mark(z0), z1)), mark(x2)), MARK(sieve(cons(z0, z1)))) MARK(filter(x0, sieve(s(z0)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(s(mark(z0))), mark(x2)), MARK(sieve(s(z0)))) MARK(filter(filter(z0, z1, z2), sieve(x1), x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(sieve(z0), sieve(x1), x2)) -> c11(A__FILTER(a__sieve(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(nats(z0), sieve(x1), x2)) -> c11(A__FILTER(a__nats(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(zprimes, sieve(x1), x2)) -> c11(A__FILTER(a__zprimes, a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(cons(z0, z1), sieve(x1), x2)) -> c11(A__FILTER(cons(mark(z0), z1), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(x0, sieve(x1), x2)) -> c11(MARK(sieve(x1))) MARK(filter(x0, sieve(0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(0), mark(x2))) MARK(filter(0, sieve(x1), x2)) -> c11(MARK(sieve(x1))) MARK(filter(s(z0), sieve(x1), x2)) -> c11(MARK(sieve(x1))) MARK(filter(x0, nats(filter(z0, z1, z2)), x2)) -> c11(A__FILTER(mark(x0), a__nats(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(nats(filter(z0, z1, z2)))) MARK(filter(x0, nats(sieve(z0)), x2)) -> c11(A__FILTER(mark(x0), a__nats(a__sieve(mark(z0))), mark(x2)), MARK(nats(sieve(z0)))) MARK(filter(x0, nats(nats(z0)), x2)) -> c11(A__FILTER(mark(x0), a__nats(a__nats(mark(z0))), mark(x2)), MARK(nats(nats(z0)))) MARK(filter(x0, nats(zprimes), x2)) -> c11(A__FILTER(mark(x0), a__nats(a__zprimes), mark(x2)), MARK(nats(zprimes))) MARK(filter(x0, nats(cons(z0, z1)), x2)) -> c11(A__FILTER(mark(x0), a__nats(cons(mark(z0), z1)), mark(x2)), MARK(nats(cons(z0, z1)))) MARK(filter(x0, nats(0), x2)) -> c11(A__FILTER(mark(x0), a__nats(0), mark(x2)), MARK(nats(0))) MARK(filter(x0, nats(s(z0)), x2)) -> c11(A__FILTER(mark(x0), a__nats(s(mark(z0))), mark(x2)), MARK(nats(s(z0)))) MARK(filter(filter(z0, z1, z2), nats(x1), x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(sieve(z0), nats(x1), x2)) -> c11(A__FILTER(a__sieve(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(nats(z0), nats(x1), x2)) -> c11(A__FILTER(a__nats(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(zprimes, nats(x1), x2)) -> c11(A__FILTER(a__zprimes, a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(cons(z0, z1), nats(x1), x2)) -> c11(A__FILTER(cons(mark(z0), z1), a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(x0, nats(x1), x2)) -> c11(MARK(nats(x1))) MARK(filter(0, nats(x1), x2)) -> c11(MARK(nats(x1))) MARK(filter(s(z0), nats(x1), x2)) -> c11(MARK(nats(x1))) MARK(filter(x0, zprimes, x1)) -> c11(A__FILTER(mark(x0), a__sieve(a__nats(s(s(0)))), mark(x1)), MARK(zprimes)) MARK(filter(filter(z0, z1, z2), zprimes, x1)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(sieve(z0), zprimes, x1)) -> c11(A__FILTER(a__sieve(mark(z0)), a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(nats(z0), zprimes, x1)) -> c11(A__FILTER(a__nats(mark(z0)), a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(zprimes, zprimes, x1)) -> c11(A__FILTER(a__zprimes, a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(cons(z0, z1), zprimes, x1)) -> c11(A__FILTER(cons(mark(z0), z1), a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(x0, zprimes, x1)) -> c11(MARK(zprimes)) MARK(filter(0, zprimes, x1)) -> c11(MARK(zprimes)) MARK(filter(s(z0), zprimes, x1)) -> c11(MARK(zprimes)) MARK(filter(x0, s(filter(z0, z1, z2)), x2)) -> c11(A__FILTER(mark(x0), s(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(s(filter(z0, z1, z2)))) MARK(filter(x0, s(sieve(z0)), x2)) -> c11(A__FILTER(mark(x0), s(a__sieve(mark(z0))), mark(x2)), MARK(s(sieve(z0)))) MARK(filter(x0, s(nats(z0)), x2)) -> c11(A__FILTER(mark(x0), s(a__nats(mark(z0))), mark(x2)), MARK(s(nats(z0)))) MARK(filter(x0, s(zprimes), x2)) -> c11(A__FILTER(mark(x0), s(a__zprimes), mark(x2)), MARK(s(zprimes))) MARK(filter(x0, s(cons(z0, z1)), x2)) -> c11(A__FILTER(mark(x0), s(cons(mark(z0), z1)), mark(x2)), MARK(s(cons(z0, z1)))) MARK(filter(x0, s(0), x2)) -> c11(A__FILTER(mark(x0), s(0), mark(x2)), MARK(s(0))) MARK(filter(x0, s(s(z0)), x2)) -> c11(A__FILTER(mark(x0), s(s(mark(z0))), mark(x2)), MARK(s(s(z0)))) MARK(filter(filter(z0, z1, z2), s(x1), x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), s(mark(x1)), mark(x2)), MARK(s(x1))) MARK(filter(sieve(z0), s(x1), x2)) -> c11(A__FILTER(a__sieve(mark(z0)), s(mark(x1)), mark(x2)), MARK(s(x1))) MARK(filter(nats(z0), s(x1), x2)) -> c11(A__FILTER(a__nats(mark(z0)), s(mark(x1)), mark(x2)), MARK(s(x1))) MARK(filter(zprimes, s(x1), x2)) -> c11(A__FILTER(a__zprimes, s(mark(x1)), mark(x2)), MARK(s(x1))) MARK(filter(cons(z0, z1), s(x1), x2)) -> c11(A__FILTER(cons(mark(z0), z1), s(mark(x1)), mark(x2)), MARK(s(x1))) MARK(filter(x0, s(x1), x2)) -> c11(MARK(s(x1))) MARK(filter(0, s(x1), x2)) -> c11(MARK(s(x1))) MARK(filter(s(z0), s(x1), x2)) -> c11(MARK(s(x1))) MARK(filter(filter(x0, x1, filter(z0, z1, z2)), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, sieve(z0)), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), a__sieve(mark(z0))), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, nats(z0)), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), a__nats(mark(z0))), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, zprimes), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), a__zprimes), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, cons(z0, z1)), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), cons(mark(z0), z1)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, 0), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), 0), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, s(z0)), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), s(mark(z0))), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, filter(z0, z1, z2), x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, sieve(z0), x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), a__sieve(mark(z0)), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, nats(z0), x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), a__nats(mark(z0)), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, zprimes, x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), a__zprimes, mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, cons(z0, z1), x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), cons(mark(z0), z1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, 0, x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), 0, mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, s(z0), x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), s(mark(z0)), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(filter(z0, z1, z2), x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(sieve(z0), x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(a__sieve(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(nats(z0), x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(a__nats(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(zprimes, x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(a__zprimes, mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(cons(z0, z1), x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(cons(mark(z0), z1), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(0, x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(0, mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(s(z0), x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(s(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, x2), cons(z0, z1), x4)) -> c11(MARK(cons(z0, z1))) MARK(filter(filter(x0, x1, x2), x3, x4)) -> c11(MARK(x3)) MARK(filter(sieve(filter(z0, z1, z2)), x1, x2)) -> c11(A__FILTER(a__sieve(a__filter(mark(z0), mark(z1), mark(z2))), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(sieve(z0)), x1, x2)) -> c11(A__FILTER(a__sieve(a__sieve(mark(z0))), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(nats(z0)), x1, x2)) -> c11(A__FILTER(a__sieve(a__nats(mark(z0))), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(zprimes), x1, x2)) -> c11(A__FILTER(a__sieve(a__zprimes), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(cons(z0, z1)), x1, x2)) -> c11(A__FILTER(a__sieve(cons(mark(z0), z1)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(0), x1, x2)) -> c11(A__FILTER(a__sieve(0), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(s(z0)), x1, x2)) -> c11(A__FILTER(a__sieve(s(mark(z0))), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(x0), cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(sieve(x0), x1, x2)) -> c11(MARK(x1)) MARK(filter(nats(x0), cons(z0, z1), x2)) -> c11(A__FILTER(a__nats(mark(x0)), cons(mark(z0), z1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(nats(x0), x1, x2)) -> c11(A__FILTER(cons(mark(mark(x0)), nats(s(mark(x0)))), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(x0), x1, x2)) -> c11(A__FILTER(nats(mark(x0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(filter(z0, z1, z2)), x1, x2)) -> c11(A__FILTER(a__nats(a__filter(mark(z0), mark(z1), mark(z2))), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(sieve(z0)), x1, x2)) -> c11(A__FILTER(a__nats(a__sieve(mark(z0))), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(nats(z0)), x1, x2)) -> c11(A__FILTER(a__nats(a__nats(mark(z0))), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(zprimes), x1, x2)) -> c11(A__FILTER(a__nats(a__zprimes), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(cons(z0, z1)), x1, x2)) -> c11(A__FILTER(a__nats(cons(mark(z0), z1)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(0), x1, x2)) -> c11(A__FILTER(a__nats(0), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(s(z0)), x1, x2)) -> c11(A__FILTER(a__nats(s(mark(z0))), mark(x1), mark(x2)), MARK(x1)) S tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(x0, 0, filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, sieve(x2))) -> c10(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x0)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, sieve(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, sieve(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, sieve(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(0)), MARK(x0)) MARK(filter(x0, x1, sieve(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), sieve(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, sieve(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, sieve(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__sieve(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, sieve(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, sieve(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, sieve(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), a__sieve(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, sieve(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__sieve(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), sieve(x2))) -> c10(MARK(x0)) MARK(filter(x0, 0, sieve(x2))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, sieve(x2))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(mark(x2)), nats(s(mark(x2))))), MARK(x0)) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), nats(mark(x2))), MARK(x0)) MARK(filter(x0, x1, nats(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, nats(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, nats(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, nats(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, nats(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, nats(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(0)), MARK(x0)) MARK(filter(x0, x1, nats(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), nats(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), nats(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), nats(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, nats(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), nats(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, nats(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__nats(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, nats(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, nats(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, nats(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), a__nats(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, nats(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__nats(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), nats(x2))) -> c10(MARK(x0)) MARK(filter(x0, 0, nats(x2))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, nats(x2))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, zprimes)) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(s(s(0))))), MARK(x0)) MARK(filter(x0, x1, zprimes)) -> c10(A__FILTER(mark(x0), mark(x1), zprimes), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), zprimes)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__zprimes), MARK(x0)) MARK(filter(x0, sieve(z0), zprimes)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__zprimes), MARK(x0)) 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MARK(filter(cons(z0, z1), filter(x1, x2, x3), x4)) -> c11(A__FILTER(cons(mark(z0), z1), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(x0, filter(x1, x2, x3), x4)) -> c11(MARK(filter(x1, x2, x3))) MARK(filter(0, filter(x1, x2, x3), x4)) -> c11(MARK(filter(x1, x2, x3))) MARK(filter(s(z0), filter(x1, x2, x3), x4)) -> c11(MARK(filter(x1, x2, x3))) MARK(filter(x0, sieve(filter(z0, z1, z2)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(sieve(filter(z0, z1, z2)))) MARK(filter(x0, sieve(sieve(z0)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(a__sieve(mark(z0))), mark(x2)), MARK(sieve(sieve(z0)))) MARK(filter(x0, sieve(nats(z0)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(a__nats(mark(z0))), mark(x2)), MARK(sieve(nats(z0)))) MARK(filter(x0, sieve(zprimes), x2)) -> c11(A__FILTER(mark(x0), a__sieve(a__zprimes), mark(x2)), MARK(sieve(zprimes))) MARK(filter(x0, sieve(cons(z0, z1)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(cons(mark(z0), z1)), mark(x2)), MARK(sieve(cons(z0, z1)))) MARK(filter(x0, sieve(s(z0)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(s(mark(z0))), mark(x2)), MARK(sieve(s(z0)))) MARK(filter(filter(z0, z1, z2), sieve(x1), x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(sieve(z0), sieve(x1), x2)) -> c11(A__FILTER(a__sieve(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(nats(z0), sieve(x1), x2)) -> c11(A__FILTER(a__nats(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(zprimes, sieve(x1), x2)) -> c11(A__FILTER(a__zprimes, a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(cons(z0, z1), sieve(x1), x2)) -> c11(A__FILTER(cons(mark(z0), z1), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(x0, sieve(x1), x2)) -> c11(MARK(sieve(x1))) MARK(filter(x0, sieve(0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(0), mark(x2))) MARK(filter(0, sieve(x1), x2)) -> c11(MARK(sieve(x1))) MARK(filter(s(z0), sieve(x1), x2)) -> c11(MARK(sieve(x1))) MARK(filter(x0, nats(filter(z0, z1, z2)), x2)) -> c11(A__FILTER(mark(x0), a__nats(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(nats(filter(z0, z1, z2)))) MARK(filter(x0, nats(sieve(z0)), x2)) -> c11(A__FILTER(mark(x0), a__nats(a__sieve(mark(z0))), mark(x2)), MARK(nats(sieve(z0)))) MARK(filter(x0, nats(nats(z0)), x2)) -> c11(A__FILTER(mark(x0), a__nats(a__nats(mark(z0))), mark(x2)), MARK(nats(nats(z0)))) MARK(filter(x0, nats(zprimes), x2)) -> c11(A__FILTER(mark(x0), a__nats(a__zprimes), mark(x2)), MARK(nats(zprimes))) MARK(filter(x0, nats(cons(z0, z1)), x2)) -> c11(A__FILTER(mark(x0), a__nats(cons(mark(z0), z1)), mark(x2)), MARK(nats(cons(z0, z1)))) MARK(filter(x0, nats(0), x2)) -> c11(A__FILTER(mark(x0), a__nats(0), mark(x2)), MARK(nats(0))) MARK(filter(x0, nats(s(z0)), x2)) -> c11(A__FILTER(mark(x0), a__nats(s(mark(z0))), mark(x2)), MARK(nats(s(z0)))) MARK(filter(filter(z0, z1, z2), nats(x1), x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(sieve(z0), nats(x1), x2)) -> c11(A__FILTER(a__sieve(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(nats(z0), nats(x1), x2)) -> c11(A__FILTER(a__nats(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(zprimes, nats(x1), x2)) -> c11(A__FILTER(a__zprimes, a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(cons(z0, z1), nats(x1), x2)) -> c11(A__FILTER(cons(mark(z0), z1), a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(x0, nats(x1), x2)) -> c11(MARK(nats(x1))) MARK(filter(0, nats(x1), x2)) -> c11(MARK(nats(x1))) MARK(filter(s(z0), nats(x1), x2)) -> c11(MARK(nats(x1))) MARK(filter(x0, zprimes, x1)) -> c11(A__FILTER(mark(x0), a__sieve(a__nats(s(s(0)))), mark(x1)), MARK(zprimes)) MARK(filter(filter(z0, z1, z2), zprimes, x1)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(sieve(z0), zprimes, x1)) -> c11(A__FILTER(a__sieve(mark(z0)), a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(nats(z0), zprimes, x1)) -> c11(A__FILTER(a__nats(mark(z0)), a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(zprimes, zprimes, x1)) -> c11(A__FILTER(a__zprimes, a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(cons(z0, z1), zprimes, x1)) -> c11(A__FILTER(cons(mark(z0), z1), a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(x0, zprimes, x1)) -> c11(MARK(zprimes)) MARK(filter(0, zprimes, x1)) -> c11(MARK(zprimes)) MARK(filter(s(z0), zprimes, x1)) -> c11(MARK(zprimes)) MARK(filter(x0, s(filter(z0, z1, z2)), x2)) -> c11(A__FILTER(mark(x0), s(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(s(filter(z0, z1, z2)))) MARK(filter(x0, s(sieve(z0)), x2)) -> c11(A__FILTER(mark(x0), s(a__sieve(mark(z0))), mark(x2)), MARK(s(sieve(z0)))) MARK(filter(x0, s(nats(z0)), x2)) -> c11(A__FILTER(mark(x0), s(a__nats(mark(z0))), mark(x2)), MARK(s(nats(z0)))) MARK(filter(x0, s(zprimes), x2)) -> c11(A__FILTER(mark(x0), s(a__zprimes), mark(x2)), MARK(s(zprimes))) MARK(filter(x0, s(cons(z0, z1)), x2)) -> c11(A__FILTER(mark(x0), s(cons(mark(z0), z1)), mark(x2)), MARK(s(cons(z0, z1)))) MARK(filter(x0, s(0), x2)) -> c11(A__FILTER(mark(x0), s(0), mark(x2)), MARK(s(0))) MARK(filter(x0, s(s(z0)), x2)) -> c11(A__FILTER(mark(x0), s(s(mark(z0))), mark(x2)), MARK(s(s(z0)))) MARK(filter(filter(z0, z1, z2), s(x1), x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), s(mark(x1)), mark(x2)), MARK(s(x1))) MARK(filter(sieve(z0), s(x1), x2)) -> c11(A__FILTER(a__sieve(mark(z0)), s(mark(x1)), mark(x2)), MARK(s(x1))) MARK(filter(nats(z0), s(x1), x2)) -> c11(A__FILTER(a__nats(mark(z0)), s(mark(x1)), mark(x2)), MARK(s(x1))) MARK(filter(zprimes, s(x1), x2)) -> c11(A__FILTER(a__zprimes, s(mark(x1)), mark(x2)), MARK(s(x1))) MARK(filter(cons(z0, z1), s(x1), x2)) -> c11(A__FILTER(cons(mark(z0), z1), s(mark(x1)), mark(x2)), MARK(s(x1))) MARK(filter(x0, s(x1), x2)) -> c11(MARK(s(x1))) MARK(filter(0, s(x1), x2)) -> c11(MARK(s(x1))) MARK(filter(s(z0), s(x1), x2)) -> c11(MARK(s(x1))) MARK(filter(filter(x0, x1, filter(z0, z1, z2)), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, sieve(z0)), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), a__sieve(mark(z0))), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, nats(z0)), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), a__nats(mark(z0))), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, zprimes), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), a__zprimes), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, cons(z0, z1)), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), cons(mark(z0), z1)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, 0), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), 0), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, s(z0)), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), s(mark(z0))), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, filter(z0, z1, z2), x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, sieve(z0), x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), a__sieve(mark(z0)), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, nats(z0), x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), a__nats(mark(z0)), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, zprimes, x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), a__zprimes, mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, cons(z0, z1), x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), cons(mark(z0), z1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, 0, x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), 0, mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, s(z0), x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), s(mark(z0)), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(filter(z0, z1, z2), x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(sieve(z0), x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(a__sieve(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(nats(z0), x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(a__nats(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(zprimes, x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(a__zprimes, mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(cons(z0, z1), x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(cons(mark(z0), z1), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(0, x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(0, mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(s(z0), x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(s(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, x2), cons(z0, z1), x4)) -> c11(MARK(cons(z0, z1))) MARK(filter(filter(x0, x1, x2), x3, x4)) -> c11(MARK(x3)) MARK(filter(sieve(filter(z0, z1, z2)), x1, x2)) -> c11(A__FILTER(a__sieve(a__filter(mark(z0), mark(z1), mark(z2))), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(sieve(z0)), x1, x2)) -> c11(A__FILTER(a__sieve(a__sieve(mark(z0))), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(nats(z0)), x1, x2)) -> c11(A__FILTER(a__sieve(a__nats(mark(z0))), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(zprimes), x1, x2)) -> c11(A__FILTER(a__sieve(a__zprimes), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(cons(z0, z1)), x1, x2)) -> c11(A__FILTER(a__sieve(cons(mark(z0), z1)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(0), x1, x2)) -> c11(A__FILTER(a__sieve(0), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(s(z0)), x1, x2)) -> c11(A__FILTER(a__sieve(s(mark(z0))), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(x0), cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(sieve(x0), x1, x2)) -> c11(MARK(x1)) MARK(filter(nats(x0), cons(z0, z1), x2)) -> c11(A__FILTER(a__nats(mark(x0)), cons(mark(z0), z1), mark(x2)), MARK(cons(z0, z1))) MARK(filter(nats(x0), x1, x2)) -> c11(A__FILTER(cons(mark(mark(x0)), nats(s(mark(x0)))), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(x0), x1, x2)) -> c11(A__FILTER(nats(mark(x0)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(filter(z0, z1, z2)), x1, x2)) -> c11(A__FILTER(a__nats(a__filter(mark(z0), mark(z1), mark(z2))), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(sieve(z0)), x1, x2)) -> c11(A__FILTER(a__nats(a__sieve(mark(z0))), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(nats(z0)), x1, x2)) -> c11(A__FILTER(a__nats(a__nats(mark(z0))), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(zprimes), x1, x2)) -> c11(A__FILTER(a__nats(a__zprimes), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(cons(z0, z1)), x1, x2)) -> c11(A__FILTER(a__nats(cons(mark(z0), z1)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(0), x1, x2)) -> c11(A__FILTER(a__nats(0), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(s(z0)), x1, x2)) -> c11(A__FILTER(a__nats(s(mark(z0))), mark(x1), mark(x2)), MARK(x1)) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c14_2, c15_1, c16_1, c18_1, c8_1, c10_1, c11_2, c11_1, c12_2, c12_1, c13_2, c13_1, c8_2, c10_2 ---------------------------------------- (223) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (224) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(x0, 0, filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, sieve(x2))) -> c10(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x0)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, sieve(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, sieve(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, sieve(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(0)), MARK(x0)) MARK(filter(x0, x1, sieve(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), sieve(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, sieve(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, sieve(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__sieve(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, sieve(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, sieve(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, sieve(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), a__sieve(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, sieve(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__sieve(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), sieve(x2))) -> c10(MARK(x0)) MARK(filter(x0, 0, sieve(x2))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, sieve(x2))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(mark(x2)), nats(s(mark(x2))))), MARK(x0)) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), nats(mark(x2))), MARK(x0)) MARK(filter(x0, x1, nats(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, nats(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, nats(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, nats(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, nats(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, nats(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(0)), MARK(x0)) MARK(filter(x0, x1, nats(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), nats(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), nats(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), nats(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, nats(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), nats(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, nats(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__nats(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, nats(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, nats(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, nats(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), a__nats(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, nats(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__nats(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), nats(x2))) -> c10(MARK(x0)) MARK(filter(x0, 0, nats(x2))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, nats(x2))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, zprimes)) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(s(s(0))))), MARK(x0)) MARK(filter(x0, x1, zprimes)) -> c10(A__FILTER(mark(x0), mark(x1), zprimes), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), zprimes)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__zprimes), MARK(x0)) MARK(filter(x0, sieve(z0), zprimes)) -> c10(A__FILTER(mark(x0), 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c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(nats(z0), x1, sieve(x2))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(zprimes, x1, sieve(x2))) -> c11(A__FILTER(a__zprimes, mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(cons(z0, z1), x1, sieve(x2))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(x0, cons(z0, z1), sieve(x2))) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, sieve(x2))) -> c11(MARK(x1)) MARK(filter(s(z0), x1, sieve(x2))) -> c11(MARK(x1)) MARK(filter(x0, x1, nats(x2))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(mark(x2)), nats(s(mark(x2))))), MARK(x1)) MARK(filter(x0, x1, nats(x2))) -> c11(A__FILTER(mark(x0), mark(x1), nats(mark(x2))), MARK(x1)) MARK(filter(x0, x1, nats(filter(z0, z1, z2)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x1)) MARK(filter(x0, x1, nats(sieve(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__sieve(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, nats(nats(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__nats(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, nats(zprimes))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__zprimes)), MARK(x1)) MARK(filter(x0, x1, nats(cons(z0, z1)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(cons(mark(z0), z1))), MARK(x1)) MARK(filter(x0, x1, nats(0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(0)), MARK(x1)) MARK(filter(x0, x1, nats(s(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(s(mark(z0)))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), nats(x2))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), nats(x2))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), a__nats(mark(x2))), MARK(sieve(z0))) MARK(filter(x0, nats(z0), nats(x2))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), a__nats(mark(x2))), MARK(nats(z0))) MARK(filter(x0, zprimes, nats(x2))) -> c11(A__FILTER(mark(x0), a__zprimes, a__nats(mark(x2))), MARK(zprimes)) MARK(filter(x0, s(z0), nats(x2))) -> c11(A__FILTER(mark(x0), s(mark(z0)), a__nats(mark(x2))), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, nats(x2))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(sieve(z0), x1, nats(x2))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(nats(z0), x1, nats(x2))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(zprimes, x1, nats(x2))) -> c11(A__FILTER(a__zprimes, mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(cons(z0, z1), x1, nats(x2))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(x0, cons(z0, z1), nats(x2))) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, nats(x2))) -> c11(MARK(x1)) MARK(filter(s(z0), x1, nats(x2))) -> c11(MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(s(s(0))))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), zprimes), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), zprimes)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__zprimes), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), zprimes)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), a__zprimes), MARK(sieve(z0))) MARK(filter(x0, nats(z0), zprimes)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), a__zprimes), MARK(nats(z0))) MARK(filter(x0, zprimes, zprimes)) -> c11(A__FILTER(mark(x0), a__zprimes, a__zprimes), MARK(zprimes)) MARK(filter(x0, s(z0), zprimes)) -> c11(A__FILTER(mark(x0), s(mark(z0)), a__zprimes), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, zprimes)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__zprimes), MARK(x1)) MARK(filter(sieve(z0), x1, zprimes)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), 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mark(x2)), MARK(sieve(cons(z0, z1)))) MARK(filter(x0, sieve(s(z0)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(s(mark(z0))), mark(x2)), MARK(sieve(s(z0)))) MARK(filter(filter(z0, z1, z2), sieve(x1), x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(sieve(z0), sieve(x1), x2)) -> c11(A__FILTER(a__sieve(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(nats(z0), sieve(x1), x2)) -> c11(A__FILTER(a__nats(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(zprimes, sieve(x1), x2)) -> c11(A__FILTER(a__zprimes, a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(cons(z0, z1), sieve(x1), x2)) -> c11(A__FILTER(cons(mark(z0), z1), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(x0, sieve(x1), x2)) -> c11(MARK(sieve(x1))) MARK(filter(x0, sieve(0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(0), mark(x2))) MARK(filter(0, sieve(x1), x2)) -> c11(MARK(sieve(x1))) MARK(filter(s(z0), sieve(x1), x2)) -> c11(MARK(sieve(x1))) MARK(filter(x0, nats(filter(z0, z1, z2)), x2)) -> c11(A__FILTER(mark(x0), a__nats(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(nats(filter(z0, z1, z2)))) MARK(filter(x0, nats(sieve(z0)), x2)) -> c11(A__FILTER(mark(x0), a__nats(a__sieve(mark(z0))), mark(x2)), MARK(nats(sieve(z0)))) MARK(filter(x0, nats(nats(z0)), x2)) -> c11(A__FILTER(mark(x0), a__nats(a__nats(mark(z0))), mark(x2)), MARK(nats(nats(z0)))) MARK(filter(x0, nats(zprimes), x2)) -> c11(A__FILTER(mark(x0), a__nats(a__zprimes), mark(x2)), MARK(nats(zprimes))) MARK(filter(x0, nats(cons(z0, z1)), x2)) -> c11(A__FILTER(mark(x0), a__nats(cons(mark(z0), z1)), mark(x2)), MARK(nats(cons(z0, z1)))) MARK(filter(x0, nats(0), x2)) -> c11(A__FILTER(mark(x0), a__nats(0), mark(x2)), MARK(nats(0))) MARK(filter(x0, nats(s(z0)), x2)) -> c11(A__FILTER(mark(x0), a__nats(s(mark(z0))), mark(x2)), MARK(nats(s(z0)))) MARK(filter(filter(z0, z1, z2), nats(x1), x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(sieve(z0), nats(x1), x2)) -> c11(A__FILTER(a__sieve(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(nats(z0), nats(x1), x2)) -> c11(A__FILTER(a__nats(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(zprimes, nats(x1), x2)) -> c11(A__FILTER(a__zprimes, a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(cons(z0, z1), nats(x1), x2)) -> c11(A__FILTER(cons(mark(z0), z1), a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(x0, nats(x1), x2)) -> c11(MARK(nats(x1))) MARK(filter(0, nats(x1), x2)) -> c11(MARK(nats(x1))) MARK(filter(s(z0), nats(x1), x2)) -> c11(MARK(nats(x1))) MARK(filter(x0, zprimes, x1)) -> c11(A__FILTER(mark(x0), a__sieve(a__nats(s(s(0)))), mark(x1)), MARK(zprimes)) MARK(filter(filter(z0, z1, z2), zprimes, x1)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(sieve(z0), zprimes, x1)) -> c11(A__FILTER(a__sieve(mark(z0)), a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(nats(z0), zprimes, x1)) -> c11(A__FILTER(a__nats(mark(z0)), a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(zprimes, zprimes, x1)) -> c11(A__FILTER(a__zprimes, a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(cons(z0, z1), zprimes, x1)) -> c11(A__FILTER(cons(mark(z0), z1), a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(x0, zprimes, x1)) -> c11(MARK(zprimes)) MARK(filter(0, zprimes, x1)) -> c11(MARK(zprimes)) MARK(filter(s(z0), zprimes, x1)) -> c11(MARK(zprimes)) MARK(filter(x0, s(filter(z0, z1, z2)), x2)) -> c11(A__FILTER(mark(x0), s(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(s(filter(z0, z1, z2)))) MARK(filter(x0, s(sieve(z0)), x2)) -> c11(A__FILTER(mark(x0), s(a__sieve(mark(z0))), mark(x2)), MARK(s(sieve(z0)))) MARK(filter(x0, s(nats(z0)), x2)) -> c11(A__FILTER(mark(x0), s(a__nats(mark(z0))), mark(x2)), MARK(s(nats(z0)))) MARK(filter(x0, s(zprimes), x2)) -> c11(A__FILTER(mark(x0), s(a__zprimes), mark(x2)), MARK(s(zprimes))) MARK(filter(x0, s(cons(z0, z1)), x2)) -> c11(A__FILTER(mark(x0), s(cons(mark(z0), z1)), mark(x2)), MARK(s(cons(z0, z1)))) MARK(filter(x0, s(0), x2)) -> c11(A__FILTER(mark(x0), s(0), mark(x2)), MARK(s(0))) MARK(filter(x0, s(s(z0)), x2)) -> c11(A__FILTER(mark(x0), s(s(mark(z0))), mark(x2)), MARK(s(s(z0)))) MARK(filter(filter(z0, z1, z2), s(x1), x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), s(mark(x1)), mark(x2)), MARK(s(x1))) MARK(filter(sieve(z0), s(x1), x2)) -> c11(A__FILTER(a__sieve(mark(z0)), s(mark(x1)), mark(x2)), MARK(s(x1))) MARK(filter(nats(z0), s(x1), x2)) -> c11(A__FILTER(a__nats(mark(z0)), s(mark(x1)), mark(x2)), MARK(s(x1))) MARK(filter(zprimes, s(x1), x2)) -> c11(A__FILTER(a__zprimes, s(mark(x1)), mark(x2)), MARK(s(x1))) MARK(filter(cons(z0, z1), s(x1), x2)) -> c11(A__FILTER(cons(mark(z0), z1), s(mark(x1)), mark(x2)), MARK(s(x1))) MARK(filter(x0, s(x1), x2)) -> c11(MARK(s(x1))) MARK(filter(0, s(x1), x2)) -> c11(MARK(s(x1))) MARK(filter(s(z0), s(x1), x2)) -> c11(MARK(s(x1))) MARK(filter(filter(x0, x1, filter(z0, z1, z2)), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, sieve(z0)), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), a__sieve(mark(z0))), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, nats(z0)), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), a__nats(mark(z0))), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, zprimes), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), a__zprimes), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, cons(z0, z1)), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), cons(mark(z0), z1)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, 0), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), 0), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, s(z0)), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), s(mark(z0))), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, filter(z0, z1, z2), x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, sieve(z0), x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), a__sieve(mark(z0)), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, nats(z0), x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), a__nats(mark(z0)), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, zprimes, x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), a__zprimes, mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, cons(z0, z1), x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), cons(mark(z0), z1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, 0, x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), 0, mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, s(z0), x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), s(mark(z0)), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(filter(z0, z1, z2), x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(sieve(z0), x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(a__sieve(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(nats(z0), x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(a__nats(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(zprimes, x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(a__zprimes, mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(cons(z0, z1), x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(cons(mark(z0), z1), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(0, x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(0, mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(s(z0), x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(s(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, x2), cons(z0, z1), x4)) -> c11(MARK(cons(z0, z1))) MARK(filter(filter(x0, x1, x2), x3, x4)) -> c11(MARK(x3)) MARK(filter(sieve(filter(z0, z1, z2)), x1, x2)) -> c11(A__FILTER(a__sieve(a__filter(mark(z0), mark(z1), mark(z2))), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(sieve(z0)), x1, x2)) -> c11(A__FILTER(a__sieve(a__sieve(mark(z0))), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(nats(z0)), x1, x2)) -> c11(A__FILTER(a__sieve(a__nats(mark(z0))), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(zprimes), x1, x2)) -> c11(A__FILTER(a__sieve(a__zprimes), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(cons(z0, z1)), x1, x2)) -> c11(A__FILTER(a__sieve(cons(mark(z0), z1)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(0), x1, x2)) -> c11(A__FILTER(a__sieve(0), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(s(z0)), x1, x2)) -> c11(A__FILTER(a__sieve(s(mark(z0))), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(x0), cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(sieve(x0), x1, x2)) -> c11(MARK(x1)) MARK(filter(nats(x0), x1, x2)) -> c11(A__FILTER(cons(mark(mark(x0)), nats(s(mark(x0)))), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(filter(z0, z1, z2)), x1, x2)) -> c11(A__FILTER(a__nats(a__filter(mark(z0), mark(z1), mark(z2))), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(sieve(z0)), x1, x2)) -> c11(A__FILTER(a__nats(a__sieve(mark(z0))), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(nats(z0)), x1, x2)) -> c11(A__FILTER(a__nats(a__nats(mark(z0))), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(zprimes), x1, x2)) -> c11(A__FILTER(a__nats(a__zprimes), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(cons(z0, z1)), x1, x2)) -> c11(A__FILTER(a__nats(cons(mark(z0), z1)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(0), x1, x2)) -> c11(A__FILTER(a__nats(0), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(s(z0)), x1, x2)) -> c11(A__FILTER(a__nats(s(mark(z0))), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(x0), cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(nats(x0), x1, x2)) -> c11(MARK(x1)) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c14_2, c15_1, c16_1, c18_1, c8_1, c10_1, c11_2, c11_1, c12_2, c12_1, c13_2, c13_1, c8_2, c10_2 ---------------------------------------- (225) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(filter(zprimes, x1, x2)) -> c11(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x1)) by MARK(filter(zprimes, x0, filter(z0, z1, z2))) -> c11(A__FILTER(a__zprimes, mark(x0), a__filter(mark(z0), mark(z1), mark(z2))), MARK(x0)) MARK(filter(zprimes, x0, sieve(z0))) -> c11(A__FILTER(a__zprimes, mark(x0), a__sieve(mark(z0))), MARK(x0)) MARK(filter(zprimes, x0, nats(z0))) -> c11(A__FILTER(a__zprimes, mark(x0), a__nats(mark(z0))), MARK(x0)) MARK(filter(zprimes, x0, zprimes)) -> c11(A__FILTER(a__zprimes, mark(x0), a__zprimes), MARK(x0)) MARK(filter(zprimes, x0, cons(z0, z1))) -> c11(A__FILTER(a__zprimes, mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(zprimes, x0, 0)) -> c11(A__FILTER(a__zprimes, mark(x0), 0), MARK(x0)) MARK(filter(zprimes, x0, s(z0))) -> c11(A__FILTER(a__zprimes, mark(x0), s(mark(z0))), MARK(x0)) MARK(filter(zprimes, filter(z0, z1, z2), x1)) -> c11(A__FILTER(a__zprimes, a__filter(mark(z0), mark(z1), mark(z2)), mark(x1)), MARK(filter(z0, z1, z2))) MARK(filter(zprimes, sieve(z0), x1)) -> c11(A__FILTER(a__zprimes, a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(zprimes, nats(z0), x1)) -> c11(A__FILTER(a__zprimes, a__nats(mark(z0)), mark(x1)), MARK(nats(z0))) MARK(filter(zprimes, zprimes, x1)) -> c11(A__FILTER(a__zprimes, a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(zprimes, cons(z0, z1), x1)) -> c11(A__FILTER(a__zprimes, cons(mark(z0), z1), mark(x1)), MARK(cons(z0, z1))) MARK(filter(zprimes, 0, x1)) -> c11(A__FILTER(a__zprimes, 0, mark(x1)), MARK(0)) MARK(filter(zprimes, s(z0), x1)) -> c11(A__FILTER(a__zprimes, s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(zprimes, x0, x1)) -> c11(A__FILTER(a__sieve(a__nats(s(s(0)))), mark(x0), mark(x1)), MARK(x0)) MARK(filter(zprimes, x0, x1)) -> c11(A__FILTER(zprimes, mark(x0), mark(x1)), MARK(x0)) ---------------------------------------- (226) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(x0, 0, filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, sieve(x2))) -> c10(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x0)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, sieve(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, sieve(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, sieve(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(0)), MARK(x0)) MARK(filter(x0, x1, sieve(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), sieve(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, sieve(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, sieve(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__sieve(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, sieve(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, sieve(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, sieve(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), a__sieve(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, sieve(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__sieve(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), sieve(x2))) -> c10(MARK(x0)) MARK(filter(x0, 0, sieve(x2))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, sieve(x2))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(mark(x2)), nats(s(mark(x2))))), MARK(x0)) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), nats(mark(x2))), MARK(x0)) MARK(filter(x0, x1, nats(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, nats(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, nats(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, nats(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, nats(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, nats(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(0)), MARK(x0)) MARK(filter(x0, x1, nats(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), nats(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), nats(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), nats(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, nats(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), nats(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, nats(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__nats(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, nats(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, nats(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, nats(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), a__nats(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, nats(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__nats(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), 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a__nats(mark(z0)), a__sieve(mark(x2))), MARK(nats(z0))) MARK(filter(x0, zprimes, sieve(x2))) -> c11(A__FILTER(mark(x0), a__zprimes, a__sieve(mark(x2))), MARK(zprimes)) MARK(filter(x0, s(z0), sieve(x2))) -> c11(A__FILTER(mark(x0), s(mark(z0)), a__sieve(mark(x2))), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, sieve(x2))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(sieve(z0), x1, sieve(x2))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(nats(z0), x1, sieve(x2))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(zprimes, x1, sieve(x2))) -> c11(A__FILTER(a__zprimes, mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(cons(z0, z1), x1, sieve(x2))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__sieve(mark(x2))), MARK(x1)) MARK(filter(x0, cons(z0, z1), sieve(x2))) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, sieve(x2))) -> c11(MARK(x1)) MARK(filter(s(z0), x1, sieve(x2))) -> c11(MARK(x1)) MARK(filter(x0, x1, nats(x2))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(mark(x2)), nats(s(mark(x2))))), MARK(x1)) MARK(filter(x0, x1, nats(x2))) -> c11(A__FILTER(mark(x0), mark(x1), nats(mark(x2))), MARK(x1)) MARK(filter(x0, x1, nats(filter(z0, z1, z2)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x1)) MARK(filter(x0, x1, nats(sieve(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__sieve(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, nats(nats(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__nats(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, nats(zprimes))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__zprimes)), MARK(x1)) MARK(filter(x0, x1, nats(cons(z0, z1)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(cons(mark(z0), z1))), MARK(x1)) MARK(filter(x0, x1, nats(0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(0)), MARK(x1)) MARK(filter(x0, x1, nats(s(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(s(mark(z0)))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), nats(x2))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), nats(x2))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), a__nats(mark(x2))), MARK(sieve(z0))) MARK(filter(x0, nats(z0), nats(x2))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), a__nats(mark(x2))), MARK(nats(z0))) MARK(filter(x0, zprimes, nats(x2))) -> c11(A__FILTER(mark(x0), a__zprimes, a__nats(mark(x2))), MARK(zprimes)) MARK(filter(x0, s(z0), nats(x2))) -> c11(A__FILTER(mark(x0), s(mark(z0)), a__nats(mark(x2))), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, nats(x2))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(sieve(z0), x1, nats(x2))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(nats(z0), x1, nats(x2))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(zprimes, x1, nats(x2))) -> c11(A__FILTER(a__zprimes, mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(cons(z0, z1), x1, nats(x2))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(x0, cons(z0, z1), nats(x2))) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, nats(x2))) -> c11(MARK(x1)) MARK(filter(s(z0), x1, nats(x2))) -> c11(MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(s(s(0))))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), zprimes), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), zprimes)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__zprimes), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), zprimes)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), a__zprimes), MARK(sieve(z0))) MARK(filter(x0, nats(z0), zprimes)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), a__zprimes), MARK(nats(z0))) MARK(filter(x0, zprimes, zprimes)) -> c11(A__FILTER(mark(x0), a__zprimes, a__zprimes), MARK(zprimes)) MARK(filter(x0, s(z0), zprimes)) -> c11(A__FILTER(mark(x0), s(mark(z0)), a__zprimes), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, zprimes)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__zprimes), MARK(x1)) MARK(filter(sieve(z0), x1, zprimes)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__zprimes), MARK(x1)) MARK(filter(nats(z0), x1, zprimes)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__zprimes), MARK(x1)) MARK(filter(zprimes, x1, zprimes)) -> c11(A__FILTER(a__zprimes, mark(x1), a__zprimes), MARK(x1)) MARK(filter(cons(z0, z1), x1, zprimes)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, cons(z0, z1), zprimes)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, zprimes)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, zprimes)) -> c11(MARK(x1)) MARK(filter(x0, x1, 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cons(x2, x3))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), cons(mark(x2), x3)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), cons(x2, x3))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), cons(mark(x2), x3)), MARK(nats(z0))) MARK(filter(x0, zprimes, cons(x2, x3))) -> c11(A__FILTER(mark(x0), a__zprimes, cons(mark(x2), x3)), MARK(zprimes)) MARK(filter(x0, s(z0), cons(x2, x3))) -> c11(A__FILTER(mark(x0), s(mark(z0)), cons(mark(x2), x3)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, cons(x2, x3))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(sieve(z0), x1, cons(x2, x3))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(nats(z0), x1, cons(x2, x3))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(zprimes, x1, cons(x2, x3))) -> c11(A__FILTER(a__zprimes, mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(cons(z0, z1), x1, cons(x2, x3))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(x0, cons(z0, z1), cons(x2, x3))) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, cons(x2, x3))) -> c11(MARK(x1)) MARK(filter(s(z0), x1, cons(x2, x3))) -> c11(MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), 0)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), 0), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), 0)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), 0), MARK(sieve(z0))) MARK(filter(x0, nats(z0), 0)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), 0), MARK(nats(z0))) MARK(filter(x0, zprimes, 0)) -> c11(A__FILTER(mark(x0), a__zprimes, 0), MARK(zprimes)) MARK(filter(x0, s(z0), 0)) -> c11(A__FILTER(mark(x0), s(mark(z0)), 0), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, 0)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), 0), MARK(x1)) MARK(filter(sieve(z0), x1, 0)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), 0), MARK(x1)) MARK(filter(nats(z0), x1, 0)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), 0), MARK(x1)) MARK(filter(zprimes, x1, 0)) -> c11(A__FILTER(a__zprimes, mark(x1), 0), MARK(x1)) MARK(filter(cons(z0, z1), x1, 0)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), 0), MARK(x1)) MARK(filter(x0, cons(z0, z1), 0)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, 0)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, 0)) -> c11(MARK(x1)) MARK(filter(x0, x1, s(filter(z0, z1, z2)))) -> c11(A__FILTER(mark(x0), mark(x1), s(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x1)) MARK(filter(x0, x1, s(sieve(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), s(a__sieve(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, s(nats(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), s(a__nats(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, s(zprimes))) -> c11(A__FILTER(mark(x0), mark(x1), s(a__zprimes)), MARK(x1)) MARK(filter(x0, x1, s(cons(z0, z1)))) -> c11(A__FILTER(mark(x0), mark(x1), s(cons(mark(z0), z1))), MARK(x1)) MARK(filter(x0, x1, s(0))) -> c11(A__FILTER(mark(x0), mark(x1), 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mark(x1), s(mark(x2))), MARK(x1)) MARK(filter(zprimes, x1, s(x2))) -> c11(A__FILTER(a__zprimes, mark(x1), s(mark(x2))), MARK(x1)) MARK(filter(cons(z0, z1), x1, s(x2))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), s(mark(x2))), MARK(x1)) MARK(filter(x0, cons(z0, z1), s(x2))) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, s(x2))) -> c11(MARK(x1)) MARK(filter(s(z0), x1, s(x2))) -> c11(MARK(x1)) MARK(filter(x0, filter(x1, x2, filter(z0, z1, z2)), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__filter(mark(z0), mark(z1), mark(z2))), mark(x4)), MARK(filter(x1, x2, filter(z0, z1, z2)))) MARK(filter(x0, filter(x1, x2, sieve(z0)), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__sieve(mark(z0))), mark(x4)), MARK(filter(x1, x2, sieve(z0)))) MARK(filter(x0, filter(x1, x2, nats(z0)), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__nats(mark(z0))), mark(x4)), MARK(filter(x1, x2, nats(z0)))) MARK(filter(x0, filter(x1, x2, zprimes), x4)) -> 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MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(x0, 0, filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, sieve(x2))) -> c10(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x0)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, sieve(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, sieve(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, sieve(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(0)), MARK(x0)) MARK(filter(x0, x1, sieve(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), sieve(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, sieve(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, sieve(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__sieve(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, sieve(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, sieve(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, sieve(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), a__sieve(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, sieve(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__sieve(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), sieve(x2))) -> c10(MARK(x0)) MARK(filter(x0, 0, sieve(x2))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, sieve(x2))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(mark(x2)), nats(s(mark(x2))))), MARK(x0)) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), nats(mark(x2))), MARK(x0)) MARK(filter(x0, x1, nats(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, nats(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, nats(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, nats(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, nats(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, nats(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(0)), MARK(x0)) MARK(filter(x0, x1, nats(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), nats(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), nats(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), nats(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, nats(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), nats(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, nats(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__nats(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, nats(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, nats(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, nats(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), a__nats(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, nats(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__nats(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), 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MARK(filter(s(z0), x1, sieve(x2))) -> c11(MARK(x1)) MARK(filter(x0, x1, nats(x2))) -> c11(A__FILTER(mark(x0), mark(x1), cons(mark(mark(x2)), nats(s(mark(x2))))), MARK(x1)) MARK(filter(x0, x1, nats(x2))) -> c11(A__FILTER(mark(x0), mark(x1), nats(mark(x2))), MARK(x1)) MARK(filter(x0, x1, nats(filter(z0, z1, z2)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x1)) MARK(filter(x0, x1, nats(sieve(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__sieve(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, nats(nats(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__nats(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, nats(zprimes))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(a__zprimes)), MARK(x1)) MARK(filter(x0, x1, nats(cons(z0, z1)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(cons(mark(z0), z1))), MARK(x1)) MARK(filter(x0, x1, nats(0))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(0)), MARK(x1)) MARK(filter(x0, x1, nats(s(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), a__nats(s(mark(z0)))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), nats(x2))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), nats(x2))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), a__nats(mark(x2))), MARK(sieve(z0))) MARK(filter(x0, nats(z0), nats(x2))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), a__nats(mark(x2))), MARK(nats(z0))) MARK(filter(x0, zprimes, nats(x2))) -> c11(A__FILTER(mark(x0), a__zprimes, a__nats(mark(x2))), MARK(zprimes)) MARK(filter(x0, s(z0), nats(x2))) -> c11(A__FILTER(mark(x0), s(mark(z0)), a__nats(mark(x2))), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, nats(x2))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(sieve(z0), x1, nats(x2))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(nats(z0), x1, nats(x2))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(zprimes, x1, nats(x2))) -> c11(A__FILTER(a__zprimes, mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(cons(z0, z1), x1, nats(x2))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__nats(mark(x2))), MARK(x1)) MARK(filter(x0, cons(z0, z1), nats(x2))) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, nats(x2))) -> c11(MARK(x1)) MARK(filter(s(z0), x1, nats(x2))) -> c11(MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(s(s(0))))), MARK(x1)) MARK(filter(x0, x1, zprimes)) -> c11(A__FILTER(mark(x0), mark(x1), zprimes), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), zprimes)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__zprimes), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), zprimes)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), a__zprimes), MARK(sieve(z0))) MARK(filter(x0, nats(z0), zprimes)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), a__zprimes), MARK(nats(z0))) MARK(filter(x0, zprimes, zprimes)) -> c11(A__FILTER(mark(x0), a__zprimes, a__zprimes), MARK(zprimes)) MARK(filter(x0, s(z0), zprimes)) -> c11(A__FILTER(mark(x0), s(mark(z0)), a__zprimes), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, zprimes)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__zprimes), MARK(x1)) MARK(filter(sieve(z0), x1, zprimes)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), a__zprimes), MARK(x1)) MARK(filter(nats(z0), x1, zprimes)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), a__zprimes), MARK(x1)) MARK(filter(zprimes, x1, zprimes)) -> c11(A__FILTER(a__zprimes, mark(x1), a__zprimes), MARK(x1)) MARK(filter(cons(z0, z1), x1, zprimes)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), a__zprimes), MARK(x1)) MARK(filter(x0, cons(z0, z1), zprimes)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, zprimes)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, zprimes)) -> c11(MARK(x1)) MARK(filter(x0, x1, cons(filter(z0, z1, z2), x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(a__filter(mark(z0), mark(z1), mark(z2)), x3)), MARK(x1)) MARK(filter(x0, x1, cons(sieve(z0), x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(a__sieve(mark(z0)), x3)), MARK(x1)) MARK(filter(x0, x1, cons(nats(z0), x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(a__nats(mark(z0)), x3)), MARK(x1)) MARK(filter(x0, x1, cons(zprimes, x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(a__zprimes, x3)), MARK(x1)) MARK(filter(x0, x1, cons(cons(z0, z1), x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(cons(mark(z0), z1), x3)), MARK(x1)) MARK(filter(x0, x1, cons(0, x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(0, x3)), MARK(x1)) MARK(filter(x0, x1, cons(s(z0), x3))) -> c11(A__FILTER(mark(x0), mark(x1), cons(s(mark(z0)), x3)), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), cons(x2, x3))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), cons(mark(x2), x3)), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), cons(x2, x3))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), cons(mark(x2), x3)), MARK(sieve(z0))) MARK(filter(x0, nats(z0), cons(x2, x3))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), cons(mark(x2), x3)), MARK(nats(z0))) MARK(filter(x0, zprimes, cons(x2, x3))) -> c11(A__FILTER(mark(x0), a__zprimes, cons(mark(x2), x3)), MARK(zprimes)) MARK(filter(x0, s(z0), cons(x2, x3))) -> c11(A__FILTER(mark(x0), s(mark(z0)), cons(mark(x2), x3)), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, cons(x2, x3))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(sieve(z0), x1, cons(x2, x3))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(nats(z0), x1, cons(x2, x3))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(zprimes, x1, cons(x2, x3))) -> c11(A__FILTER(a__zprimes, mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(cons(z0, z1), x1, cons(x2, x3))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), cons(mark(x2), x3)), MARK(x1)) MARK(filter(x0, cons(z0, z1), cons(x2, x3))) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, cons(x2, x3))) -> c11(MARK(x1)) MARK(filter(s(z0), x1, cons(x2, x3))) -> c11(MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), 0)) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), 0), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), 0)) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), 0), MARK(sieve(z0))) MARK(filter(x0, nats(z0), 0)) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), 0), MARK(nats(z0))) MARK(filter(x0, zprimes, 0)) -> c11(A__FILTER(mark(x0), a__zprimes, 0), MARK(zprimes)) MARK(filter(x0, s(z0), 0)) -> c11(A__FILTER(mark(x0), s(mark(z0)), 0), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, 0)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), 0), MARK(x1)) MARK(filter(sieve(z0), x1, 0)) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), 0), MARK(x1)) MARK(filter(nats(z0), x1, 0)) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), 0), MARK(x1)) MARK(filter(zprimes, x1, 0)) -> c11(A__FILTER(a__zprimes, mark(x1), 0), MARK(x1)) MARK(filter(cons(z0, z1), x1, 0)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), 0), MARK(x1)) MARK(filter(x0, cons(z0, z1), 0)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, 0)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, 0)) -> c11(MARK(x1)) MARK(filter(x0, x1, s(filter(z0, z1, z2)))) -> c11(A__FILTER(mark(x0), mark(x1), s(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x1)) MARK(filter(x0, x1, s(sieve(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), s(a__sieve(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, s(nats(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), s(a__nats(mark(z0)))), MARK(x1)) MARK(filter(x0, x1, s(zprimes))) -> c11(A__FILTER(mark(x0), mark(x1), s(a__zprimes)), MARK(x1)) MARK(filter(x0, x1, s(cons(z0, z1)))) -> c11(A__FILTER(mark(x0), mark(x1), s(cons(mark(z0), z1))), MARK(x1)) MARK(filter(x0, x1, s(0))) -> c11(A__FILTER(mark(x0), mark(x1), s(0)), MARK(x1)) MARK(filter(x0, x1, s(s(z0)))) -> c11(A__FILTER(mark(x0), mark(x1), s(s(mark(z0)))), MARK(x1)) MARK(filter(x0, filter(z0, z1, z2), s(x2))) -> c11(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), s(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, sieve(z0), s(x2))) -> c11(A__FILTER(mark(x0), a__sieve(mark(z0)), s(mark(x2))), MARK(sieve(z0))) MARK(filter(x0, nats(z0), s(x2))) -> c11(A__FILTER(mark(x0), a__nats(mark(z0)), s(mark(x2))), MARK(nats(z0))) MARK(filter(x0, zprimes, s(x2))) -> c11(A__FILTER(mark(x0), a__zprimes, s(mark(x2))), MARK(zprimes)) MARK(filter(x0, s(z0), s(x2))) -> c11(A__FILTER(mark(x0), s(mark(z0)), s(mark(x2))), MARK(s(z0))) MARK(filter(filter(z0, z1, z2), x1, s(x2))) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), s(mark(x2))), MARK(x1)) MARK(filter(sieve(z0), x1, s(x2))) -> c11(A__FILTER(a__sieve(mark(z0)), mark(x1), s(mark(x2))), MARK(x1)) MARK(filter(nats(z0), x1, s(x2))) -> c11(A__FILTER(a__nats(mark(z0)), mark(x1), s(mark(x2))), MARK(x1)) MARK(filter(zprimes, x1, s(x2))) -> c11(A__FILTER(a__zprimes, mark(x1), s(mark(x2))), MARK(x1)) MARK(filter(cons(z0, z1), x1, s(x2))) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), s(mark(x2))), MARK(x1)) MARK(filter(x0, cons(z0, z1), s(x2))) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, s(x2))) -> c11(MARK(x1)) MARK(filter(s(z0), x1, s(x2))) -> c11(MARK(x1)) MARK(filter(x0, filter(x1, x2, filter(z0, z1, z2)), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__filter(mark(z0), mark(z1), mark(z2))), mark(x4)), MARK(filter(x1, x2, filter(z0, z1, z2)))) MARK(filter(x0, filter(x1, x2, sieve(z0)), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__sieve(mark(z0))), mark(x4)), MARK(filter(x1, x2, sieve(z0)))) MARK(filter(x0, filter(x1, x2, nats(z0)), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__nats(mark(z0))), mark(x4)), MARK(filter(x1, x2, nats(z0)))) MARK(filter(x0, filter(x1, x2, zprimes), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), a__zprimes), mark(x4)), MARK(filter(x1, x2, zprimes))) MARK(filter(x0, filter(x1, x2, cons(z0, z1)), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), cons(mark(z0), z1)), mark(x4)), MARK(filter(x1, x2, cons(z0, z1)))) MARK(filter(x0, filter(x1, x2, 0), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), 0), mark(x4)), MARK(filter(x1, x2, 0))) MARK(filter(x0, filter(x1, x2, s(z0)), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), mark(x2), s(mark(z0))), mark(x4)), MARK(filter(x1, x2, s(z0)))) MARK(filter(x0, filter(x1, filter(z0, z1, z2), x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), a__filter(mark(z0), mark(z1), mark(z2)), mark(x3)), mark(x4)), MARK(filter(x1, filter(z0, z1, z2), x3))) MARK(filter(x0, filter(x1, sieve(z0), x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), a__sieve(mark(z0)), mark(x3)), mark(x4)), MARK(filter(x1, sieve(z0), x3))) MARK(filter(x0, filter(x1, nats(z0), x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), a__nats(mark(z0)), mark(x3)), mark(x4)), MARK(filter(x1, nats(z0), x3))) MARK(filter(x0, filter(x1, zprimes, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), a__zprimes, mark(x3)), mark(x4)), MARK(filter(x1, zprimes, x3))) MARK(filter(x0, filter(x1, cons(z0, z1), x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), cons(mark(z0), z1), mark(x3)), mark(x4)), MARK(filter(x1, cons(z0, z1), x3))) MARK(filter(x0, filter(x1, 0, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), 0, mark(x3)), mark(x4)), MARK(filter(x1, 0, x3))) MARK(filter(x0, filter(x1, s(z0), x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(mark(x1), s(mark(z0)), mark(x3)), mark(x4)), MARK(filter(x1, s(z0), x3))) MARK(filter(x0, filter(filter(z0, z1, z2), x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x2), mark(x3)), mark(x4)), MARK(filter(filter(z0, z1, z2), x2, x3))) MARK(filter(x0, filter(sieve(z0), x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(a__sieve(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(filter(sieve(z0), x2, x3))) MARK(filter(x0, filter(nats(z0), x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(a__nats(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(filter(nats(z0), x2, x3))) MARK(filter(x0, filter(zprimes, x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(a__zprimes, mark(x2), mark(x3)), mark(x4)), MARK(filter(zprimes, x2, x3))) MARK(filter(x0, filter(cons(z0, z1), x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(cons(mark(z0), z1), mark(x2), mark(x3)), mark(x4)), MARK(filter(cons(z0, z1), x2, x3))) MARK(filter(x0, filter(0, x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(0, mark(x2), mark(x3)), mark(x4)), MARK(filter(0, x2, x3))) MARK(filter(x0, filter(s(z0), x2, x3), x4)) -> c11(A__FILTER(mark(x0), a__filter(s(mark(z0)), mark(x2), mark(x3)), mark(x4)), MARK(filter(s(z0), x2, x3))) MARK(filter(filter(z0, z1, z2), filter(x1, x2, x3), x4)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(sieve(z0), filter(x1, x2, x3), x4)) -> c11(A__FILTER(a__sieve(mark(z0)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(nats(z0), filter(x1, x2, x3), x4)) -> c11(A__FILTER(a__nats(mark(z0)), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(zprimes, filter(x1, x2, x3), x4)) -> c11(A__FILTER(a__zprimes, a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(cons(z0, z1), filter(x1, x2, x3), x4)) -> c11(A__FILTER(cons(mark(z0), z1), a__filter(mark(x1), mark(x2), mark(x3)), mark(x4)), MARK(filter(x1, x2, x3))) MARK(filter(x0, filter(x1, x2, x3), x4)) -> c11(MARK(filter(x1, x2, x3))) MARK(filter(0, filter(x1, x2, x3), x4)) -> c11(MARK(filter(x1, x2, x3))) MARK(filter(s(z0), filter(x1, x2, x3), x4)) -> c11(MARK(filter(x1, x2, x3))) MARK(filter(x0, sieve(filter(z0, z1, z2)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(sieve(filter(z0, z1, z2)))) MARK(filter(x0, sieve(sieve(z0)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(a__sieve(mark(z0))), mark(x2)), MARK(sieve(sieve(z0)))) MARK(filter(x0, sieve(nats(z0)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(a__nats(mark(z0))), mark(x2)), MARK(sieve(nats(z0)))) MARK(filter(x0, sieve(zprimes), x2)) -> c11(A__FILTER(mark(x0), a__sieve(a__zprimes), mark(x2)), MARK(sieve(zprimes))) MARK(filter(x0, sieve(cons(z0, z1)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(cons(mark(z0), z1)), mark(x2)), MARK(sieve(cons(z0, z1)))) MARK(filter(x0, sieve(s(z0)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(s(mark(z0))), mark(x2)), MARK(sieve(s(z0)))) MARK(filter(filter(z0, z1, z2), sieve(x1), x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(sieve(z0), sieve(x1), x2)) -> c11(A__FILTER(a__sieve(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(nats(z0), sieve(x1), x2)) -> c11(A__FILTER(a__nats(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(zprimes, sieve(x1), x2)) -> c11(A__FILTER(a__zprimes, a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(cons(z0, z1), sieve(x1), x2)) -> c11(A__FILTER(cons(mark(z0), z1), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(x0, sieve(x1), x2)) -> c11(MARK(sieve(x1))) MARK(filter(x0, sieve(0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(0), mark(x2))) MARK(filter(0, sieve(x1), x2)) -> c11(MARK(sieve(x1))) MARK(filter(s(z0), sieve(x1), x2)) -> c11(MARK(sieve(x1))) MARK(filter(x0, nats(filter(z0, z1, z2)), x2)) -> c11(A__FILTER(mark(x0), a__nats(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(nats(filter(z0, z1, z2)))) MARK(filter(x0, nats(sieve(z0)), x2)) -> c11(A__FILTER(mark(x0), a__nats(a__sieve(mark(z0))), mark(x2)), MARK(nats(sieve(z0)))) MARK(filter(x0, nats(nats(z0)), x2)) -> c11(A__FILTER(mark(x0), a__nats(a__nats(mark(z0))), mark(x2)), MARK(nats(nats(z0)))) MARK(filter(x0, nats(zprimes), x2)) -> c11(A__FILTER(mark(x0), a__nats(a__zprimes), mark(x2)), MARK(nats(zprimes))) MARK(filter(x0, nats(cons(z0, z1)), x2)) -> c11(A__FILTER(mark(x0), a__nats(cons(mark(z0), z1)), mark(x2)), MARK(nats(cons(z0, z1)))) MARK(filter(x0, nats(0), x2)) -> c11(A__FILTER(mark(x0), a__nats(0), mark(x2)), MARK(nats(0))) MARK(filter(x0, nats(s(z0)), x2)) -> c11(A__FILTER(mark(x0), a__nats(s(mark(z0))), mark(x2)), MARK(nats(s(z0)))) MARK(filter(filter(z0, z1, z2), nats(x1), x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(sieve(z0), nats(x1), x2)) -> c11(A__FILTER(a__sieve(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(nats(z0), nats(x1), x2)) -> c11(A__FILTER(a__nats(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(zprimes, nats(x1), x2)) -> c11(A__FILTER(a__zprimes, a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(cons(z0, z1), nats(x1), x2)) -> c11(A__FILTER(cons(mark(z0), z1), a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(x0, nats(x1), x2)) -> c11(MARK(nats(x1))) MARK(filter(0, nats(x1), x2)) -> c11(MARK(nats(x1))) MARK(filter(s(z0), nats(x1), x2)) -> c11(MARK(nats(x1))) MARK(filter(x0, zprimes, x1)) -> c11(A__FILTER(mark(x0), a__sieve(a__nats(s(s(0)))), mark(x1)), MARK(zprimes)) MARK(filter(filter(z0, z1, z2), zprimes, x1)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(sieve(z0), zprimes, x1)) -> c11(A__FILTER(a__sieve(mark(z0)), a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(nats(z0), zprimes, x1)) -> c11(A__FILTER(a__nats(mark(z0)), a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(zprimes, zprimes, x1)) -> c11(A__FILTER(a__zprimes, a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(cons(z0, z1), zprimes, x1)) -> c11(A__FILTER(cons(mark(z0), z1), a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(x0, zprimes, x1)) -> c11(MARK(zprimes)) MARK(filter(0, zprimes, x1)) -> c11(MARK(zprimes)) MARK(filter(s(z0), zprimes, x1)) -> c11(MARK(zprimes)) MARK(filter(x0, s(filter(z0, z1, z2)), x2)) -> c11(A__FILTER(mark(x0), s(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(s(filter(z0, z1, z2)))) MARK(filter(x0, s(sieve(z0)), x2)) -> c11(A__FILTER(mark(x0), s(a__sieve(mark(z0))), mark(x2)), MARK(s(sieve(z0)))) MARK(filter(x0, s(nats(z0)), x2)) -> c11(A__FILTER(mark(x0), s(a__nats(mark(z0))), mark(x2)), MARK(s(nats(z0)))) MARK(filter(x0, s(zprimes), x2)) -> c11(A__FILTER(mark(x0), s(a__zprimes), mark(x2)), MARK(s(zprimes))) MARK(filter(x0, s(cons(z0, z1)), x2)) -> c11(A__FILTER(mark(x0), s(cons(mark(z0), z1)), mark(x2)), MARK(s(cons(z0, z1)))) MARK(filter(x0, s(0), x2)) -> c11(A__FILTER(mark(x0), s(0), mark(x2)), MARK(s(0))) MARK(filter(x0, s(s(z0)), x2)) -> c11(A__FILTER(mark(x0), s(s(mark(z0))), mark(x2)), MARK(s(s(z0)))) MARK(filter(filter(z0, z1, z2), s(x1), x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), s(mark(x1)), mark(x2)), MARK(s(x1))) MARK(filter(sieve(z0), s(x1), x2)) -> c11(A__FILTER(a__sieve(mark(z0)), s(mark(x1)), mark(x2)), MARK(s(x1))) MARK(filter(nats(z0), s(x1), x2)) -> c11(A__FILTER(a__nats(mark(z0)), s(mark(x1)), mark(x2)), MARK(s(x1))) MARK(filter(zprimes, s(x1), x2)) -> c11(A__FILTER(a__zprimes, s(mark(x1)), mark(x2)), MARK(s(x1))) MARK(filter(cons(z0, z1), s(x1), x2)) -> c11(A__FILTER(cons(mark(z0), z1), s(mark(x1)), mark(x2)), MARK(s(x1))) MARK(filter(x0, s(x1), x2)) -> c11(MARK(s(x1))) MARK(filter(0, s(x1), x2)) -> c11(MARK(s(x1))) MARK(filter(s(z0), s(x1), x2)) -> c11(MARK(s(x1))) MARK(filter(filter(x0, x1, filter(z0, z1, z2)), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, sieve(z0)), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), a__sieve(mark(z0))), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, nats(z0)), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), a__nats(mark(z0))), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, zprimes), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), a__zprimes), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, cons(z0, z1)), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), cons(mark(z0), z1)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, 0), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), 0), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, s(z0)), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), s(mark(z0))), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, filter(z0, z1, z2), x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, sieve(z0), x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), a__sieve(mark(z0)), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, nats(z0), x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), a__nats(mark(z0)), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, zprimes, x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), a__zprimes, mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, cons(z0, z1), x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), cons(mark(z0), z1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, 0, x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), 0, mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, s(z0), x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), s(mark(z0)), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(filter(z0, z1, z2), x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(sieve(z0), x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(a__sieve(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(nats(z0), x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(a__nats(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(zprimes, x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(a__zprimes, mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(cons(z0, z1), x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(cons(mark(z0), z1), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(0, x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(0, mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(s(z0), x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(s(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, x2), cons(z0, z1), x4)) -> c11(MARK(cons(z0, z1))) MARK(filter(filter(x0, x1, x2), x3, x4)) -> c11(MARK(x3)) MARK(filter(sieve(filter(z0, z1, z2)), x1, x2)) -> c11(A__FILTER(a__sieve(a__filter(mark(z0), mark(z1), mark(z2))), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(sieve(z0)), x1, x2)) -> c11(A__FILTER(a__sieve(a__sieve(mark(z0))), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(nats(z0)), x1, x2)) -> c11(A__FILTER(a__sieve(a__nats(mark(z0))), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(zprimes), x1, x2)) -> c11(A__FILTER(a__sieve(a__zprimes), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(cons(z0, z1)), x1, x2)) -> c11(A__FILTER(a__sieve(cons(mark(z0), z1)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(0), x1, x2)) -> c11(A__FILTER(a__sieve(0), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(s(z0)), x1, x2)) -> c11(A__FILTER(a__sieve(s(mark(z0))), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(x0), cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(sieve(x0), x1, x2)) -> c11(MARK(x1)) MARK(filter(nats(x0), x1, x2)) -> c11(A__FILTER(cons(mark(mark(x0)), nats(s(mark(x0)))), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(filter(z0, z1, z2)), x1, x2)) -> c11(A__FILTER(a__nats(a__filter(mark(z0), mark(z1), mark(z2))), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(sieve(z0)), x1, x2)) -> c11(A__FILTER(a__nats(a__sieve(mark(z0))), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(nats(z0)), x1, x2)) -> c11(A__FILTER(a__nats(a__nats(mark(z0))), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(zprimes), x1, x2)) -> c11(A__FILTER(a__nats(a__zprimes), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(cons(z0, z1)), x1, x2)) -> c11(A__FILTER(a__nats(cons(mark(z0), z1)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(0), x1, x2)) -> c11(A__FILTER(a__nats(0), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(s(z0)), x1, x2)) -> c11(A__FILTER(a__nats(s(mark(z0))), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(x0), cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(nats(x0), x1, x2)) -> c11(MARK(x1)) MARK(filter(zprimes, cons(z0, z1), x1)) -> c11(A__FILTER(a__zprimes, cons(mark(z0), z1), mark(x1)), MARK(cons(z0, z1))) MARK(filter(zprimes, 0, x1)) -> c11(A__FILTER(a__zprimes, 0, mark(x1)), MARK(0)) MARK(filter(zprimes, x0, x1)) -> c11(A__FILTER(a__sieve(a__nats(s(s(0)))), mark(x0), mark(x1)), MARK(x0)) MARK(filter(zprimes, x0, x1)) -> c11(A__FILTER(zprimes, mark(x0), mark(x1)), MARK(x0)) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c14_2, c15_1, c16_1, c18_1, c8_1, c10_1, c11_2, c11_1, c12_2, c12_1, c13_2, c13_1, c8_2, c10_2 ---------------------------------------- (227) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(filter(zprimes, 0, x1)) -> c11(A__FILTER(a__zprimes, 0, mark(x1)), MARK(0)) ---------------------------------------- (228) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) MARK(filter(x0, x1, cons(z0, z1))) -> c12(A__FILTER(mark(x0), mark(x1), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, x1, s(z0))) -> c12(A__FILTER(mark(x0), mark(x1), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, filter(z0, z1, z2), x2)) -> c12(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), MARK(x2)) MARK(filter(x0, sieve(z0), x2)) -> c12(A__FILTER(mark(x0), a__sieve(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, nats(z0), x2)) -> c12(A__FILTER(mark(x0), a__nats(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(x0, zprimes, x2)) -> c12(A__FILTER(mark(x0), a__zprimes, mark(x2)), MARK(x2)) MARK(filter(x0, s(z0), x2)) -> c12(A__FILTER(mark(x0), s(mark(z0)), mark(x2)), MARK(x2)) MARK(filter(filter(z0, z1, z2), x1, x2)) -> c12(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(sieve(z0), x1, x2)) -> c12(A__FILTER(a__sieve(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(nats(z0), x1, x2)) -> c12(A__FILTER(a__nats(mark(z0)), mark(x1), mark(x2)), MARK(x2)) MARK(filter(zprimes, x1, x2)) -> c12(A__FILTER(a__zprimes, mark(x1), mark(x2)), MARK(x2)) MARK(filter(cons(z0, z1), x1, x2)) -> c12(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x2)) MARK(filter(x0, x1, 0)) -> c12(A__FILTER(mark(x0), mark(x1), 0)) MARK(filter(x0, cons(z0, z1), x2)) -> c12(MARK(x2)) MARK(filter(x0, 0, x2)) -> c12(MARK(x2)) MARK(filter(0, x1, x2)) -> c12(MARK(x2)) MARK(filter(s(z0), x1, x2)) -> c12(MARK(x2)) MARK(sieve(filter(z0, z1, z2))) -> c13(A__SIEVE(a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(sieve(sieve(z0))) -> c13(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(nats(z0))) -> c13(A__SIEVE(a__nats(mark(z0))), MARK(nats(z0))) MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) MARK(sieve(cons(z0, z1))) -> c13(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c13(MARK(s(z0))) A__ZPRIMES -> c8(A__SIEVE(cons(s(mark(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) MARK(filter(x0, x1, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), a__zprimes)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, 0))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), 0)), MARK(x0)) MARK(filter(x0, x1, filter(x2, x3, s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), mark(x3), s(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, filter(x2, filter(z0, z1, z2), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__filter(mark(z0), mark(z1), mark(z2)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, sieve(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__sieve(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, nats(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__nats(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, zprimes, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), a__zprimes, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, cons(z0, z1), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), cons(mark(z0), z1), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, 0, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), 0, mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(x2, s(z0), x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(mark(x2), s(mark(z0)), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(filter(z0, z1, z2), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(sieve(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__sieve(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(nats(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__nats(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(zprimes, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(a__zprimes, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(cons(z0, z1), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(cons(mark(z0), z1), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(0, x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(0, mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, x1, filter(s(z0), x3, x4))) -> c10(A__FILTER(mark(x0), mark(x1), a__filter(s(mark(z0)), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, sieve(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, nats(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, zprimes, filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), a__zprimes, a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(x0, s(z0), filter(x2, x3, x4))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__filter(mark(x2), mark(x3), mark(x4))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(nats(z0))) MARK(filter(zprimes, x1, filter(x2, x3, x4))) -> c10(A__FILTER(a__zprimes, mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, filter(x2, x3, x4))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__filter(mark(x2), mark(x3), mark(x4))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(x0, 0, filter(x2, x3, x4))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, filter(x2, x3, x4))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, sieve(x2))) -> c10(A__FILTER(mark(x0), mark(x1), sieve(mark(x2))), MARK(x0)) MARK(filter(x0, x1, sieve(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, sieve(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, sieve(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, sieve(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, sieve(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(0)), MARK(x0)) MARK(filter(x0, x1, sieve(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), sieve(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, sieve(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__sieve(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), sieve(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__sieve(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, sieve(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__sieve(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, sieve(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, sieve(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__sieve(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, sieve(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), a__sieve(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, sieve(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__sieve(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), sieve(x2))) -> c10(MARK(x0)) MARK(filter(x0, 0, sieve(x2))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, sieve(x2))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), cons(mark(mark(x2)), nats(s(mark(x2))))), MARK(x0)) MARK(filter(x0, x1, nats(x2))) -> c10(A__FILTER(mark(x0), mark(x1), nats(mark(x2))), MARK(x0)) MARK(filter(x0, x1, nats(filter(z0, z1, z2)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__filter(mark(z0), mark(z1), mark(z2)))), MARK(x0)) MARK(filter(x0, x1, nats(sieve(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, nats(nats(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__nats(mark(z0)))), MARK(x0)) MARK(filter(x0, x1, nats(zprimes))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(a__zprimes)), MARK(x0)) MARK(filter(x0, x1, nats(cons(z0, z1)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, x1, nats(0))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(0)), MARK(x0)) MARK(filter(x0, x1, nats(s(z0)))) -> c10(A__FILTER(mark(x0), mark(x1), a__nats(s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), nats(x2))) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, sieve(z0), nats(x2))) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, nats(z0), nats(x2))) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, zprimes, nats(x2))) -> c10(A__FILTER(mark(x0), a__zprimes, a__nats(mark(x2))), MARK(x0)) MARK(filter(x0, s(z0), nats(x2))) -> c10(A__FILTER(mark(x0), s(mark(z0)), a__nats(mark(x2))), MARK(x0)) MARK(filter(filter(z0, z1, z2), x1, nats(x2))) -> c10(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), a__nats(mark(x2))), MARK(filter(z0, z1, z2))) MARK(filter(sieve(z0), x1, nats(x2))) -> c10(A__FILTER(a__sieve(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(sieve(z0))) MARK(filter(nats(z0), x1, nats(x2))) -> c10(A__FILTER(a__nats(mark(z0)), mark(x1), a__nats(mark(x2))), MARK(nats(z0))) MARK(filter(zprimes, x1, nats(x2))) -> c10(A__FILTER(a__zprimes, mark(x1), a__nats(mark(x2))), MARK(zprimes)) MARK(filter(cons(z0, z1), x1, nats(x2))) -> c10(A__FILTER(cons(mark(z0), z1), mark(x1), a__nats(mark(x2))), MARK(cons(z0, z1))) MARK(filter(x0, cons(z0, z1), nats(x2))) -> c10(MARK(x0)) MARK(filter(x0, 0, nats(x2))) -> c10(MARK(x0)) MARK(filter(s(z0), x1, nats(x2))) -> c10(MARK(s(z0))) MARK(filter(x0, x1, zprimes)) -> c10(A__FILTER(mark(x0), mark(x1), a__sieve(a__nats(s(s(0))))), MARK(x0)) MARK(filter(x0, x1, zprimes)) -> c10(A__FILTER(mark(x0), mark(x1), zprimes), MARK(x0)) MARK(filter(x0, filter(z0, z1, z2), zprimes)) -> c10(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), a__zprimes), MARK(x0)) MARK(filter(x0, sieve(z0), zprimes)) -> c10(A__FILTER(mark(x0), a__sieve(mark(z0)), a__zprimes), MARK(x0)) MARK(filter(x0, nats(z0), zprimes)) -> c10(A__FILTER(mark(x0), a__nats(mark(z0)), a__zprimes), MARK(x0)) MARK(filter(x0, zprimes, 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mark(x2)), MARK(sieve(cons(z0, z1)))) MARK(filter(x0, sieve(s(z0)), x2)) -> c11(A__FILTER(mark(x0), a__sieve(s(mark(z0))), mark(x2)), MARK(sieve(s(z0)))) MARK(filter(filter(z0, z1, z2), sieve(x1), x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(sieve(z0), sieve(x1), x2)) -> c11(A__FILTER(a__sieve(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(nats(z0), sieve(x1), x2)) -> c11(A__FILTER(a__nats(mark(z0)), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(zprimes, sieve(x1), x2)) -> c11(A__FILTER(a__zprimes, a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(cons(z0, z1), sieve(x1), x2)) -> c11(A__FILTER(cons(mark(z0), z1), a__sieve(mark(x1)), mark(x2)), MARK(sieve(x1))) MARK(filter(x0, sieve(x1), x2)) -> c11(MARK(sieve(x1))) MARK(filter(x0, sieve(0), x2)) -> c11(A__FILTER(mark(x0), a__sieve(0), mark(x2))) MARK(filter(0, sieve(x1), x2)) -> c11(MARK(sieve(x1))) MARK(filter(s(z0), sieve(x1), x2)) -> c11(MARK(sieve(x1))) MARK(filter(x0, nats(filter(z0, z1, z2)), x2)) -> c11(A__FILTER(mark(x0), a__nats(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(nats(filter(z0, z1, z2)))) MARK(filter(x0, nats(sieve(z0)), x2)) -> c11(A__FILTER(mark(x0), a__nats(a__sieve(mark(z0))), mark(x2)), MARK(nats(sieve(z0)))) MARK(filter(x0, nats(nats(z0)), x2)) -> c11(A__FILTER(mark(x0), a__nats(a__nats(mark(z0))), mark(x2)), MARK(nats(nats(z0)))) MARK(filter(x0, nats(zprimes), x2)) -> c11(A__FILTER(mark(x0), a__nats(a__zprimes), mark(x2)), MARK(nats(zprimes))) MARK(filter(x0, nats(cons(z0, z1)), x2)) -> c11(A__FILTER(mark(x0), a__nats(cons(mark(z0), z1)), mark(x2)), MARK(nats(cons(z0, z1)))) MARK(filter(x0, nats(0), x2)) -> c11(A__FILTER(mark(x0), a__nats(0), mark(x2)), MARK(nats(0))) MARK(filter(x0, nats(s(z0)), x2)) -> c11(A__FILTER(mark(x0), a__nats(s(mark(z0))), mark(x2)), MARK(nats(s(z0)))) MARK(filter(filter(z0, z1, z2), nats(x1), x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(sieve(z0), nats(x1), x2)) -> c11(A__FILTER(a__sieve(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(nats(z0), nats(x1), x2)) -> c11(A__FILTER(a__nats(mark(z0)), a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(zprimes, nats(x1), x2)) -> c11(A__FILTER(a__zprimes, a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(cons(z0, z1), nats(x1), x2)) -> c11(A__FILTER(cons(mark(z0), z1), a__nats(mark(x1)), mark(x2)), MARK(nats(x1))) MARK(filter(x0, nats(x1), x2)) -> c11(MARK(nats(x1))) MARK(filter(0, nats(x1), x2)) -> c11(MARK(nats(x1))) MARK(filter(s(z0), nats(x1), x2)) -> c11(MARK(nats(x1))) MARK(filter(x0, zprimes, x1)) -> c11(A__FILTER(mark(x0), a__sieve(a__nats(s(s(0)))), mark(x1)), MARK(zprimes)) MARK(filter(filter(z0, z1, z2), zprimes, x1)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(sieve(z0), zprimes, x1)) -> c11(A__FILTER(a__sieve(mark(z0)), a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(nats(z0), zprimes, x1)) -> c11(A__FILTER(a__nats(mark(z0)), a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(zprimes, zprimes, x1)) -> c11(A__FILTER(a__zprimes, a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(cons(z0, z1), zprimes, x1)) -> c11(A__FILTER(cons(mark(z0), z1), a__zprimes, mark(x1)), MARK(zprimes)) MARK(filter(x0, zprimes, x1)) -> c11(MARK(zprimes)) MARK(filter(0, zprimes, x1)) -> c11(MARK(zprimes)) MARK(filter(s(z0), zprimes, x1)) -> c11(MARK(zprimes)) MARK(filter(x0, s(filter(z0, z1, z2)), x2)) -> c11(A__FILTER(mark(x0), s(a__filter(mark(z0), mark(z1), mark(z2))), mark(x2)), MARK(s(filter(z0, z1, z2)))) MARK(filter(x0, s(sieve(z0)), x2)) -> c11(A__FILTER(mark(x0), s(a__sieve(mark(z0))), mark(x2)), MARK(s(sieve(z0)))) MARK(filter(x0, s(nats(z0)), x2)) -> c11(A__FILTER(mark(x0), s(a__nats(mark(z0))), mark(x2)), MARK(s(nats(z0)))) MARK(filter(x0, s(zprimes), x2)) -> c11(A__FILTER(mark(x0), s(a__zprimes), mark(x2)), MARK(s(zprimes))) MARK(filter(x0, s(cons(z0, z1)), x2)) -> c11(A__FILTER(mark(x0), s(cons(mark(z0), z1)), mark(x2)), MARK(s(cons(z0, z1)))) MARK(filter(x0, s(0), x2)) -> c11(A__FILTER(mark(x0), s(0), mark(x2)), MARK(s(0))) MARK(filter(x0, s(s(z0)), x2)) -> c11(A__FILTER(mark(x0), s(s(mark(z0))), mark(x2)), MARK(s(s(z0)))) MARK(filter(filter(z0, z1, z2), s(x1), x2)) -> c11(A__FILTER(a__filter(mark(z0), mark(z1), mark(z2)), s(mark(x1)), mark(x2)), MARK(s(x1))) MARK(filter(sieve(z0), s(x1), x2)) -> c11(A__FILTER(a__sieve(mark(z0)), s(mark(x1)), mark(x2)), MARK(s(x1))) MARK(filter(nats(z0), s(x1), x2)) -> c11(A__FILTER(a__nats(mark(z0)), s(mark(x1)), mark(x2)), MARK(s(x1))) MARK(filter(zprimes, s(x1), x2)) -> c11(A__FILTER(a__zprimes, s(mark(x1)), mark(x2)), MARK(s(x1))) MARK(filter(cons(z0, z1), s(x1), x2)) -> c11(A__FILTER(cons(mark(z0), z1), s(mark(x1)), mark(x2)), MARK(s(x1))) MARK(filter(x0, s(x1), x2)) -> c11(MARK(s(x1))) MARK(filter(0, s(x1), x2)) -> c11(MARK(s(x1))) MARK(filter(s(z0), s(x1), x2)) -> c11(MARK(s(x1))) MARK(filter(filter(x0, x1, filter(z0, z1, z2)), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, sieve(z0)), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), a__sieve(mark(z0))), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, nats(z0)), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), a__nats(mark(z0))), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, zprimes), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), a__zprimes), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, cons(z0, z1)), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), cons(mark(z0), z1)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, 0), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), 0), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, s(z0)), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), mark(x1), s(mark(z0))), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, filter(z0, z1, z2), x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), a__filter(mark(z0), mark(z1), mark(z2)), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, sieve(z0), x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), a__sieve(mark(z0)), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, nats(z0), x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), a__nats(mark(z0)), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, zprimes, x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), a__zprimes, mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, cons(z0, z1), x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), cons(mark(z0), z1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, 0, x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), 0, mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, s(z0), x2), x3, x4)) -> c11(A__FILTER(a__filter(mark(x0), s(mark(z0)), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(filter(z0, z1, z2), x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(a__filter(mark(z0), mark(z1), mark(z2)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(sieve(z0), x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(a__sieve(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(nats(z0), x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(a__nats(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(zprimes, x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(a__zprimes, mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(cons(z0, z1), x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(cons(mark(z0), z1), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(0, x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(0, mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(s(z0), x1, x2), x3, x4)) -> c11(A__FILTER(a__filter(s(mark(z0)), mark(x1), mark(x2)), mark(x3), mark(x4)), MARK(x3)) MARK(filter(filter(x0, x1, x2), cons(z0, z1), x4)) -> c11(MARK(cons(z0, z1))) MARK(filter(filter(x0, x1, x2), x3, x4)) -> c11(MARK(x3)) MARK(filter(sieve(filter(z0, z1, z2)), x1, x2)) -> c11(A__FILTER(a__sieve(a__filter(mark(z0), mark(z1), mark(z2))), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(sieve(z0)), x1, x2)) -> c11(A__FILTER(a__sieve(a__sieve(mark(z0))), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(nats(z0)), x1, x2)) -> c11(A__FILTER(a__sieve(a__nats(mark(z0))), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(zprimes), x1, x2)) -> c11(A__FILTER(a__sieve(a__zprimes), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(cons(z0, z1)), x1, x2)) -> c11(A__FILTER(a__sieve(cons(mark(z0), z1)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(0), x1, x2)) -> c11(A__FILTER(a__sieve(0), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(s(z0)), x1, x2)) -> c11(A__FILTER(a__sieve(s(mark(z0))), mark(x1), mark(x2)), MARK(x1)) MARK(filter(sieve(x0), cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(sieve(x0), x1, x2)) -> c11(MARK(x1)) MARK(filter(nats(x0), x1, x2)) -> c11(A__FILTER(cons(mark(mark(x0)), nats(s(mark(x0)))), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(filter(z0, z1, z2)), x1, x2)) -> c11(A__FILTER(a__nats(a__filter(mark(z0), mark(z1), mark(z2))), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(sieve(z0)), x1, x2)) -> c11(A__FILTER(a__nats(a__sieve(mark(z0))), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(nats(z0)), x1, x2)) -> c11(A__FILTER(a__nats(a__nats(mark(z0))), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(zprimes), x1, x2)) -> c11(A__FILTER(a__nats(a__zprimes), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(cons(z0, z1)), x1, x2)) -> c11(A__FILTER(a__nats(cons(mark(z0), z1)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(0), x1, x2)) -> c11(A__FILTER(a__nats(0), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(s(z0)), x1, x2)) -> c11(A__FILTER(a__nats(s(mark(z0))), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(x0), cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(nats(x0), x1, x2)) -> c11(MARK(x1)) MARK(filter(zprimes, cons(z0, z1), x1)) -> c11(A__FILTER(a__zprimes, cons(mark(z0), z1), mark(x1)), MARK(cons(z0, z1))) MARK(filter(zprimes, x0, x1)) -> c11(A__FILTER(a__sieve(a__nats(s(s(0)))), mark(x0), mark(x1)), MARK(x0)) MARK(filter(zprimes, x0, x1)) -> c11(A__FILTER(zprimes, mark(x0), mark(x1)), MARK(x0)) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c14_2, c15_1, c16_1, c18_1, c8_1, c10_1, c11_2, c11_1, c12_2, c12_1, c13_2, c13_1, c8_2, c10_2 ---------------------------------------- (229) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (230) Obligation: Complexity Dependency Tuples Problem Rules: a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) a__filter(z0, z1, z2) -> filter(z0, z1, z2) a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) a__sieve(z0) -> sieve(z0) a__nats(z0) -> cons(mark(z0), nats(s(z0))) a__nats(z0) -> nats(z0) a__zprimes -> a__sieve(a__nats(s(s(0)))) a__zprimes -> zprimes mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) mark(sieve(z0)) -> a__sieve(mark(z0)) mark(nats(z0)) -> a__nats(mark(z0)) mark(zprimes) -> a__zprimes mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(0) -> 0 mark(s(z0)) -> s(mark(z0)) Tuples: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) A__NATS(z0) -> c6(MARK(z0)) MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) MARK(zprimes) -> c15(A__ZPRIMES) MARK(cons(z0, z1)) -> c16(MARK(z0)) MARK(s(z0)) -> c18(MARK(z0)) A__ZPRIMES -> c8(A__NATS(s(s(0)))) MARK(filter(x0, cons(z0, z1), x2)) -> c10(MARK(x0)) MARK(filter(x0, 0, x2)) -> c10(MARK(x0)) MARK(filter(s(z0), x1, x2)) -> c10(MARK(s(z0))) MARK(filter(cons(z0, z1), x1, x2)) -> c11(A__FILTER(cons(mark(z0), z1), mark(x1), mark(x2)), MARK(x1)) MARK(filter(x0, cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(0, x1, x2)) -> c11(MARK(x1)) MARK(filter(s(z0), x1, x2)) -> c11(MARK(x1)) MARK(filter(x0, x1, filter(z0, z1, z2))) -> c12(A__FILTER(mark(x0), mark(x1), a__filter(mark(z0), mark(z1), mark(z2))), MARK(filter(z0, z1, z2))) MARK(filter(x0, x1, sieve(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, x1, nats(z0))) -> c12(A__FILTER(mark(x0), mark(x1), a__nats(mark(z0))), MARK(nats(z0))) MARK(filter(x0, x1, zprimes)) -> c12(A__FILTER(mark(x0), mark(x1), a__zprimes), MARK(zprimes)) 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c11(A__FILTER(a__nats(a__zprimes), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(cons(z0, z1)), x1, x2)) -> c11(A__FILTER(a__nats(cons(mark(z0), z1)), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(0), x1, x2)) -> c11(A__FILTER(a__nats(0), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(s(z0)), x1, x2)) -> c11(A__FILTER(a__nats(s(mark(z0))), mark(x1), mark(x2)), MARK(x1)) MARK(filter(nats(x0), cons(z0, z1), x2)) -> c11(MARK(cons(z0, z1))) MARK(filter(nats(x0), x1, x2)) -> c11(MARK(x1)) MARK(filter(zprimes, x0, x1)) -> c11(A__FILTER(a__sieve(a__nats(s(s(0)))), mark(x0), mark(x1)), MARK(x0)) MARK(filter(zprimes, cons(z0, z1), x1)) -> c11(MARK(cons(z0, z1))) MARK(filter(zprimes, x0, x1)) -> c11(MARK(x0)) K tuples:none Defined Rule Symbols: a__filter_3, a__sieve_1, a__nats_1, a__zprimes, mark_1 Defined Pair Symbols: A__FILTER_3, A__SIEVE_1, A__NATS_1, MARK_1, A__ZPRIMES Compound Symbols: c1_1, c4_1, c6_1, c14_2, c15_1, c16_1, c18_1, c8_1, c10_1, c11_2, c11_1, c12_2, c12_1, c13_2, c13_1, c8_2, c10_2