KILLED proof of input_Z4stlZXFpJ.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). (0) CpxTRS (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxTRS (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (4) CpxTRS (5) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (6) CdtProblem (7) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CpxRelTRS (9) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (10) TRS for Loop Detection (11) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (12) CpxRelTRS (13) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (14) typed CpxTrs (15) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms] (16) CpxWeightedTrs (17) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (18) CpxTypedWeightedTrs (19) CompletionProof [UPPER BOUND(ID), 0 ms] (20) CpxTypedWeightedCompleteTrs (21) NarrowingProof [BOTH BOUNDS(ID, ID), 94 ms] (22) CpxTypedWeightedCompleteTrs (23) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (24) CpxRNTS (25) SimplificationProof [BOTH BOUNDS(ID, ID), 7 ms] (26) CpxRNTS (27) CompletionProof [UPPER BOUND(ID), 0 ms] (28) CpxTypedWeightedCompleteTrs (29) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (30) CpxRNTS (31) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (32) CdtProblem (33) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (34) CdtProblem (35) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms] (36) CpxRelTRS (37) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (38) CpxTRS (39) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms] (40) CpxWeightedTrs (41) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (42) CpxTypedWeightedTrs (43) CompletionProof [UPPER BOUND(ID), 0 ms] (44) CpxTypedWeightedCompleteTrs (45) NarrowingProof [BOTH BOUNDS(ID, ID), 2192 ms] (46) CpxTypedWeightedCompleteTrs (47) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (48) CpxRNTS (49) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (50) CpxRNTS (51) CompletionProof [UPPER BOUND(ID), 0 ms] (52) CpxTypedWeightedCompleteTrs (53) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (54) CpxRNTS (55) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (56) CdtProblem (57) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (58) CdtProblem (59) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (60) CdtProblem (61) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (62) CdtProblem (63) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (64) CdtProblem (65) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (66) CdtProblem (67) 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), 18 ms] (72) CdtProblem (73) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (74) CdtProblem (75) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (76) CdtProblem (77) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (78) CdtProblem (79) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (80) CdtProblem (81) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 9 ms] (82) CdtProblem (83) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 27 ms] (84) CdtProblem (85) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (86) CdtProblem (87) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 12 ms] (88) CdtProblem (89) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 36 ms] (90) CdtProblem (91) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (92) CdtProblem (93) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (94) CdtProblem (95) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 98 ms] (96) CdtProblem (97) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (98) CdtProblem (99) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (100) CdtProblem (101) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 51 ms] (102) CdtProblem (103) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (104) CdtProblem (105) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (106) CdtProblem (107) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 60 ms] (108) CdtProblem (109) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (110) CdtProblem (111) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 2 ms] (112) CdtProblem (113) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 22 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), 74 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), 164 ms] (126) CdtProblem (127) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (128) CdtProblem (129) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (130) CdtProblem (131) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 173 ms] (132) CdtProblem (133) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (134) CdtProblem (135) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 18 ms] (136) CdtProblem (137) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 120 ms] (138) CdtProblem (139) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 17 ms] (140) CdtProblem (141) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 39 ms] (142) CdtProblem (143) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 136 ms] (144) CdtProblem (145) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (146) CdtProblem (147) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (148) CdtProblem (149) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 200 ms] (150) CdtProblem (151) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (152) CdtProblem (153) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (154) CdtProblem (155) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 335 ms] (156) CdtProblem (157) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (158) CdtProblem (159) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 87 ms] (160) CdtProblem (161) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (162) CdtProblem (163) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 115 ms] (164) CdtProblem (165) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 18 ms] (166) CdtProblem (167) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 133 ms] (168) CdtProblem (169) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (170) CdtProblem (171) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 148 ms] (172) CdtProblem (173) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 30 ms] (174) CdtProblem (175) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 432 ms] (176) CdtProblem (177) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 40 ms] (178) CdtProblem (179) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 14 ms] (180) CdtProblem (181) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 338 ms] (182) CdtProblem (183) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 58 ms] (184) CdtProblem (185) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 37 ms] (186) CdtProblem (187) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 360 ms] (188) CdtProblem (189) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 32 ms] (190) CdtProblem (191) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 48 ms] (192) CdtProblem (193) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 291 ms] (194) CdtProblem (195) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 91 ms] (196) CdtProblem (197) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 66 ms] (198) CdtProblem (199) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 359 ms] (200) CdtProblem (201) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 17 ms] (202) CdtProblem (203) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 81 ms] (204) CdtProblem (205) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 262 ms] (206) CdtProblem (207) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 57 ms] (208) CdtProblem (209) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 126 ms] (210) CdtProblem (211) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 449 ms] (212) CdtProblem (213) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 101 ms] (214) CdtProblem (215) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 136 ms] (216) CdtProblem (217) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 672 ms] (218) CdtProblem (219) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 58 ms] (220) CdtProblem (221) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 291 ms] (222) CdtProblem (223) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 91 ms] (224) CdtProblem (225) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 250 ms] (226) CdtProblem (227) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 84 ms] (228) CdtProblem (229) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 220 ms] (230) CdtProblem (231) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 123 ms] (232) CdtProblem (233) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 285 ms] (234) CdtProblem (235) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 101 ms] (236) CdtProblem (237) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 787 ms] (238) CdtProblem (239) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 189 ms] (240) CdtProblem (241) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 44 ms] (242) CdtProblem (243) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 565 ms] (244) CdtProblem (245) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 127 ms] (246) CdtProblem (247) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 100 ms] (248) CdtProblem (249) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 707 ms] (250) CdtProblem (251) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 245 ms] (252) CdtProblem (253) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 154 ms] (254) CdtProblem (255) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 469 ms] (256) CdtProblem (257) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 146 ms] (258) CdtProblem (259) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 140 ms] (260) 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) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (6) 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 ---------------------------------------- (7) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID)) Converted S to standard rules, and D \ S as well as R to relative rules. ---------------------------------------- (8) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: 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)) The (relative) TRS S consists of the following 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)) Rewrite Strategy: INNERMOST ---------------------------------------- (9) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) Transformed a relative TRS into a decreasing-loop problem. ---------------------------------------- (10) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: A__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)) The (relative) TRS S consists of the following 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)) Rewrite Strategy: INNERMOST ---------------------------------------- (11) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (12) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: A__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)) The (relative) TRS S consists of the following 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)) Rewrite Strategy: INNERMOST ---------------------------------------- (13) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Inferred types. ---------------------------------------- (14) Obligation: Innermost TRS: Rules: 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)) 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)) Types: A__FILTER :: cons:0':s:filter:sieve:nats:zprimes -> cons:0':s:filter:sieve:nats:zprimes -> cons:0':s:filter:sieve:nats:zprimes -> c:c1:c2 cons :: cons:0':s:filter:sieve:nats:zprimes -> cons:0':s:filter:sieve:nats:zprimes -> cons:0':s:filter:sieve:nats:zprimes 0' :: cons:0':s:filter:sieve:nats:zprimes c :: c:c1:c2 s :: cons:0':s:filter:sieve:nats:zprimes -> cons:0':s:filter:sieve:nats:zprimes c1 :: c10:c11:c12:c13:c14:c15:c16:c17:c18 -> c:c1:c2 MARK :: cons:0':s:filter:sieve:nats:zprimes -> c10:c11:c12:c13:c14:c15:c16:c17:c18 c2 :: c:c1:c2 A__SIEVE :: cons:0':s:filter:sieve:nats:zprimes -> c3:c4:c5 c3 :: c3:c4:c5 c4 :: c10:c11:c12:c13:c14:c15:c16:c17:c18 -> c3:c4:c5 c5 :: c3:c4:c5 A__NATS :: cons:0':s:filter:sieve:nats:zprimes -> c6:c7 c6 :: c10:c11:c12:c13:c14:c15:c16:c17:c18 -> c6:c7 c7 :: c6:c7 A__ZPRIMES :: c8:c9 c8 :: c3:c4:c5 -> c6:c7 -> c8:c9 a__nats :: cons:0':s:filter:sieve:nats:zprimes -> cons:0':s:filter:sieve:nats:zprimes c9 :: c8:c9 filter :: cons:0':s:filter:sieve:nats:zprimes -> cons:0':s:filter:sieve:nats:zprimes -> cons:0':s:filter:sieve:nats:zprimes -> cons:0':s:filter:sieve:nats:zprimes c10 :: c:c1:c2 -> c10:c11:c12:c13:c14:c15:c16:c17:c18 -> c10:c11:c12:c13:c14:c15:c16:c17:c18 mark :: cons:0':s:filter:sieve:nats:zprimes -> cons:0':s:filter:sieve:nats:zprimes c11 :: c:c1:c2 -> c10:c11:c12:c13:c14:c15:c16:c17:c18 -> c10:c11:c12:c13:c14:c15:c16:c17:c18 c12 :: c:c1:c2 -> c10:c11:c12:c13:c14:c15:c16:c17:c18 -> c10:c11:c12:c13:c14:c15:c16:c17:c18 sieve :: cons:0':s:filter:sieve:nats:zprimes -> cons:0':s:filter:sieve:nats:zprimes c13 :: c3:c4:c5 -> c10:c11:c12:c13:c14:c15:c16:c17:c18 -> c10:c11:c12:c13:c14:c15:c16:c17:c18 nats :: cons:0':s:filter:sieve:nats:zprimes -> cons:0':s:filter:sieve:nats:zprimes c14 :: c6:c7 -> c10:c11:c12:c13:c14:c15:c16:c17:c18 -> c10:c11:c12:c13:c14:c15:c16:c17:c18 zprimes :: cons:0':s:filter:sieve:nats:zprimes c15 :: c8:c9 -> c10:c11:c12:c13:c14:c15:c16:c17:c18 c16 :: c10:c11:c12:c13:c14:c15:c16:c17:c18 -> c10:c11:c12:c13:c14:c15:c16:c17:c18 c17 :: c10:c11:c12:c13:c14:c15:c16:c17:c18 c18 :: c10:c11:c12:c13:c14:c15:c16:c17:c18 -> c10:c11:c12:c13:c14:c15:c16:c17:c18 a__filter :: cons:0':s:filter:sieve:nats:zprimes -> cons:0':s:filter:sieve:nats:zprimes -> cons:0':s:filter:sieve:nats:zprimes -> cons:0':s:filter:sieve:nats:zprimes a__sieve :: cons:0':s:filter:sieve:nats:zprimes -> cons:0':s:filter:sieve:nats:zprimes a__zprimes :: cons:0':s:filter:sieve:nats:zprimes hole_c:c1:c21_19 :: c:c1:c2 hole_cons:0':s:filter:sieve:nats:zprimes2_19 :: cons:0':s:filter:sieve:nats:zprimes hole_c10:c11:c12:c13:c14:c15:c16:c17:c183_19 :: c10:c11:c12:c13:c14:c15:c16:c17:c18 hole_c3:c4:c54_19 :: c3:c4:c5 hole_c6:c75_19 :: c6:c7 hole_c8:c96_19 :: c8:c9 gen_cons:0':s:filter:sieve:nats:zprimes7_19 :: Nat -> cons:0':s:filter:sieve:nats:zprimes gen_c10:c11:c12:c13:c14:c15:c16:c17:c188_19 :: Nat -> c10:c11:c12:c13:c14:c15:c16:c17:c18 ---------------------------------------- (15) RelTrsToWeightedTrsProof (UPPER BOUND(ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (16) 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 ---------------------------------------- (17) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (18) 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 ---------------------------------------- (19) 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 ---------------------------------------- (20) 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 ---------------------------------------- (21) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (22) 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 ---------------------------------------- (23) 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 ---------------------------------------- (24) 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 ---------------------------------------- (25) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (26) 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 ---------------------------------------- (27) 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 ---------------------------------------- (28) 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 ---------------------------------------- (29) 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 ---------------------------------------- (30) 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. ---------------------------------------- (31) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (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), 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 ---------------------------------------- (33) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 7 trailing nodes: A__SIEVE(z0) -> c5 A__ZPRIMES -> c9 A__FILTER(cons(z0, z1), 0, z2) -> c A__SIEVE(cons(0, z0)) -> c3 MARK(0) -> c17 A__FILTER(z0, z1, z2) -> c2 A__NATS(z0) -> c7 ---------------------------------------- (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)) 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 ---------------------------------------- (35) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID)) Converted S to standard rules, and D \ S as well as R to relative rules. ---------------------------------------- (36) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: A__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)) The (relative) TRS S consists of the following 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)) Rewrite Strategy: INNERMOST ---------------------------------------- (37) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (38) Obligation: The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: A__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)) 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)) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (39) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (40) 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(z0, z1), s(z2), z3) -> c1(MARK(z0)) [1] A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) [1] A__NATS(z0) -> c6(MARK(z0)) [1] A__ZPRIMES -> c8(A__SIEVE(a__nats(s(s(0)))), A__NATS(s(s(0)))) [1] MARK(filter(z0, z1, z2)) -> c10(A__FILTER(mark(z0), mark(z1), mark(z2)), MARK(z0)) [1] MARK(filter(z0, z1, z2)) -> c11(A__FILTER(mark(z0), mark(z1), mark(z2)), MARK(z1)) [1] MARK(filter(z0, z1, z2)) -> c12(A__FILTER(mark(z0), mark(z1), mark(z2)), MARK(z2)) [1] MARK(sieve(z0)) -> c13(A__SIEVE(mark(z0)), MARK(z0)) [1] MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) [1] MARK(zprimes) -> c15(A__ZPRIMES) [1] MARK(cons(z0, z1)) -> c16(MARK(z0)) [1] MARK(s(z0)) -> c18(MARK(z0)) [1] a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) [0] a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) [0] a__filter(z0, z1, z2) -> filter(z0, z1, z2) [0] a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) [0] a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) [0] a__sieve(z0) -> sieve(z0) [0] a__nats(z0) -> cons(mark(z0), nats(s(z0))) [0] a__nats(z0) -> nats(z0) [0] a__zprimes -> a__sieve(a__nats(s(s(0)))) [0] a__zprimes -> zprimes [0] mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) [0] mark(sieve(z0)) -> a__sieve(mark(z0)) [0] mark(nats(z0)) -> a__nats(mark(z0)) [0] mark(zprimes) -> a__zprimes [0] mark(cons(z0, z1)) -> cons(mark(z0), z1) [0] mark(0) -> 0 [0] mark(s(z0)) -> s(mark(z0)) [0] Rewrite Strategy: INNERMOST ---------------------------------------- (41) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (42) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: A__FILTER(cons(z0, z1), s(z2), z3) -> c1(MARK(z0)) [1] A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) [1] A__NATS(z0) -> c6(MARK(z0)) [1] A__ZPRIMES -> c8(A__SIEVE(a__nats(s(s(0)))), A__NATS(s(s(0)))) [1] MARK(filter(z0, z1, z2)) -> c10(A__FILTER(mark(z0), mark(z1), mark(z2)), MARK(z0)) [1] MARK(filter(z0, z1, z2)) -> c11(A__FILTER(mark(z0), mark(z1), mark(z2)), MARK(z1)) [1] MARK(filter(z0, z1, z2)) -> c12(A__FILTER(mark(z0), mark(z1), mark(z2)), MARK(z2)) [1] MARK(sieve(z0)) -> c13(A__SIEVE(mark(z0)), MARK(z0)) [1] MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) [1] MARK(zprimes) -> c15(A__ZPRIMES) [1] MARK(cons(z0, z1)) -> c16(MARK(z0)) [1] MARK(s(z0)) -> c18(MARK(z0)) [1] a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) [0] a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) [0] a__filter(z0, z1, z2) -> filter(z0, z1, z2) [0] a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) [0] a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) [0] a__sieve(z0) -> sieve(z0) [0] a__nats(z0) -> cons(mark(z0), nats(s(z0))) [0] a__nats(z0) -> nats(z0) [0] a__zprimes -> a__sieve(a__nats(s(s(0)))) [0] a__zprimes -> zprimes [0] mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) [0] mark(sieve(z0)) -> a__sieve(mark(z0)) [0] mark(nats(z0)) -> a__nats(mark(z0)) [0] mark(zprimes) -> a__zprimes [0] mark(cons(z0, z1)) -> cons(mark(z0), z1) [0] mark(0) -> 0 [0] mark(s(z0)) -> s(mark(z0)) [0] The TRS has the following type information: A__FILTER :: cons:s:0:filter:sieve:nats:zprimes -> cons:s:0:filter:sieve:nats:zprimes -> cons:s:0:filter:sieve:nats:zprimes -> c1 cons :: cons:s:0:filter:sieve:nats:zprimes -> cons:s:0:filter:sieve:nats:zprimes -> cons:s:0:filter:sieve:nats:zprimes s :: cons:s:0:filter:sieve:nats:zprimes -> cons:s:0:filter:sieve:nats:zprimes c1 :: c10:c11:c12:c13:c14:c15:c16:c18 -> c1 MARK :: cons:s:0:filter:sieve:nats:zprimes -> c10:c11:c12:c13:c14:c15:c16:c18 A__SIEVE :: cons:s:0:filter:sieve:nats:zprimes -> c4 c4 :: c10:c11:c12:c13:c14:c15:c16:c18 -> c4 A__NATS :: cons:s:0:filter:sieve:nats:zprimes -> c6 c6 :: c10:c11:c12:c13:c14:c15:c16:c18 -> c6 A__ZPRIMES :: c8 c8 :: c4 -> c6 -> c8 a__nats :: cons:s:0:filter:sieve:nats:zprimes -> cons:s:0:filter:sieve:nats:zprimes 0 :: cons:s:0:filter:sieve:nats:zprimes filter :: cons:s:0:filter:sieve:nats:zprimes -> cons:s:0:filter:sieve:nats:zprimes -> cons:s:0:filter:sieve:nats:zprimes -> cons:s:0:filter:sieve:nats:zprimes c10 :: c1 -> c10:c11:c12:c13:c14:c15:c16:c18 -> c10:c11:c12:c13:c14:c15:c16:c18 mark :: cons:s:0:filter:sieve:nats:zprimes -> cons:s:0:filter:sieve:nats:zprimes c11 :: c1 -> c10:c11:c12:c13:c14:c15:c16:c18 -> c10:c11:c12:c13:c14:c15:c16:c18 c12 :: c1 -> c10:c11:c12:c13:c14:c15:c16:c18 -> c10:c11:c12:c13:c14:c15:c16:c18 sieve :: cons:s:0:filter:sieve:nats:zprimes -> cons:s:0:filter:sieve:nats:zprimes c13 :: c4 -> c10:c11:c12:c13:c14:c15:c16:c18 -> c10:c11:c12:c13:c14:c15:c16:c18 nats :: cons:s:0:filter:sieve:nats:zprimes -> cons:s:0:filter:sieve:nats:zprimes c14 :: c6 -> c10:c11:c12:c13:c14:c15:c16:c18 -> c10:c11:c12:c13:c14:c15:c16:c18 zprimes :: cons:s:0:filter:sieve:nats:zprimes c15 :: c8 -> c10:c11:c12:c13:c14:c15:c16:c18 c16 :: c10:c11:c12:c13:c14:c15:c16:c18 -> c10:c11:c12:c13:c14:c15:c16:c18 c18 :: c10:c11:c12:c13:c14:c15:c16:c18 -> c10:c11:c12:c13:c14:c15:c16:c18 a__filter :: cons:s:0:filter:sieve:nats:zprimes -> cons:s:0:filter:sieve:nats:zprimes -> cons:s:0:filter:sieve:nats:zprimes -> cons:s:0:filter:sieve:nats:zprimes a__sieve :: cons:s:0:filter:sieve:nats:zprimes -> cons:s:0:filter:sieve:nats:zprimes a__zprimes :: cons:s:0:filter:sieve:nats:zprimes Rewrite Strategy: INNERMOST ---------------------------------------- (43) CompletionProof (UPPER BOUND(ID)) The transformation into a RNTS is sound, since: (a) The obligation is a constructor system where every type has a constant constructor, (b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols: A__FILTER_3 A__SIEVE_1 A__NATS_1 A__ZPRIMES MARK_1 (c) The following functions are completely defined: a__filter_3 a__sieve_1 a__nats_1 a__zprimes mark_1 Due to the following rules being added: a__filter(v0, v1, v2) -> null_a__filter [0] a__sieve(v0) -> null_a__sieve [0] a__nats(v0) -> null_a__nats [0] a__zprimes -> null_a__zprimes [0] mark(v0) -> null_mark [0] And the following fresh constants: null_a__filter, null_a__sieve, null_a__nats, null_a__zprimes, null_mark, const, const1, const2, const3, const4 ---------------------------------------- (44) 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(z0, z1), s(z2), z3) -> c1(MARK(z0)) [1] A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) [1] A__NATS(z0) -> c6(MARK(z0)) [1] A__ZPRIMES -> c8(A__SIEVE(a__nats(s(s(0)))), A__NATS(s(s(0)))) [1] MARK(filter(z0, z1, z2)) -> c10(A__FILTER(mark(z0), mark(z1), mark(z2)), MARK(z0)) [1] MARK(filter(z0, z1, z2)) -> c11(A__FILTER(mark(z0), mark(z1), mark(z2)), MARK(z1)) [1] MARK(filter(z0, z1, z2)) -> c12(A__FILTER(mark(z0), mark(z1), mark(z2)), MARK(z2)) [1] MARK(sieve(z0)) -> c13(A__SIEVE(mark(z0)), MARK(z0)) [1] MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) [1] MARK(zprimes) -> c15(A__ZPRIMES) [1] MARK(cons(z0, z1)) -> c16(MARK(z0)) [1] MARK(s(z0)) -> c18(MARK(z0)) [1] a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) [0] a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) [0] a__filter(z0, z1, z2) -> filter(z0, z1, z2) [0] a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) [0] a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) [0] a__sieve(z0) -> sieve(z0) [0] a__nats(z0) -> cons(mark(z0), nats(s(z0))) [0] a__nats(z0) -> nats(z0) [0] a__zprimes -> a__sieve(a__nats(s(s(0)))) [0] a__zprimes -> zprimes [0] mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) [0] mark(sieve(z0)) -> a__sieve(mark(z0)) [0] mark(nats(z0)) -> a__nats(mark(z0)) [0] mark(zprimes) -> a__zprimes [0] mark(cons(z0, z1)) -> cons(mark(z0), z1) [0] mark(0) -> 0 [0] mark(s(z0)) -> s(mark(z0)) [0] a__filter(v0, v1, v2) -> null_a__filter [0] a__sieve(v0) -> null_a__sieve [0] a__nats(v0) -> null_a__nats [0] a__zprimes -> null_a__zprimes [0] mark(v0) -> null_mark [0] The TRS has the following type information: A__FILTER :: cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark -> cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark -> cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark -> c1 cons :: cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark -> cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark -> cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark s :: cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark -> cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark c1 :: c10:c11:c12:c13:c14:c15:c16:c18 -> c1 MARK :: cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark -> c10:c11:c12:c13:c14:c15:c16:c18 A__SIEVE :: cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark -> c4 c4 :: c10:c11:c12:c13:c14:c15:c16:c18 -> c4 A__NATS :: cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark -> c6 c6 :: c10:c11:c12:c13:c14:c15:c16:c18 -> c6 A__ZPRIMES :: c8 c8 :: c4 -> c6 -> c8 a__nats :: cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark -> cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark 0 :: cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark filter :: cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark -> cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark -> cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark -> cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark c10 :: c1 -> c10:c11:c12:c13:c14:c15:c16:c18 -> c10:c11:c12:c13:c14:c15:c16:c18 mark :: cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark -> cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark c11 :: c1 -> c10:c11:c12:c13:c14:c15:c16:c18 -> c10:c11:c12:c13:c14:c15:c16:c18 c12 :: c1 -> c10:c11:c12:c13:c14:c15:c16:c18 -> c10:c11:c12:c13:c14:c15:c16:c18 sieve :: cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark -> cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark c13 :: c4 -> c10:c11:c12:c13:c14:c15:c16:c18 -> c10:c11:c12:c13:c14:c15:c16:c18 nats :: cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark -> cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark c14 :: c6 -> c10:c11:c12:c13:c14:c15:c16:c18 -> c10:c11:c12:c13:c14:c15:c16:c18 zprimes :: cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark c15 :: c8 -> c10:c11:c12:c13:c14:c15:c16:c18 c16 :: c10:c11:c12:c13:c14:c15:c16:c18 -> c10:c11:c12:c13:c14:c15:c16:c18 c18 :: c10:c11:c12:c13:c14:c15:c16:c18 -> c10:c11:c12:c13:c14:c15:c16:c18 a__filter :: cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark -> cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark -> cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark -> cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark a__sieve :: cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark -> cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark a__zprimes :: cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark null_a__filter :: cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark null_a__sieve :: cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark null_a__nats :: cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark null_a__zprimes :: cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark null_mark :: cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark const :: c1 const1 :: c10:c11:c12:c13:c14:c15:c16:c18 const2 :: c4 const3 :: c6 const4 :: c8 Rewrite Strategy: INNERMOST ---------------------------------------- (45) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (46) 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(z0, z1), s(z2), z3) -> c1(MARK(z0)) [1] A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) [1] A__NATS(z0) -> c6(MARK(z0)) [1] A__ZPRIMES -> c8(A__SIEVE(cons(mark(s(s(0))), nats(s(s(s(0)))))), A__NATS(s(s(0)))) [1] A__ZPRIMES -> c8(A__SIEVE(nats(s(s(0)))), A__NATS(s(s(0)))) [1] A__ZPRIMES -> c8(A__SIEVE(null_a__nats), A__NATS(s(s(0)))) [1] MARK(filter(z0, z1, z2)) -> c10(A__FILTER(mark(z0), mark(z1), mark(z2)), MARK(z0)) [1] MARK(filter(z0, z1, z2)) -> c11(A__FILTER(mark(z0), mark(z1), mark(z2)), MARK(z1)) [1] MARK(filter(z0, z1, z2)) -> c12(A__FILTER(mark(z0), mark(z1), mark(z2)), MARK(z2)) [1] MARK(sieve(filter(z01094, z1437, z2218))) -> c13(A__SIEVE(a__filter(mark(z01094), mark(z1437), mark(z2218))), MARK(filter(z01094, z1437, z2218))) [1] MARK(sieve(sieve(z01095))) -> c13(A__SIEVE(a__sieve(mark(z01095))), MARK(sieve(z01095))) [1] MARK(sieve(nats(z01096))) -> c13(A__SIEVE(a__nats(mark(z01096))), MARK(nats(z01096))) [1] MARK(sieve(zprimes)) -> c13(A__SIEVE(a__zprimes), MARK(zprimes)) [1] MARK(sieve(cons(z01097, z1438))) -> c13(A__SIEVE(cons(mark(z01097), z1438)), MARK(cons(z01097, z1438))) [1] MARK(sieve(0)) -> c13(A__SIEVE(0), MARK(0)) [1] MARK(sieve(s(z01098))) -> c13(A__SIEVE(s(mark(z01098))), MARK(s(z01098))) [1] MARK(sieve(z0)) -> c13(A__SIEVE(null_mark), MARK(z0)) [1] MARK(nats(filter(z01099, z1439, z2219))) -> c14(A__NATS(a__filter(mark(z01099), mark(z1439), mark(z2219))), MARK(filter(z01099, z1439, z2219))) [1] MARK(nats(sieve(z01100))) -> c14(A__NATS(a__sieve(mark(z01100))), MARK(sieve(z01100))) [1] MARK(nats(nats(z01101))) -> c14(A__NATS(a__nats(mark(z01101))), MARK(nats(z01101))) [1] MARK(nats(zprimes)) -> c14(A__NATS(a__zprimes), MARK(zprimes)) [1] MARK(nats(cons(z01102, z1440))) -> c14(A__NATS(cons(mark(z01102), z1440)), MARK(cons(z01102, z1440))) [1] MARK(nats(0)) -> c14(A__NATS(0), MARK(0)) [1] MARK(nats(s(z01103))) -> c14(A__NATS(s(mark(z01103))), MARK(s(z01103))) [1] MARK(nats(z0)) -> c14(A__NATS(null_mark), MARK(z0)) [1] MARK(zprimes) -> c15(A__ZPRIMES) [1] MARK(cons(z0, z1)) -> c16(MARK(z0)) [1] MARK(s(z0)) -> c18(MARK(z0)) [1] a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) [0] a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) [0] a__filter(z0, z1, z2) -> filter(z0, z1, z2) [0] a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) [0] a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) [0] a__sieve(z0) -> sieve(z0) [0] a__nats(z0) -> cons(mark(z0), nats(s(z0))) [0] a__nats(z0) -> nats(z0) [0] a__zprimes -> a__sieve(cons(mark(s(s(0))), nats(s(s(s(0)))))) [0] a__zprimes -> a__sieve(nats(s(s(0)))) [0] a__zprimes -> a__sieve(null_a__nats) [0] a__zprimes -> zprimes [0] mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) [0] mark(sieve(filter(z01469, z1587, z2293))) -> a__sieve(a__filter(mark(z01469), mark(z1587), mark(z2293))) [0] mark(sieve(sieve(z01470))) -> a__sieve(a__sieve(mark(z01470))) [0] mark(sieve(nats(z01471))) -> a__sieve(a__nats(mark(z01471))) [0] mark(sieve(zprimes)) -> a__sieve(a__zprimes) [0] mark(sieve(cons(z01472, z1588))) -> a__sieve(cons(mark(z01472), z1588)) [0] mark(sieve(0)) -> a__sieve(0) [0] mark(sieve(s(z01473))) -> a__sieve(s(mark(z01473))) [0] mark(sieve(z0)) -> a__sieve(null_mark) [0] mark(nats(filter(z01474, z1589, z2294))) -> a__nats(a__filter(mark(z01474), mark(z1589), mark(z2294))) [0] mark(nats(sieve(z01475))) -> a__nats(a__sieve(mark(z01475))) [0] mark(nats(nats(z01476))) -> a__nats(a__nats(mark(z01476))) [0] mark(nats(zprimes)) -> a__nats(a__zprimes) [0] mark(nats(cons(z01477, z1590))) -> a__nats(cons(mark(z01477), z1590)) [0] mark(nats(0)) -> a__nats(0) [0] mark(nats(s(z01478))) -> a__nats(s(mark(z01478))) [0] mark(nats(z0)) -> a__nats(null_mark) [0] mark(zprimes) -> a__zprimes [0] mark(cons(z0, z1)) -> cons(mark(z0), z1) [0] mark(0) -> 0 [0] mark(s(z0)) -> s(mark(z0)) [0] a__filter(v0, v1, v2) -> null_a__filter [0] a__sieve(v0) -> null_a__sieve [0] a__nats(v0) -> null_a__nats [0] a__zprimes -> null_a__zprimes [0] mark(v0) -> null_mark [0] The TRS has the following type information: A__FILTER :: cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark -> cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark -> cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark -> c1 cons :: cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark -> cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark -> cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark s :: cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark -> cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark c1 :: c10:c11:c12:c13:c14:c15:c16:c18 -> c1 MARK :: cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark -> c10:c11:c12:c13:c14:c15:c16:c18 A__SIEVE :: cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark -> c4 c4 :: c10:c11:c12:c13:c14:c15:c16:c18 -> c4 A__NATS :: cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark -> c6 c6 :: c10:c11:c12:c13:c14:c15:c16:c18 -> c6 A__ZPRIMES :: c8 c8 :: c4 -> c6 -> c8 a__nats :: cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark -> cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark 0 :: cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark filter :: cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark -> cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark -> cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark -> cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark c10 :: c1 -> c10:c11:c12:c13:c14:c15:c16:c18 -> c10:c11:c12:c13:c14:c15:c16:c18 mark :: cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark -> cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark c11 :: c1 -> c10:c11:c12:c13:c14:c15:c16:c18 -> c10:c11:c12:c13:c14:c15:c16:c18 c12 :: c1 -> c10:c11:c12:c13:c14:c15:c16:c18 -> c10:c11:c12:c13:c14:c15:c16:c18 sieve :: cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark -> cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark c13 :: c4 -> c10:c11:c12:c13:c14:c15:c16:c18 -> c10:c11:c12:c13:c14:c15:c16:c18 nats :: cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark -> cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark c14 :: c6 -> c10:c11:c12:c13:c14:c15:c16:c18 -> c10:c11:c12:c13:c14:c15:c16:c18 zprimes :: cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark c15 :: c8 -> c10:c11:c12:c13:c14:c15:c16:c18 c16 :: c10:c11:c12:c13:c14:c15:c16:c18 -> c10:c11:c12:c13:c14:c15:c16:c18 c18 :: c10:c11:c12:c13:c14:c15:c16:c18 -> c10:c11:c12:c13:c14:c15:c16:c18 a__filter :: cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark -> cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark -> cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark -> cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark a__sieve :: cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark -> cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark a__zprimes :: cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark null_a__filter :: cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark null_a__sieve :: cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark null_a__nats :: cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark null_a__zprimes :: cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark null_mark :: cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark const :: c1 const1 :: c10:c11:c12:c13:c14:c15:c16:c18 const2 :: c4 const3 :: c6 const4 :: c8 Rewrite Strategy: INNERMOST ---------------------------------------- (47) 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 null_a__filter => 0 null_a__sieve => 0 null_a__nats => 0 null_a__zprimes => 0 null_mark => 0 const => 0 const1 => 0 const2 => 0 const3 => 0 const4 => 0 ---------------------------------------- (48) Obligation: Complexity RNTS consisting of the following rules: A__FILTER(z, z', z'') -{ 1 }-> 1 + MARK(z0) :|: z1 >= 0, z0 >= 0, z'' = z3, z = 1 + z0 + z1, z' = 1 + z2, z2 >= 0, z3 >= 0 A__NATS(z) -{ 1 }-> 1 + MARK(z0) :|: z = z0, z0 >= 0 A__SIEVE(z) -{ 1 }-> 1 + MARK(z0) :|: z1 >= 0, z = 1 + (1 + z0) + z1, z0 >= 0 A__ZPRIMES -{ 1 }-> 1 + A__SIEVE(0) + A__NATS(1 + (1 + 0)) :|: A__ZPRIMES -{ 1 }-> 1 + A__SIEVE(1 + (1 + (1 + 0))) + A__NATS(1 + (1 + 0)) :|: A__ZPRIMES -{ 1 }-> 1 + A__SIEVE(1 + mark(1 + (1 + 0)) + (1 + (1 + (1 + (1 + 0))))) + A__NATS(1 + (1 + 0)) :|: MARK(z) -{ 1 }-> 1 + MARK(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 MARK(z) -{ 1 }-> 1 + MARK(z0) :|: z = 1 + z0, z0 >= 0 MARK(z) -{ 1 }-> 1 + A__ZPRIMES :|: z = 1 MARK(z) -{ 1 }-> 1 + A__SIEVE(a__zprimes) + MARK(1) :|: z = 1 + 1 MARK(z) -{ 1 }-> 1 + A__SIEVE(a__sieve(mark(z01095))) + MARK(1 + z01095) :|: z01095 >= 0, z = 1 + (1 + z01095) MARK(z) -{ 1 }-> 1 + A__SIEVE(a__nats(mark(z01096))) + MARK(1 + z01096) :|: z = 1 + (1 + z01096), z01096 >= 0 MARK(z) -{ 1 }-> 1 + A__SIEVE(a__filter(mark(z01094), mark(z1437), mark(z2218))) + MARK(1 + z01094 + z1437 + z2218) :|: z2218 >= 0, z01094 >= 0, z1437 >= 0, z = 1 + (1 + z01094 + z1437 + z2218) MARK(z) -{ 1 }-> 1 + A__SIEVE(0) + MARK(z0) :|: z = 1 + z0, z0 >= 0 MARK(z) -{ 1 }-> 1 + A__SIEVE(0) + MARK(0) :|: z = 1 + 0 MARK(z) -{ 1 }-> 1 + A__SIEVE(1 + mark(z01098)) + MARK(1 + z01098) :|: z = 1 + (1 + z01098), z01098 >= 0 MARK(z) -{ 1 }-> 1 + A__SIEVE(1 + mark(z01097) + z1438) + MARK(1 + z01097 + z1438) :|: z1438 >= 0, z = 1 + (1 + z01097 + z1438), z01097 >= 0 MARK(z) -{ 1 }-> 1 + A__NATS(a__zprimes) + MARK(1) :|: z = 1 + 1 MARK(z) -{ 1 }-> 1 + A__NATS(a__sieve(mark(z01100))) + MARK(1 + z01100) :|: z01100 >= 0, z = 1 + (1 + z01100) MARK(z) -{ 1 }-> 1 + A__NATS(a__nats(mark(z01101))) + MARK(1 + z01101) :|: z = 1 + (1 + z01101), z01101 >= 0 MARK(z) -{ 1 }-> 1 + A__NATS(a__filter(mark(z01099), mark(z1439), mark(z2219))) + MARK(1 + z01099 + z1439 + z2219) :|: z01099 >= 0, z1439 >= 0, z = 1 + (1 + z01099 + z1439 + z2219), z2219 >= 0 MARK(z) -{ 1 }-> 1 + A__NATS(0) + MARK(z0) :|: z = 1 + z0, z0 >= 0 MARK(z) -{ 1 }-> 1 + A__NATS(0) + MARK(0) :|: z = 1 + 0 MARK(z) -{ 1 }-> 1 + A__NATS(1 + mark(z01103)) + MARK(1 + z01103) :|: z01103 >= 0, z = 1 + (1 + z01103) MARK(z) -{ 1 }-> 1 + A__NATS(1 + mark(z01102) + z1440) + MARK(1 + z01102 + z1440) :|: z = 1 + (1 + z01102 + z1440), z1440 >= 0, z01102 >= 0 MARK(z) -{ 1 }-> 1 + A__FILTER(mark(z0), mark(z1), mark(z2)) + MARK(z0) :|: z1 >= 0, z = 1 + z0 + z1 + z2, z0 >= 0, z2 >= 0 MARK(z) -{ 1 }-> 1 + A__FILTER(mark(z0), mark(z1), mark(z2)) + MARK(z1) :|: z1 >= 0, z = 1 + z0 + z1 + z2, z0 >= 0, z2 >= 0 MARK(z) -{ 1 }-> 1 + A__FILTER(mark(z0), mark(z1), mark(z2)) + MARK(z2) :|: z1 >= 0, z = 1 + z0 + z1 + z2, z0 >= 0, z2 >= 0 a__filter(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 a__filter(z, z', z'') -{ 0 }-> 1 + mark(z0) + (1 + z1 + z2 + z3) :|: z1 >= 0, z0 >= 0, z'' = z3, z = 1 + z0 + z1, z' = 1 + z2, z2 >= 0, z3 >= 0 a__filter(z, z', z'') -{ 0 }-> 1 + 0 + (1 + z1 + z2 + z2) :|: z'' = z2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z' = 0 a__filter(z, z', z'') -{ 0 }-> 1 + z0 + z1 + z2 :|: z'' = z2, z = z0, z1 >= 0, z' = z1, z0 >= 0, z2 >= 0 a__nats(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 a__nats(z) -{ 0 }-> 1 + z0 :|: z = z0, z0 >= 0 a__nats(z) -{ 0 }-> 1 + mark(z0) + (1 + (1 + z0)) :|: z = z0, z0 >= 0 a__sieve(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 a__sieve(z) -{ 0 }-> 1 + z0 :|: z = z0, z0 >= 0 a__sieve(z) -{ 0 }-> 1 + 0 + (1 + z0) :|: z0 >= 0, z = 1 + 0 + z0 a__sieve(z) -{ 0 }-> 1 + (1 + mark(z0)) + (1 + (1 + z1 + z0 + z0)) :|: z1 >= 0, z = 1 + (1 + z0) + z1, z0 >= 0 a__zprimes -{ 0 }-> a__sieve(0) :|: a__zprimes -{ 0 }-> a__sieve(1 + (1 + (1 + 0))) :|: a__zprimes -{ 0 }-> a__sieve(1 + mark(1 + (1 + 0)) + (1 + (1 + (1 + (1 + 0))))) :|: a__zprimes -{ 0 }-> 1 :|: a__zprimes -{ 0 }-> 0 :|: mark(z) -{ 0 }-> a__zprimes :|: z = 1 mark(z) -{ 0 }-> a__sieve(a__zprimes) :|: z = 1 + 1 mark(z) -{ 0 }-> a__sieve(a__sieve(mark(z01470))) :|: z01470 >= 0, z = 1 + (1 + z01470) mark(z) -{ 0 }-> a__sieve(a__nats(mark(z01471))) :|: z01471 >= 0, z = 1 + (1 + z01471) mark(z) -{ 0 }-> a__sieve(a__filter(mark(z01469), mark(z1587), mark(z2293))) :|: z1587 >= 0, z = 1 + (1 + z01469 + z1587 + z2293), z01469 >= 0, z2293 >= 0 mark(z) -{ 0 }-> a__sieve(0) :|: z = 1 + 0 mark(z) -{ 0 }-> a__sieve(0) :|: z = 1 + z0, z0 >= 0 mark(z) -{ 0 }-> a__sieve(1 + mark(z01473)) :|: z01473 >= 0, z = 1 + (1 + z01473) mark(z) -{ 0 }-> a__sieve(1 + mark(z01472) + z1588) :|: z1588 >= 0, z = 1 + (1 + z01472 + z1588), z01472 >= 0 mark(z) -{ 0 }-> a__nats(a__zprimes) :|: z = 1 + 1 mark(z) -{ 0 }-> a__nats(a__sieve(mark(z01475))) :|: z = 1 + (1 + z01475), z01475 >= 0 mark(z) -{ 0 }-> a__nats(a__nats(mark(z01476))) :|: z = 1 + (1 + z01476), z01476 >= 0 mark(z) -{ 0 }-> a__nats(a__filter(mark(z01474), mark(z1589), mark(z2294))) :|: z = 1 + (1 + z01474 + z1589 + z2294), z01474 >= 0, z2294 >= 0, z1589 >= 0 mark(z) -{ 0 }-> a__nats(0) :|: z = 1 + 0 mark(z) -{ 0 }-> a__nats(0) :|: z = 1 + z0, z0 >= 0 mark(z) -{ 0 }-> a__nats(1 + mark(z01478)) :|: z = 1 + (1 + z01478), z01478 >= 0 mark(z) -{ 0 }-> a__nats(1 + mark(z01477) + z1590) :|: z01477 >= 0, z1590 >= 0, z = 1 + (1 + z01477 + z1590) mark(z) -{ 0 }-> a__filter(mark(z0), mark(z1), mark(z2)) :|: z1 >= 0, z = 1 + z0 + z1 + z2, z0 >= 0, z2 >= 0 mark(z) -{ 0 }-> 0 :|: z = 0 mark(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 mark(z) -{ 0 }-> 1 + mark(z0) :|: z = 1 + z0, z0 >= 0 mark(z) -{ 0 }-> 1 + mark(z0) + z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 ---------------------------------------- (49) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (50) Obligation: Complexity RNTS consisting of the following rules: A__FILTER(z, z', z'') -{ 1 }-> 1 + MARK(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' - 1 >= 0, z'' >= 0 A__NATS(z) -{ 1 }-> 1 + MARK(z) :|: z >= 0 A__SIEVE(z) -{ 1 }-> 1 + MARK(z0) :|: z1 >= 0, z = 1 + (1 + z0) + z1, z0 >= 0 A__ZPRIMES -{ 1 }-> 1 + A__SIEVE(0) + A__NATS(1 + (1 + 0)) :|: A__ZPRIMES -{ 1 }-> 1 + A__SIEVE(1 + (1 + (1 + 0))) + A__NATS(1 + (1 + 0)) :|: A__ZPRIMES -{ 1 }-> 1 + A__SIEVE(1 + mark(1 + (1 + 0)) + (1 + (1 + (1 + (1 + 0))))) + A__NATS(1 + (1 + 0)) :|: MARK(z) -{ 1 }-> 1 + MARK(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 MARK(z) -{ 1 }-> 1 + MARK(z - 1) :|: z - 1 >= 0 MARK(z) -{ 1 }-> 1 + A__ZPRIMES :|: z = 1 MARK(z) -{ 1 }-> 1 + A__SIEVE(a__zprimes) + MARK(1) :|: z = 1 + 1 MARK(z) -{ 1 }-> 1 + A__SIEVE(a__sieve(mark(z - 2))) + MARK(1 + (z - 2)) :|: z - 2 >= 0 MARK(z) -{ 1 }-> 1 + A__SIEVE(a__nats(mark(z - 2))) + MARK(1 + (z - 2)) :|: z - 2 >= 0 MARK(z) -{ 1 }-> 1 + A__SIEVE(a__filter(mark(z01094), mark(z1437), mark(z2218))) + MARK(1 + z01094 + z1437 + z2218) :|: z2218 >= 0, z01094 >= 0, z1437 >= 0, z = 1 + (1 + z01094 + z1437 + z2218) MARK(z) -{ 1 }-> 1 + A__SIEVE(0) + MARK(0) :|: z = 1 + 0 MARK(z) -{ 1 }-> 1 + A__SIEVE(0) + MARK(z - 1) :|: z - 1 >= 0 MARK(z) -{ 1 }-> 1 + A__SIEVE(1 + mark(z - 2)) + MARK(1 + (z - 2)) :|: z - 2 >= 0 MARK(z) -{ 1 }-> 1 + A__SIEVE(1 + mark(z01097) + z1438) + MARK(1 + z01097 + z1438) :|: z1438 >= 0, z = 1 + (1 + z01097 + z1438), z01097 >= 0 MARK(z) -{ 1 }-> 1 + A__NATS(a__zprimes) + MARK(1) :|: z = 1 + 1 MARK(z) -{ 1 }-> 1 + A__NATS(a__sieve(mark(z - 2))) + MARK(1 + (z - 2)) :|: z - 2 >= 0 MARK(z) -{ 1 }-> 1 + A__NATS(a__nats(mark(z - 2))) + MARK(1 + (z - 2)) :|: z - 2 >= 0 MARK(z) -{ 1 }-> 1 + A__NATS(a__filter(mark(z01099), mark(z1439), mark(z2219))) + MARK(1 + z01099 + z1439 + z2219) :|: z01099 >= 0, z1439 >= 0, z = 1 + (1 + z01099 + z1439 + z2219), z2219 >= 0 MARK(z) -{ 1 }-> 1 + A__NATS(0) + MARK(0) :|: z = 1 + 0 MARK(z) -{ 1 }-> 1 + A__NATS(0) + MARK(z - 1) :|: z - 1 >= 0 MARK(z) -{ 1 }-> 1 + A__NATS(1 + mark(z - 2)) + MARK(1 + (z - 2)) :|: z - 2 >= 0 MARK(z) -{ 1 }-> 1 + A__NATS(1 + mark(z01102) + z1440) + MARK(1 + z01102 + z1440) :|: z = 1 + (1 + z01102 + z1440), z1440 >= 0, z01102 >= 0 MARK(z) -{ 1 }-> 1 + A__FILTER(mark(z0), mark(z1), mark(z2)) + MARK(z0) :|: z1 >= 0, z = 1 + z0 + z1 + z2, z0 >= 0, z2 >= 0 MARK(z) -{ 1 }-> 1 + A__FILTER(mark(z0), mark(z1), mark(z2)) + MARK(z1) :|: z1 >= 0, z = 1 + z0 + z1 + z2, z0 >= 0, z2 >= 0 MARK(z) -{ 1 }-> 1 + A__FILTER(mark(z0), mark(z1), mark(z2)) + MARK(z2) :|: z1 >= 0, z = 1 + z0 + z1 + z2, z0 >= 0, z2 >= 0 a__filter(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 a__filter(z, z', z'') -{ 0 }-> 1 + mark(z0) + (1 + z1 + (z' - 1) + z'') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' - 1 >= 0, z'' >= 0 a__filter(z, z', z'') -{ 0 }-> 1 + 0 + (1 + z1 + z'' + z'') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z'' >= 0, z' = 0 a__filter(z, z', z'') -{ 0 }-> 1 + z + z' + z'' :|: z' >= 0, z >= 0, z'' >= 0 a__nats(z) -{ 0 }-> 0 :|: z >= 0 a__nats(z) -{ 0 }-> 1 + z :|: z >= 0 a__nats(z) -{ 0 }-> 1 + mark(z) + (1 + (1 + z)) :|: z >= 0 a__sieve(z) -{ 0 }-> 0 :|: z >= 0 a__sieve(z) -{ 0 }-> 1 + z :|: z >= 0 a__sieve(z) -{ 0 }-> 1 + 0 + (1 + (z - 1)) :|: z - 1 >= 0 a__sieve(z) -{ 0 }-> 1 + (1 + mark(z0)) + (1 + (1 + z1 + z0 + z0)) :|: z1 >= 0, z = 1 + (1 + z0) + z1, z0 >= 0 a__zprimes -{ 0 }-> a__sieve(0) :|: a__zprimes -{ 0 }-> a__sieve(1 + (1 + (1 + 0))) :|: a__zprimes -{ 0 }-> a__sieve(1 + mark(1 + (1 + 0)) + (1 + (1 + (1 + (1 + 0))))) :|: a__zprimes -{ 0 }-> 1 :|: a__zprimes -{ 0 }-> 0 :|: mark(z) -{ 0 }-> a__zprimes :|: z = 1 mark(z) -{ 0 }-> a__sieve(a__zprimes) :|: z = 1 + 1 mark(z) -{ 0 }-> a__sieve(a__sieve(mark(z - 2))) :|: z - 2 >= 0 mark(z) -{ 0 }-> a__sieve(a__nats(mark(z - 2))) :|: z - 2 >= 0 mark(z) -{ 0 }-> a__sieve(a__filter(mark(z01469), mark(z1587), mark(z2293))) :|: z1587 >= 0, z = 1 + (1 + z01469 + z1587 + z2293), z01469 >= 0, z2293 >= 0 mark(z) -{ 0 }-> a__sieve(0) :|: z = 1 + 0 mark(z) -{ 0 }-> a__sieve(0) :|: z - 1 >= 0 mark(z) -{ 0 }-> a__sieve(1 + mark(z - 2)) :|: z - 2 >= 0 mark(z) -{ 0 }-> a__sieve(1 + mark(z01472) + z1588) :|: z1588 >= 0, z = 1 + (1 + z01472 + z1588), z01472 >= 0 mark(z) -{ 0 }-> a__nats(a__zprimes) :|: z = 1 + 1 mark(z) -{ 0 }-> a__nats(a__sieve(mark(z - 2))) :|: z - 2 >= 0 mark(z) -{ 0 }-> a__nats(a__nats(mark(z - 2))) :|: z - 2 >= 0 mark(z) -{ 0 }-> a__nats(a__filter(mark(z01474), mark(z1589), mark(z2294))) :|: z = 1 + (1 + z01474 + z1589 + z2294), z01474 >= 0, z2294 >= 0, z1589 >= 0 mark(z) -{ 0 }-> a__nats(0) :|: z = 1 + 0 mark(z) -{ 0 }-> a__nats(0) :|: z - 1 >= 0 mark(z) -{ 0 }-> a__nats(1 + mark(z - 2)) :|: z - 2 >= 0 mark(z) -{ 0 }-> a__nats(1 + mark(z01477) + z1590) :|: z01477 >= 0, z1590 >= 0, z = 1 + (1 + z01477 + z1590) mark(z) -{ 0 }-> a__filter(mark(z0), mark(z1), mark(z2)) :|: z1 >= 0, z = 1 + z0 + z1 + z2, z0 >= 0, z2 >= 0 mark(z) -{ 0 }-> 0 :|: z = 0 mark(z) -{ 0 }-> 0 :|: z >= 0 mark(z) -{ 0 }-> 1 + mark(z - 1) :|: z - 1 >= 0 mark(z) -{ 0 }-> 1 + mark(z0) + z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 ---------------------------------------- (51) CompletionProof (UPPER BOUND(ID)) The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added: a__filter(v0, v1, v2) -> null_a__filter [0] a__sieve(v0) -> null_a__sieve [0] a__nats(v0) -> null_a__nats [0] a__zprimes -> null_a__zprimes [0] mark(v0) -> null_mark [0] A__FILTER(v0, v1, v2) -> null_A__FILTER [0] A__SIEVE(v0) -> null_A__SIEVE [0] MARK(v0) -> null_MARK [0] And the following fresh constants: null_a__filter, null_a__sieve, null_a__nats, null_a__zprimes, null_mark, null_A__FILTER, null_A__SIEVE, null_MARK, const, const1 ---------------------------------------- (52) 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(z0, z1), s(z2), z3) -> c1(MARK(z0)) [1] A__SIEVE(cons(s(z0), z1)) -> c4(MARK(z0)) [1] A__NATS(z0) -> c6(MARK(z0)) [1] A__ZPRIMES -> c8(A__SIEVE(a__nats(s(s(0)))), A__NATS(s(s(0)))) [1] MARK(filter(z0, z1, z2)) -> c10(A__FILTER(mark(z0), mark(z1), mark(z2)), MARK(z0)) [1] MARK(filter(z0, z1, z2)) -> c11(A__FILTER(mark(z0), mark(z1), mark(z2)), MARK(z1)) [1] MARK(filter(z0, z1, z2)) -> c12(A__FILTER(mark(z0), mark(z1), mark(z2)), MARK(z2)) [1] MARK(sieve(z0)) -> c13(A__SIEVE(mark(z0)), MARK(z0)) [1] MARK(nats(z0)) -> c14(A__NATS(mark(z0)), MARK(z0)) [1] MARK(zprimes) -> c15(A__ZPRIMES) [1] MARK(cons(z0, z1)) -> c16(MARK(z0)) [1] MARK(s(z0)) -> c18(MARK(z0)) [1] a__filter(cons(z0, z1), 0, z2) -> cons(0, filter(z1, z2, z2)) [0] a__filter(cons(z0, z1), s(z2), z3) -> cons(mark(z0), filter(z1, z2, z3)) [0] a__filter(z0, z1, z2) -> filter(z0, z1, z2) [0] a__sieve(cons(0, z0)) -> cons(0, sieve(z0)) [0] a__sieve(cons(s(z0), z1)) -> cons(s(mark(z0)), sieve(filter(z1, z0, z0))) [0] a__sieve(z0) -> sieve(z0) [0] a__nats(z0) -> cons(mark(z0), nats(s(z0))) [0] a__nats(z0) -> nats(z0) [0] a__zprimes -> a__sieve(a__nats(s(s(0)))) [0] a__zprimes -> zprimes [0] mark(filter(z0, z1, z2)) -> a__filter(mark(z0), mark(z1), mark(z2)) [0] mark(sieve(z0)) -> a__sieve(mark(z0)) [0] mark(nats(z0)) -> a__nats(mark(z0)) [0] mark(zprimes) -> a__zprimes [0] mark(cons(z0, z1)) -> cons(mark(z0), z1) [0] mark(0) -> 0 [0] mark(s(z0)) -> s(mark(z0)) [0] a__filter(v0, v1, v2) -> null_a__filter [0] a__sieve(v0) -> null_a__sieve [0] a__nats(v0) -> null_a__nats [0] a__zprimes -> null_a__zprimes [0] mark(v0) -> null_mark [0] A__FILTER(v0, v1, v2) -> null_A__FILTER [0] A__SIEVE(v0) -> null_A__SIEVE [0] MARK(v0) -> null_MARK [0] The TRS has the following type information: A__FILTER :: cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark -> cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark -> cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark -> c1:null_A__FILTER cons :: cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark -> cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark -> cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark s :: cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark -> cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark c1 :: c10:c11:c12:c13:c14:c15:c16:c18:null_MARK -> c1:null_A__FILTER MARK :: cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark -> c10:c11:c12:c13:c14:c15:c16:c18:null_MARK A__SIEVE :: cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark -> c4:null_A__SIEVE c4 :: c10:c11:c12:c13:c14:c15:c16:c18:null_MARK -> c4:null_A__SIEVE A__NATS :: cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark -> c6 c6 :: c10:c11:c12:c13:c14:c15:c16:c18:null_MARK -> c6 A__ZPRIMES :: c8 c8 :: c4:null_A__SIEVE -> c6 -> c8 a__nats :: cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark -> cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark 0 :: cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark filter :: cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark -> cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark -> cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark -> cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark c10 :: c1:null_A__FILTER -> c10:c11:c12:c13:c14:c15:c16:c18:null_MARK -> c10:c11:c12:c13:c14:c15:c16:c18:null_MARK mark :: cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark -> cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark c11 :: c1:null_A__FILTER -> c10:c11:c12:c13:c14:c15:c16:c18:null_MARK -> c10:c11:c12:c13:c14:c15:c16:c18:null_MARK c12 :: c1:null_A__FILTER -> c10:c11:c12:c13:c14:c15:c16:c18:null_MARK -> c10:c11:c12:c13:c14:c15:c16:c18:null_MARK sieve :: cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark -> cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark c13 :: c4:null_A__SIEVE -> c10:c11:c12:c13:c14:c15:c16:c18:null_MARK -> c10:c11:c12:c13:c14:c15:c16:c18:null_MARK nats :: cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark -> cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark c14 :: c6 -> c10:c11:c12:c13:c14:c15:c16:c18:null_MARK -> c10:c11:c12:c13:c14:c15:c16:c18:null_MARK zprimes :: cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark c15 :: c8 -> c10:c11:c12:c13:c14:c15:c16:c18:null_MARK c16 :: c10:c11:c12:c13:c14:c15:c16:c18:null_MARK -> c10:c11:c12:c13:c14:c15:c16:c18:null_MARK c18 :: c10:c11:c12:c13:c14:c15:c16:c18:null_MARK -> c10:c11:c12:c13:c14:c15:c16:c18:null_MARK a__filter :: cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark -> cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark -> cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark -> cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark a__sieve :: cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark -> cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark a__zprimes :: cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark null_a__filter :: cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark null_a__sieve :: cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark null_a__nats :: cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark null_a__zprimes :: cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark null_mark :: cons:s:0:filter:sieve:nats:zprimes:null_a__filter:null_a__sieve:null_a__nats:null_a__zprimes:null_mark null_A__FILTER :: c1:null_A__FILTER null_A__SIEVE :: c4:null_A__SIEVE null_MARK :: c10:c11:c12:c13:c14:c15:c16:c18:null_MARK const :: c6 const1 :: c8 Rewrite Strategy: INNERMOST ---------------------------------------- (53) 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 null_a__filter => 0 null_a__sieve => 0 null_a__nats => 0 null_a__zprimes => 0 null_mark => 0 null_A__FILTER => 0 null_A__SIEVE => 0 null_MARK => 0 const => 0 const1 => 0 ---------------------------------------- (54) Obligation: Complexity RNTS consisting of the following rules: A__FILTER(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 A__FILTER(z, z', z'') -{ 1 }-> 1 + MARK(z0) :|: z1 >= 0, z0 >= 0, z'' = z3, z = 1 + z0 + z1, z' = 1 + z2, z2 >= 0, z3 >= 0 A__NATS(z) -{ 1 }-> 1 + MARK(z0) :|: z = z0, z0 >= 0 A__SIEVE(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 A__SIEVE(z) -{ 1 }-> 1 + MARK(z0) :|: z1 >= 0, z = 1 + (1 + z0) + z1, z0 >= 0 A__ZPRIMES -{ 1 }-> 1 + A__SIEVE(a__nats(1 + (1 + 0))) + A__NATS(1 + (1 + 0)) :|: MARK(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 MARK(z) -{ 1 }-> 1 + MARK(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 MARK(z) -{ 1 }-> 1 + MARK(z0) :|: z = 1 + z0, z0 >= 0 MARK(z) -{ 1 }-> 1 + A__ZPRIMES :|: z = 1 MARK(z) -{ 1 }-> 1 + A__SIEVE(mark(z0)) + MARK(z0) :|: z = 1 + z0, z0 >= 0 MARK(z) -{ 1 }-> 1 + A__NATS(mark(z0)) + MARK(z0) :|: z = 1 + z0, z0 >= 0 MARK(z) -{ 1 }-> 1 + A__FILTER(mark(z0), mark(z1), mark(z2)) + MARK(z0) :|: z1 >= 0, z = 1 + z0 + z1 + z2, z0 >= 0, z2 >= 0 MARK(z) -{ 1 }-> 1 + A__FILTER(mark(z0), mark(z1), mark(z2)) + MARK(z1) :|: z1 >= 0, z = 1 + z0 + z1 + z2, z0 >= 0, z2 >= 0 MARK(z) -{ 1 }-> 1 + A__FILTER(mark(z0), mark(z1), mark(z2)) + MARK(z2) :|: z1 >= 0, z = 1 + z0 + z1 + z2, z0 >= 0, z2 >= 0 a__filter(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 a__filter(z, z', z'') -{ 0 }-> 1 + mark(z0) + (1 + z1 + z2 + z3) :|: z1 >= 0, z0 >= 0, z'' = z3, z = 1 + z0 + z1, z' = 1 + z2, z2 >= 0, z3 >= 0 a__filter(z, z', z'') -{ 0 }-> 1 + 0 + (1 + z1 + z2 + z2) :|: z'' = z2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z' = 0 a__filter(z, z', z'') -{ 0 }-> 1 + z0 + z1 + z2 :|: z'' = z2, z = z0, z1 >= 0, z' = z1, z0 >= 0, z2 >= 0 a__nats(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 a__nats(z) -{ 0 }-> 1 + z0 :|: z = z0, z0 >= 0 a__nats(z) -{ 0 }-> 1 + mark(z0) + (1 + (1 + z0)) :|: z = z0, z0 >= 0 a__sieve(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 a__sieve(z) -{ 0 }-> 1 + z0 :|: z = z0, z0 >= 0 a__sieve(z) -{ 0 }-> 1 + 0 + (1 + z0) :|: z0 >= 0, z = 1 + 0 + z0 a__sieve(z) -{ 0 }-> 1 + (1 + mark(z0)) + (1 + (1 + z1 + z0 + z0)) :|: z1 >= 0, z = 1 + (1 + z0) + z1, z0 >= 0 a__zprimes -{ 0 }-> a__sieve(a__nats(1 + (1 + 0))) :|: a__zprimes -{ 0 }-> 1 :|: a__zprimes -{ 0 }-> 0 :|: mark(z) -{ 0 }-> a__zprimes :|: z = 1 mark(z) -{ 0 }-> a__sieve(mark(z0)) :|: z = 1 + z0, z0 >= 0 mark(z) -{ 0 }-> a__nats(mark(z0)) :|: z = 1 + z0, z0 >= 0 mark(z) -{ 0 }-> a__filter(mark(z0), mark(z1), mark(z2)) :|: z1 >= 0, z = 1 + z0 + z1 + z2, z0 >= 0, z2 >= 0 mark(z) -{ 0 }-> 0 :|: z = 0 mark(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 mark(z) -{ 0 }-> 1 + mark(z0) :|: z = 1 + z0, z0 >= 0 mark(z) -{ 0 }-> 1 + mark(z0) + z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (55) 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)))) ---------------------------------------- (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(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 ---------------------------------------- (57) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 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(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 ---------------------------------------- (59) 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))) ---------------------------------------- (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(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 ---------------------------------------- (61) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(filter(0, x1, x2)) -> c10(A__FILTER(0, mark(x1), 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(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 ---------------------------------------- (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(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 ---------------------------------------- (65) 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)) ---------------------------------------- (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(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 ---------------------------------------- (67) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(filter(x0, 0, x2)) -> c11(A__FILTER(mark(x0), 0, 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(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 ---------------------------------------- (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(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 ---------------------------------------- (71) 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)) ---------------------------------------- (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(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 ---------------------------------------- (73) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 5 trailing tuple parts ---------------------------------------- (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(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 ---------------------------------------- (75) 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))) ---------------------------------------- (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__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 ---------------------------------------- (77) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(sieve(0)) -> c13(A__SIEVE(0), MARK(0)) ---------------------------------------- (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__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 ---------------------------------------- (79) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (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__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 ---------------------------------------- (81) 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)))) ---------------------------------------- (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, 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 ---------------------------------------- (83) 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))) ---------------------------------------- (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, 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 ---------------------------------------- (85) 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)) ---------------------------------------- (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, 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 ---------------------------------------- (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, 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 ---------------------------------------- (89) 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))) ---------------------------------------- (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, 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)) -> 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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), 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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 ---------------------------------------- (91) 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)) ---------------------------------------- (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, 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, 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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 ---------------------------------------- (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, 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 ---------------------------------------- (95) 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))) ---------------------------------------- (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, 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 ---------------------------------------- (97) 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)) ---------------------------------------- (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, 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(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(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 ---------------------------------------- (99) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 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, 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 ---------------------------------------- (101) 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))) ---------------------------------------- (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, 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), 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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), 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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 ---------------------------------------- (103) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(filter(0, x1, zprimes)) -> c10(A__FILTER(0, mark(x1), a__zprimes), 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, 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 ---------------------------------------- (105) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 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, 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, 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), 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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))) 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, 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))) ---------------------------------------- (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, 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)) 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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 ---------------------------------------- (109) 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)) ---------------------------------------- (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, 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)), 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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 ---------------------------------------- (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, 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 ---------------------------------------- (113) 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))) ---------------------------------------- (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, 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), 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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(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))) 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)) 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))) -> 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(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 ---------------------------------------- (115) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(filter(0, x1, 0)) -> c10(A__FILTER(0, mark(x1), 0), 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, 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), 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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, 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 ---------------------------------------- (117) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 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, 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), 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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))) 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, 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))) ---------------------------------------- (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(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))) 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 ---------------------------------------- (121) 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)) ---------------------------------------- (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(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, 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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), 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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, 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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 ---------------------------------------- (123) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 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(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 ---------------------------------------- (125) 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))) ---------------------------------------- (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(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 ---------------------------------------- (127) 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)) ---------------------------------------- (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(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 ---------------------------------------- (129) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (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(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 ---------------------------------------- (131) 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))) ---------------------------------------- (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(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 ---------------------------------------- (133) 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)) ---------------------------------------- (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(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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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(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 ---------------------------------------- (135) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 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(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 ---------------------------------------- (137) 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))) ---------------------------------------- (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(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 ---------------------------------------- (139) 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)) ---------------------------------------- (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(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 ---------------------------------------- (141) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (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, 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 ---------------------------------------- (143) 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))) ---------------------------------------- (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, 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 ---------------------------------------- (145) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(filter(0, zprimes, x1)) -> c10(A__FILTER(0, a__zprimes, mark(x1)), MARK(0)) ---------------------------------------- (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, 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 ---------------------------------------- (147) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (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, 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 ---------------------------------------- (149) 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))) ---------------------------------------- (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(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 ---------------------------------------- (151) 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)) ---------------------------------------- (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(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 ---------------------------------------- (153) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (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(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 ---------------------------------------- (155) 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))) ---------------------------------------- (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(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 ---------------------------------------- (157) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (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(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 ---------------------------------------- (159) 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)))) ---------------------------------------- (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(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 ---------------------------------------- (161) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 4 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(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 ---------------------------------------- (163) 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))) ---------------------------------------- (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(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 ---------------------------------------- (165) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (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(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 ---------------------------------------- (167) 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)) ---------------------------------------- (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(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 ---------------------------------------- (169) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (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(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 ---------------------------------------- (171) 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))) ---------------------------------------- (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, 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 ---------------------------------------- (173) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 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, 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 ---------------------------------------- (175) 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)) ---------------------------------------- (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, 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 ---------------------------------------- (177) 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)) ---------------------------------------- (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, 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 ---------------------------------------- (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, 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 ---------------------------------------- (181) 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)) ---------------------------------------- (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, 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 ---------------------------------------- (183) 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)) ---------------------------------------- (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, 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 ---------------------------------------- (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, 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 ---------------------------------------- (187) 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)) ---------------------------------------- (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, 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 ---------------------------------------- (189) 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)) ---------------------------------------- (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, 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 ---------------------------------------- (191) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (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, 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 ---------------------------------------- (193) 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)) ---------------------------------------- (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, 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 ---------------------------------------- (195) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(filter(x0, 0, zprimes)) -> c11(A__FILTER(mark(x0), 0, a__zprimes), MARK(0)) ---------------------------------------- (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, 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 ---------------------------------------- (197) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 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, 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 ---------------------------------------- (199) 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)) ---------------------------------------- (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, 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 ---------------------------------------- (201) 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)) ---------------------------------------- (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, 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 ---------------------------------------- (203) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (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(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 ---------------------------------------- (205) 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)) ---------------------------------------- (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(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 ---------------------------------------- (207) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(filter(x0, 0, 0)) -> c11(A__FILTER(mark(x0), 0, 0), MARK(0)) ---------------------------------------- (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(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 ---------------------------------------- (209) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (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(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 ---------------------------------------- (211) 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)) ---------------------------------------- (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(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 ---------------------------------------- (213) 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)) ---------------------------------------- (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(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 ---------------------------------------- (215) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (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(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 ---------------------------------------- (217) 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))) ---------------------------------------- (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(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 ---------------------------------------- (219) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (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(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 ---------------------------------------- (221) 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))) ---------------------------------------- (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(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 ---------------------------------------- (223) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 4 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(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 ---------------------------------------- (225) 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))) ---------------------------------------- (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(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 ---------------------------------------- (227) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing tuple parts ---------------------------------------- (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(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 ---------------------------------------- (229) 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)) ---------------------------------------- (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(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 ---------------------------------------- (231) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (232) 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 ---------------------------------------- (233) 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))) ---------------------------------------- (234) 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 ---------------------------------------- (235) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (236) 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 ---------------------------------------- (237) 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)) ---------------------------------------- (238) 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 ---------------------------------------- (239) 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)) ---------------------------------------- (240) 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 ---------------------------------------- (241) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (242) 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 ---------------------------------------- (243) 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)) ---------------------------------------- (244) 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 ---------------------------------------- (245) 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)) ---------------------------------------- (246) 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 ---------------------------------------- (247) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (248) 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 ---------------------------------------- (249) 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)) ---------------------------------------- (250) 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 ---------------------------------------- (251) 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)) ---------------------------------------- (252) 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 ---------------------------------------- (253) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (254) 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 ---------------------------------------- (255) 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)) ---------------------------------------- (256) 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 ---------------------------------------- (257) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(filter(zprimes, 0, x1)) -> c11(A__FILTER(a__zprimes, 0, mark(x1)), MARK(0)) ---------------------------------------- (258) 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 ---------------------------------------- (259) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (260) 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