KILLED proof of input_YxNmss0l3a.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).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), 67 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), 127 ms] (22) CpxTypedWeightedCompleteTrs (23) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 18 ms] (24) CpxRNTS (25) SimplificationProof [BOTH BOUNDS(ID, ID), 11 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) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (36) CdtProblem (37) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms] (38) CpxRelTRS (39) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (40) CpxTRS (41) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms] (42) CpxWeightedTrs (43) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (44) CpxTypedWeightedTrs (45) CompletionProof [UPPER BOUND(ID), 0 ms] (46) CpxTypedWeightedCompleteTrs (47) NarrowingProof [BOTH BOUNDS(ID, ID), 305 ms] (48) CpxTypedWeightedCompleteTrs (49) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (50) CpxRNTS (51) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (52) CpxRNTS (53) CompletionProof [UPPER BOUND(ID), 0 ms] (54) CpxTypedWeightedCompleteTrs (55) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (56) CpxRNTS (57) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (58) CdtProblem (59) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (60) CdtProblem (61) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (62) CdtProblem (63) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (64) CdtProblem (65) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (66) CdtProblem (67) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (68) CdtProblem (69) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (70) CdtProblem (71) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (72) CdtProblem (73) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (74) CdtProblem (75) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (76) CdtProblem (77) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (78) CdtProblem (79) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 5 ms] (80) CdtProblem (81) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (82) CdtProblem (83) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (84) CdtProblem (85) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (86) CdtProblem (87) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (88) CdtProblem (89) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (90) CdtProblem (91) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 20 ms] (92) CdtProblem (93) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (94) CdtProblem (95) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (96) CdtProblem (97) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (98) CdtProblem (99) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (100) CdtProblem (101) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (102) CdtProblem (103) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (104) CdtProblem (105) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 5 ms] (106) CdtProblem (107) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (108) CdtProblem (109) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (110) CdtProblem (111) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (112) CdtProblem (113) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (114) CdtProblem (115) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (116) CdtProblem (117) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (118) CdtProblem (119) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 15 ms] (120) CdtProblem (121) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (122) CdtProblem (123) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 37 ms] (124) CdtProblem (125) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (126) CdtProblem (127) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 14 ms] (128) CdtProblem (129) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (130) CdtProblem (131) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (132) CdtProblem (133) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (134) CdtProblem (135) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (136) CdtProblem (137) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (138) CdtProblem (139) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (140) CdtProblem (141) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (142) CdtProblem (143) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (144) CdtProblem (145) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (146) CdtProblem (147) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 32 ms] (148) CdtProblem (149) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (150) CdtProblem (151) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 64 ms] (152) CdtProblem (153) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (154) CdtProblem (155) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (156) CdtProblem (157) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 6 ms] (158) CdtProblem (159) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (160) CdtProblem (161) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (162) CdtProblem (163) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 60 ms] (164) CdtProblem (165) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (166) CdtProblem (167) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 81 ms] (168) CdtProblem (169) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (170) CdtProblem (171) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 25 ms] (172) CdtProblem (173) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (174) CdtProblem (175) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 6 ms] (176) CdtProblem (177) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (178) CdtProblem (179) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 30 ms] (180) CdtProblem (181) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (182) CdtProblem (183) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 46 ms] (184) CdtProblem (185) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (186) CdtProblem (187) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 74 ms] (188) CdtProblem (189) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (190) CdtProblem (191) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 90 ms] (192) CdtProblem (193) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (194) CdtProblem (195) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 79 ms] (196) CdtProblem (197) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (198) CdtProblem (199) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 33 ms] (200) CdtProblem (201) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (202) CdtProblem (203) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 47 ms] (204) CdtProblem (205) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (206) CdtProblem (207) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 60 ms] (208) CdtProblem (209) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (210) CdtProblem (211) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 138 ms] (212) CdtProblem (213) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (214) CdtProblem (215) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (216) CdtProblem (217) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 263 ms] (218) CdtProblem (219) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 26 ms] (220) CdtProblem (221) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 4 ms] (222) CdtProblem (223) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 299 ms] (224) CdtProblem (225) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (226) CdtProblem (227) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 23 ms] (228) CdtProblem (229) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 268 ms] (230) CdtProblem (231) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (232) CdtProblem (233) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 27 ms] (234) CdtProblem (235) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 361 ms] (236) CdtProblem (237) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (238) CdtProblem (239) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 13 ms] (240) CdtProblem (241) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 324 ms] (242) CdtProblem (243) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 12 ms] (244) CdtProblem (245) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 26 ms] (246) CdtProblem (247) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 519 ms] (248) CdtProblem (249) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 38 ms] (250) CdtProblem (251) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 37 ms] (252) 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__primes -> a__sieve(a__from(s(s(0)))) a__from(X) -> cons(mark(X), from(s(X))) a__head(cons(X, Y)) -> mark(X) a__tail(cons(X, Y)) -> mark(Y) a__if(true, X, Y) -> mark(X) a__if(false, X, Y) -> mark(Y) a__filter(s(s(X)), cons(Y, Z)) -> a__if(divides(s(s(mark(X))), mark(Y)), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y)))) a__sieve(cons(X, Y)) -> cons(mark(X), filter(X, sieve(Y))) mark(primes) -> a__primes mark(sieve(X)) -> a__sieve(mark(X)) mark(from(X)) -> a__from(mark(X)) mark(head(X)) -> a__head(mark(X)) mark(tail(X)) -> a__tail(mark(X)) mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3) mark(filter(X1, X2)) -> a__filter(mark(X1), mark(X2)) mark(s(X)) -> s(mark(X)) mark(0) -> 0 mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(true) -> true mark(false) -> false mark(divides(X1, X2)) -> divides(mark(X1), mark(X2)) a__primes -> primes a__sieve(X) -> sieve(X) a__from(X) -> from(X) a__head(X) -> head(X) a__tail(X) -> tail(X) a__if(X1, X2, X3) -> if(X1, X2, X3) a__filter(X1, X2) -> filter(X1, X2) 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__primes -> a__sieve(a__from(s(s(0')))) a__from(X) -> cons(mark(X), from(s(X))) a__head(cons(X, Y)) -> mark(X) a__tail(cons(X, Y)) -> mark(Y) a__if(true, X, Y) -> mark(X) a__if(false, X, Y) -> mark(Y) a__filter(s(s(X)), cons(Y, Z)) -> a__if(divides(s(s(mark(X))), mark(Y)), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y)))) a__sieve(cons(X, Y)) -> cons(mark(X), filter(X, sieve(Y))) mark(primes) -> a__primes mark(sieve(X)) -> a__sieve(mark(X)) mark(from(X)) -> a__from(mark(X)) mark(head(X)) -> a__head(mark(X)) mark(tail(X)) -> a__tail(mark(X)) mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3) mark(filter(X1, X2)) -> a__filter(mark(X1), mark(X2)) mark(s(X)) -> s(mark(X)) mark(0') -> 0' mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(true) -> true mark(false) -> false mark(divides(X1, X2)) -> divides(mark(X1), mark(X2)) a__primes -> primes a__sieve(X) -> sieve(X) a__from(X) -> from(X) a__head(X) -> head(X) a__tail(X) -> tail(X) a__if(X1, X2, X3) -> if(X1, X2, X3) a__filter(X1, X2) -> filter(X1, X2) 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__primes -> a__sieve(a__from(s(s(0)))) a__from(X) -> cons(mark(X), from(s(X))) a__head(cons(X, Y)) -> mark(X) a__tail(cons(X, Y)) -> mark(Y) a__if(true, X, Y) -> mark(X) a__if(false, X, Y) -> mark(Y) a__filter(s(s(X)), cons(Y, Z)) -> a__if(divides(s(s(mark(X))), mark(Y)), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y)))) a__sieve(cons(X, Y)) -> cons(mark(X), filter(X, sieve(Y))) mark(primes) -> a__primes mark(sieve(X)) -> a__sieve(mark(X)) mark(from(X)) -> a__from(mark(X)) mark(head(X)) -> a__head(mark(X)) mark(tail(X)) -> a__tail(mark(X)) mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3) mark(filter(X1, X2)) -> a__filter(mark(X1), mark(X2)) mark(s(X)) -> s(mark(X)) mark(0) -> 0 mark(cons(X1, X2)) -> cons(mark(X1), X2) mark(true) -> true mark(false) -> false mark(divides(X1, X2)) -> divides(mark(X1), mark(X2)) a__primes -> primes a__sieve(X) -> sieve(X) a__from(X) -> from(X) a__head(X) -> head(X) a__tail(X) -> tail(X) a__if(X1, X2, X3) -> if(X1, X2, X3) a__filter(X1, X2) -> filter(X1, X2) 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__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__PRIMES -> c(A__SIEVE(a__from(s(s(0)))), A__FROM(s(s(0)))) A__PRIMES -> c1 A__FROM(z0) -> c2(MARK(z0)) A__FROM(z0) -> c3 A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__HEAD(z0) -> c5 A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__TAIL(z0) -> c7 A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__IF(z0, z1, z2) -> c10 A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(A__IF(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))), MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(A__IF(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))), MARK(z1)) A__FILTER(z0, z1) -> c13 A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) A__SIEVE(z0) -> c15 MARK(primes) -> c16(A__PRIMES) MARK(sieve(z0)) -> c17(A__SIEVE(mark(z0)), MARK(z0)) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(head(z0)) -> c19(A__HEAD(mark(z0)), MARK(z0)) MARK(tail(z0)) -> c20(A__TAIL(mark(z0)), MARK(z0)) MARK(if(z0, z1, z2)) -> c21(A__IF(mark(z0), z1, z2), MARK(z0)) MARK(filter(z0, z1)) -> c22(A__FILTER(mark(z0), mark(z1)), MARK(z0)) MARK(filter(z0, z1)) -> c23(A__FILTER(mark(z0), mark(z1)), MARK(z1)) MARK(s(z0)) -> c24(MARK(z0)) MARK(0) -> c25 MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(true) -> c27 MARK(false) -> c28 MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) S tuples: A__PRIMES -> c(A__SIEVE(a__from(s(s(0)))), A__FROM(s(s(0)))) A__PRIMES -> c1 A__FROM(z0) -> c2(MARK(z0)) A__FROM(z0) -> c3 A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__HEAD(z0) -> c5 A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__TAIL(z0) -> c7 A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__IF(z0, z1, z2) -> c10 A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(A__IF(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))), MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(A__IF(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))), MARK(z1)) A__FILTER(z0, z1) -> c13 A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) A__SIEVE(z0) -> c15 MARK(primes) -> c16(A__PRIMES) MARK(sieve(z0)) -> c17(A__SIEVE(mark(z0)), MARK(z0)) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(head(z0)) -> c19(A__HEAD(mark(z0)), MARK(z0)) MARK(tail(z0)) -> c20(A__TAIL(mark(z0)), MARK(z0)) MARK(if(z0, z1, z2)) -> c21(A__IF(mark(z0), z1, z2), MARK(z0)) MARK(filter(z0, z1)) -> c22(A__FILTER(mark(z0), mark(z1)), MARK(z0)) MARK(filter(z0, z1)) -> c23(A__FILTER(mark(z0), mark(z1)), MARK(z1)) MARK(s(z0)) -> c24(MARK(z0)) MARK(0) -> c25 MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(true) -> c27 MARK(false) -> c28 MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__PRIMES, A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__FILTER_2, A__SIEVE_1, MARK_1 Compound Symbols: c_2, c1, c2_1, c3, c4_1, c5, c6_1, c7, c8_1, c9_1, c10, c11_2, c12_2, c13, c14_1, c15, c16_1, c17_2, c18_2, c19_2, c20_2, c21_2, c22_2, c23_2, c24_1, c25, c26_1, c27, c28, c29_1, c30_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__PRIMES -> c(A__SIEVE(a__from(s(s(0)))), A__FROM(s(s(0)))) A__PRIMES -> c1 A__FROM(z0) -> c2(MARK(z0)) A__FROM(z0) -> c3 A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__HEAD(z0) -> c5 A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__TAIL(z0) -> c7 A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__IF(z0, z1, z2) -> c10 A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(A__IF(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))), MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(A__IF(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))), MARK(z1)) A__FILTER(z0, z1) -> c13 A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) A__SIEVE(z0) -> c15 MARK(primes) -> c16(A__PRIMES) MARK(sieve(z0)) -> c17(A__SIEVE(mark(z0)), MARK(z0)) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(head(z0)) -> c19(A__HEAD(mark(z0)), MARK(z0)) MARK(tail(z0)) -> c20(A__TAIL(mark(z0)), MARK(z0)) MARK(if(z0, z1, z2)) -> c21(A__IF(mark(z0), z1, z2), MARK(z0)) MARK(filter(z0, z1)) -> c22(A__FILTER(mark(z0), mark(z1)), MARK(z0)) MARK(filter(z0, z1)) -> c23(A__FILTER(mark(z0), mark(z1)), MARK(z1)) MARK(s(z0)) -> c24(MARK(z0)) MARK(0) -> c25 MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(true) -> c27 MARK(false) -> c28 MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) The (relative) TRS S consists of the following rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) 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__PRIMES -> c(A__SIEVE(a__from(s(s(0)))), A__FROM(s(s(0)))) A__PRIMES -> c1 A__FROM(z0) -> c2(MARK(z0)) A__FROM(z0) -> c3 A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__HEAD(z0) -> c5 A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__TAIL(z0) -> c7 A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__IF(z0, z1, z2) -> c10 A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(A__IF(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))), MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(A__IF(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))), MARK(z1)) A__FILTER(z0, z1) -> c13 A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) A__SIEVE(z0) -> c15 MARK(primes) -> c16(A__PRIMES) MARK(sieve(z0)) -> c17(A__SIEVE(mark(z0)), MARK(z0)) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(head(z0)) -> c19(A__HEAD(mark(z0)), MARK(z0)) MARK(tail(z0)) -> c20(A__TAIL(mark(z0)), MARK(z0)) MARK(if(z0, z1, z2)) -> c21(A__IF(mark(z0), z1, z2), MARK(z0)) MARK(filter(z0, z1)) -> c22(A__FILTER(mark(z0), mark(z1)), MARK(z0)) MARK(filter(z0, z1)) -> c23(A__FILTER(mark(z0), mark(z1)), MARK(z1)) MARK(s(z0)) -> c24(MARK(z0)) MARK(0) -> c25 MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(true) -> c27 MARK(false) -> c28 MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) The (relative) TRS S consists of the following rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) 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__PRIMES -> c(A__SIEVE(a__from(s(s(0')))), A__FROM(s(s(0')))) A__PRIMES -> c1 A__FROM(z0) -> c2(MARK(z0)) A__FROM(z0) -> c3 A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__HEAD(z0) -> c5 A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__TAIL(z0) -> c7 A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__IF(z0, z1, z2) -> c10 A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(A__IF(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))), MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(A__IF(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))), MARK(z1)) A__FILTER(z0, z1) -> c13 A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) A__SIEVE(z0) -> c15 MARK(primes) -> c16(A__PRIMES) MARK(sieve(z0)) -> c17(A__SIEVE(mark(z0)), MARK(z0)) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(head(z0)) -> c19(A__HEAD(mark(z0)), MARK(z0)) MARK(tail(z0)) -> c20(A__TAIL(mark(z0)), MARK(z0)) MARK(if(z0, z1, z2)) -> c21(A__IF(mark(z0), z1, z2), MARK(z0)) MARK(filter(z0, z1)) -> c22(A__FILTER(mark(z0), mark(z1)), MARK(z0)) MARK(filter(z0, z1)) -> c23(A__FILTER(mark(z0), mark(z1)), MARK(z1)) MARK(s(z0)) -> c24(MARK(z0)) MARK(0') -> c25 MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(true) -> c27 MARK(false) -> c28 MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) The (relative) TRS S consists of the following rules: a__primes -> a__sieve(a__from(s(s(0')))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0') -> 0' mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Rewrite Strategy: INNERMOST ---------------------------------------- (13) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Inferred types. ---------------------------------------- (14) Obligation: Innermost TRS: Rules: A__PRIMES -> c(A__SIEVE(a__from(s(s(0')))), A__FROM(s(s(0')))) A__PRIMES -> c1 A__FROM(z0) -> c2(MARK(z0)) A__FROM(z0) -> c3 A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__HEAD(z0) -> c5 A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__TAIL(z0) -> c7 A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__IF(z0, z1, z2) -> c10 A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(A__IF(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))), MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(A__IF(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))), MARK(z1)) A__FILTER(z0, z1) -> c13 A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) A__SIEVE(z0) -> c15 MARK(primes) -> c16(A__PRIMES) MARK(sieve(z0)) -> c17(A__SIEVE(mark(z0)), MARK(z0)) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(head(z0)) -> c19(A__HEAD(mark(z0)), MARK(z0)) MARK(tail(z0)) -> c20(A__TAIL(mark(z0)), MARK(z0)) MARK(if(z0, z1, z2)) -> c21(A__IF(mark(z0), z1, z2), MARK(z0)) MARK(filter(z0, z1)) -> c22(A__FILTER(mark(z0), mark(z1)), MARK(z0)) MARK(filter(z0, z1)) -> c23(A__FILTER(mark(z0), mark(z1)), MARK(z1)) MARK(s(z0)) -> c24(MARK(z0)) MARK(0') -> c25 MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(true) -> c27 MARK(false) -> c28 MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) a__primes -> a__sieve(a__from(s(s(0')))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0') -> 0' mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Types: A__PRIMES :: c:c1 c :: c14:c15 -> c2:c3 -> c:c1 A__SIEVE :: 0':s:cons:true:false:divides:filter:sieve:primes:from:head:tail:if -> c14:c15 a__from :: 0':s:cons:true:false:divides:filter:sieve:primes:from:head:tail:if -> 0':s:cons:true:false:divides:filter:sieve:primes:from:head:tail:if s :: 0':s:cons:true:false:divides:filter:sieve:primes:from:head:tail:if -> 0':s:cons:true:false:divides:filter:sieve:primes:from:head:tail:if 0' :: 0':s:cons:true:false:divides:filter:sieve:primes:from:head:tail:if A__FROM :: 0':s:cons:true:false:divides:filter:sieve:primes:from:head:tail:if -> c2:c3 c1 :: c:c1 c2 :: c16:c17:c18:c19:c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30 -> c2:c3 MARK :: 0':s:cons:true:false:divides:filter:sieve:primes:from:head:tail:if -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30 c3 :: c2:c3 A__HEAD :: 0':s:cons:true:false:divides:filter:sieve:primes:from:head:tail:if -> c4:c5 cons :: 0':s:cons:true:false:divides:filter:sieve:primes:from:head:tail:if -> 0':s:cons:true:false:divides:filter:sieve:primes:from:head:tail:if -> 0':s:cons:true:false:divides:filter:sieve:primes:from:head:tail:if c4 :: c16:c17:c18:c19:c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30 -> c4:c5 c5 :: c4:c5 A__TAIL :: 0':s:cons:true:false:divides:filter:sieve:primes:from:head:tail:if -> c6:c7 c6 :: c16:c17:c18:c19:c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30 -> c6:c7 c7 :: c6:c7 A__IF :: 0':s:cons:true:false:divides:filter:sieve:primes:from:head:tail:if -> 0':s:cons:true:false:divides:filter:sieve:primes:from:head:tail:if -> 0':s:cons:true:false:divides:filter:sieve:primes:from:head:tail:if -> c8:c9:c10 true :: 0':s:cons:true:false:divides:filter:sieve:primes:from:head:tail:if c8 :: c16:c17:c18:c19:c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30 -> c8:c9:c10 false :: 0':s:cons:true:false:divides:filter:sieve:primes:from:head:tail:if c9 :: c16:c17:c18:c19:c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30 -> c8:c9:c10 c10 :: c8:c9:c10 A__FILTER :: 0':s:cons:true:false:divides:filter:sieve:primes:from:head:tail:if -> 0':s:cons:true:false:divides:filter:sieve:primes:from:head:tail:if -> c11:c12:c13 c11 :: c8:c9:c10 -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30 -> c11:c12:c13 divides :: 0':s:cons:true:false:divides:filter:sieve:primes:from:head:tail:if -> 0':s:cons:true:false:divides:filter:sieve:primes:from:head:tail:if -> 0':s:cons:true:false:divides:filter:sieve:primes:from:head:tail:if mark :: 0':s:cons:true:false:divides:filter:sieve:primes:from:head:tail:if -> 0':s:cons:true:false:divides:filter:sieve:primes:from:head:tail:if filter :: 0':s:cons:true:false:divides:filter:sieve:primes:from:head:tail:if -> 0':s:cons:true:false:divides:filter:sieve:primes:from:head:tail:if -> 0':s:cons:true:false:divides:filter:sieve:primes:from:head:tail:if sieve :: 0':s:cons:true:false:divides:filter:sieve:primes:from:head:tail:if -> 0':s:cons:true:false:divides:filter:sieve:primes:from:head:tail:if c12 :: c8:c9:c10 -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30 -> c11:c12:c13 c13 :: c11:c12:c13 c14 :: c16:c17:c18:c19:c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30 -> c14:c15 c15 :: c14:c15 primes :: 0':s:cons:true:false:divides:filter:sieve:primes:from:head:tail:if c16 :: c:c1 -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30 c17 :: c14:c15 -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30 -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30 from :: 0':s:cons:true:false:divides:filter:sieve:primes:from:head:tail:if -> 0':s:cons:true:false:divides:filter:sieve:primes:from:head:tail:if c18 :: c2:c3 -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30 -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30 head :: 0':s:cons:true:false:divides:filter:sieve:primes:from:head:tail:if -> 0':s:cons:true:false:divides:filter:sieve:primes:from:head:tail:if c19 :: c4:c5 -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30 -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30 tail :: 0':s:cons:true:false:divides:filter:sieve:primes:from:head:tail:if -> 0':s:cons:true:false:divides:filter:sieve:primes:from:head:tail:if c20 :: c6:c7 -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30 -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30 if :: 0':s:cons:true:false:divides:filter:sieve:primes:from:head:tail:if -> 0':s:cons:true:false:divides:filter:sieve:primes:from:head:tail:if -> 0':s:cons:true:false:divides:filter:sieve:primes:from:head:tail:if -> 0':s:cons:true:false:divides:filter:sieve:primes:from:head:tail:if c21 :: c8:c9:c10 -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30 -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30 c22 :: c11:c12:c13 -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30 -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30 c23 :: c11:c12:c13 -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30 -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30 c24 :: c16:c17:c18:c19:c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30 -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30 c25 :: c16:c17:c18:c19:c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30 c26 :: c16:c17:c18:c19:c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30 -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30 c27 :: c16:c17:c18:c19:c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30 c28 :: c16:c17:c18:c19:c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30 c29 :: c16:c17:c18:c19:c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30 -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30 c30 :: c16:c17:c18:c19:c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30 -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30 a__primes :: 0':s:cons:true:false:divides:filter:sieve:primes:from:head:tail:if a__sieve :: 0':s:cons:true:false:divides:filter:sieve:primes:from:head:tail:if -> 0':s:cons:true:false:divides:filter:sieve:primes:from:head:tail:if a__head :: 0':s:cons:true:false:divides:filter:sieve:primes:from:head:tail:if -> 0':s:cons:true:false:divides:filter:sieve:primes:from:head:tail:if a__tail :: 0':s:cons:true:false:divides:filter:sieve:primes:from:head:tail:if -> 0':s:cons:true:false:divides:filter:sieve:primes:from:head:tail:if a__if :: 0':s:cons:true:false:divides:filter:sieve:primes:from:head:tail:if -> 0':s:cons:true:false:divides:filter:sieve:primes:from:head:tail:if -> 0':s:cons:true:false:divides:filter:sieve:primes:from:head:tail:if -> 0':s:cons:true:false:divides:filter:sieve:primes:from:head:tail:if a__filter :: 0':s:cons:true:false:divides:filter:sieve:primes:from:head:tail:if -> 0':s:cons:true:false:divides:filter:sieve:primes:from:head:tail:if -> 0':s:cons:true:false:divides:filter:sieve:primes:from:head:tail:if hole_c:c11_31 :: c:c1 hole_c14:c152_31 :: c14:c15 hole_c2:c33_31 :: c2:c3 hole_0':s:cons:true:false:divides:filter:sieve:primes:from:head:tail:if4_31 :: 0':s:cons:true:false:divides:filter:sieve:primes:from:head:tail:if hole_c16:c17:c18:c19:c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c305_31 :: c16:c17:c18:c19:c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30 hole_c4:c56_31 :: c4:c5 hole_c6:c77_31 :: c6:c7 hole_c8:c9:c108_31 :: c8:c9:c10 hole_c11:c12:c139_31 :: c11:c12:c13 gen_0':s:cons:true:false:divides:filter:sieve:primes:from:head:tail:if10_31 :: Nat -> 0':s:cons:true:false:divides:filter:sieve:primes:from:head:tail:if gen_c16:c17:c18:c19:c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c3011_31 :: Nat -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c25:c26:c27:c28:c29:c30 ---------------------------------------- (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__primes -> a__sieve(a__from(s(s(0)))) [1] a__from(X) -> cons(mark(X), from(s(X))) [1] a__head(cons(X, Y)) -> mark(X) [1] a__tail(cons(X, Y)) -> mark(Y) [1] a__if(true, X, Y) -> mark(X) [1] a__if(false, X, Y) -> mark(Y) [1] a__filter(s(s(X)), cons(Y, Z)) -> a__if(divides(s(s(mark(X))), mark(Y)), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y)))) [1] a__sieve(cons(X, Y)) -> cons(mark(X), filter(X, sieve(Y))) [1] mark(primes) -> a__primes [1] mark(sieve(X)) -> a__sieve(mark(X)) [1] mark(from(X)) -> a__from(mark(X)) [1] mark(head(X)) -> a__head(mark(X)) [1] mark(tail(X)) -> a__tail(mark(X)) [1] mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3) [1] mark(filter(X1, X2)) -> a__filter(mark(X1), mark(X2)) [1] mark(s(X)) -> s(mark(X)) [1] mark(0) -> 0 [1] mark(cons(X1, X2)) -> cons(mark(X1), X2) [1] mark(true) -> true [1] mark(false) -> false [1] mark(divides(X1, X2)) -> divides(mark(X1), mark(X2)) [1] a__primes -> primes [1] a__sieve(X) -> sieve(X) [1] a__from(X) -> from(X) [1] a__head(X) -> head(X) [1] a__tail(X) -> tail(X) [1] a__if(X1, X2, X3) -> if(X1, X2, X3) [1] a__filter(X1, X2) -> filter(X1, X2) [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__primes -> a__sieve(a__from(s(s(0)))) [1] a__from(X) -> cons(mark(X), from(s(X))) [1] a__head(cons(X, Y)) -> mark(X) [1] a__tail(cons(X, Y)) -> mark(Y) [1] a__if(true, X, Y) -> mark(X) [1] a__if(false, X, Y) -> mark(Y) [1] a__filter(s(s(X)), cons(Y, Z)) -> a__if(divides(s(s(mark(X))), mark(Y)), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y)))) [1] a__sieve(cons(X, Y)) -> cons(mark(X), filter(X, sieve(Y))) [1] mark(primes) -> a__primes [1] mark(sieve(X)) -> a__sieve(mark(X)) [1] mark(from(X)) -> a__from(mark(X)) [1] mark(head(X)) -> a__head(mark(X)) [1] mark(tail(X)) -> a__tail(mark(X)) [1] mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3) [1] mark(filter(X1, X2)) -> a__filter(mark(X1), mark(X2)) [1] mark(s(X)) -> s(mark(X)) [1] mark(0) -> 0 [1] mark(cons(X1, X2)) -> cons(mark(X1), X2) [1] mark(true) -> true [1] mark(false) -> false [1] mark(divides(X1, X2)) -> divides(mark(X1), mark(X2)) [1] a__primes -> primes [1] a__sieve(X) -> sieve(X) [1] a__from(X) -> from(X) [1] a__head(X) -> head(X) [1] a__tail(X) -> tail(X) [1] a__if(X1, X2, X3) -> if(X1, X2, X3) [1] a__filter(X1, X2) -> filter(X1, X2) [1] The TRS has the following type information: a__primes :: 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if a__sieve :: 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if -> 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if a__from :: 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if -> 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if s :: 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if -> 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if 0 :: 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if cons :: 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if -> 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if -> 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if mark :: 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if -> 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if from :: 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if -> 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if a__head :: 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if -> 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if a__tail :: 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if -> 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if a__if :: 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if -> 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if -> 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if -> 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if true :: 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if false :: 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if a__filter :: 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if -> 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if -> 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if divides :: 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if -> 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if -> 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if filter :: 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if -> 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if -> 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if sieve :: 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if -> 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if primes :: 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if head :: 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if -> 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if tail :: 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if -> 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if if :: 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if -> 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if -> 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if -> 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if 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__from_1 a__sieve_1 a__filter_2 a__tail_1 a__primes a__if_3 a__head_1 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__primes -> a__sieve(a__from(s(s(0)))) [1] a__from(X) -> cons(mark(X), from(s(X))) [1] a__head(cons(X, Y)) -> mark(X) [1] a__tail(cons(X, Y)) -> mark(Y) [1] a__if(true, X, Y) -> mark(X) [1] a__if(false, X, Y) -> mark(Y) [1] a__filter(s(s(X)), cons(Y, Z)) -> a__if(divides(s(s(mark(X))), mark(Y)), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y)))) [1] a__sieve(cons(X, Y)) -> cons(mark(X), filter(X, sieve(Y))) [1] mark(primes) -> a__primes [1] mark(sieve(X)) -> a__sieve(mark(X)) [1] mark(from(X)) -> a__from(mark(X)) [1] mark(head(X)) -> a__head(mark(X)) [1] mark(tail(X)) -> a__tail(mark(X)) [1] mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3) [1] mark(filter(X1, X2)) -> a__filter(mark(X1), mark(X2)) [1] mark(s(X)) -> s(mark(X)) [1] mark(0) -> 0 [1] mark(cons(X1, X2)) -> cons(mark(X1), X2) [1] mark(true) -> true [1] mark(false) -> false [1] mark(divides(X1, X2)) -> divides(mark(X1), mark(X2)) [1] a__primes -> primes [1] a__sieve(X) -> sieve(X) [1] a__from(X) -> from(X) [1] a__head(X) -> head(X) [1] a__tail(X) -> tail(X) [1] a__if(X1, X2, X3) -> if(X1, X2, X3) [1] a__filter(X1, X2) -> filter(X1, X2) [1] The TRS has the following type information: a__primes :: 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if a__sieve :: 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if -> 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if a__from :: 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if -> 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if s :: 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if -> 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if 0 :: 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if cons :: 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if -> 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if -> 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if mark :: 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if -> 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if from :: 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if -> 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if a__head :: 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if -> 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if a__tail :: 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if -> 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if a__if :: 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if -> 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if -> 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if -> 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if true :: 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if false :: 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if a__filter :: 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if -> 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if -> 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if divides :: 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if -> 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if -> 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if filter :: 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if -> 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if -> 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if sieve :: 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if -> 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if primes :: 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if head :: 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if -> 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if tail :: 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if -> 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if if :: 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if -> 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if -> 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if -> 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if 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__primes -> a__sieve(cons(mark(s(s(0))), from(s(s(s(0)))))) [2] a__primes -> a__sieve(from(s(s(0)))) [2] a__from(X) -> cons(mark(X), from(s(X))) [1] a__head(cons(X, Y)) -> mark(X) [1] a__tail(cons(X, Y)) -> mark(Y) [1] a__if(true, X, Y) -> mark(X) [1] a__if(false, X, Y) -> mark(Y) [1] a__filter(s(s(X)), cons(Y, Z)) -> a__if(divides(s(s(mark(X))), mark(Y)), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y)))) [1] a__sieve(cons(X, Y)) -> cons(mark(X), filter(X, sieve(Y))) [1] mark(primes) -> a__primes [1] mark(sieve(primes)) -> a__sieve(a__primes) [2] mark(sieve(sieve(X88))) -> a__sieve(a__sieve(mark(X88))) [2] mark(sieve(from(X89))) -> a__sieve(a__from(mark(X89))) [2] mark(sieve(head(X90))) -> a__sieve(a__head(mark(X90))) [2] mark(sieve(tail(X91))) -> a__sieve(a__tail(mark(X91))) [2] mark(sieve(if(X158, X256, X320))) -> a__sieve(a__if(mark(X158), X256, X320)) [2] mark(sieve(filter(X159, X257))) -> a__sieve(a__filter(mark(X159), mark(X257))) [2] mark(sieve(s(X92))) -> a__sieve(s(mark(X92))) [2] mark(sieve(0)) -> a__sieve(0) [2] mark(sieve(cons(X160, X258))) -> a__sieve(cons(mark(X160), X258)) [2] mark(sieve(true)) -> a__sieve(true) [2] mark(sieve(false)) -> a__sieve(false) [2] mark(sieve(divides(X161, X259))) -> a__sieve(divides(mark(X161), mark(X259))) [2] mark(from(primes)) -> a__from(a__primes) [2] mark(from(sieve(X93))) -> a__from(a__sieve(mark(X93))) [2] mark(from(from(X94))) -> a__from(a__from(mark(X94))) [2] mark(from(head(X95))) -> a__from(a__head(mark(X95))) [2] mark(from(tail(X96))) -> a__from(a__tail(mark(X96))) [2] mark(from(if(X162, X260, X321))) -> a__from(a__if(mark(X162), X260, X321)) [2] mark(from(filter(X163, X261))) -> a__from(a__filter(mark(X163), mark(X261))) [2] mark(from(s(X97))) -> a__from(s(mark(X97))) [2] mark(from(0)) -> a__from(0) [2] mark(from(cons(X164, X262))) -> a__from(cons(mark(X164), X262)) [2] mark(from(true)) -> a__from(true) [2] mark(from(false)) -> a__from(false) [2] mark(from(divides(X165, X263))) -> a__from(divides(mark(X165), mark(X263))) [2] mark(head(primes)) -> a__head(a__primes) [2] mark(head(sieve(X98))) -> a__head(a__sieve(mark(X98))) [2] mark(head(from(X99))) -> a__head(a__from(mark(X99))) [2] mark(head(head(X100))) -> a__head(a__head(mark(X100))) [2] mark(head(tail(X101))) -> a__head(a__tail(mark(X101))) [2] mark(head(if(X166, X264, X322))) -> a__head(a__if(mark(X166), X264, X322)) [2] mark(head(filter(X167, X265))) -> a__head(a__filter(mark(X167), mark(X265))) [2] mark(head(s(X102))) -> a__head(s(mark(X102))) [2] mark(head(0)) -> a__head(0) [2] mark(head(cons(X168, X266))) -> a__head(cons(mark(X168), X266)) [2] mark(head(true)) -> a__head(true) [2] mark(head(false)) -> a__head(false) [2] mark(head(divides(X169, X267))) -> a__head(divides(mark(X169), mark(X267))) [2] mark(tail(primes)) -> a__tail(a__primes) [2] mark(tail(sieve(X103))) -> a__tail(a__sieve(mark(X103))) [2] mark(tail(from(X104))) -> a__tail(a__from(mark(X104))) [2] mark(tail(head(X105))) -> a__tail(a__head(mark(X105))) [2] mark(tail(tail(X106))) -> a__tail(a__tail(mark(X106))) [2] mark(tail(if(X170, X268, X323))) -> a__tail(a__if(mark(X170), X268, X323)) [2] mark(tail(filter(X171, X269))) -> a__tail(a__filter(mark(X171), mark(X269))) [2] mark(tail(s(X107))) -> a__tail(s(mark(X107))) [2] mark(tail(0)) -> a__tail(0) [2] mark(tail(cons(X172, X270))) -> a__tail(cons(mark(X172), X270)) [2] mark(tail(true)) -> a__tail(true) [2] mark(tail(false)) -> a__tail(false) [2] mark(tail(divides(X173, X271))) -> a__tail(divides(mark(X173), mark(X271))) [2] mark(if(primes, X2, X3)) -> a__if(a__primes, X2, X3) [2] mark(if(sieve(X108), X2, X3)) -> a__if(a__sieve(mark(X108)), X2, X3) [2] mark(if(from(X109), X2, X3)) -> a__if(a__from(mark(X109)), X2, X3) [2] mark(if(head(X174), X2, X3)) -> a__if(a__head(mark(X174)), X2, X3) [2] mark(if(tail(X175), X2, X3)) -> a__if(a__tail(mark(X175)), X2, X3) [2] mark(if(if(X176, X272, X324), X2, X3)) -> a__if(a__if(mark(X176), X272, X324), X2, X3) [2] mark(if(filter(X177, X273), X2, X3)) -> a__if(a__filter(mark(X177), mark(X273)), X2, X3) [2] mark(if(s(X178), X2, X3)) -> a__if(s(mark(X178)), X2, X3) [2] mark(if(0, X2, X3)) -> a__if(0, X2, X3) [2] mark(if(cons(X179, X274), X2, X3)) -> a__if(cons(mark(X179), X274), X2, X3) [2] mark(if(true, X2, X3)) -> a__if(true, X2, X3) [2] mark(if(false, X2, X3)) -> a__if(false, X2, X3) [2] mark(if(divides(X180, X275), X2, X3)) -> a__if(divides(mark(X180), mark(X275)), X2, X3) [2] mark(filter(X1, X2)) -> a__filter(mark(X1), mark(X2)) [1] mark(s(X)) -> s(mark(X)) [1] mark(0) -> 0 [1] mark(cons(X1, X2)) -> cons(mark(X1), X2) [1] mark(true) -> true [1] mark(false) -> false [1] mark(divides(X1, X2)) -> divides(mark(X1), mark(X2)) [1] a__primes -> primes [1] a__sieve(X) -> sieve(X) [1] a__from(X) -> from(X) [1] a__head(X) -> head(X) [1] a__tail(X) -> tail(X) [1] a__if(X1, X2, X3) -> if(X1, X2, X3) [1] a__filter(X1, X2) -> filter(X1, X2) [1] The TRS has the following type information: a__primes :: 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if a__sieve :: 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if -> 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if a__from :: 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if -> 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if s :: 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if -> 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if 0 :: 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if cons :: 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if -> 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if -> 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if mark :: 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if -> 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if from :: 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if -> 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if a__head :: 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if -> 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if a__tail :: 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if -> 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if a__if :: 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if -> 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if -> 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if -> 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if true :: 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if false :: 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if a__filter :: 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if -> 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if -> 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if divides :: 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if -> 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if -> 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if filter :: 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if -> 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if -> 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if sieve :: 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if -> 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if primes :: 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if head :: 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if -> 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if tail :: 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if -> 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if if :: 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if -> 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if -> 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if -> 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if 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 true => 3 false => 1 primes => 2 ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: a__filter(z, z') -{ 1 }-> a__if(1 + (1 + (1 + mark(X))) + mark(Y), 1 + (1 + (1 + X)) + Z, 1 + Y + (1 + X + (1 + Y))) :|: Z >= 0, z = 1 + (1 + X), Y >= 0, X >= 0, z' = 1 + Y + Z a__filter(z, z') -{ 1 }-> 1 + X1 + X2 :|: X1 >= 0, X2 >= 0, z = X1, z' = X2 a__from(z) -{ 1 }-> 1 + X :|: X >= 0, z = X a__from(z) -{ 1 }-> 1 + mark(X) + (1 + (1 + X)) :|: X >= 0, z = X a__head(z) -{ 1 }-> mark(X) :|: Y >= 0, z = 1 + X + Y, X >= 0 a__head(z) -{ 1 }-> 1 + X :|: X >= 0, z = X a__if(z, z', z'') -{ 1 }-> mark(X) :|: z' = X, z = 3, Y >= 0, z'' = Y, X >= 0 a__if(z, z', z'') -{ 1 }-> mark(Y) :|: z' = X, Y >= 0, z = 1, z'' = Y, X >= 0 a__if(z, z', z'') -{ 1 }-> 1 + X1 + X2 + X3 :|: X1 >= 0, X3 >= 0, X2 >= 0, z = X1, z' = X2, z'' = X3 a__primes -{ 2 }-> a__sieve(1 + (1 + (1 + 0))) :|: a__primes -{ 2 }-> a__sieve(1 + mark(1 + (1 + 0)) + (1 + (1 + (1 + (1 + 0))))) :|: a__primes -{ 1 }-> 2 :|: a__sieve(z) -{ 1 }-> 1 + X :|: X >= 0, z = X a__sieve(z) -{ 1 }-> 1 + mark(X) + (1 + X + (1 + Y)) :|: Y >= 0, z = 1 + X + Y, X >= 0 a__tail(z) -{ 1 }-> mark(Y) :|: Y >= 0, z = 1 + X + Y, X >= 0 a__tail(z) -{ 1 }-> 1 + X :|: X >= 0, z = X mark(z) -{ 2 }-> a__tail(a__tail(mark(X106))) :|: z = 1 + (1 + X106), X106 >= 0 mark(z) -{ 2 }-> a__tail(a__sieve(mark(X103))) :|: X103 >= 0, z = 1 + (1 + X103) mark(z) -{ 2 }-> a__tail(a__primes) :|: z = 1 + 2 mark(z) -{ 2 }-> a__tail(a__if(mark(X170), X268, X323)) :|: z = 1 + (1 + X170 + X268 + X323), X323 >= 0, X268 >= 0, X170 >= 0 mark(z) -{ 2 }-> a__tail(a__head(mark(X105))) :|: z = 1 + (1 + X105), X105 >= 0 mark(z) -{ 2 }-> a__tail(a__from(mark(X104))) :|: z = 1 + (1 + X104), X104 >= 0 mark(z) -{ 2 }-> a__tail(a__filter(mark(X171), mark(X269))) :|: X269 >= 0, z = 1 + (1 + X171 + X269), X171 >= 0 mark(z) -{ 2 }-> a__tail(3) :|: z = 1 + 3 mark(z) -{ 2 }-> a__tail(1) :|: z = 1 + 1 mark(z) -{ 2 }-> a__tail(0) :|: z = 1 + 0 mark(z) -{ 2 }-> a__tail(1 + mark(X107)) :|: z = 1 + (1 + X107), X107 >= 0 mark(z) -{ 2 }-> a__tail(1 + mark(X172) + X270) :|: X172 >= 0, z = 1 + (1 + X172 + X270), X270 >= 0 mark(z) -{ 2 }-> a__tail(1 + mark(X173) + mark(X271)) :|: X173 >= 0, X271 >= 0, z = 1 + (1 + X173 + X271) mark(z) -{ 2 }-> a__sieve(a__tail(mark(X91))) :|: X91 >= 0, z = 1 + (1 + X91) mark(z) -{ 2 }-> a__sieve(a__sieve(mark(X88))) :|: X88 >= 0, z = 1 + (1 + X88) mark(z) -{ 2 }-> a__sieve(a__primes) :|: z = 1 + 2 mark(z) -{ 2 }-> a__sieve(a__if(mark(X158), X256, X320)) :|: X320 >= 0, z = 1 + (1 + X158 + X256 + X320), X256 >= 0, X158 >= 0 mark(z) -{ 2 }-> a__sieve(a__head(mark(X90))) :|: z = 1 + (1 + X90), X90 >= 0 mark(z) -{ 2 }-> a__sieve(a__from(mark(X89))) :|: X89 >= 0, z = 1 + (1 + X89) mark(z) -{ 2 }-> a__sieve(a__filter(mark(X159), mark(X257))) :|: z = 1 + (1 + X159 + X257), X159 >= 0, X257 >= 0 mark(z) -{ 2 }-> a__sieve(3) :|: z = 1 + 3 mark(z) -{ 2 }-> a__sieve(1) :|: z = 1 + 1 mark(z) -{ 2 }-> a__sieve(0) :|: z = 1 + 0 mark(z) -{ 2 }-> a__sieve(1 + mark(X92)) :|: X92 >= 0, z = 1 + (1 + X92) mark(z) -{ 2 }-> a__sieve(1 + mark(X160) + X258) :|: z = 1 + (1 + X160 + X258), X160 >= 0, X258 >= 0 mark(z) -{ 2 }-> a__sieve(1 + mark(X161) + mark(X259)) :|: X259 >= 0, z = 1 + (1 + X161 + X259), X161 >= 0 mark(z) -{ 1 }-> a__primes :|: z = 2 mark(z) -{ 2 }-> a__if(a__tail(mark(X175)), X2, X3) :|: X175 >= 0, X3 >= 0, z = 1 + (1 + X175) + X2 + X3, X2 >= 0 mark(z) -{ 2 }-> a__if(a__sieve(mark(X108)), X2, X3) :|: z = 1 + (1 + X108) + X2 + X3, X3 >= 0, X2 >= 0, X108 >= 0 mark(z) -{ 2 }-> a__if(a__primes, X2, X3) :|: z = 1 + 2 + X2 + X3, X3 >= 0, X2 >= 0 mark(z) -{ 2 }-> a__if(a__if(mark(X176), X272, X324), X2, X3) :|: X176 >= 0, X272 >= 0, z = 1 + (1 + X176 + X272 + X324) + X2 + X3, X324 >= 0, X3 >= 0, X2 >= 0 mark(z) -{ 2 }-> a__if(a__head(mark(X174)), X2, X3) :|: X174 >= 0, X3 >= 0, X2 >= 0, z = 1 + (1 + X174) + X2 + X3 mark(z) -{ 2 }-> a__if(a__from(mark(X109)), X2, X3) :|: z = 1 + (1 + X109) + X2 + X3, X3 >= 0, X2 >= 0, X109 >= 0 mark(z) -{ 2 }-> a__if(a__filter(mark(X177), mark(X273)), X2, X3) :|: X177 >= 0, X273 >= 0, X3 >= 0, z = 1 + (1 + X177 + X273) + X2 + X3, X2 >= 0 mark(z) -{ 2 }-> a__if(3, X2, X3) :|: z = 1 + 3 + X2 + X3, X3 >= 0, X2 >= 0 mark(z) -{ 2 }-> a__if(1, X2, X3) :|: z = 1 + 1 + X2 + X3, X3 >= 0, X2 >= 0 mark(z) -{ 2 }-> a__if(0, X2, X3) :|: z = 1 + 0 + X2 + X3, X3 >= 0, X2 >= 0 mark(z) -{ 2 }-> a__if(1 + mark(X178), X2, X3) :|: z = 1 + (1 + X178) + X2 + X3, X3 >= 0, X2 >= 0, X178 >= 0 mark(z) -{ 2 }-> a__if(1 + mark(X179) + X274, X2, X3) :|: z = 1 + (1 + X179 + X274) + X2 + X3, X274 >= 0, X3 >= 0, X2 >= 0, X179 >= 0 mark(z) -{ 2 }-> a__if(1 + mark(X180) + mark(X275), X2, X3) :|: X275 >= 0, z = 1 + (1 + X180 + X275) + X2 + X3, X3 >= 0, X2 >= 0, X180 >= 0 mark(z) -{ 2 }-> a__head(a__tail(mark(X101))) :|: z = 1 + (1 + X101), X101 >= 0 mark(z) -{ 2 }-> a__head(a__sieve(mark(X98))) :|: X98 >= 0, z = 1 + (1 + X98) mark(z) -{ 2 }-> a__head(a__primes) :|: z = 1 + 2 mark(z) -{ 2 }-> a__head(a__if(mark(X166), X264, X322)) :|: X166 >= 0, X322 >= 0, X264 >= 0, z = 1 + (1 + X166 + X264 + X322) mark(z) -{ 2 }-> a__head(a__head(mark(X100))) :|: z = 1 + (1 + X100), X100 >= 0 mark(z) -{ 2 }-> a__head(a__from(mark(X99))) :|: X99 >= 0, z = 1 + (1 + X99) mark(z) -{ 2 }-> a__head(a__filter(mark(X167), mark(X265))) :|: X265 >= 0, z = 1 + (1 + X167 + X265), X167 >= 0 mark(z) -{ 2 }-> a__head(3) :|: z = 1 + 3 mark(z) -{ 2 }-> a__head(1) :|: z = 1 + 1 mark(z) -{ 2 }-> a__head(0) :|: z = 1 + 0 mark(z) -{ 2 }-> a__head(1 + mark(X102)) :|: X102 >= 0, z = 1 + (1 + X102) mark(z) -{ 2 }-> a__head(1 + mark(X168) + X266) :|: X168 >= 0, z = 1 + (1 + X168 + X266), X266 >= 0 mark(z) -{ 2 }-> a__head(1 + mark(X169) + mark(X267)) :|: z = 1 + (1 + X169 + X267), X267 >= 0, X169 >= 0 mark(z) -{ 2 }-> a__from(a__tail(mark(X96))) :|: X96 >= 0, z = 1 + (1 + X96) mark(z) -{ 2 }-> a__from(a__sieve(mark(X93))) :|: z = 1 + (1 + X93), X93 >= 0 mark(z) -{ 2 }-> a__from(a__primes) :|: z = 1 + 2 mark(z) -{ 2 }-> a__from(a__if(mark(X162), X260, X321)) :|: X162 >= 0, X321 >= 0, z = 1 + (1 + X162 + X260 + X321), X260 >= 0 mark(z) -{ 2 }-> a__from(a__head(mark(X95))) :|: X95 >= 0, z = 1 + (1 + X95) mark(z) -{ 2 }-> a__from(a__from(mark(X94))) :|: z = 1 + (1 + X94), X94 >= 0 mark(z) -{ 2 }-> a__from(a__filter(mark(X163), mark(X261))) :|: X261 >= 0, X163 >= 0, z = 1 + (1 + X163 + X261) mark(z) -{ 2 }-> a__from(3) :|: z = 1 + 3 mark(z) -{ 2 }-> a__from(1) :|: z = 1 + 1 mark(z) -{ 2 }-> a__from(0) :|: z = 1 + 0 mark(z) -{ 2 }-> a__from(1 + mark(X97)) :|: X97 >= 0, z = 1 + (1 + X97) mark(z) -{ 2 }-> a__from(1 + mark(X164) + X262) :|: z = 1 + (1 + X164 + X262), X262 >= 0, X164 >= 0 mark(z) -{ 2 }-> a__from(1 + mark(X165) + mark(X263)) :|: X165 >= 0, X263 >= 0, z = 1 + (1 + X165 + X263) mark(z) -{ 1 }-> a__filter(mark(X1), mark(X2)) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2 mark(z) -{ 1 }-> 3 :|: z = 3 mark(z) -{ 1 }-> 1 :|: z = 1 mark(z) -{ 1 }-> 0 :|: z = 0 mark(z) -{ 1 }-> 1 + mark(X) :|: z = 1 + X, X >= 0 mark(z) -{ 1 }-> 1 + mark(X1) + X2 :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2 mark(z) -{ 1 }-> 1 + mark(X1) + mark(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') -{ 1 }-> a__if(1 + (1 + (1 + mark(z - 2))) + mark(Y), 1 + (1 + (1 + (z - 2))) + Z, 1 + Y + (1 + (z - 2) + (1 + Y))) :|: Z >= 0, Y >= 0, z - 2 >= 0, z' = 1 + Y + Z a__filter(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0 a__from(z) -{ 1 }-> 1 + z :|: z >= 0 a__from(z) -{ 1 }-> 1 + mark(z) + (1 + (1 + z)) :|: z >= 0 a__head(z) -{ 1 }-> mark(X) :|: Y >= 0, z = 1 + X + Y, X >= 0 a__head(z) -{ 1 }-> 1 + z :|: z >= 0 a__if(z, z', z'') -{ 1 }-> mark(z') :|: z = 3, z'' >= 0, z' >= 0 a__if(z, z', z'') -{ 1 }-> mark(z'') :|: z'' >= 0, z = 1, z' >= 0 a__if(z, z', z'') -{ 1 }-> 1 + z + z' + z'' :|: z >= 0, z'' >= 0, z' >= 0 a__primes -{ 2 }-> a__sieve(1 + (1 + (1 + 0))) :|: a__primes -{ 2 }-> a__sieve(1 + mark(1 + (1 + 0)) + (1 + (1 + (1 + (1 + 0))))) :|: a__primes -{ 1 }-> 2 :|: a__sieve(z) -{ 1 }-> 1 + z :|: z >= 0 a__sieve(z) -{ 1 }-> 1 + mark(X) + (1 + X + (1 + Y)) :|: Y >= 0, z = 1 + X + Y, X >= 0 a__tail(z) -{ 1 }-> mark(Y) :|: Y >= 0, z = 1 + X + Y, X >= 0 a__tail(z) -{ 1 }-> 1 + z :|: z >= 0 mark(z) -{ 2 }-> a__tail(a__tail(mark(z - 2))) :|: z - 2 >= 0 mark(z) -{ 2 }-> a__tail(a__sieve(mark(z - 2))) :|: z - 2 >= 0 mark(z) -{ 2 }-> a__tail(a__primes) :|: z = 1 + 2 mark(z) -{ 2 }-> a__tail(a__if(mark(X170), X268, X323)) :|: z = 1 + (1 + X170 + X268 + X323), X323 >= 0, X268 >= 0, X170 >= 0 mark(z) -{ 2 }-> a__tail(a__head(mark(z - 2))) :|: z - 2 >= 0 mark(z) -{ 2 }-> a__tail(a__from(mark(z - 2))) :|: z - 2 >= 0 mark(z) -{ 2 }-> a__tail(a__filter(mark(X171), mark(X269))) :|: X269 >= 0, z = 1 + (1 + X171 + X269), X171 >= 0 mark(z) -{ 2 }-> a__tail(3) :|: z = 1 + 3 mark(z) -{ 2 }-> a__tail(1) :|: z = 1 + 1 mark(z) -{ 2 }-> a__tail(0) :|: z = 1 + 0 mark(z) -{ 2 }-> a__tail(1 + mark(z - 2)) :|: z - 2 >= 0 mark(z) -{ 2 }-> a__tail(1 + mark(X172) + X270) :|: X172 >= 0, z = 1 + (1 + X172 + X270), X270 >= 0 mark(z) -{ 2 }-> a__tail(1 + mark(X173) + mark(X271)) :|: X173 >= 0, X271 >= 0, z = 1 + (1 + X173 + X271) mark(z) -{ 2 }-> a__sieve(a__tail(mark(z - 2))) :|: z - 2 >= 0 mark(z) -{ 2 }-> a__sieve(a__sieve(mark(z - 2))) :|: z - 2 >= 0 mark(z) -{ 2 }-> a__sieve(a__primes) :|: z = 1 + 2 mark(z) -{ 2 }-> a__sieve(a__if(mark(X158), X256, X320)) :|: X320 >= 0, z = 1 + (1 + X158 + X256 + X320), X256 >= 0, X158 >= 0 mark(z) -{ 2 }-> a__sieve(a__head(mark(z - 2))) :|: z - 2 >= 0 mark(z) -{ 2 }-> a__sieve(a__from(mark(z - 2))) :|: z - 2 >= 0 mark(z) -{ 2 }-> a__sieve(a__filter(mark(X159), mark(X257))) :|: z = 1 + (1 + X159 + X257), X159 >= 0, X257 >= 0 mark(z) -{ 2 }-> a__sieve(3) :|: z = 1 + 3 mark(z) -{ 2 }-> a__sieve(1) :|: z = 1 + 1 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(X160) + X258) :|: z = 1 + (1 + X160 + X258), X160 >= 0, X258 >= 0 mark(z) -{ 2 }-> a__sieve(1 + mark(X161) + mark(X259)) :|: X259 >= 0, z = 1 + (1 + X161 + X259), X161 >= 0 mark(z) -{ 1 }-> a__primes :|: z = 2 mark(z) -{ 2 }-> a__if(a__tail(mark(X175)), X2, X3) :|: X175 >= 0, X3 >= 0, z = 1 + (1 + X175) + X2 + X3, X2 >= 0 mark(z) -{ 2 }-> a__if(a__sieve(mark(X108)), X2, X3) :|: z = 1 + (1 + X108) + X2 + X3, X3 >= 0, X2 >= 0, X108 >= 0 mark(z) -{ 2 }-> a__if(a__primes, X2, X3) :|: z = 1 + 2 + X2 + X3, X3 >= 0, X2 >= 0 mark(z) -{ 2 }-> a__if(a__if(mark(X176), X272, X324), X2, X3) :|: X176 >= 0, X272 >= 0, z = 1 + (1 + X176 + X272 + X324) + X2 + X3, X324 >= 0, X3 >= 0, X2 >= 0 mark(z) -{ 2 }-> a__if(a__head(mark(X174)), X2, X3) :|: X174 >= 0, X3 >= 0, X2 >= 0, z = 1 + (1 + X174) + X2 + X3 mark(z) -{ 2 }-> a__if(a__from(mark(X109)), X2, X3) :|: z = 1 + (1 + X109) + X2 + X3, X3 >= 0, X2 >= 0, X109 >= 0 mark(z) -{ 2 }-> a__if(a__filter(mark(X177), mark(X273)), X2, X3) :|: X177 >= 0, X273 >= 0, X3 >= 0, z = 1 + (1 + X177 + X273) + X2 + X3, X2 >= 0 mark(z) -{ 2 }-> a__if(3, X2, X3) :|: z = 1 + 3 + X2 + X3, X3 >= 0, X2 >= 0 mark(z) -{ 2 }-> a__if(1, X2, X3) :|: z = 1 + 1 + X2 + X3, X3 >= 0, X2 >= 0 mark(z) -{ 2 }-> a__if(0, X2, X3) :|: z = 1 + 0 + X2 + X3, X3 >= 0, X2 >= 0 mark(z) -{ 2 }-> a__if(1 + mark(X178), X2, X3) :|: z = 1 + (1 + X178) + X2 + X3, X3 >= 0, X2 >= 0, X178 >= 0 mark(z) -{ 2 }-> a__if(1 + mark(X179) + X274, X2, X3) :|: z = 1 + (1 + X179 + X274) + X2 + X3, X274 >= 0, X3 >= 0, X2 >= 0, X179 >= 0 mark(z) -{ 2 }-> a__if(1 + mark(X180) + mark(X275), X2, X3) :|: X275 >= 0, z = 1 + (1 + X180 + X275) + X2 + X3, X3 >= 0, X2 >= 0, X180 >= 0 mark(z) -{ 2 }-> a__head(a__tail(mark(z - 2))) :|: z - 2 >= 0 mark(z) -{ 2 }-> a__head(a__sieve(mark(z - 2))) :|: z - 2 >= 0 mark(z) -{ 2 }-> a__head(a__primes) :|: z = 1 + 2 mark(z) -{ 2 }-> a__head(a__if(mark(X166), X264, X322)) :|: X166 >= 0, X322 >= 0, X264 >= 0, z = 1 + (1 + X166 + X264 + X322) mark(z) -{ 2 }-> a__head(a__head(mark(z - 2))) :|: z - 2 >= 0 mark(z) -{ 2 }-> a__head(a__from(mark(z - 2))) :|: z - 2 >= 0 mark(z) -{ 2 }-> a__head(a__filter(mark(X167), mark(X265))) :|: X265 >= 0, z = 1 + (1 + X167 + X265), X167 >= 0 mark(z) -{ 2 }-> a__head(3) :|: z = 1 + 3 mark(z) -{ 2 }-> a__head(1) :|: z = 1 + 1 mark(z) -{ 2 }-> a__head(0) :|: z = 1 + 0 mark(z) -{ 2 }-> a__head(1 + mark(z - 2)) :|: z - 2 >= 0 mark(z) -{ 2 }-> a__head(1 + mark(X168) + X266) :|: X168 >= 0, z = 1 + (1 + X168 + X266), X266 >= 0 mark(z) -{ 2 }-> a__head(1 + mark(X169) + mark(X267)) :|: z = 1 + (1 + X169 + X267), X267 >= 0, X169 >= 0 mark(z) -{ 2 }-> a__from(a__tail(mark(z - 2))) :|: z - 2 >= 0 mark(z) -{ 2 }-> a__from(a__sieve(mark(z - 2))) :|: z - 2 >= 0 mark(z) -{ 2 }-> a__from(a__primes) :|: z = 1 + 2 mark(z) -{ 2 }-> a__from(a__if(mark(X162), X260, X321)) :|: X162 >= 0, X321 >= 0, z = 1 + (1 + X162 + X260 + X321), X260 >= 0 mark(z) -{ 2 }-> a__from(a__head(mark(z - 2))) :|: z - 2 >= 0 mark(z) -{ 2 }-> a__from(a__from(mark(z - 2))) :|: z - 2 >= 0 mark(z) -{ 2 }-> a__from(a__filter(mark(X163), mark(X261))) :|: X261 >= 0, X163 >= 0, z = 1 + (1 + X163 + X261) mark(z) -{ 2 }-> a__from(3) :|: z = 1 + 3 mark(z) -{ 2 }-> a__from(1) :|: z = 1 + 1 mark(z) -{ 2 }-> a__from(0) :|: z = 1 + 0 mark(z) -{ 2 }-> a__from(1 + mark(z - 2)) :|: z - 2 >= 0 mark(z) -{ 2 }-> a__from(1 + mark(X164) + X262) :|: z = 1 + (1 + X164 + X262), X262 >= 0, X164 >= 0 mark(z) -{ 2 }-> a__from(1 + mark(X165) + mark(X263)) :|: X165 >= 0, X263 >= 0, z = 1 + (1 + X165 + X263) mark(z) -{ 1 }-> a__filter(mark(X1), mark(X2)) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2 mark(z) -{ 1 }-> 3 :|: z = 3 mark(z) -{ 1 }-> 1 :|: z = 1 mark(z) -{ 1 }-> 0 :|: z = 0 mark(z) -{ 1 }-> 1 + mark(z - 1) :|: z - 1 >= 0 mark(z) -{ 1 }-> 1 + mark(X1) + X2 :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2 mark(z) -{ 1 }-> 1 + mark(X1) + mark(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__primes -> a__sieve(a__from(s(s(0)))) [1] a__from(X) -> cons(mark(X), from(s(X))) [1] a__head(cons(X, Y)) -> mark(X) [1] a__tail(cons(X, Y)) -> mark(Y) [1] a__if(true, X, Y) -> mark(X) [1] a__if(false, X, Y) -> mark(Y) [1] a__filter(s(s(X)), cons(Y, Z)) -> a__if(divides(s(s(mark(X))), mark(Y)), filter(s(s(X)), Z), cons(Y, filter(X, sieve(Y)))) [1] a__sieve(cons(X, Y)) -> cons(mark(X), filter(X, sieve(Y))) [1] mark(primes) -> a__primes [1] mark(sieve(X)) -> a__sieve(mark(X)) [1] mark(from(X)) -> a__from(mark(X)) [1] mark(head(X)) -> a__head(mark(X)) [1] mark(tail(X)) -> a__tail(mark(X)) [1] mark(if(X1, X2, X3)) -> a__if(mark(X1), X2, X3) [1] mark(filter(X1, X2)) -> a__filter(mark(X1), mark(X2)) [1] mark(s(X)) -> s(mark(X)) [1] mark(0) -> 0 [1] mark(cons(X1, X2)) -> cons(mark(X1), X2) [1] mark(true) -> true [1] mark(false) -> false [1] mark(divides(X1, X2)) -> divides(mark(X1), mark(X2)) [1] a__primes -> primes [1] a__sieve(X) -> sieve(X) [1] a__from(X) -> from(X) [1] a__head(X) -> head(X) [1] a__tail(X) -> tail(X) [1] a__if(X1, X2, X3) -> if(X1, X2, X3) [1] a__filter(X1, X2) -> filter(X1, X2) [1] The TRS has the following type information: a__primes :: 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if a__sieve :: 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if -> 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if a__from :: 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if -> 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if s :: 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if -> 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if 0 :: 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if cons :: 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if -> 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if -> 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if mark :: 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if -> 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if from :: 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if -> 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if a__head :: 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if -> 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if a__tail :: 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if -> 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if a__if :: 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if -> 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if -> 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if -> 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if true :: 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if false :: 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if a__filter :: 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if -> 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if -> 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if divides :: 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if -> 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if -> 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if filter :: 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if -> 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if -> 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if sieve :: 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if -> 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if primes :: 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if head :: 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if -> 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if tail :: 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if -> 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if if :: 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if -> 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if -> 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if -> 0:s:from:cons:true:false:divides:filter:sieve:primes:head:tail:if 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 true => 3 false => 1 primes => 2 ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: a__filter(z, z') -{ 1 }-> a__if(1 + (1 + (1 + mark(X))) + mark(Y), 1 + (1 + (1 + X)) + Z, 1 + Y + (1 + X + (1 + Y))) :|: Z >= 0, z = 1 + (1 + X), Y >= 0, X >= 0, z' = 1 + Y + Z a__filter(z, z') -{ 1 }-> 1 + X1 + X2 :|: X1 >= 0, X2 >= 0, z = X1, z' = X2 a__from(z) -{ 1 }-> 1 + X :|: X >= 0, z = X a__from(z) -{ 1 }-> 1 + mark(X) + (1 + (1 + X)) :|: X >= 0, z = X a__head(z) -{ 1 }-> mark(X) :|: Y >= 0, z = 1 + X + Y, X >= 0 a__head(z) -{ 1 }-> 1 + X :|: X >= 0, z = X a__if(z, z', z'') -{ 1 }-> mark(X) :|: z' = X, z = 3, Y >= 0, z'' = Y, X >= 0 a__if(z, z', z'') -{ 1 }-> mark(Y) :|: z' = X, Y >= 0, z = 1, z'' = Y, X >= 0 a__if(z, z', z'') -{ 1 }-> 1 + X1 + X2 + X3 :|: X1 >= 0, X3 >= 0, X2 >= 0, z = X1, z' = X2, z'' = X3 a__primes -{ 1 }-> a__sieve(a__from(1 + (1 + 0))) :|: a__primes -{ 1 }-> 2 :|: a__sieve(z) -{ 1 }-> 1 + X :|: X >= 0, z = X a__sieve(z) -{ 1 }-> 1 + mark(X) + (1 + X + (1 + Y)) :|: Y >= 0, z = 1 + X + Y, X >= 0 a__tail(z) -{ 1 }-> mark(Y) :|: Y >= 0, z = 1 + X + Y, X >= 0 a__tail(z) -{ 1 }-> 1 + X :|: X >= 0, z = X mark(z) -{ 1 }-> a__tail(mark(X)) :|: z = 1 + X, X >= 0 mark(z) -{ 1 }-> a__sieve(mark(X)) :|: z = 1 + X, X >= 0 mark(z) -{ 1 }-> a__primes :|: z = 2 mark(z) -{ 1 }-> a__if(mark(X1), X2, X3) :|: X1 >= 0, X3 >= 0, z = 1 + X1 + X2 + X3, X2 >= 0 mark(z) -{ 1 }-> a__head(mark(X)) :|: z = 1 + X, X >= 0 mark(z) -{ 1 }-> a__from(mark(X)) :|: z = 1 + X, X >= 0 mark(z) -{ 1 }-> a__filter(mark(X1), mark(X2)) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2 mark(z) -{ 1 }-> 3 :|: z = 3 mark(z) -{ 1 }-> 1 :|: z = 1 mark(z) -{ 1 }-> 0 :|: z = 0 mark(z) -{ 1 }-> 1 + mark(X) :|: z = 1 + X, X >= 0 mark(z) -{ 1 }-> 1 + mark(X1) + X2 :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2 mark(z) -{ 1 }-> 1 + mark(X1) + mark(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__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__PRIMES -> c(A__SIEVE(a__from(s(s(0)))), A__FROM(s(s(0)))) A__PRIMES -> c1 A__FROM(z0) -> c2(MARK(z0)) A__FROM(z0) -> c3 A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__HEAD(z0) -> c5 A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__TAIL(z0) -> c7 A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__IF(z0, z1, z2) -> c10 A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(A__IF(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))), MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(A__IF(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))), MARK(z1)) A__FILTER(z0, z1) -> c13 A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) A__SIEVE(z0) -> c15 MARK(primes) -> c16(A__PRIMES) MARK(sieve(z0)) -> c17(A__SIEVE(mark(z0)), MARK(z0)) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(head(z0)) -> c19(A__HEAD(mark(z0)), MARK(z0)) MARK(tail(z0)) -> c20(A__TAIL(mark(z0)), MARK(z0)) MARK(if(z0, z1, z2)) -> c21(A__IF(mark(z0), z1, z2), MARK(z0)) MARK(filter(z0, z1)) -> c22(A__FILTER(mark(z0), mark(z1)), MARK(z0)) MARK(filter(z0, z1)) -> c23(A__FILTER(mark(z0), mark(z1)), MARK(z1)) MARK(s(z0)) -> c24(MARK(z0)) MARK(0) -> c25 MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(true) -> c27 MARK(false) -> c28 MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) S tuples: A__PRIMES -> c(A__SIEVE(a__from(s(s(0)))), A__FROM(s(s(0)))) A__PRIMES -> c1 A__FROM(z0) -> c2(MARK(z0)) A__FROM(z0) -> c3 A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__HEAD(z0) -> c5 A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__TAIL(z0) -> c7 A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__IF(z0, z1, z2) -> c10 A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(A__IF(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))), MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(A__IF(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))), MARK(z1)) A__FILTER(z0, z1) -> c13 A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) A__SIEVE(z0) -> c15 MARK(primes) -> c16(A__PRIMES) MARK(sieve(z0)) -> c17(A__SIEVE(mark(z0)), MARK(z0)) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(head(z0)) -> c19(A__HEAD(mark(z0)), MARK(z0)) MARK(tail(z0)) -> c20(A__TAIL(mark(z0)), MARK(z0)) MARK(if(z0, z1, z2)) -> c21(A__IF(mark(z0), z1, z2), MARK(z0)) MARK(filter(z0, z1)) -> c22(A__FILTER(mark(z0), mark(z1)), MARK(z0)) MARK(filter(z0, z1)) -> c23(A__FILTER(mark(z0), mark(z1)), MARK(z1)) MARK(s(z0)) -> c24(MARK(z0)) MARK(0) -> c25 MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(true) -> c27 MARK(false) -> c28 MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__PRIMES, A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__FILTER_2, A__SIEVE_1, MARK_1 Compound Symbols: c_2, c1, c2_1, c3, c4_1, c5, c6_1, c7, c8_1, c9_1, c10, c11_2, c12_2, c13, c14_1, c15, c16_1, c17_2, c18_2, c19_2, c20_2, c21_2, c22_2, c23_2, c24_1, c25, c26_1, c27, c28, c29_1, c30_1 ---------------------------------------- (33) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 10 trailing nodes: A__FILTER(z0, z1) -> c13 A__PRIMES -> c1 A__TAIL(z0) -> c7 A__FROM(z0) -> c3 MARK(true) -> c27 A__SIEVE(z0) -> c15 A__IF(z0, z1, z2) -> c10 MARK(0) -> c25 A__HEAD(z0) -> c5 MARK(false) -> c28 ---------------------------------------- (34) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__PRIMES -> c(A__SIEVE(a__from(s(s(0)))), A__FROM(s(s(0)))) A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(A__IF(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))), MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(A__IF(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))), MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(sieve(z0)) -> c17(A__SIEVE(mark(z0)), MARK(z0)) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(head(z0)) -> c19(A__HEAD(mark(z0)), MARK(z0)) MARK(tail(z0)) -> c20(A__TAIL(mark(z0)), MARK(z0)) MARK(if(z0, z1, z2)) -> c21(A__IF(mark(z0), z1, z2), MARK(z0)) MARK(filter(z0, z1)) -> c22(A__FILTER(mark(z0), mark(z1)), MARK(z0)) MARK(filter(z0, z1)) -> c23(A__FILTER(mark(z0), mark(z1)), MARK(z1)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) S tuples: A__PRIMES -> c(A__SIEVE(a__from(s(s(0)))), A__FROM(s(s(0)))) A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(A__IF(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))), MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(A__IF(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))), MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(sieve(z0)) -> c17(A__SIEVE(mark(z0)), MARK(z0)) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(head(z0)) -> c19(A__HEAD(mark(z0)), MARK(z0)) MARK(tail(z0)) -> c20(A__TAIL(mark(z0)), MARK(z0)) MARK(if(z0, z1, z2)) -> c21(A__IF(mark(z0), z1, z2), MARK(z0)) MARK(filter(z0, z1)) -> c22(A__FILTER(mark(z0), mark(z1)), MARK(z0)) MARK(filter(z0, z1)) -> c23(A__FILTER(mark(z0), mark(z1)), MARK(z1)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__PRIMES, A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__FILTER_2, A__SIEVE_1, MARK_1 Compound Symbols: c_2, c2_1, c4_1, c6_1, c8_1, c9_1, c11_2, c12_2, c14_1, c16_1, c17_2, c18_2, c19_2, c20_2, c21_2, c22_2, c23_2, c24_1, c26_1, c29_1, c30_1 ---------------------------------------- (35) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (36) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__PRIMES -> c(A__SIEVE(a__from(s(s(0)))), A__FROM(s(s(0)))) A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(sieve(z0)) -> c17(A__SIEVE(mark(z0)), MARK(z0)) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(head(z0)) -> c19(A__HEAD(mark(z0)), MARK(z0)) MARK(tail(z0)) -> c20(A__TAIL(mark(z0)), MARK(z0)) MARK(if(z0, z1, z2)) -> c21(A__IF(mark(z0), z1, z2), MARK(z0)) MARK(filter(z0, z1)) -> c22(A__FILTER(mark(z0), mark(z1)), MARK(z0)) MARK(filter(z0, z1)) -> c23(A__FILTER(mark(z0), mark(z1)), MARK(z1)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) S tuples: A__PRIMES -> c(A__SIEVE(a__from(s(s(0)))), A__FROM(s(s(0)))) A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(sieve(z0)) -> c17(A__SIEVE(mark(z0)), MARK(z0)) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(head(z0)) -> c19(A__HEAD(mark(z0)), MARK(z0)) MARK(tail(z0)) -> c20(A__TAIL(mark(z0)), MARK(z0)) MARK(if(z0, z1, z2)) -> c21(A__IF(mark(z0), z1, z2), MARK(z0)) MARK(filter(z0, z1)) -> c22(A__FILTER(mark(z0), mark(z1)), MARK(z0)) MARK(filter(z0, z1)) -> c23(A__FILTER(mark(z0), mark(z1)), MARK(z1)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__PRIMES, A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2 Compound Symbols: c_2, c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c17_2, c18_2, c19_2, c20_2, c21_2, c22_2, c23_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1 ---------------------------------------- (37) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID)) Converted S to standard rules, and D \ S as well as R to relative rules. ---------------------------------------- (38) 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__PRIMES -> c(A__SIEVE(a__from(s(s(0)))), A__FROM(s(s(0)))) A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(sieve(z0)) -> c17(A__SIEVE(mark(z0)), MARK(z0)) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(head(z0)) -> c19(A__HEAD(mark(z0)), MARK(z0)) MARK(tail(z0)) -> c20(A__TAIL(mark(z0)), MARK(z0)) MARK(if(z0, z1, z2)) -> c21(A__IF(mark(z0), z1, z2), MARK(z0)) MARK(filter(z0, z1)) -> c22(A__FILTER(mark(z0), mark(z1)), MARK(z0)) MARK(filter(z0, z1)) -> c23(A__FILTER(mark(z0), mark(z1)), MARK(z1)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) The (relative) TRS S consists of the following rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Rewrite Strategy: INNERMOST ---------------------------------------- (39) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (40) 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__PRIMES -> c(A__SIEVE(a__from(s(s(0)))), A__FROM(s(s(0)))) A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(sieve(z0)) -> c17(A__SIEVE(mark(z0)), MARK(z0)) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(head(z0)) -> c19(A__HEAD(mark(z0)), MARK(z0)) MARK(tail(z0)) -> c20(A__TAIL(mark(z0)), MARK(z0)) MARK(if(z0, z1, z2)) -> c21(A__IF(mark(z0), z1, z2), MARK(z0)) MARK(filter(z0, z1)) -> c22(A__FILTER(mark(z0), mark(z1)), MARK(z0)) MARK(filter(z0, z1)) -> c23(A__FILTER(mark(z0), mark(z1)), MARK(z1)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) S is empty. Rewrite Strategy: INNERMOST ---------------------------------------- (41) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (42) 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__PRIMES -> c(A__SIEVE(a__from(s(s(0)))), A__FROM(s(s(0)))) [1] A__FROM(z0) -> c2(MARK(z0)) [1] A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) [1] A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) [1] A__IF(true, z0, z1) -> c8(MARK(z0)) [1] A__IF(false, z0, z1) -> c9(MARK(z1)) [1] A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) [1] MARK(primes) -> c16(A__PRIMES) [1] MARK(sieve(z0)) -> c17(A__SIEVE(mark(z0)), MARK(z0)) [1] MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) [1] MARK(head(z0)) -> c19(A__HEAD(mark(z0)), MARK(z0)) [1] MARK(tail(z0)) -> c20(A__TAIL(mark(z0)), MARK(z0)) [1] MARK(if(z0, z1, z2)) -> c21(A__IF(mark(z0), z1, z2), MARK(z0)) [1] MARK(filter(z0, z1)) -> c22(A__FILTER(mark(z0), mark(z1)), MARK(z0)) [1] MARK(filter(z0, z1)) -> c23(A__FILTER(mark(z0), mark(z1)), MARK(z1)) [1] MARK(s(z0)) -> c24(MARK(z0)) [1] MARK(cons(z0, z1)) -> c26(MARK(z0)) [1] MARK(divides(z0, z1)) -> c29(MARK(z0)) [1] MARK(divides(z0, z1)) -> c30(MARK(z1)) [1] A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) [1] A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) [1] a__primes -> a__sieve(a__from(s(s(0)))) [0] a__primes -> primes [0] a__from(z0) -> cons(mark(z0), from(s(z0))) [0] a__from(z0) -> from(z0) [0] a__head(cons(z0, z1)) -> mark(z0) [0] a__head(z0) -> head(z0) [0] a__tail(cons(z0, z1)) -> mark(z1) [0] a__tail(z0) -> tail(z0) [0] a__if(true, z0, z1) -> mark(z0) [0] a__if(false, z0, z1) -> mark(z1) [0] a__if(z0, z1, z2) -> if(z0, z1, z2) [0] a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) [0] a__filter(z0, z1) -> filter(z0, z1) [0] a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) [0] a__sieve(z0) -> sieve(z0) [0] mark(primes) -> a__primes [0] mark(sieve(z0)) -> a__sieve(mark(z0)) [0] mark(from(z0)) -> a__from(mark(z0)) [0] mark(head(z0)) -> a__head(mark(z0)) [0] mark(tail(z0)) -> a__tail(mark(z0)) [0] mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) [0] mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) [0] mark(s(z0)) -> s(mark(z0)) [0] mark(0) -> 0 [0] mark(cons(z0, z1)) -> cons(mark(z0), z1) [0] mark(true) -> true [0] mark(false) -> false [0] mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) [0] Rewrite Strategy: INNERMOST ---------------------------------------- (43) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (44) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: A__PRIMES -> c(A__SIEVE(a__from(s(s(0)))), A__FROM(s(s(0)))) [1] A__FROM(z0) -> c2(MARK(z0)) [1] A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) [1] A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) [1] A__IF(true, z0, z1) -> c8(MARK(z0)) [1] A__IF(false, z0, z1) -> c9(MARK(z1)) [1] A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) [1] MARK(primes) -> c16(A__PRIMES) [1] MARK(sieve(z0)) -> c17(A__SIEVE(mark(z0)), MARK(z0)) [1] MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) [1] MARK(head(z0)) -> c19(A__HEAD(mark(z0)), MARK(z0)) [1] MARK(tail(z0)) -> c20(A__TAIL(mark(z0)), MARK(z0)) [1] MARK(if(z0, z1, z2)) -> c21(A__IF(mark(z0), z1, z2), MARK(z0)) [1] MARK(filter(z0, z1)) -> c22(A__FILTER(mark(z0), mark(z1)), MARK(z0)) [1] MARK(filter(z0, z1)) -> c23(A__FILTER(mark(z0), mark(z1)), MARK(z1)) [1] MARK(s(z0)) -> c24(MARK(z0)) [1] MARK(cons(z0, z1)) -> c26(MARK(z0)) [1] MARK(divides(z0, z1)) -> c29(MARK(z0)) [1] MARK(divides(z0, z1)) -> c30(MARK(z1)) [1] A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) [1] A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) [1] a__primes -> a__sieve(a__from(s(s(0)))) [0] a__primes -> primes [0] a__from(z0) -> cons(mark(z0), from(s(z0))) [0] a__from(z0) -> from(z0) [0] a__head(cons(z0, z1)) -> mark(z0) [0] a__head(z0) -> head(z0) [0] a__tail(cons(z0, z1)) -> mark(z1) [0] a__tail(z0) -> tail(z0) [0] a__if(true, z0, z1) -> mark(z0) [0] a__if(false, z0, z1) -> mark(z1) [0] a__if(z0, z1, z2) -> if(z0, z1, z2) [0] a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) [0] a__filter(z0, z1) -> filter(z0, z1) [0] a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) [0] a__sieve(z0) -> sieve(z0) [0] mark(primes) -> a__primes [0] mark(sieve(z0)) -> a__sieve(mark(z0)) [0] mark(from(z0)) -> a__from(mark(z0)) [0] mark(head(z0)) -> a__head(mark(z0)) [0] mark(tail(z0)) -> a__tail(mark(z0)) [0] mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) [0] mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) [0] mark(s(z0)) -> s(mark(z0)) [0] mark(0) -> 0 [0] mark(cons(z0, z1)) -> cons(mark(z0), z1) [0] mark(true) -> true [0] mark(false) -> false [0] mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) [0] The TRS has the following type information: A__PRIMES :: c c :: c14 -> c2 -> c A__SIEVE :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides -> c14 a__from :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides s :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides 0 :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides A__FROM :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides -> c2 c2 :: c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 -> c2 MARK :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 A__HEAD :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides -> c4 cons :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides c4 :: c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 -> c4 A__TAIL :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides -> c6 c6 :: c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 -> c6 A__IF :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides -> c8:c9 true :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides c8 :: c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 -> c8:c9 false :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides c9 :: c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 -> c8:c9 c14 :: c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 -> c14 primes :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides c16 :: c -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 sieve :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides c17 :: c14 -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 mark :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides from :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides c18 :: c2 -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 head :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides c19 :: c4 -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 tail :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides c20 :: c6 -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 if :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides c21 :: c8:c9 -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 filter :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides c22 :: c11:c12 -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 A__FILTER :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides -> c11:c12 c23 :: c11:c12 -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 c24 :: c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 c26 :: c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 divides :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides c29 :: c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 c30 :: c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 c11 :: c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 -> c11:c12 c12 :: c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 -> c11:c12 a__primes :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides a__sieve :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides a__head :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides a__tail :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides a__if :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides a__filter :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides Rewrite Strategy: INNERMOST ---------------------------------------- (45) 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__PRIMES A__FROM_1 A__HEAD_1 A__TAIL_1 A__IF_3 A__SIEVE_1 MARK_1 A__FILTER_2 (c) The following functions are completely defined: a__primes a__from_1 a__head_1 a__tail_1 a__if_3 a__filter_2 a__sieve_1 mark_1 Due to the following rules being added: a__primes -> null_a__primes [0] a__from(v0) -> null_a__from [0] a__head(v0) -> null_a__head [0] a__tail(v0) -> null_a__tail [0] a__if(v0, v1, v2) -> null_a__if [0] a__filter(v0, v1) -> null_a__filter [0] a__sieve(v0) -> null_a__sieve [0] mark(v0) -> null_mark [0] And the following fresh constants: null_a__primes, null_a__from, null_a__head, null_a__tail, null_a__if, null_a__filter, null_a__sieve, null_mark, const, const1, const2, const3, const4, const5, const6, const7 ---------------------------------------- (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__PRIMES -> c(A__SIEVE(a__from(s(s(0)))), A__FROM(s(s(0)))) [1] A__FROM(z0) -> c2(MARK(z0)) [1] A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) [1] A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) [1] A__IF(true, z0, z1) -> c8(MARK(z0)) [1] A__IF(false, z0, z1) -> c9(MARK(z1)) [1] A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) [1] MARK(primes) -> c16(A__PRIMES) [1] MARK(sieve(z0)) -> c17(A__SIEVE(mark(z0)), MARK(z0)) [1] MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) [1] MARK(head(z0)) -> c19(A__HEAD(mark(z0)), MARK(z0)) [1] MARK(tail(z0)) -> c20(A__TAIL(mark(z0)), MARK(z0)) [1] MARK(if(z0, z1, z2)) -> c21(A__IF(mark(z0), z1, z2), MARK(z0)) [1] MARK(filter(z0, z1)) -> c22(A__FILTER(mark(z0), mark(z1)), MARK(z0)) [1] MARK(filter(z0, z1)) -> c23(A__FILTER(mark(z0), mark(z1)), MARK(z1)) [1] MARK(s(z0)) -> c24(MARK(z0)) [1] MARK(cons(z0, z1)) -> c26(MARK(z0)) [1] MARK(divides(z0, z1)) -> c29(MARK(z0)) [1] MARK(divides(z0, z1)) -> c30(MARK(z1)) [1] A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) [1] A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) [1] a__primes -> a__sieve(a__from(s(s(0)))) [0] a__primes -> primes [0] a__from(z0) -> cons(mark(z0), from(s(z0))) [0] a__from(z0) -> from(z0) [0] a__head(cons(z0, z1)) -> mark(z0) [0] a__head(z0) -> head(z0) [0] a__tail(cons(z0, z1)) -> mark(z1) [0] a__tail(z0) -> tail(z0) [0] a__if(true, z0, z1) -> mark(z0) [0] a__if(false, z0, z1) -> mark(z1) [0] a__if(z0, z1, z2) -> if(z0, z1, z2) [0] a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) [0] a__filter(z0, z1) -> filter(z0, z1) [0] a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) [0] a__sieve(z0) -> sieve(z0) [0] mark(primes) -> a__primes [0] mark(sieve(z0)) -> a__sieve(mark(z0)) [0] mark(from(z0)) -> a__from(mark(z0)) [0] mark(head(z0)) -> a__head(mark(z0)) [0] mark(tail(z0)) -> a__tail(mark(z0)) [0] mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) [0] mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) [0] mark(s(z0)) -> s(mark(z0)) [0] mark(0) -> 0 [0] mark(cons(z0, z1)) -> cons(mark(z0), z1) [0] mark(true) -> true [0] mark(false) -> false [0] mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) [0] a__primes -> null_a__primes [0] a__from(v0) -> null_a__from [0] a__head(v0) -> null_a__head [0] a__tail(v0) -> null_a__tail [0] a__if(v0, v1, v2) -> null_a__if [0] a__filter(v0, v1) -> null_a__filter [0] a__sieve(v0) -> null_a__sieve [0] mark(v0) -> null_mark [0] The TRS has the following type information: A__PRIMES :: c c :: c14 -> c2 -> c A__SIEVE :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> c14 a__from :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark s :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark 0 :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark A__FROM :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> c2 c2 :: c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 -> c2 MARK :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 A__HEAD :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> c4 cons :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark c4 :: c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 -> c4 A__TAIL :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> c6 c6 :: c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 -> c6 A__IF :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> c8:c9 true :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark c8 :: c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 -> c8:c9 false :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark c9 :: c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 -> c8:c9 c14 :: c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 -> c14 primes :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark c16 :: c -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 sieve :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark c17 :: c14 -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 mark :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark from :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark c18 :: c2 -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 head :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark c19 :: c4 -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 tail :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark c20 :: c6 -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 if :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark c21 :: c8:c9 -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 filter :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark c22 :: c11:c12 -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 A__FILTER :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> c11:c12 c23 :: c11:c12 -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 c24 :: c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 c26 :: c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 divides :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark c29 :: c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 c30 :: c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 c11 :: c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 -> c11:c12 c12 :: c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 -> c11:c12 a__primes :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark a__sieve :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark a__head :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark a__tail :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark a__if :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark a__filter :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark null_a__primes :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark null_a__from :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark null_a__head :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark null_a__tail :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark null_a__if :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark null_a__filter :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark null_a__sieve :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark null_mark :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark const :: c const1 :: c14 const2 :: c2 const3 :: c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 const4 :: c4 const5 :: c6 const6 :: c8:c9 const7 :: c11:c12 Rewrite Strategy: INNERMOST ---------------------------------------- (47) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (48) 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__PRIMES -> c(A__SIEVE(cons(mark(s(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) [1] A__PRIMES -> c(A__SIEVE(from(s(s(0)))), A__FROM(s(s(0)))) [1] A__PRIMES -> c(A__SIEVE(null_a__from), A__FROM(s(s(0)))) [1] A__FROM(z0) -> c2(MARK(z0)) [1] A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) [1] A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) [1] A__IF(true, z0, z1) -> c8(MARK(z0)) [1] A__IF(false, z0, z1) -> c9(MARK(z1)) [1] A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) [1] MARK(primes) -> c16(A__PRIMES) [1] MARK(sieve(primes)) -> c17(A__SIEVE(a__primes), MARK(primes)) [1] MARK(sieve(sieve(z0'))) -> c17(A__SIEVE(a__sieve(mark(z0'))), MARK(sieve(z0'))) [1] MARK(sieve(from(z0''))) -> c17(A__SIEVE(a__from(mark(z0''))), MARK(from(z0''))) [1] MARK(sieve(head(z01))) -> c17(A__SIEVE(a__head(mark(z01))), MARK(head(z01))) [1] MARK(sieve(tail(z02))) -> c17(A__SIEVE(a__tail(mark(z02))), MARK(tail(z02))) [1] MARK(sieve(if(z03, z1', z2'))) -> c17(A__SIEVE(a__if(mark(z03), z1', z2')), MARK(if(z03, z1', z2'))) [1] MARK(sieve(filter(z04, z1''))) -> c17(A__SIEVE(a__filter(mark(z04), mark(z1''))), MARK(filter(z04, z1''))) [1] MARK(sieve(s(z05))) -> c17(A__SIEVE(s(mark(z05))), MARK(s(z05))) [1] MARK(sieve(0)) -> c17(A__SIEVE(0), MARK(0)) [1] MARK(sieve(cons(z06, z11))) -> c17(A__SIEVE(cons(mark(z06), z11)), MARK(cons(z06, z11))) [1] MARK(sieve(true)) -> c17(A__SIEVE(true), MARK(true)) [1] MARK(sieve(false)) -> c17(A__SIEVE(false), MARK(false)) [1] MARK(sieve(divides(z07, z12))) -> c17(A__SIEVE(divides(mark(z07), mark(z12))), MARK(divides(z07, z12))) [1] MARK(sieve(z0)) -> c17(A__SIEVE(null_mark), MARK(z0)) [1] MARK(from(primes)) -> c18(A__FROM(a__primes), MARK(primes)) [1] MARK(from(sieve(z08))) -> c18(A__FROM(a__sieve(mark(z08))), MARK(sieve(z08))) [1] MARK(from(from(z09))) -> c18(A__FROM(a__from(mark(z09))), MARK(from(z09))) [1] MARK(from(head(z010))) -> c18(A__FROM(a__head(mark(z010))), MARK(head(z010))) [1] MARK(from(tail(z011))) -> c18(A__FROM(a__tail(mark(z011))), MARK(tail(z011))) [1] MARK(from(if(z012, z13, z2''))) -> c18(A__FROM(a__if(mark(z012), z13, z2'')), MARK(if(z012, z13, z2''))) [1] MARK(from(filter(z013, z14))) -> c18(A__FROM(a__filter(mark(z013), mark(z14))), MARK(filter(z013, z14))) [1] MARK(from(s(z014))) -> c18(A__FROM(s(mark(z014))), MARK(s(z014))) [1] MARK(from(0)) -> c18(A__FROM(0), MARK(0)) [1] MARK(from(cons(z015, z15))) -> c18(A__FROM(cons(mark(z015), z15)), MARK(cons(z015, z15))) [1] MARK(from(true)) -> c18(A__FROM(true), MARK(true)) [1] MARK(from(false)) -> c18(A__FROM(false), MARK(false)) [1] MARK(from(divides(z016, z16))) -> c18(A__FROM(divides(mark(z016), mark(z16))), MARK(divides(z016, z16))) [1] MARK(from(z0)) -> c18(A__FROM(null_mark), MARK(z0)) [1] MARK(head(primes)) -> c19(A__HEAD(a__primes), MARK(primes)) [1] MARK(head(sieve(z017))) -> c19(A__HEAD(a__sieve(mark(z017))), MARK(sieve(z017))) [1] MARK(head(from(z018))) -> c19(A__HEAD(a__from(mark(z018))), MARK(from(z018))) [1] MARK(head(head(z019))) -> c19(A__HEAD(a__head(mark(z019))), MARK(head(z019))) [1] MARK(head(tail(z020))) -> c19(A__HEAD(a__tail(mark(z020))), MARK(tail(z020))) [1] MARK(head(if(z021, z17, z21))) -> c19(A__HEAD(a__if(mark(z021), z17, z21)), MARK(if(z021, z17, z21))) [1] MARK(head(filter(z022, z18))) -> c19(A__HEAD(a__filter(mark(z022), mark(z18))), MARK(filter(z022, z18))) [1] MARK(head(s(z023))) -> c19(A__HEAD(s(mark(z023))), MARK(s(z023))) [1] MARK(head(0)) -> c19(A__HEAD(0), MARK(0)) [1] MARK(head(cons(z024, z19))) -> c19(A__HEAD(cons(mark(z024), z19)), MARK(cons(z024, z19))) [1] MARK(head(true)) -> c19(A__HEAD(true), MARK(true)) [1] MARK(head(false)) -> c19(A__HEAD(false), MARK(false)) [1] MARK(head(divides(z025, z110))) -> c19(A__HEAD(divides(mark(z025), mark(z110))), MARK(divides(z025, z110))) [1] MARK(head(z0)) -> c19(A__HEAD(null_mark), MARK(z0)) [1] MARK(tail(primes)) -> c20(A__TAIL(a__primes), MARK(primes)) [1] MARK(tail(sieve(z026))) -> c20(A__TAIL(a__sieve(mark(z026))), MARK(sieve(z026))) [1] MARK(tail(from(z027))) -> c20(A__TAIL(a__from(mark(z027))), MARK(from(z027))) [1] MARK(tail(head(z028))) -> c20(A__TAIL(a__head(mark(z028))), MARK(head(z028))) [1] MARK(tail(tail(z029))) -> c20(A__TAIL(a__tail(mark(z029))), MARK(tail(z029))) [1] MARK(tail(if(z030, z111, z22))) -> c20(A__TAIL(a__if(mark(z030), z111, z22)), MARK(if(z030, z111, z22))) [1] MARK(tail(filter(z031, z112))) -> c20(A__TAIL(a__filter(mark(z031), mark(z112))), MARK(filter(z031, z112))) [1] MARK(tail(s(z032))) -> c20(A__TAIL(s(mark(z032))), MARK(s(z032))) [1] MARK(tail(0)) -> c20(A__TAIL(0), MARK(0)) [1] MARK(tail(cons(z033, z113))) -> c20(A__TAIL(cons(mark(z033), z113)), MARK(cons(z033, z113))) [1] MARK(tail(true)) -> c20(A__TAIL(true), MARK(true)) [1] MARK(tail(false)) -> c20(A__TAIL(false), MARK(false)) [1] MARK(tail(divides(z034, z114))) -> c20(A__TAIL(divides(mark(z034), mark(z114))), MARK(divides(z034, z114))) [1] MARK(tail(z0)) -> c20(A__TAIL(null_mark), MARK(z0)) [1] MARK(if(primes, z1, z2)) -> c21(A__IF(a__primes, z1, z2), MARK(primes)) [1] MARK(if(sieve(z035), z1, z2)) -> c21(A__IF(a__sieve(mark(z035)), z1, z2), MARK(sieve(z035))) [1] MARK(if(from(z036), z1, z2)) -> c21(A__IF(a__from(mark(z036)), z1, z2), MARK(from(z036))) [1] MARK(if(head(z037), z1, z2)) -> c21(A__IF(a__head(mark(z037)), z1, z2), MARK(head(z037))) [1] MARK(if(tail(z038), z1, z2)) -> c21(A__IF(a__tail(mark(z038)), z1, z2), MARK(tail(z038))) [1] MARK(if(if(z039, z115, z23), z1, z2)) -> c21(A__IF(a__if(mark(z039), z115, z23), z1, z2), MARK(if(z039, z115, z23))) [1] MARK(if(filter(z040, z116), z1, z2)) -> c21(A__IF(a__filter(mark(z040), mark(z116)), z1, z2), MARK(filter(z040, z116))) [1] MARK(if(s(z041), z1, z2)) -> c21(A__IF(s(mark(z041)), z1, z2), MARK(s(z041))) [1] MARK(if(0, z1, z2)) -> c21(A__IF(0, z1, z2), MARK(0)) [1] MARK(if(cons(z042, z117), z1, z2)) -> c21(A__IF(cons(mark(z042), z117), z1, z2), MARK(cons(z042, z117))) [1] MARK(if(true, z1, z2)) -> c21(A__IF(true, z1, z2), MARK(true)) [1] MARK(if(false, z1, z2)) -> c21(A__IF(false, z1, z2), MARK(false)) [1] MARK(if(divides(z043, z118), z1, z2)) -> c21(A__IF(divides(mark(z043), mark(z118)), z1, z2), MARK(divides(z043, z118))) [1] MARK(if(z0, z1, z2)) -> c21(A__IF(null_mark, z1, z2), MARK(z0)) [1] MARK(filter(z0, z1)) -> c22(A__FILTER(mark(z0), mark(z1)), MARK(z0)) [1] MARK(filter(z0, z1)) -> c23(A__FILTER(mark(z0), mark(z1)), MARK(z1)) [1] MARK(s(z0)) -> c24(MARK(z0)) [1] MARK(cons(z0, z1)) -> c26(MARK(z0)) [1] MARK(divides(z0, z1)) -> c29(MARK(z0)) [1] MARK(divides(z0, z1)) -> c30(MARK(z1)) [1] A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) [1] A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) [1] a__primes -> a__sieve(cons(mark(s(s(0))), from(s(s(s(0)))))) [0] a__primes -> a__sieve(from(s(s(0)))) [0] a__primes -> a__sieve(null_a__from) [0] a__primes -> primes [0] a__from(z0) -> cons(mark(z0), from(s(z0))) [0] a__from(z0) -> from(z0) [0] a__head(cons(z0, z1)) -> mark(z0) [0] a__head(z0) -> head(z0) [0] a__tail(cons(z0, z1)) -> mark(z1) [0] a__tail(z0) -> tail(z0) [0] a__if(true, z0, z1) -> mark(z0) [0] a__if(false, z0, z1) -> mark(z1) [0] a__if(z0, z1, z2) -> if(z0, z1, z2) [0] a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) [0] a__filter(z0, z1) -> filter(z0, z1) [0] a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) [0] a__sieve(z0) -> sieve(z0) [0] mark(primes) -> a__primes [0] mark(sieve(primes)) -> a__sieve(a__primes) [0] mark(sieve(sieve(z0449))) -> a__sieve(a__sieve(mark(z0449))) [0] mark(sieve(from(z0450))) -> a__sieve(a__from(mark(z0450))) [0] mark(sieve(head(z0451))) -> a__sieve(a__head(mark(z0451))) [0] mark(sieve(tail(z0452))) -> a__sieve(a__tail(mark(z0452))) [0] mark(sieve(if(z0453, z1199, z249))) -> a__sieve(a__if(mark(z0453), z1199, z249)) [0] mark(sieve(filter(z0454, z1200))) -> a__sieve(a__filter(mark(z0454), mark(z1200))) [0] mark(sieve(s(z0455))) -> a__sieve(s(mark(z0455))) [0] mark(sieve(0)) -> a__sieve(0) [0] mark(sieve(cons(z0456, z1201))) -> a__sieve(cons(mark(z0456), z1201)) [0] mark(sieve(true)) -> a__sieve(true) [0] mark(sieve(false)) -> a__sieve(false) [0] mark(sieve(divides(z0457, z1202))) -> a__sieve(divides(mark(z0457), mark(z1202))) [0] mark(sieve(z0)) -> a__sieve(null_mark) [0] mark(from(primes)) -> a__from(a__primes) [0] mark(from(sieve(z0458))) -> a__from(a__sieve(mark(z0458))) [0] mark(from(from(z0459))) -> a__from(a__from(mark(z0459))) [0] mark(from(head(z0460))) -> a__from(a__head(mark(z0460))) [0] mark(from(tail(z0461))) -> a__from(a__tail(mark(z0461))) [0] mark(from(if(z0462, z1203, z250))) -> a__from(a__if(mark(z0462), z1203, z250)) [0] mark(from(filter(z0463, z1204))) -> a__from(a__filter(mark(z0463), mark(z1204))) [0] mark(from(s(z0464))) -> a__from(s(mark(z0464))) [0] mark(from(0)) -> a__from(0) [0] mark(from(cons(z0465, z1205))) -> a__from(cons(mark(z0465), z1205)) [0] mark(from(true)) -> a__from(true) [0] mark(from(false)) -> a__from(false) [0] mark(from(divides(z0466, z1206))) -> a__from(divides(mark(z0466), mark(z1206))) [0] mark(from(z0)) -> a__from(null_mark) [0] mark(head(primes)) -> a__head(a__primes) [0] mark(head(sieve(z0467))) -> a__head(a__sieve(mark(z0467))) [0] mark(head(from(z0468))) -> a__head(a__from(mark(z0468))) [0] mark(head(head(z0469))) -> a__head(a__head(mark(z0469))) [0] mark(head(tail(z0470))) -> a__head(a__tail(mark(z0470))) [0] mark(head(if(z0471, z1207, z251))) -> a__head(a__if(mark(z0471), z1207, z251)) [0] mark(head(filter(z0472, z1208))) -> a__head(a__filter(mark(z0472), mark(z1208))) [0] mark(head(s(z0473))) -> a__head(s(mark(z0473))) [0] mark(head(0)) -> a__head(0) [0] mark(head(cons(z0474, z1209))) -> a__head(cons(mark(z0474), z1209)) [0] mark(head(true)) -> a__head(true) [0] mark(head(false)) -> a__head(false) [0] mark(head(divides(z0475, z1210))) -> a__head(divides(mark(z0475), mark(z1210))) [0] mark(head(z0)) -> a__head(null_mark) [0] mark(tail(primes)) -> a__tail(a__primes) [0] mark(tail(sieve(z0476))) -> a__tail(a__sieve(mark(z0476))) [0] mark(tail(from(z0477))) -> a__tail(a__from(mark(z0477))) [0] mark(tail(head(z0478))) -> a__tail(a__head(mark(z0478))) [0] mark(tail(tail(z0479))) -> a__tail(a__tail(mark(z0479))) [0] mark(tail(if(z0480, z1211, z252))) -> a__tail(a__if(mark(z0480), z1211, z252)) [0] mark(tail(filter(z0481, z1212))) -> a__tail(a__filter(mark(z0481), mark(z1212))) [0] mark(tail(s(z0482))) -> a__tail(s(mark(z0482))) [0] mark(tail(0)) -> a__tail(0) [0] mark(tail(cons(z0483, z1213))) -> a__tail(cons(mark(z0483), z1213)) [0] mark(tail(true)) -> a__tail(true) [0] mark(tail(false)) -> a__tail(false) [0] mark(tail(divides(z0484, z1214))) -> a__tail(divides(mark(z0484), mark(z1214))) [0] mark(tail(z0)) -> a__tail(null_mark) [0] mark(if(primes, z1, z2)) -> a__if(a__primes, z1, z2) [0] mark(if(sieve(z0485), z1, z2)) -> a__if(a__sieve(mark(z0485)), z1, z2) [0] mark(if(from(z0486), z1, z2)) -> a__if(a__from(mark(z0486)), z1, z2) [0] mark(if(head(z0487), z1, z2)) -> a__if(a__head(mark(z0487)), z1, z2) [0] mark(if(tail(z0488), z1, z2)) -> a__if(a__tail(mark(z0488)), z1, z2) [0] mark(if(if(z0489, z1215, z253), z1, z2)) -> a__if(a__if(mark(z0489), z1215, z253), z1, z2) [0] mark(if(filter(z0490, z1216), z1, z2)) -> a__if(a__filter(mark(z0490), mark(z1216)), z1, z2) [0] mark(if(s(z0491), z1, z2)) -> a__if(s(mark(z0491)), z1, z2) [0] mark(if(0, z1, z2)) -> a__if(0, z1, z2) [0] mark(if(cons(z0492, z1217), z1, z2)) -> a__if(cons(mark(z0492), z1217), z1, z2) [0] mark(if(true, z1, z2)) -> a__if(true, z1, z2) [0] mark(if(false, z1, z2)) -> a__if(false, z1, z2) [0] mark(if(divides(z0493, z1218), z1, z2)) -> a__if(divides(mark(z0493), mark(z1218)), z1, z2) [0] mark(if(z0, z1, z2)) -> a__if(null_mark, z1, z2) [0] mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) [0] mark(s(z0)) -> s(mark(z0)) [0] mark(0) -> 0 [0] mark(cons(z0, z1)) -> cons(mark(z0), z1) [0] mark(true) -> true [0] mark(false) -> false [0] mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) [0] a__primes -> null_a__primes [0] a__from(v0) -> null_a__from [0] a__head(v0) -> null_a__head [0] a__tail(v0) -> null_a__tail [0] a__if(v0, v1, v2) -> null_a__if [0] a__filter(v0, v1) -> null_a__filter [0] a__sieve(v0) -> null_a__sieve [0] mark(v0) -> null_mark [0] The TRS has the following type information: A__PRIMES :: c c :: c14 -> c2 -> c A__SIEVE :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> c14 a__from :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark s :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark 0 :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark A__FROM :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> c2 c2 :: c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 -> c2 MARK :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 A__HEAD :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> c4 cons :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark c4 :: c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 -> c4 A__TAIL :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> c6 c6 :: c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 -> c6 A__IF :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> c8:c9 true :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark c8 :: c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 -> c8:c9 false :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark c9 :: c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 -> c8:c9 c14 :: c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 -> c14 primes :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark c16 :: c -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 sieve :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark c17 :: c14 -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 mark :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark from :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark c18 :: c2 -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 head :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark c19 :: c4 -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 tail :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark c20 :: c6 -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 if :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark c21 :: c8:c9 -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 filter :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark c22 :: c11:c12 -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 A__FILTER :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> c11:c12 c23 :: c11:c12 -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 c24 :: c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 c26 :: c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 divides :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark c29 :: c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 c30 :: c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 c11 :: c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 -> c11:c12 c12 :: c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 -> c11:c12 a__primes :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark a__sieve :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark a__head :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark a__tail :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark a__if :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark a__filter :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark null_a__primes :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark null_a__from :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark null_a__head :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark null_a__tail :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark null_a__if :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark null_a__filter :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark null_a__sieve :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark null_mark :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark const :: c const1 :: c14 const2 :: c2 const3 :: c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30 const4 :: c4 const5 :: c6 const6 :: c8:c9 const7 :: c11:c12 Rewrite Strategy: INNERMOST ---------------------------------------- (49) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: 0 => 0 true => 3 false => 1 primes => 2 null_a__primes => 0 null_a__from => 0 null_a__head => 0 null_a__tail => 0 null_a__if => 0 null_a__filter => 0 null_a__sieve => 0 null_mark => 0 const => 0 const1 => 0 const2 => 0 const3 => 0 const4 => 0 const5 => 0 const6 => 0 const7 => 0 ---------------------------------------- (50) Obligation: Complexity RNTS consisting of the following rules: A__FILTER(z, z') -{ 1 }-> 1 + MARK(z0) :|: z1 >= 0, z' = 1 + z1 + z2, z0 >= 0, z = 1 + (1 + z0), z2 >= 0 A__FILTER(z, z') -{ 1 }-> 1 + MARK(z1) :|: z1 >= 0, z' = 1 + z1 + z2, z0 >= 0, z = 1 + (1 + z0), z2 >= 0 A__FROM(z) -{ 1 }-> 1 + MARK(z0) :|: z = z0, z0 >= 0 A__HEAD(z) -{ 1 }-> 1 + MARK(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 A__IF(z, z', z'') -{ 1 }-> 1 + MARK(z0) :|: z = 3, z1 >= 0, z0 >= 0, z' = z0, z'' = z1 A__IF(z, z', z'') -{ 1 }-> 1 + MARK(z1) :|: z1 >= 0, z = 1, z0 >= 0, z' = z0, z'' = z1 A__PRIMES -{ 1 }-> 1 + A__SIEVE(0) + A__FROM(1 + (1 + 0)) :|: A__PRIMES -{ 1 }-> 1 + A__SIEVE(1 + (1 + (1 + 0))) + A__FROM(1 + (1 + 0)) :|: A__PRIMES -{ 1 }-> 1 + A__SIEVE(1 + mark(1 + (1 + 0)) + (1 + (1 + (1 + (1 + 0))))) + A__FROM(1 + (1 + 0)) :|: A__SIEVE(z) -{ 1 }-> 1 + MARK(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 A__TAIL(z) -{ 1 }-> 1 + MARK(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 MARK(z) -{ 1 }-> 1 + MARK(z0) :|: z = 1 + z0, z0 >= 0 MARK(z) -{ 1 }-> 1 + MARK(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 MARK(z) -{ 1 }-> 1 + MARK(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 MARK(z) -{ 1 }-> 1 + A__PRIMES :|: z = 2 MARK(z) -{ 1 }-> 1 + A__TAIL(a__tail(mark(z029))) + MARK(1 + z029) :|: z029 >= 0, z = 1 + (1 + z029) MARK(z) -{ 1 }-> 1 + A__TAIL(a__sieve(mark(z026))) + MARK(1 + z026) :|: z = 1 + (1 + z026), z026 >= 0 MARK(z) -{ 1 }-> 1 + A__TAIL(a__primes) + MARK(2) :|: z = 1 + 2 MARK(z) -{ 1 }-> 1 + A__TAIL(a__if(mark(z030), z111, z22)) + MARK(1 + z030 + z111 + z22) :|: z = 1 + (1 + z030 + z111 + z22), z111 >= 0, z22 >= 0, z030 >= 0 MARK(z) -{ 1 }-> 1 + A__TAIL(a__head(mark(z028))) + MARK(1 + z028) :|: z028 >= 0, z = 1 + (1 + z028) MARK(z) -{ 1 }-> 1 + A__TAIL(a__from(mark(z027))) + MARK(1 + z027) :|: z027 >= 0, z = 1 + (1 + z027) MARK(z) -{ 1 }-> 1 + A__TAIL(a__filter(mark(z031), mark(z112))) + MARK(1 + z031 + z112) :|: z031 >= 0, z112 >= 0, z = 1 + (1 + z031 + z112) MARK(z) -{ 1 }-> 1 + A__TAIL(3) + MARK(3) :|: z = 1 + 3 MARK(z) -{ 1 }-> 1 + A__TAIL(1) + MARK(1) :|: z = 1 + 1 MARK(z) -{ 1 }-> 1 + A__TAIL(0) + MARK(z0) :|: z = 1 + z0, z0 >= 0 MARK(z) -{ 1 }-> 1 + A__TAIL(0) + MARK(0) :|: z = 1 + 0 MARK(z) -{ 1 }-> 1 + A__TAIL(1 + mark(z032)) + MARK(1 + z032) :|: z032 >= 0, z = 1 + (1 + z032) MARK(z) -{ 1 }-> 1 + A__TAIL(1 + mark(z033) + z113) + MARK(1 + z033 + z113) :|: z = 1 + (1 + z033 + z113), z033 >= 0, z113 >= 0 MARK(z) -{ 1 }-> 1 + A__TAIL(1 + mark(z034) + mark(z114)) + MARK(1 + z034 + z114) :|: z114 >= 0, z034 >= 0, z = 1 + (1 + z034 + z114) MARK(z) -{ 1 }-> 1 + A__SIEVE(a__tail(mark(z02))) + MARK(1 + z02) :|: z = 1 + (1 + z02), z02 >= 0 MARK(z) -{ 1 }-> 1 + A__SIEVE(a__sieve(mark(z0'))) + MARK(1 + z0') :|: z = 1 + (1 + z0'), z0' >= 0 MARK(z) -{ 1 }-> 1 + A__SIEVE(a__primes) + MARK(2) :|: z = 1 + 2 MARK(z) -{ 1 }-> 1 + A__SIEVE(a__if(mark(z03), z1', z2')) + MARK(1 + z03 + z1' + z2') :|: z = 1 + (1 + z03 + z1' + z2'), z1' >= 0, z2' >= 0, z03 >= 0 MARK(z) -{ 1 }-> 1 + A__SIEVE(a__head(mark(z01))) + MARK(1 + z01) :|: z = 1 + (1 + z01), z01 >= 0 MARK(z) -{ 1 }-> 1 + A__SIEVE(a__from(mark(z0''))) + MARK(1 + z0'') :|: z = 1 + (1 + z0''), z0'' >= 0 MARK(z) -{ 1 }-> 1 + A__SIEVE(a__filter(mark(z04), mark(z1''))) + MARK(1 + z04 + z1'') :|: z04 >= 0, z = 1 + (1 + z04 + z1''), z1'' >= 0 MARK(z) -{ 1 }-> 1 + A__SIEVE(3) + MARK(3) :|: z = 1 + 3 MARK(z) -{ 1 }-> 1 + A__SIEVE(1) + MARK(1) :|: z = 1 + 1 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(z05)) + MARK(1 + z05) :|: z05 >= 0, z = 1 + (1 + z05) MARK(z) -{ 1 }-> 1 + A__SIEVE(1 + mark(z06) + z11) + MARK(1 + z06 + z11) :|: z11 >= 0, z06 >= 0, z = 1 + (1 + z06 + z11) MARK(z) -{ 1 }-> 1 + A__SIEVE(1 + mark(z07) + mark(z12)) + MARK(1 + z07 + z12) :|: z07 >= 0, z12 >= 0, z = 1 + (1 + z07 + z12) MARK(z) -{ 1 }-> 1 + A__IF(a__tail(mark(z038)), z1, z2) + MARK(1 + z038) :|: z = 1 + (1 + z038) + z1 + z2, z1 >= 0, z038 >= 0, z2 >= 0 MARK(z) -{ 1 }-> 1 + A__IF(a__sieve(mark(z035)), z1, z2) + MARK(1 + z035) :|: z035 >= 0, z1 >= 0, z = 1 + (1 + z035) + z1 + z2, z2 >= 0 MARK(z) -{ 1 }-> 1 + A__IF(a__primes, z1, z2) + MARK(2) :|: z1 >= 0, z = 1 + 2 + z1 + z2, z2 >= 0 MARK(z) -{ 1 }-> 1 + A__IF(a__if(mark(z039), z115, z23), z1, z2) + MARK(1 + z039 + z115 + z23) :|: z1 >= 0, z039 >= 0, z23 >= 0, z = 1 + (1 + z039 + z115 + z23) + z1 + z2, z115 >= 0, z2 >= 0 MARK(z) -{ 1 }-> 1 + A__IF(a__head(mark(z037)), z1, z2) + MARK(1 + z037) :|: z1 >= 0, z = 1 + (1 + z037) + z1 + z2, z037 >= 0, z2 >= 0 MARK(z) -{ 1 }-> 1 + A__IF(a__from(mark(z036)), z1, z2) + MARK(1 + z036) :|: z036 >= 0, z1 >= 0, z = 1 + (1 + z036) + z1 + z2, z2 >= 0 MARK(z) -{ 1 }-> 1 + A__IF(a__filter(mark(z040), mark(z116)), z1, z2) + MARK(1 + z040 + z116) :|: z1 >= 0, z = 1 + (1 + z040 + z116) + z1 + z2, z040 >= 0, z2 >= 0, z116 >= 0 MARK(z) -{ 1 }-> 1 + A__IF(3, z1, z2) + MARK(3) :|: z1 >= 0, z = 1 + 3 + z1 + z2, z2 >= 0 MARK(z) -{ 1 }-> 1 + A__IF(1, z1, z2) + MARK(1) :|: z1 >= 0, z = 1 + 1 + z1 + z2, z2 >= 0 MARK(z) -{ 1 }-> 1 + A__IF(0, z1, z2) + MARK(z0) :|: z1 >= 0, z = 1 + z0 + z1 + z2, z0 >= 0, z2 >= 0 MARK(z) -{ 1 }-> 1 + A__IF(0, z1, z2) + MARK(0) :|: z1 >= 0, z = 1 + 0 + z1 + z2, z2 >= 0 MARK(z) -{ 1 }-> 1 + A__IF(1 + mark(z041), z1, z2) + MARK(1 + z041) :|: z = 1 + (1 + z041) + z1 + z2, z1 >= 0, z041 >= 0, z2 >= 0 MARK(z) -{ 1 }-> 1 + A__IF(1 + mark(z042) + z117, z1, z2) + MARK(1 + z042 + z117) :|: z = 1 + (1 + z042 + z117) + z1 + z2, z042 >= 0, z117 >= 0, z1 >= 0, z2 >= 0 MARK(z) -{ 1 }-> 1 + A__IF(1 + mark(z043) + mark(z118), z1, z2) + MARK(1 + z043 + z118) :|: z1 >= 0, z118 >= 0, z043 >= 0, z = 1 + (1 + z043 + z118) + z1 + z2, z2 >= 0 MARK(z) -{ 1 }-> 1 + A__HEAD(a__tail(mark(z020))) + MARK(1 + z020) :|: z = 1 + (1 + z020), z020 >= 0 MARK(z) -{ 1 }-> 1 + A__HEAD(a__sieve(mark(z017))) + MARK(1 + z017) :|: z = 1 + (1 + z017), z017 >= 0 MARK(z) -{ 1 }-> 1 + A__HEAD(a__primes) + MARK(2) :|: z = 1 + 2 MARK(z) -{ 1 }-> 1 + A__HEAD(a__if(mark(z021), z17, z21)) + MARK(1 + z021 + z17 + z21) :|: z021 >= 0, z21 >= 0, z17 >= 0, z = 1 + (1 + z021 + z17 + z21) MARK(z) -{ 1 }-> 1 + A__HEAD(a__head(mark(z019))) + MARK(1 + z019) :|: z019 >= 0, z = 1 + (1 + z019) MARK(z) -{ 1 }-> 1 + A__HEAD(a__from(mark(z018))) + MARK(1 + z018) :|: z = 1 + (1 + z018), z018 >= 0 MARK(z) -{ 1 }-> 1 + A__HEAD(a__filter(mark(z022), mark(z18))) + MARK(1 + z022 + z18) :|: z18 >= 0, z022 >= 0, z = 1 + (1 + z022 + z18) MARK(z) -{ 1 }-> 1 + A__HEAD(3) + MARK(3) :|: z = 1 + 3 MARK(z) -{ 1 }-> 1 + A__HEAD(1) + MARK(1) :|: z = 1 + 1 MARK(z) -{ 1 }-> 1 + A__HEAD(0) + MARK(z0) :|: z = 1 + z0, z0 >= 0 MARK(z) -{ 1 }-> 1 + A__HEAD(0) + MARK(0) :|: z = 1 + 0 MARK(z) -{ 1 }-> 1 + A__HEAD(1 + mark(z023)) + MARK(1 + z023) :|: z023 >= 0, z = 1 + (1 + z023) MARK(z) -{ 1 }-> 1 + A__HEAD(1 + mark(z024) + z19) + MARK(1 + z024 + z19) :|: z19 >= 0, z024 >= 0, z = 1 + (1 + z024 + z19) MARK(z) -{ 1 }-> 1 + A__HEAD(1 + mark(z025) + mark(z110)) + MARK(1 + z025 + z110) :|: z025 >= 0, z110 >= 0, z = 1 + (1 + z025 + z110) MARK(z) -{ 1 }-> 1 + A__FROM(a__tail(mark(z011))) + MARK(1 + z011) :|: z = 1 + (1 + z011), z011 >= 0 MARK(z) -{ 1 }-> 1 + A__FROM(a__sieve(mark(z08))) + MARK(1 + z08) :|: z08 >= 0, z = 1 + (1 + z08) MARK(z) -{ 1 }-> 1 + A__FROM(a__primes) + MARK(2) :|: z = 1 + 2 MARK(z) -{ 1 }-> 1 + A__FROM(a__if(mark(z012), z13, z2'')) + MARK(1 + z012 + z13 + z2'') :|: z = 1 + (1 + z012 + z13 + z2''), z012 >= 0, z2'' >= 0, z13 >= 0 MARK(z) -{ 1 }-> 1 + A__FROM(a__head(mark(z010))) + MARK(1 + z010) :|: z = 1 + (1 + z010), z010 >= 0 MARK(z) -{ 1 }-> 1 + A__FROM(a__from(mark(z09))) + MARK(1 + z09) :|: z = 1 + (1 + z09), z09 >= 0 MARK(z) -{ 1 }-> 1 + A__FROM(a__filter(mark(z013), mark(z14))) + MARK(1 + z013 + z14) :|: z = 1 + (1 + z013 + z14), z013 >= 0, z14 >= 0 MARK(z) -{ 1 }-> 1 + A__FROM(3) + MARK(3) :|: z = 1 + 3 MARK(z) -{ 1 }-> 1 + A__FROM(1) + MARK(1) :|: z = 1 + 1 MARK(z) -{ 1 }-> 1 + A__FROM(0) + MARK(z0) :|: z = 1 + z0, z0 >= 0 MARK(z) -{ 1 }-> 1 + A__FROM(0) + MARK(0) :|: z = 1 + 0 MARK(z) -{ 1 }-> 1 + A__FROM(1 + mark(z014)) + MARK(1 + z014) :|: z = 1 + (1 + z014), z014 >= 0 MARK(z) -{ 1 }-> 1 + A__FROM(1 + mark(z015) + z15) + MARK(1 + z015 + z15) :|: z015 >= 0, z15 >= 0, z = 1 + (1 + z015 + z15) MARK(z) -{ 1 }-> 1 + A__FROM(1 + mark(z016) + mark(z16)) + MARK(1 + z016 + z16) :|: z = 1 + (1 + z016 + z16), z016 >= 0, z16 >= 0 MARK(z) -{ 1 }-> 1 + A__FILTER(mark(z0), mark(z1)) + MARK(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 MARK(z) -{ 1 }-> 1 + A__FILTER(mark(z0), mark(z1)) + MARK(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 a__filter(z, z') -{ 0 }-> a__if(1 + (1 + (1 + mark(z0))) + mark(z1), 1 + (1 + (1 + z0)) + z2, 1 + z1 + (1 + z0 + (1 + z1))) :|: z1 >= 0, z' = 1 + z1 + z2, z0 >= 0, z = 1 + (1 + z0), z2 >= 0 a__filter(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 a__filter(z, z') -{ 0 }-> 1 + z0 + z1 :|: z = z0, z1 >= 0, z' = z1, z0 >= 0 a__from(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 a__from(z) -{ 0 }-> 1 + z0 :|: z = z0, z0 >= 0 a__from(z) -{ 0 }-> 1 + mark(z0) + (1 + (1 + z0)) :|: z = z0, z0 >= 0 a__head(z) -{ 0 }-> mark(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 a__head(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 a__head(z) -{ 0 }-> 1 + z0 :|: z = z0, z0 >= 0 a__if(z, z', z'') -{ 0 }-> mark(z0) :|: z = 3, z1 >= 0, z0 >= 0, z' = z0, z'' = z1 a__if(z, z', z'') -{ 0 }-> mark(z1) :|: z1 >= 0, z = 1, z0 >= 0, z' = z0, z'' = z1 a__if(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 a__if(z, z', z'') -{ 0 }-> 1 + z0 + z1 + z2 :|: z'' = z2, z = z0, z1 >= 0, z' = z1, z0 >= 0, z2 >= 0 a__primes -{ 0 }-> a__sieve(0) :|: a__primes -{ 0 }-> a__sieve(1 + (1 + (1 + 0))) :|: a__primes -{ 0 }-> a__sieve(1 + mark(1 + (1 + 0)) + (1 + (1 + (1 + (1 + 0))))) :|: a__primes -{ 0 }-> 2 :|: a__primes -{ 0 }-> 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 + mark(z0) + (1 + z0 + (1 + z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 a__tail(z) -{ 0 }-> mark(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 a__tail(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 a__tail(z) -{ 0 }-> 1 + z0 :|: z = z0, z0 >= 0 mark(z) -{ 0 }-> a__tail(a__tail(mark(z0479))) :|: z = 1 + (1 + z0479), z0479 >= 0 mark(z) -{ 0 }-> a__tail(a__sieve(mark(z0476))) :|: z = 1 + (1 + z0476), z0476 >= 0 mark(z) -{ 0 }-> a__tail(a__primes) :|: z = 1 + 2 mark(z) -{ 0 }-> a__tail(a__if(mark(z0480), z1211, z252)) :|: z = 1 + (1 + z0480 + z1211 + z252), z252 >= 0, z0480 >= 0, z1211 >= 0 mark(z) -{ 0 }-> a__tail(a__head(mark(z0478))) :|: z = 1 + (1 + z0478), z0478 >= 0 mark(z) -{ 0 }-> a__tail(a__from(mark(z0477))) :|: z = 1 + (1 + z0477), z0477 >= 0 mark(z) -{ 0 }-> a__tail(a__filter(mark(z0481), mark(z1212))) :|: z1212 >= 0, z0481 >= 0, z = 1 + (1 + z0481 + z1212) mark(z) -{ 0 }-> a__tail(3) :|: z = 1 + 3 mark(z) -{ 0 }-> a__tail(1) :|: z = 1 + 1 mark(z) -{ 0 }-> a__tail(0) :|: z = 1 + 0 mark(z) -{ 0 }-> a__tail(0) :|: z = 1 + z0, z0 >= 0 mark(z) -{ 0 }-> a__tail(1 + mark(z0482)) :|: z = 1 + (1 + z0482), z0482 >= 0 mark(z) -{ 0 }-> a__tail(1 + mark(z0483) + z1213) :|: z1213 >= 0, z0483 >= 0, z = 1 + (1 + z0483 + z1213) mark(z) -{ 0 }-> a__tail(1 + mark(z0484) + mark(z1214)) :|: z1214 >= 0, z0484 >= 0, z = 1 + (1 + z0484 + z1214) mark(z) -{ 0 }-> a__sieve(a__tail(mark(z0452))) :|: z = 1 + (1 + z0452), z0452 >= 0 mark(z) -{ 0 }-> a__sieve(a__sieve(mark(z0449))) :|: z0449 >= 0, z = 1 + (1 + z0449) mark(z) -{ 0 }-> a__sieve(a__primes) :|: z = 1 + 2 mark(z) -{ 0 }-> a__sieve(a__if(mark(z0453), z1199, z249)) :|: z = 1 + (1 + z0453 + z1199 + z249), z249 >= 0, z1199 >= 0, z0453 >= 0 mark(z) -{ 0 }-> a__sieve(a__head(mark(z0451))) :|: z = 1 + (1 + z0451), z0451 >= 0 mark(z) -{ 0 }-> a__sieve(a__from(mark(z0450))) :|: z0450 >= 0, z = 1 + (1 + z0450) mark(z) -{ 0 }-> a__sieve(a__filter(mark(z0454), mark(z1200))) :|: z = 1 + (1 + z0454 + z1200), z0454 >= 0, z1200 >= 0 mark(z) -{ 0 }-> a__sieve(3) :|: z = 1 + 3 mark(z) -{ 0 }-> a__sieve(1) :|: z = 1 + 1 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(z0455)) :|: z0455 >= 0, z = 1 + (1 + z0455) mark(z) -{ 0 }-> a__sieve(1 + mark(z0456) + z1201) :|: z1201 >= 0, z0456 >= 0, z = 1 + (1 + z0456 + z1201) mark(z) -{ 0 }-> a__sieve(1 + mark(z0457) + mark(z1202)) :|: z1202 >= 0, z0457 >= 0, z = 1 + (1 + z0457 + z1202) mark(z) -{ 0 }-> a__primes :|: z = 2 mark(z) -{ 0 }-> a__if(a__tail(mark(z0488)), z1, z2) :|: z = 1 + (1 + z0488) + z1 + z2, z1 >= 0, z0488 >= 0, z2 >= 0 mark(z) -{ 0 }-> a__if(a__sieve(mark(z0485)), z1, z2) :|: z0485 >= 0, z1 >= 0, z2 >= 0, z = 1 + (1 + z0485) + z1 + z2 mark(z) -{ 0 }-> a__if(a__primes, z1, z2) :|: z1 >= 0, z = 1 + 2 + z1 + z2, z2 >= 0 mark(z) -{ 0 }-> a__if(a__if(mark(z0489), z1215, z253), z1, z2) :|: z0489 >= 0, z1 >= 0, z = 1 + (1 + z0489 + z1215 + z253) + z1 + z2, z1215 >= 0, z253 >= 0, z2 >= 0 mark(z) -{ 0 }-> a__if(a__head(mark(z0487)), z1, z2) :|: z1 >= 0, z = 1 + (1 + z0487) + z1 + z2, z0487 >= 0, z2 >= 0 mark(z) -{ 0 }-> a__if(a__from(mark(z0486)), z1, z2) :|: z1 >= 0, z0486 >= 0, z = 1 + (1 + z0486) + z1 + z2, z2 >= 0 mark(z) -{ 0 }-> a__if(a__filter(mark(z0490), mark(z1216)), z1, z2) :|: z = 1 + (1 + z0490 + z1216) + z1 + z2, z1216 >= 0, z1 >= 0, z0490 >= 0, z2 >= 0 mark(z) -{ 0 }-> a__if(3, z1, z2) :|: z1 >= 0, z = 1 + 3 + z1 + z2, z2 >= 0 mark(z) -{ 0 }-> a__if(1, z1, z2) :|: z1 >= 0, z = 1 + 1 + z1 + z2, z2 >= 0 mark(z) -{ 0 }-> a__if(0, z1, z2) :|: z1 >= 0, z = 1 + 0 + z1 + z2, z2 >= 0 mark(z) -{ 0 }-> a__if(0, z1, z2) :|: z1 >= 0, z = 1 + z0 + z1 + z2, z0 >= 0, z2 >= 0 mark(z) -{ 0 }-> a__if(1 + mark(z0491), z1, z2) :|: z1 >= 0, z = 1 + (1 + z0491) + z1 + z2, z0491 >= 0, z2 >= 0 mark(z) -{ 0 }-> a__if(1 + mark(z0492) + z1217, z1, z2) :|: z1 >= 0, z1217 >= 0, z = 1 + (1 + z0492 + z1217) + z1 + z2, z2 >= 0, z0492 >= 0 mark(z) -{ 0 }-> a__if(1 + mark(z0493) + mark(z1218), z1, z2) :|: z1 >= 0, z = 1 + (1 + z0493 + z1218) + z1 + z2, z1218 >= 0, z0493 >= 0, z2 >= 0 mark(z) -{ 0 }-> a__head(a__tail(mark(z0470))) :|: z0470 >= 0, z = 1 + (1 + z0470) mark(z) -{ 0 }-> a__head(a__sieve(mark(z0467))) :|: z = 1 + (1 + z0467), z0467 >= 0 mark(z) -{ 0 }-> a__head(a__primes) :|: z = 1 + 2 mark(z) -{ 0 }-> a__head(a__if(mark(z0471), z1207, z251)) :|: z0471 >= 0, z = 1 + (1 + z0471 + z1207 + z251), z251 >= 0, z1207 >= 0 mark(z) -{ 0 }-> a__head(a__head(mark(z0469))) :|: z = 1 + (1 + z0469), z0469 >= 0 mark(z) -{ 0 }-> a__head(a__from(mark(z0468))) :|: z = 1 + (1 + z0468), z0468 >= 0 mark(z) -{ 0 }-> a__head(a__filter(mark(z0472), mark(z1208))) :|: z = 1 + (1 + z0472 + z1208), z1208 >= 0, z0472 >= 0 mark(z) -{ 0 }-> a__head(3) :|: z = 1 + 3 mark(z) -{ 0 }-> a__head(1) :|: z = 1 + 1 mark(z) -{ 0 }-> a__head(0) :|: z = 1 + 0 mark(z) -{ 0 }-> a__head(0) :|: z = 1 + z0, z0 >= 0 mark(z) -{ 0 }-> a__head(1 + mark(z0473)) :|: z0473 >= 0, z = 1 + (1 + z0473) mark(z) -{ 0 }-> a__head(1 + mark(z0474) + z1209) :|: z1209 >= 0, z0474 >= 0, z = 1 + (1 + z0474 + z1209) mark(z) -{ 0 }-> a__head(1 + mark(z0475) + mark(z1210)) :|: z0475 >= 0, z = 1 + (1 + z0475 + z1210), z1210 >= 0 mark(z) -{ 0 }-> a__from(a__tail(mark(z0461))) :|: z = 1 + (1 + z0461), z0461 >= 0 mark(z) -{ 0 }-> a__from(a__sieve(mark(z0458))) :|: z0458 >= 0, z = 1 + (1 + z0458) mark(z) -{ 0 }-> a__from(a__primes) :|: z = 1 + 2 mark(z) -{ 0 }-> a__from(a__if(mark(z0462), z1203, z250)) :|: z250 >= 0, z1203 >= 0, z = 1 + (1 + z0462 + z1203 + z250), z0462 >= 0 mark(z) -{ 0 }-> a__from(a__head(mark(z0460))) :|: z0460 >= 0, z = 1 + (1 + z0460) mark(z) -{ 0 }-> a__from(a__from(mark(z0459))) :|: z = 1 + (1 + z0459), z0459 >= 0 mark(z) -{ 0 }-> a__from(a__filter(mark(z0463), mark(z1204))) :|: z = 1 + (1 + z0463 + z1204), z1204 >= 0, z0463 >= 0 mark(z) -{ 0 }-> a__from(3) :|: z = 1 + 3 mark(z) -{ 0 }-> a__from(1) :|: z = 1 + 1 mark(z) -{ 0 }-> a__from(0) :|: z = 1 + 0 mark(z) -{ 0 }-> a__from(0) :|: z = 1 + z0, z0 >= 0 mark(z) -{ 0 }-> a__from(1 + mark(z0464)) :|: z0464 >= 0, z = 1 + (1 + z0464) mark(z) -{ 0 }-> a__from(1 + mark(z0465) + z1205) :|: z1205 >= 0, z = 1 + (1 + z0465 + z1205), z0465 >= 0 mark(z) -{ 0 }-> a__from(1 + mark(z0466) + mark(z1206)) :|: z = 1 + (1 + z0466 + z1206), z1206 >= 0, z0466 >= 0 mark(z) -{ 0 }-> a__filter(mark(z0), mark(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 mark(z) -{ 0 }-> 3 :|: z = 3 mark(z) -{ 0 }-> 1 :|: z = 1 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 mark(z) -{ 0 }-> 1 + mark(z0) + mark(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 ---------------------------------------- (51) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (52) Obligation: Complexity RNTS consisting of the following rules: A__FILTER(z, z') -{ 1 }-> 1 + MARK(z1) :|: z1 >= 0, z' = 1 + z1 + z2, z - 2 >= 0, z2 >= 0 A__FILTER(z, z') -{ 1 }-> 1 + MARK(z - 2) :|: z1 >= 0, z' = 1 + z1 + z2, z - 2 >= 0, z2 >= 0 A__FROM(z) -{ 1 }-> 1 + MARK(z) :|: z >= 0 A__HEAD(z) -{ 1 }-> 1 + MARK(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 A__IF(z, z', z'') -{ 1 }-> 1 + MARK(z') :|: z = 3, z'' >= 0, z' >= 0 A__IF(z, z', z'') -{ 1 }-> 1 + MARK(z'') :|: z'' >= 0, z = 1, z' >= 0 A__PRIMES -{ 1 }-> 1 + A__SIEVE(0) + A__FROM(1 + (1 + 0)) :|: A__PRIMES -{ 1 }-> 1 + A__SIEVE(1 + (1 + (1 + 0))) + A__FROM(1 + (1 + 0)) :|: A__PRIMES -{ 1 }-> 1 + A__SIEVE(1 + mark(1 + (1 + 0)) + (1 + (1 + (1 + (1 + 0))))) + A__FROM(1 + (1 + 0)) :|: A__SIEVE(z) -{ 1 }-> 1 + MARK(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 A__TAIL(z) -{ 1 }-> 1 + MARK(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 MARK(z) -{ 1 }-> 1 + MARK(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 MARK(z) -{ 1 }-> 1 + MARK(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 MARK(z) -{ 1 }-> 1 + MARK(z - 1) :|: z - 1 >= 0 MARK(z) -{ 1 }-> 1 + A__PRIMES :|: z = 2 MARK(z) -{ 1 }-> 1 + A__TAIL(a__tail(mark(z - 2))) + MARK(1 + (z - 2)) :|: z - 2 >= 0 MARK(z) -{ 1 }-> 1 + A__TAIL(a__sieve(mark(z - 2))) + MARK(1 + (z - 2)) :|: z - 2 >= 0 MARK(z) -{ 1 }-> 1 + A__TAIL(a__primes) + MARK(2) :|: z = 1 + 2 MARK(z) -{ 1 }-> 1 + A__TAIL(a__if(mark(z030), z111, z22)) + MARK(1 + z030 + z111 + z22) :|: z = 1 + (1 + z030 + z111 + z22), z111 >= 0, z22 >= 0, z030 >= 0 MARK(z) -{ 1 }-> 1 + A__TAIL(a__head(mark(z - 2))) + MARK(1 + (z - 2)) :|: z - 2 >= 0 MARK(z) -{ 1 }-> 1 + A__TAIL(a__from(mark(z - 2))) + MARK(1 + (z - 2)) :|: z - 2 >= 0 MARK(z) -{ 1 }-> 1 + A__TAIL(a__filter(mark(z031), mark(z112))) + MARK(1 + z031 + z112) :|: z031 >= 0, z112 >= 0, z = 1 + (1 + z031 + z112) MARK(z) -{ 1 }-> 1 + A__TAIL(3) + MARK(3) :|: z = 1 + 3 MARK(z) -{ 1 }-> 1 + A__TAIL(1) + MARK(1) :|: z = 1 + 1 MARK(z) -{ 1 }-> 1 + A__TAIL(0) + MARK(0) :|: z = 1 + 0 MARK(z) -{ 1 }-> 1 + A__TAIL(0) + MARK(z - 1) :|: z - 1 >= 0 MARK(z) -{ 1 }-> 1 + A__TAIL(1 + mark(z - 2)) + MARK(1 + (z - 2)) :|: z - 2 >= 0 MARK(z) -{ 1 }-> 1 + A__TAIL(1 + mark(z033) + z113) + MARK(1 + z033 + z113) :|: z = 1 + (1 + z033 + z113), z033 >= 0, z113 >= 0 MARK(z) -{ 1 }-> 1 + A__TAIL(1 + mark(z034) + mark(z114)) + MARK(1 + z034 + z114) :|: z114 >= 0, z034 >= 0, z = 1 + (1 + z034 + z114) MARK(z) -{ 1 }-> 1 + A__SIEVE(a__tail(mark(z - 2))) + MARK(1 + (z - 2)) :|: z - 2 >= 0 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__primes) + MARK(2) :|: z = 1 + 2 MARK(z) -{ 1 }-> 1 + A__SIEVE(a__if(mark(z03), z1', z2')) + MARK(1 + z03 + z1' + z2') :|: z = 1 + (1 + z03 + z1' + z2'), z1' >= 0, z2' >= 0, z03 >= 0 MARK(z) -{ 1 }-> 1 + A__SIEVE(a__head(mark(z - 2))) + MARK(1 + (z - 2)) :|: z - 2 >= 0 MARK(z) -{ 1 }-> 1 + A__SIEVE(a__from(mark(z - 2))) + MARK(1 + (z - 2)) :|: z - 2 >= 0 MARK(z) -{ 1 }-> 1 + A__SIEVE(a__filter(mark(z04), mark(z1''))) + MARK(1 + z04 + z1'') :|: z04 >= 0, z = 1 + (1 + z04 + z1''), z1'' >= 0 MARK(z) -{ 1 }-> 1 + A__SIEVE(3) + MARK(3) :|: z = 1 + 3 MARK(z) -{ 1 }-> 1 + A__SIEVE(1) + MARK(1) :|: z = 1 + 1 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(z06) + z11) + MARK(1 + z06 + z11) :|: z11 >= 0, z06 >= 0, z = 1 + (1 + z06 + z11) MARK(z) -{ 1 }-> 1 + A__SIEVE(1 + mark(z07) + mark(z12)) + MARK(1 + z07 + z12) :|: z07 >= 0, z12 >= 0, z = 1 + (1 + z07 + z12) MARK(z) -{ 1 }-> 1 + A__IF(a__tail(mark(z038)), z1, z2) + MARK(1 + z038) :|: z = 1 + (1 + z038) + z1 + z2, z1 >= 0, z038 >= 0, z2 >= 0 MARK(z) -{ 1 }-> 1 + A__IF(a__sieve(mark(z035)), z1, z2) + MARK(1 + z035) :|: z035 >= 0, z1 >= 0, z = 1 + (1 + z035) + z1 + z2, z2 >= 0 MARK(z) -{ 1 }-> 1 + A__IF(a__primes, z1, z2) + MARK(2) :|: z1 >= 0, z = 1 + 2 + z1 + z2, z2 >= 0 MARK(z) -{ 1 }-> 1 + A__IF(a__if(mark(z039), z115, z23), z1, z2) + MARK(1 + z039 + z115 + z23) :|: z1 >= 0, z039 >= 0, z23 >= 0, z = 1 + (1 + z039 + z115 + z23) + z1 + z2, z115 >= 0, z2 >= 0 MARK(z) -{ 1 }-> 1 + A__IF(a__head(mark(z037)), z1, z2) + MARK(1 + z037) :|: z1 >= 0, z = 1 + (1 + z037) + z1 + z2, z037 >= 0, z2 >= 0 MARK(z) -{ 1 }-> 1 + A__IF(a__from(mark(z036)), z1, z2) + MARK(1 + z036) :|: z036 >= 0, z1 >= 0, z = 1 + (1 + z036) + z1 + z2, z2 >= 0 MARK(z) -{ 1 }-> 1 + A__IF(a__filter(mark(z040), mark(z116)), z1, z2) + MARK(1 + z040 + z116) :|: z1 >= 0, z = 1 + (1 + z040 + z116) + z1 + z2, z040 >= 0, z2 >= 0, z116 >= 0 MARK(z) -{ 1 }-> 1 + A__IF(3, z1, z2) + MARK(3) :|: z1 >= 0, z = 1 + 3 + z1 + z2, z2 >= 0 MARK(z) -{ 1 }-> 1 + A__IF(1, z1, z2) + MARK(1) :|: z1 >= 0, z = 1 + 1 + z1 + z2, z2 >= 0 MARK(z) -{ 1 }-> 1 + A__IF(0, z1, z2) + MARK(z0) :|: z1 >= 0, z = 1 + z0 + z1 + z2, z0 >= 0, z2 >= 0 MARK(z) -{ 1 }-> 1 + A__IF(0, z1, z2) + MARK(0) :|: z1 >= 0, z = 1 + 0 + z1 + z2, z2 >= 0 MARK(z) -{ 1 }-> 1 + A__IF(1 + mark(z041), z1, z2) + MARK(1 + z041) :|: z = 1 + (1 + z041) + z1 + z2, z1 >= 0, z041 >= 0, z2 >= 0 MARK(z) -{ 1 }-> 1 + A__IF(1 + mark(z042) + z117, z1, z2) + MARK(1 + z042 + z117) :|: z = 1 + (1 + z042 + z117) + z1 + z2, z042 >= 0, z117 >= 0, z1 >= 0, z2 >= 0 MARK(z) -{ 1 }-> 1 + A__IF(1 + mark(z043) + mark(z118), z1, z2) + MARK(1 + z043 + z118) :|: z1 >= 0, z118 >= 0, z043 >= 0, z = 1 + (1 + z043 + z118) + z1 + z2, z2 >= 0 MARK(z) -{ 1 }-> 1 + A__HEAD(a__tail(mark(z - 2))) + MARK(1 + (z - 2)) :|: z - 2 >= 0 MARK(z) -{ 1 }-> 1 + A__HEAD(a__sieve(mark(z - 2))) + MARK(1 + (z - 2)) :|: z - 2 >= 0 MARK(z) -{ 1 }-> 1 + A__HEAD(a__primes) + MARK(2) :|: z = 1 + 2 MARK(z) -{ 1 }-> 1 + A__HEAD(a__if(mark(z021), z17, z21)) + MARK(1 + z021 + z17 + z21) :|: z021 >= 0, z21 >= 0, z17 >= 0, z = 1 + (1 + z021 + z17 + z21) MARK(z) -{ 1 }-> 1 + A__HEAD(a__head(mark(z - 2))) + MARK(1 + (z - 2)) :|: z - 2 >= 0 MARK(z) -{ 1 }-> 1 + A__HEAD(a__from(mark(z - 2))) + MARK(1 + (z - 2)) :|: z - 2 >= 0 MARK(z) -{ 1 }-> 1 + A__HEAD(a__filter(mark(z022), mark(z18))) + MARK(1 + z022 + z18) :|: z18 >= 0, z022 >= 0, z = 1 + (1 + z022 + z18) MARK(z) -{ 1 }-> 1 + A__HEAD(3) + MARK(3) :|: z = 1 + 3 MARK(z) -{ 1 }-> 1 + A__HEAD(1) + MARK(1) :|: z = 1 + 1 MARK(z) -{ 1 }-> 1 + A__HEAD(0) + MARK(0) :|: z = 1 + 0 MARK(z) -{ 1 }-> 1 + A__HEAD(0) + MARK(z - 1) :|: z - 1 >= 0 MARK(z) -{ 1 }-> 1 + A__HEAD(1 + mark(z - 2)) + MARK(1 + (z - 2)) :|: z - 2 >= 0 MARK(z) -{ 1 }-> 1 + A__HEAD(1 + mark(z024) + z19) + MARK(1 + z024 + z19) :|: z19 >= 0, z024 >= 0, z = 1 + (1 + z024 + z19) MARK(z) -{ 1 }-> 1 + A__HEAD(1 + mark(z025) + mark(z110)) + MARK(1 + z025 + z110) :|: z025 >= 0, z110 >= 0, z = 1 + (1 + z025 + z110) MARK(z) -{ 1 }-> 1 + A__FROM(a__tail(mark(z - 2))) + MARK(1 + (z - 2)) :|: z - 2 >= 0 MARK(z) -{ 1 }-> 1 + A__FROM(a__sieve(mark(z - 2))) + MARK(1 + (z - 2)) :|: z - 2 >= 0 MARK(z) -{ 1 }-> 1 + A__FROM(a__primes) + MARK(2) :|: z = 1 + 2 MARK(z) -{ 1 }-> 1 + A__FROM(a__if(mark(z012), z13, z2'')) + MARK(1 + z012 + z13 + z2'') :|: z = 1 + (1 + z012 + z13 + z2''), z012 >= 0, z2'' >= 0, z13 >= 0 MARK(z) -{ 1 }-> 1 + A__FROM(a__head(mark(z - 2))) + MARK(1 + (z - 2)) :|: z - 2 >= 0 MARK(z) -{ 1 }-> 1 + A__FROM(a__from(mark(z - 2))) + MARK(1 + (z - 2)) :|: z - 2 >= 0 MARK(z) -{ 1 }-> 1 + A__FROM(a__filter(mark(z013), mark(z14))) + MARK(1 + z013 + z14) :|: z = 1 + (1 + z013 + z14), z013 >= 0, z14 >= 0 MARK(z) -{ 1 }-> 1 + A__FROM(3) + MARK(3) :|: z = 1 + 3 MARK(z) -{ 1 }-> 1 + A__FROM(1) + MARK(1) :|: z = 1 + 1 MARK(z) -{ 1 }-> 1 + A__FROM(0) + MARK(0) :|: z = 1 + 0 MARK(z) -{ 1 }-> 1 + A__FROM(0) + MARK(z - 1) :|: z - 1 >= 0 MARK(z) -{ 1 }-> 1 + A__FROM(1 + mark(z - 2)) + MARK(1 + (z - 2)) :|: z - 2 >= 0 MARK(z) -{ 1 }-> 1 + A__FROM(1 + mark(z015) + z15) + MARK(1 + z015 + z15) :|: z015 >= 0, z15 >= 0, z = 1 + (1 + z015 + z15) MARK(z) -{ 1 }-> 1 + A__FROM(1 + mark(z016) + mark(z16)) + MARK(1 + z016 + z16) :|: z = 1 + (1 + z016 + z16), z016 >= 0, z16 >= 0 MARK(z) -{ 1 }-> 1 + A__FILTER(mark(z0), mark(z1)) + MARK(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 MARK(z) -{ 1 }-> 1 + A__FILTER(mark(z0), mark(z1)) + MARK(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 a__filter(z, z') -{ 0 }-> a__if(1 + (1 + (1 + mark(z - 2))) + mark(z1), 1 + (1 + (1 + (z - 2))) + z2, 1 + z1 + (1 + (z - 2) + (1 + z1))) :|: z1 >= 0, z' = 1 + z1 + z2, z - 2 >= 0, z2 >= 0 a__filter(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 a__filter(z, z') -{ 0 }-> 1 + z + z' :|: z' >= 0, z >= 0 a__from(z) -{ 0 }-> 0 :|: z >= 0 a__from(z) -{ 0 }-> 1 + z :|: z >= 0 a__from(z) -{ 0 }-> 1 + mark(z) + (1 + (1 + z)) :|: z >= 0 a__head(z) -{ 0 }-> mark(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 a__head(z) -{ 0 }-> 0 :|: z >= 0 a__head(z) -{ 0 }-> 1 + z :|: z >= 0 a__if(z, z', z'') -{ 0 }-> mark(z') :|: z = 3, z'' >= 0, z' >= 0 a__if(z, z', z'') -{ 0 }-> mark(z'') :|: z'' >= 0, z = 1, z' >= 0 a__if(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 a__if(z, z', z'') -{ 0 }-> 1 + z + z' + z'' :|: z' >= 0, z >= 0, z'' >= 0 a__primes -{ 0 }-> a__sieve(0) :|: a__primes -{ 0 }-> a__sieve(1 + (1 + (1 + 0))) :|: a__primes -{ 0 }-> a__sieve(1 + mark(1 + (1 + 0)) + (1 + (1 + (1 + (1 + 0))))) :|: a__primes -{ 0 }-> 2 :|: a__primes -{ 0 }-> 0 :|: a__sieve(z) -{ 0 }-> 0 :|: z >= 0 a__sieve(z) -{ 0 }-> 1 + z :|: z >= 0 a__sieve(z) -{ 0 }-> 1 + mark(z0) + (1 + z0 + (1 + z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 a__tail(z) -{ 0 }-> mark(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 a__tail(z) -{ 0 }-> 0 :|: z >= 0 a__tail(z) -{ 0 }-> 1 + z :|: z >= 0 mark(z) -{ 0 }-> a__tail(a__tail(mark(z - 2))) :|: z - 2 >= 0 mark(z) -{ 0 }-> a__tail(a__sieve(mark(z - 2))) :|: z - 2 >= 0 mark(z) -{ 0 }-> a__tail(a__primes) :|: z = 1 + 2 mark(z) -{ 0 }-> a__tail(a__if(mark(z0480), z1211, z252)) :|: z = 1 + (1 + z0480 + z1211 + z252), z252 >= 0, z0480 >= 0, z1211 >= 0 mark(z) -{ 0 }-> a__tail(a__head(mark(z - 2))) :|: z - 2 >= 0 mark(z) -{ 0 }-> a__tail(a__from(mark(z - 2))) :|: z - 2 >= 0 mark(z) -{ 0 }-> a__tail(a__filter(mark(z0481), mark(z1212))) :|: z1212 >= 0, z0481 >= 0, z = 1 + (1 + z0481 + z1212) mark(z) -{ 0 }-> a__tail(3) :|: z = 1 + 3 mark(z) -{ 0 }-> a__tail(1) :|: z = 1 + 1 mark(z) -{ 0 }-> a__tail(0) :|: z = 1 + 0 mark(z) -{ 0 }-> a__tail(0) :|: z - 1 >= 0 mark(z) -{ 0 }-> a__tail(1 + mark(z - 2)) :|: z - 2 >= 0 mark(z) -{ 0 }-> a__tail(1 + mark(z0483) + z1213) :|: z1213 >= 0, z0483 >= 0, z = 1 + (1 + z0483 + z1213) mark(z) -{ 0 }-> a__tail(1 + mark(z0484) + mark(z1214)) :|: z1214 >= 0, z0484 >= 0, z = 1 + (1 + z0484 + z1214) mark(z) -{ 0 }-> a__sieve(a__tail(mark(z - 2))) :|: z - 2 >= 0 mark(z) -{ 0 }-> a__sieve(a__sieve(mark(z - 2))) :|: z - 2 >= 0 mark(z) -{ 0 }-> a__sieve(a__primes) :|: z = 1 + 2 mark(z) -{ 0 }-> a__sieve(a__if(mark(z0453), z1199, z249)) :|: z = 1 + (1 + z0453 + z1199 + z249), z249 >= 0, z1199 >= 0, z0453 >= 0 mark(z) -{ 0 }-> a__sieve(a__head(mark(z - 2))) :|: z - 2 >= 0 mark(z) -{ 0 }-> a__sieve(a__from(mark(z - 2))) :|: z - 2 >= 0 mark(z) -{ 0 }-> a__sieve(a__filter(mark(z0454), mark(z1200))) :|: z = 1 + (1 + z0454 + z1200), z0454 >= 0, z1200 >= 0 mark(z) -{ 0 }-> a__sieve(3) :|: z = 1 + 3 mark(z) -{ 0 }-> a__sieve(1) :|: z = 1 + 1 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(z0456) + z1201) :|: z1201 >= 0, z0456 >= 0, z = 1 + (1 + z0456 + z1201) mark(z) -{ 0 }-> a__sieve(1 + mark(z0457) + mark(z1202)) :|: z1202 >= 0, z0457 >= 0, z = 1 + (1 + z0457 + z1202) mark(z) -{ 0 }-> a__primes :|: z = 2 mark(z) -{ 0 }-> a__if(a__tail(mark(z0488)), z1, z2) :|: z = 1 + (1 + z0488) + z1 + z2, z1 >= 0, z0488 >= 0, z2 >= 0 mark(z) -{ 0 }-> a__if(a__sieve(mark(z0485)), z1, z2) :|: z0485 >= 0, z1 >= 0, z2 >= 0, z = 1 + (1 + z0485) + z1 + z2 mark(z) -{ 0 }-> a__if(a__primes, z1, z2) :|: z1 >= 0, z = 1 + 2 + z1 + z2, z2 >= 0 mark(z) -{ 0 }-> a__if(a__if(mark(z0489), z1215, z253), z1, z2) :|: z0489 >= 0, z1 >= 0, z = 1 + (1 + z0489 + z1215 + z253) + z1 + z2, z1215 >= 0, z253 >= 0, z2 >= 0 mark(z) -{ 0 }-> a__if(a__head(mark(z0487)), z1, z2) :|: z1 >= 0, z = 1 + (1 + z0487) + z1 + z2, z0487 >= 0, z2 >= 0 mark(z) -{ 0 }-> a__if(a__from(mark(z0486)), z1, z2) :|: z1 >= 0, z0486 >= 0, z = 1 + (1 + z0486) + z1 + z2, z2 >= 0 mark(z) -{ 0 }-> a__if(a__filter(mark(z0490), mark(z1216)), z1, z2) :|: z = 1 + (1 + z0490 + z1216) + z1 + z2, z1216 >= 0, z1 >= 0, z0490 >= 0, z2 >= 0 mark(z) -{ 0 }-> a__if(3, z1, z2) :|: z1 >= 0, z = 1 + 3 + z1 + z2, z2 >= 0 mark(z) -{ 0 }-> a__if(1, z1, z2) :|: z1 >= 0, z = 1 + 1 + z1 + z2, z2 >= 0 mark(z) -{ 0 }-> a__if(0, z1, z2) :|: z1 >= 0, z = 1 + 0 + z1 + z2, z2 >= 0 mark(z) -{ 0 }-> a__if(0, z1, z2) :|: z1 >= 0, z = 1 + z0 + z1 + z2, z0 >= 0, z2 >= 0 mark(z) -{ 0 }-> a__if(1 + mark(z0491), z1, z2) :|: z1 >= 0, z = 1 + (1 + z0491) + z1 + z2, z0491 >= 0, z2 >= 0 mark(z) -{ 0 }-> a__if(1 + mark(z0492) + z1217, z1, z2) :|: z1 >= 0, z1217 >= 0, z = 1 + (1 + z0492 + z1217) + z1 + z2, z2 >= 0, z0492 >= 0 mark(z) -{ 0 }-> a__if(1 + mark(z0493) + mark(z1218), z1, z2) :|: z1 >= 0, z = 1 + (1 + z0493 + z1218) + z1 + z2, z1218 >= 0, z0493 >= 0, z2 >= 0 mark(z) -{ 0 }-> a__head(a__tail(mark(z - 2))) :|: z - 2 >= 0 mark(z) -{ 0 }-> a__head(a__sieve(mark(z - 2))) :|: z - 2 >= 0 mark(z) -{ 0 }-> a__head(a__primes) :|: z = 1 + 2 mark(z) -{ 0 }-> a__head(a__if(mark(z0471), z1207, z251)) :|: z0471 >= 0, z = 1 + (1 + z0471 + z1207 + z251), z251 >= 0, z1207 >= 0 mark(z) -{ 0 }-> a__head(a__head(mark(z - 2))) :|: z - 2 >= 0 mark(z) -{ 0 }-> a__head(a__from(mark(z - 2))) :|: z - 2 >= 0 mark(z) -{ 0 }-> a__head(a__filter(mark(z0472), mark(z1208))) :|: z = 1 + (1 + z0472 + z1208), z1208 >= 0, z0472 >= 0 mark(z) -{ 0 }-> a__head(3) :|: z = 1 + 3 mark(z) -{ 0 }-> a__head(1) :|: z = 1 + 1 mark(z) -{ 0 }-> a__head(0) :|: z = 1 + 0 mark(z) -{ 0 }-> a__head(0) :|: z - 1 >= 0 mark(z) -{ 0 }-> a__head(1 + mark(z - 2)) :|: z - 2 >= 0 mark(z) -{ 0 }-> a__head(1 + mark(z0474) + z1209) :|: z1209 >= 0, z0474 >= 0, z = 1 + (1 + z0474 + z1209) mark(z) -{ 0 }-> a__head(1 + mark(z0475) + mark(z1210)) :|: z0475 >= 0, z = 1 + (1 + z0475 + z1210), z1210 >= 0 mark(z) -{ 0 }-> a__from(a__tail(mark(z - 2))) :|: z - 2 >= 0 mark(z) -{ 0 }-> a__from(a__sieve(mark(z - 2))) :|: z - 2 >= 0 mark(z) -{ 0 }-> a__from(a__primes) :|: z = 1 + 2 mark(z) -{ 0 }-> a__from(a__if(mark(z0462), z1203, z250)) :|: z250 >= 0, z1203 >= 0, z = 1 + (1 + z0462 + z1203 + z250), z0462 >= 0 mark(z) -{ 0 }-> a__from(a__head(mark(z - 2))) :|: z - 2 >= 0 mark(z) -{ 0 }-> a__from(a__from(mark(z - 2))) :|: z - 2 >= 0 mark(z) -{ 0 }-> a__from(a__filter(mark(z0463), mark(z1204))) :|: z = 1 + (1 + z0463 + z1204), z1204 >= 0, z0463 >= 0 mark(z) -{ 0 }-> a__from(3) :|: z = 1 + 3 mark(z) -{ 0 }-> a__from(1) :|: z = 1 + 1 mark(z) -{ 0 }-> a__from(0) :|: z = 1 + 0 mark(z) -{ 0 }-> a__from(0) :|: z - 1 >= 0 mark(z) -{ 0 }-> a__from(1 + mark(z - 2)) :|: z - 2 >= 0 mark(z) -{ 0 }-> a__from(1 + mark(z0465) + z1205) :|: z1205 >= 0, z = 1 + (1 + z0465 + z1205), z0465 >= 0 mark(z) -{ 0 }-> a__from(1 + mark(z0466) + mark(z1206)) :|: z = 1 + (1 + z0466 + z1206), z1206 >= 0, z0466 >= 0 mark(z) -{ 0 }-> a__filter(mark(z0), mark(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 mark(z) -{ 0 }-> 3 :|: z = 3 mark(z) -{ 0 }-> 1 :|: z = 1 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 mark(z) -{ 0 }-> 1 + mark(z0) + mark(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 ---------------------------------------- (53) 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__primes -> null_a__primes [0] a__from(v0) -> null_a__from [0] a__head(v0) -> null_a__head [0] a__tail(v0) -> null_a__tail [0] a__if(v0, v1, v2) -> null_a__if [0] a__filter(v0, v1) -> null_a__filter [0] a__sieve(v0) -> null_a__sieve [0] mark(v0) -> null_mark [0] A__HEAD(v0) -> null_A__HEAD [0] A__TAIL(v0) -> null_A__TAIL [0] A__IF(v0, v1, v2) -> null_A__IF [0] A__SIEVE(v0) -> null_A__SIEVE [0] MARK(v0) -> null_MARK [0] A__FILTER(v0, v1) -> null_A__FILTER [0] And the following fresh constants: null_a__primes, null_a__from, null_a__head, null_a__tail, null_a__if, null_a__filter, null_a__sieve, null_mark, null_A__HEAD, null_A__TAIL, null_A__IF, null_A__SIEVE, null_MARK, null_A__FILTER, const, const1 ---------------------------------------- (54) 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__PRIMES -> c(A__SIEVE(a__from(s(s(0)))), A__FROM(s(s(0)))) [1] A__FROM(z0) -> c2(MARK(z0)) [1] A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) [1] A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) [1] A__IF(true, z0, z1) -> c8(MARK(z0)) [1] A__IF(false, z0, z1) -> c9(MARK(z1)) [1] A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) [1] MARK(primes) -> c16(A__PRIMES) [1] MARK(sieve(z0)) -> c17(A__SIEVE(mark(z0)), MARK(z0)) [1] MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) [1] MARK(head(z0)) -> c19(A__HEAD(mark(z0)), MARK(z0)) [1] MARK(tail(z0)) -> c20(A__TAIL(mark(z0)), MARK(z0)) [1] MARK(if(z0, z1, z2)) -> c21(A__IF(mark(z0), z1, z2), MARK(z0)) [1] MARK(filter(z0, z1)) -> c22(A__FILTER(mark(z0), mark(z1)), MARK(z0)) [1] MARK(filter(z0, z1)) -> c23(A__FILTER(mark(z0), mark(z1)), MARK(z1)) [1] MARK(s(z0)) -> c24(MARK(z0)) [1] MARK(cons(z0, z1)) -> c26(MARK(z0)) [1] MARK(divides(z0, z1)) -> c29(MARK(z0)) [1] MARK(divides(z0, z1)) -> c30(MARK(z1)) [1] A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) [1] A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) [1] a__primes -> a__sieve(a__from(s(s(0)))) [0] a__primes -> primes [0] a__from(z0) -> cons(mark(z0), from(s(z0))) [0] a__from(z0) -> from(z0) [0] a__head(cons(z0, z1)) -> mark(z0) [0] a__head(z0) -> head(z0) [0] a__tail(cons(z0, z1)) -> mark(z1) [0] a__tail(z0) -> tail(z0) [0] a__if(true, z0, z1) -> mark(z0) [0] a__if(false, z0, z1) -> mark(z1) [0] a__if(z0, z1, z2) -> if(z0, z1, z2) [0] a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) [0] a__filter(z0, z1) -> filter(z0, z1) [0] a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) [0] a__sieve(z0) -> sieve(z0) [0] mark(primes) -> a__primes [0] mark(sieve(z0)) -> a__sieve(mark(z0)) [0] mark(from(z0)) -> a__from(mark(z0)) [0] mark(head(z0)) -> a__head(mark(z0)) [0] mark(tail(z0)) -> a__tail(mark(z0)) [0] mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) [0] mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) [0] mark(s(z0)) -> s(mark(z0)) [0] mark(0) -> 0 [0] mark(cons(z0, z1)) -> cons(mark(z0), z1) [0] mark(true) -> true [0] mark(false) -> false [0] mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) [0] a__primes -> null_a__primes [0] a__from(v0) -> null_a__from [0] a__head(v0) -> null_a__head [0] a__tail(v0) -> null_a__tail [0] a__if(v0, v1, v2) -> null_a__if [0] a__filter(v0, v1) -> null_a__filter [0] a__sieve(v0) -> null_a__sieve [0] mark(v0) -> null_mark [0] A__HEAD(v0) -> null_A__HEAD [0] A__TAIL(v0) -> null_A__TAIL [0] A__IF(v0, v1, v2) -> null_A__IF [0] A__SIEVE(v0) -> null_A__SIEVE [0] MARK(v0) -> null_MARK [0] A__FILTER(v0, v1) -> null_A__FILTER [0] The TRS has the following type information: A__PRIMES :: c c :: c14:null_A__SIEVE -> c2 -> c A__SIEVE :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> c14:null_A__SIEVE a__from :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark s :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark 0 :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark A__FROM :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> c2 c2 :: c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30:null_MARK -> c2 MARK :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30:null_MARK A__HEAD :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> c4:null_A__HEAD cons :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark c4 :: c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30:null_MARK -> c4:null_A__HEAD A__TAIL :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> c6:null_A__TAIL c6 :: c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30:null_MARK -> c6:null_A__TAIL A__IF :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> c8:c9:null_A__IF true :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark c8 :: c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30:null_MARK -> c8:c9:null_A__IF false :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark c9 :: c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30:null_MARK -> c8:c9:null_A__IF c14 :: c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30:null_MARK -> c14:null_A__SIEVE primes :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark c16 :: c -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30:null_MARK sieve :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark c17 :: c14:null_A__SIEVE -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30:null_MARK -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30:null_MARK mark :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark from :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark c18 :: c2 -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30:null_MARK -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30:null_MARK head :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark c19 :: c4:null_A__HEAD -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30:null_MARK -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30:null_MARK tail :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark c20 :: c6:null_A__TAIL -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30:null_MARK -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30:null_MARK if :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark c21 :: c8:c9:null_A__IF -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30:null_MARK -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30:null_MARK filter :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark c22 :: c11:c12:null_A__FILTER -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30:null_MARK -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30:null_MARK A__FILTER :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> c11:c12:null_A__FILTER c23 :: c11:c12:null_A__FILTER -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30:null_MARK -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30:null_MARK c24 :: c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30:null_MARK -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30:null_MARK c26 :: c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30:null_MARK -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30:null_MARK divides :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark c29 :: c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30:null_MARK -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30:null_MARK c30 :: c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30:null_MARK -> c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30:null_MARK c11 :: c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30:null_MARK -> c11:c12:null_A__FILTER c12 :: c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30:null_MARK -> c11:c12:null_A__FILTER a__primes :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark a__sieve :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark a__head :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark a__tail :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark a__if :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark a__filter :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark -> 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark null_a__primes :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark null_a__from :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark null_a__head :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark null_a__tail :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark null_a__if :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark null_a__filter :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark null_a__sieve :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark null_mark :: 0:s:cons:true:false:primes:sieve:from:head:tail:if:filter:divides:null_a__primes:null_a__from:null_a__head:null_a__tail:null_a__if:null_a__filter:null_a__sieve:null_mark null_A__HEAD :: c4:null_A__HEAD null_A__TAIL :: c6:null_A__TAIL null_A__IF :: c8:c9:null_A__IF null_A__SIEVE :: c14:null_A__SIEVE null_MARK :: c16:c17:c18:c19:c20:c21:c22:c23:c24:c26:c29:c30:null_MARK null_A__FILTER :: c11:c12:null_A__FILTER const :: c const1 :: c2 Rewrite Strategy: INNERMOST ---------------------------------------- (55) 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 true => 3 false => 1 primes => 2 null_a__primes => 0 null_a__from => 0 null_a__head => 0 null_a__tail => 0 null_a__if => 0 null_a__filter => 0 null_a__sieve => 0 null_mark => 0 null_A__HEAD => 0 null_A__TAIL => 0 null_A__IF => 0 null_A__SIEVE => 0 null_MARK => 0 null_A__FILTER => 0 const => 0 const1 => 0 ---------------------------------------- (56) Obligation: Complexity RNTS consisting of the following rules: A__FILTER(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 A__FILTER(z, z') -{ 1 }-> 1 + MARK(z0) :|: z1 >= 0, z' = 1 + z1 + z2, z0 >= 0, z = 1 + (1 + z0), z2 >= 0 A__FILTER(z, z') -{ 1 }-> 1 + MARK(z1) :|: z1 >= 0, z' = 1 + z1 + z2, z0 >= 0, z = 1 + (1 + z0), z2 >= 0 A__FROM(z) -{ 1 }-> 1 + MARK(z0) :|: z = z0, z0 >= 0 A__HEAD(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 A__HEAD(z) -{ 1 }-> 1 + MARK(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 A__IF(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 A__IF(z, z', z'') -{ 1 }-> 1 + MARK(z0) :|: z = 3, z1 >= 0, z0 >= 0, z' = z0, z'' = z1 A__IF(z, z', z'') -{ 1 }-> 1 + MARK(z1) :|: z1 >= 0, z = 1, z0 >= 0, z' = z0, z'' = z1 A__PRIMES -{ 1 }-> 1 + A__SIEVE(a__from(1 + (1 + 0))) + A__FROM(1 + (1 + 0)) :|: A__SIEVE(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 A__SIEVE(z) -{ 1 }-> 1 + MARK(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 A__TAIL(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 A__TAIL(z) -{ 1 }-> 1 + MARK(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 MARK(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 MARK(z) -{ 1 }-> 1 + MARK(z0) :|: z = 1 + z0, z0 >= 0 MARK(z) -{ 1 }-> 1 + MARK(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 MARK(z) -{ 1 }-> 1 + MARK(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 MARK(z) -{ 1 }-> 1 + A__PRIMES :|: z = 2 MARK(z) -{ 1 }-> 1 + A__TAIL(mark(z0)) + MARK(z0) :|: z = 1 + z0, z0 >= 0 MARK(z) -{ 1 }-> 1 + A__SIEVE(mark(z0)) + MARK(z0) :|: z = 1 + z0, z0 >= 0 MARK(z) -{ 1 }-> 1 + A__IF(mark(z0), z1, z2) + MARK(z0) :|: z1 >= 0, z = 1 + z0 + z1 + z2, z0 >= 0, z2 >= 0 MARK(z) -{ 1 }-> 1 + A__HEAD(mark(z0)) + MARK(z0) :|: z = 1 + z0, z0 >= 0 MARK(z) -{ 1 }-> 1 + A__FROM(mark(z0)) + MARK(z0) :|: z = 1 + z0, z0 >= 0 MARK(z) -{ 1 }-> 1 + A__FILTER(mark(z0), mark(z1)) + MARK(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 MARK(z) -{ 1 }-> 1 + A__FILTER(mark(z0), mark(z1)) + MARK(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 a__filter(z, z') -{ 0 }-> a__if(1 + (1 + (1 + mark(z0))) + mark(z1), 1 + (1 + (1 + z0)) + z2, 1 + z1 + (1 + z0 + (1 + z1))) :|: z1 >= 0, z' = 1 + z1 + z2, z0 >= 0, z = 1 + (1 + z0), z2 >= 0 a__filter(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 a__filter(z, z') -{ 0 }-> 1 + z0 + z1 :|: z = z0, z1 >= 0, z' = z1, z0 >= 0 a__from(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 a__from(z) -{ 0 }-> 1 + z0 :|: z = z0, z0 >= 0 a__from(z) -{ 0 }-> 1 + mark(z0) + (1 + (1 + z0)) :|: z = z0, z0 >= 0 a__head(z) -{ 0 }-> mark(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 a__head(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 a__head(z) -{ 0 }-> 1 + z0 :|: z = z0, z0 >= 0 a__if(z, z', z'') -{ 0 }-> mark(z0) :|: z = 3, z1 >= 0, z0 >= 0, z' = z0, z'' = z1 a__if(z, z', z'') -{ 0 }-> mark(z1) :|: z1 >= 0, z = 1, z0 >= 0, z' = z0, z'' = z1 a__if(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 a__if(z, z', z'') -{ 0 }-> 1 + z0 + z1 + z2 :|: z'' = z2, z = z0, z1 >= 0, z' = z1, z0 >= 0, z2 >= 0 a__primes -{ 0 }-> a__sieve(a__from(1 + (1 + 0))) :|: a__primes -{ 0 }-> 2 :|: a__primes -{ 0 }-> 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 + mark(z0) + (1 + z0 + (1 + z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 a__tail(z) -{ 0 }-> mark(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 a__tail(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 a__tail(z) -{ 0 }-> 1 + z0 :|: z = z0, z0 >= 0 mark(z) -{ 0 }-> a__tail(mark(z0)) :|: z = 1 + z0, z0 >= 0 mark(z) -{ 0 }-> a__sieve(mark(z0)) :|: z = 1 + z0, z0 >= 0 mark(z) -{ 0 }-> a__primes :|: z = 2 mark(z) -{ 0 }-> a__if(mark(z0), z1, z2) :|: z1 >= 0, z = 1 + z0 + z1 + z2, z0 >= 0, z2 >= 0 mark(z) -{ 0 }-> a__head(mark(z0)) :|: z = 1 + z0, z0 >= 0 mark(z) -{ 0 }-> a__from(mark(z0)) :|: z = 1 + z0, z0 >= 0 mark(z) -{ 0 }-> a__filter(mark(z0), mark(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 mark(z) -{ 0 }-> 3 :|: z = 3 mark(z) -{ 0 }-> 1 :|: z = 1 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 mark(z) -{ 0 }-> 1 + mark(z0) + mark(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (57) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace A__PRIMES -> c(A__SIEVE(a__from(s(s(0)))), A__FROM(s(s(0)))) by A__PRIMES -> c(A__SIEVE(cons(mark(s(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) A__PRIMES -> c(A__SIEVE(from(s(s(0)))), A__FROM(s(s(0)))) ---------------------------------------- (58) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(sieve(z0)) -> c17(A__SIEVE(mark(z0)), MARK(z0)) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(head(z0)) -> c19(A__HEAD(mark(z0)), MARK(z0)) MARK(tail(z0)) -> c20(A__TAIL(mark(z0)), MARK(z0)) MARK(if(z0, z1, z2)) -> c21(A__IF(mark(z0), z1, z2), MARK(z0)) MARK(filter(z0, z1)) -> c22(A__FILTER(mark(z0), mark(z1)), MARK(z0)) MARK(filter(z0, z1)) -> c23(A__FILTER(mark(z0), mark(z1)), MARK(z1)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__SIEVE(cons(mark(s(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) A__PRIMES -> c(A__SIEVE(from(s(s(0)))), A__FROM(s(s(0)))) S tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(sieve(z0)) -> c17(A__SIEVE(mark(z0)), MARK(z0)) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(head(z0)) -> c19(A__HEAD(mark(z0)), MARK(z0)) MARK(tail(z0)) -> c20(A__TAIL(mark(z0)), MARK(z0)) MARK(if(z0, z1, z2)) -> c21(A__IF(mark(z0), z1, z2), MARK(z0)) MARK(filter(z0, z1)) -> c22(A__FILTER(mark(z0), mark(z1)), MARK(z0)) MARK(filter(z0, z1)) -> c23(A__FILTER(mark(z0), mark(z1)), MARK(z1)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__SIEVE(cons(mark(s(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) A__PRIMES -> c(A__SIEVE(from(s(s(0)))), A__FROM(s(s(0)))) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c17_2, c18_2, c19_2, c20_2, c21_2, c22_2, c23_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_2 ---------------------------------------- (59) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (60) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(sieve(z0)) -> c17(A__SIEVE(mark(z0)), MARK(z0)) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(head(z0)) -> c19(A__HEAD(mark(z0)), MARK(z0)) MARK(tail(z0)) -> c20(A__TAIL(mark(z0)), MARK(z0)) MARK(if(z0, z1, z2)) -> c21(A__IF(mark(z0), z1, z2), MARK(z0)) MARK(filter(z0, z1)) -> c22(A__FILTER(mark(z0), mark(z1)), MARK(z0)) MARK(filter(z0, z1)) -> c23(A__FILTER(mark(z0), mark(z1)), MARK(z1)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__SIEVE(cons(mark(s(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) A__PRIMES -> c(A__FROM(s(s(0)))) S tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(sieve(z0)) -> c17(A__SIEVE(mark(z0)), MARK(z0)) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(head(z0)) -> c19(A__HEAD(mark(z0)), MARK(z0)) MARK(tail(z0)) -> c20(A__TAIL(mark(z0)), MARK(z0)) MARK(if(z0, z1, z2)) -> c21(A__IF(mark(z0), z1, z2), MARK(z0)) MARK(filter(z0, z1)) -> c22(A__FILTER(mark(z0), mark(z1)), MARK(z0)) MARK(filter(z0, z1)) -> c23(A__FILTER(mark(z0), mark(z1)), MARK(z1)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__SIEVE(cons(mark(s(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) A__PRIMES -> c(A__FROM(s(s(0)))) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c17_2, c18_2, c19_2, c20_2, c21_2, c22_2, c23_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_2, c_1 ---------------------------------------- (61) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(sieve(z0)) -> c17(A__SIEVE(mark(z0)), MARK(z0)) by MARK(sieve(primes)) -> c17(A__SIEVE(a__primes), MARK(primes)) MARK(sieve(sieve(z0))) -> c17(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(from(z0))) -> c17(A__SIEVE(a__from(mark(z0))), MARK(from(z0))) MARK(sieve(head(z0))) -> c17(A__SIEVE(a__head(mark(z0))), MARK(head(z0))) MARK(sieve(tail(z0))) -> c17(A__SIEVE(a__tail(mark(z0))), MARK(tail(z0))) MARK(sieve(if(z0, z1, z2))) -> c17(A__SIEVE(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(sieve(filter(z0, z1))) -> c17(A__SIEVE(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(sieve(s(z0))) -> c17(A__SIEVE(s(mark(z0))), MARK(s(z0))) MARK(sieve(0)) -> c17(A__SIEVE(0), MARK(0)) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(true)) -> c17(A__SIEVE(true), MARK(true)) MARK(sieve(false)) -> c17(A__SIEVE(false), MARK(false)) MARK(sieve(divides(z0, z1))) -> c17(A__SIEVE(divides(mark(z0), mark(z1))), MARK(divides(z0, z1))) ---------------------------------------- (62) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(head(z0)) -> c19(A__HEAD(mark(z0)), MARK(z0)) MARK(tail(z0)) -> c20(A__TAIL(mark(z0)), MARK(z0)) MARK(if(z0, z1, z2)) -> c21(A__IF(mark(z0), z1, z2), MARK(z0)) MARK(filter(z0, z1)) -> c22(A__FILTER(mark(z0), mark(z1)), MARK(z0)) MARK(filter(z0, z1)) -> c23(A__FILTER(mark(z0), mark(z1)), MARK(z1)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__SIEVE(cons(mark(s(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__primes), MARK(primes)) MARK(sieve(sieve(z0))) -> c17(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(from(z0))) -> c17(A__SIEVE(a__from(mark(z0))), MARK(from(z0))) MARK(sieve(head(z0))) -> c17(A__SIEVE(a__head(mark(z0))), MARK(head(z0))) MARK(sieve(tail(z0))) -> c17(A__SIEVE(a__tail(mark(z0))), MARK(tail(z0))) MARK(sieve(if(z0, z1, z2))) -> c17(A__SIEVE(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(sieve(filter(z0, z1))) -> c17(A__SIEVE(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(sieve(s(z0))) -> c17(A__SIEVE(s(mark(z0))), MARK(s(z0))) MARK(sieve(0)) -> c17(A__SIEVE(0), MARK(0)) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(true)) -> c17(A__SIEVE(true), MARK(true)) MARK(sieve(false)) -> c17(A__SIEVE(false), MARK(false)) MARK(sieve(divides(z0, z1))) -> c17(A__SIEVE(divides(mark(z0), mark(z1))), MARK(divides(z0, z1))) S tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(head(z0)) -> c19(A__HEAD(mark(z0)), MARK(z0)) MARK(tail(z0)) -> c20(A__TAIL(mark(z0)), MARK(z0)) MARK(if(z0, z1, z2)) -> c21(A__IF(mark(z0), z1, z2), MARK(z0)) MARK(filter(z0, z1)) -> c22(A__FILTER(mark(z0), mark(z1)), MARK(z0)) MARK(filter(z0, z1)) -> c23(A__FILTER(mark(z0), mark(z1)), MARK(z1)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__SIEVE(cons(mark(s(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__primes), MARK(primes)) MARK(sieve(sieve(z0))) -> c17(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(from(z0))) -> c17(A__SIEVE(a__from(mark(z0))), MARK(from(z0))) MARK(sieve(head(z0))) -> c17(A__SIEVE(a__head(mark(z0))), MARK(head(z0))) MARK(sieve(tail(z0))) -> c17(A__SIEVE(a__tail(mark(z0))), MARK(tail(z0))) MARK(sieve(if(z0, z1, z2))) -> c17(A__SIEVE(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(sieve(filter(z0, z1))) -> c17(A__SIEVE(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(sieve(s(z0))) -> c17(A__SIEVE(s(mark(z0))), MARK(s(z0))) MARK(sieve(0)) -> c17(A__SIEVE(0), MARK(0)) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(true)) -> c17(A__SIEVE(true), MARK(true)) MARK(sieve(false)) -> c17(A__SIEVE(false), MARK(false)) MARK(sieve(divides(z0, z1))) -> c17(A__SIEVE(divides(mark(z0), mark(z1))), MARK(divides(z0, z1))) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c18_2, c19_2, c20_2, c21_2, c22_2, c23_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_2, c_1, c17_2 ---------------------------------------- (63) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing nodes: MARK(sieve(true)) -> c17(A__SIEVE(true), MARK(true)) MARK(sieve(0)) -> c17(A__SIEVE(0), MARK(0)) MARK(sieve(false)) -> c17(A__SIEVE(false), MARK(false)) ---------------------------------------- (64) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(head(z0)) -> c19(A__HEAD(mark(z0)), MARK(z0)) MARK(tail(z0)) -> c20(A__TAIL(mark(z0)), MARK(z0)) MARK(if(z0, z1, z2)) -> c21(A__IF(mark(z0), z1, z2), MARK(z0)) MARK(filter(z0, z1)) -> c22(A__FILTER(mark(z0), mark(z1)), MARK(z0)) MARK(filter(z0, z1)) -> c23(A__FILTER(mark(z0), mark(z1)), MARK(z1)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__SIEVE(cons(mark(s(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__primes), MARK(primes)) MARK(sieve(sieve(z0))) -> c17(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(from(z0))) -> c17(A__SIEVE(a__from(mark(z0))), MARK(from(z0))) MARK(sieve(head(z0))) -> c17(A__SIEVE(a__head(mark(z0))), MARK(head(z0))) MARK(sieve(tail(z0))) -> c17(A__SIEVE(a__tail(mark(z0))), MARK(tail(z0))) MARK(sieve(if(z0, z1, z2))) -> c17(A__SIEVE(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(sieve(filter(z0, z1))) -> c17(A__SIEVE(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(sieve(s(z0))) -> c17(A__SIEVE(s(mark(z0))), MARK(s(z0))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(divides(z0, z1))) -> c17(A__SIEVE(divides(mark(z0), mark(z1))), MARK(divides(z0, z1))) S tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(head(z0)) -> c19(A__HEAD(mark(z0)), MARK(z0)) MARK(tail(z0)) -> c20(A__TAIL(mark(z0)), MARK(z0)) MARK(if(z0, z1, z2)) -> c21(A__IF(mark(z0), z1, z2), MARK(z0)) MARK(filter(z0, z1)) -> c22(A__FILTER(mark(z0), mark(z1)), MARK(z0)) MARK(filter(z0, z1)) -> c23(A__FILTER(mark(z0), mark(z1)), MARK(z1)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__SIEVE(cons(mark(s(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__primes), MARK(primes)) MARK(sieve(sieve(z0))) -> c17(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(from(z0))) -> c17(A__SIEVE(a__from(mark(z0))), MARK(from(z0))) MARK(sieve(head(z0))) -> c17(A__SIEVE(a__head(mark(z0))), MARK(head(z0))) MARK(sieve(tail(z0))) -> c17(A__SIEVE(a__tail(mark(z0))), MARK(tail(z0))) MARK(sieve(if(z0, z1, z2))) -> c17(A__SIEVE(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(sieve(filter(z0, z1))) -> c17(A__SIEVE(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(sieve(s(z0))) -> c17(A__SIEVE(s(mark(z0))), MARK(s(z0))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(divides(z0, z1))) -> c17(A__SIEVE(divides(mark(z0), mark(z1))), MARK(divides(z0, z1))) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c18_2, c19_2, c20_2, c21_2, c22_2, c23_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_2, c_1, c17_2 ---------------------------------------- (65) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (66) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(head(z0)) -> c19(A__HEAD(mark(z0)), MARK(z0)) MARK(tail(z0)) -> c20(A__TAIL(mark(z0)), MARK(z0)) MARK(if(z0, z1, z2)) -> c21(A__IF(mark(z0), z1, z2), MARK(z0)) MARK(filter(z0, z1)) -> c22(A__FILTER(mark(z0), mark(z1)), MARK(z0)) MARK(filter(z0, z1)) -> c23(A__FILTER(mark(z0), mark(z1)), MARK(z1)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__SIEVE(cons(mark(s(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__primes), MARK(primes)) MARK(sieve(sieve(z0))) -> c17(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(from(z0))) -> c17(A__SIEVE(a__from(mark(z0))), MARK(from(z0))) MARK(sieve(head(z0))) -> c17(A__SIEVE(a__head(mark(z0))), MARK(head(z0))) MARK(sieve(tail(z0))) -> c17(A__SIEVE(a__tail(mark(z0))), MARK(tail(z0))) MARK(sieve(if(z0, z1, z2))) -> c17(A__SIEVE(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(sieve(filter(z0, z1))) -> c17(A__SIEVE(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) S tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(head(z0)) -> c19(A__HEAD(mark(z0)), MARK(z0)) MARK(tail(z0)) -> c20(A__TAIL(mark(z0)), MARK(z0)) MARK(if(z0, z1, z2)) -> c21(A__IF(mark(z0), z1, z2), MARK(z0)) MARK(filter(z0, z1)) -> c22(A__FILTER(mark(z0), mark(z1)), MARK(z0)) MARK(filter(z0, z1)) -> c23(A__FILTER(mark(z0), mark(z1)), MARK(z1)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__SIEVE(cons(mark(s(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__primes), MARK(primes)) MARK(sieve(sieve(z0))) -> c17(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(from(z0))) -> c17(A__SIEVE(a__from(mark(z0))), MARK(from(z0))) MARK(sieve(head(z0))) -> c17(A__SIEVE(a__head(mark(z0))), MARK(head(z0))) MARK(sieve(tail(z0))) -> c17(A__SIEVE(a__tail(mark(z0))), MARK(tail(z0))) MARK(sieve(if(z0, z1, z2))) -> c17(A__SIEVE(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(sieve(filter(z0, z1))) -> c17(A__SIEVE(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c18_2, c19_2, c20_2, c21_2, c22_2, c23_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_2, c_1, c17_2, c17_1 ---------------------------------------- (67) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(head(z0)) -> c19(A__HEAD(mark(z0)), MARK(z0)) by MARK(head(primes)) -> c19(A__HEAD(a__primes), MARK(primes)) MARK(head(sieve(z0))) -> c19(A__HEAD(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(head(from(z0))) -> c19(A__HEAD(a__from(mark(z0))), MARK(from(z0))) MARK(head(head(z0))) -> c19(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c19(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(if(z0, z1, z2))) -> c19(A__HEAD(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(head(filter(z0, z1))) -> c19(A__HEAD(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(head(s(z0))) -> c19(A__HEAD(s(mark(z0))), MARK(s(z0))) MARK(head(0)) -> c19(A__HEAD(0), MARK(0)) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(true)) -> c19(A__HEAD(true), MARK(true)) MARK(head(false)) -> c19(A__HEAD(false), MARK(false)) MARK(head(divides(z0, z1))) -> c19(A__HEAD(divides(mark(z0), mark(z1))), MARK(divides(z0, z1))) ---------------------------------------- (68) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(tail(z0)) -> c20(A__TAIL(mark(z0)), MARK(z0)) MARK(if(z0, z1, z2)) -> c21(A__IF(mark(z0), z1, z2), MARK(z0)) MARK(filter(z0, z1)) -> c22(A__FILTER(mark(z0), mark(z1)), MARK(z0)) MARK(filter(z0, z1)) -> c23(A__FILTER(mark(z0), mark(z1)), MARK(z1)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__SIEVE(cons(mark(s(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__primes), MARK(primes)) MARK(sieve(sieve(z0))) -> c17(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(from(z0))) -> c17(A__SIEVE(a__from(mark(z0))), MARK(from(z0))) MARK(sieve(head(z0))) -> c17(A__SIEVE(a__head(mark(z0))), MARK(head(z0))) MARK(sieve(tail(z0))) -> c17(A__SIEVE(a__tail(mark(z0))), MARK(tail(z0))) MARK(sieve(if(z0, z1, z2))) -> c17(A__SIEVE(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(sieve(filter(z0, z1))) -> c17(A__SIEVE(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(primes)) -> c19(A__HEAD(a__primes), MARK(primes)) MARK(head(sieve(z0))) -> c19(A__HEAD(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(head(from(z0))) -> c19(A__HEAD(a__from(mark(z0))), MARK(from(z0))) MARK(head(head(z0))) -> c19(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c19(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(if(z0, z1, z2))) -> c19(A__HEAD(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(head(filter(z0, z1))) -> c19(A__HEAD(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(head(s(z0))) -> c19(A__HEAD(s(mark(z0))), MARK(s(z0))) MARK(head(0)) -> c19(A__HEAD(0), MARK(0)) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(true)) -> c19(A__HEAD(true), MARK(true)) MARK(head(false)) -> c19(A__HEAD(false), MARK(false)) MARK(head(divides(z0, z1))) -> c19(A__HEAD(divides(mark(z0), mark(z1))), MARK(divides(z0, z1))) S tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(tail(z0)) -> c20(A__TAIL(mark(z0)), MARK(z0)) MARK(if(z0, z1, z2)) -> c21(A__IF(mark(z0), z1, z2), MARK(z0)) MARK(filter(z0, z1)) -> c22(A__FILTER(mark(z0), mark(z1)), MARK(z0)) MARK(filter(z0, z1)) -> c23(A__FILTER(mark(z0), mark(z1)), MARK(z1)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__SIEVE(cons(mark(s(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__primes), MARK(primes)) MARK(sieve(sieve(z0))) -> c17(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(from(z0))) -> c17(A__SIEVE(a__from(mark(z0))), MARK(from(z0))) MARK(sieve(head(z0))) -> c17(A__SIEVE(a__head(mark(z0))), MARK(head(z0))) MARK(sieve(tail(z0))) -> c17(A__SIEVE(a__tail(mark(z0))), MARK(tail(z0))) MARK(sieve(if(z0, z1, z2))) -> c17(A__SIEVE(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(sieve(filter(z0, z1))) -> c17(A__SIEVE(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(primes)) -> c19(A__HEAD(a__primes), MARK(primes)) MARK(head(sieve(z0))) -> c19(A__HEAD(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(head(from(z0))) -> c19(A__HEAD(a__from(mark(z0))), MARK(from(z0))) MARK(head(head(z0))) -> c19(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c19(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(if(z0, z1, z2))) -> c19(A__HEAD(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(head(filter(z0, z1))) -> c19(A__HEAD(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(head(s(z0))) -> c19(A__HEAD(s(mark(z0))), MARK(s(z0))) MARK(head(0)) -> c19(A__HEAD(0), MARK(0)) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(true)) -> c19(A__HEAD(true), MARK(true)) MARK(head(false)) -> c19(A__HEAD(false), MARK(false)) MARK(head(divides(z0, z1))) -> c19(A__HEAD(divides(mark(z0), mark(z1))), MARK(divides(z0, z1))) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c18_2, c20_2, c21_2, c22_2, c23_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_2, c_1, c17_2, c17_1, c19_2 ---------------------------------------- (69) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing nodes: MARK(head(0)) -> c19(A__HEAD(0), MARK(0)) MARK(head(false)) -> c19(A__HEAD(false), MARK(false)) MARK(head(true)) -> c19(A__HEAD(true), MARK(true)) ---------------------------------------- (70) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(tail(z0)) -> c20(A__TAIL(mark(z0)), MARK(z0)) MARK(if(z0, z1, z2)) -> c21(A__IF(mark(z0), z1, z2), MARK(z0)) MARK(filter(z0, z1)) -> c22(A__FILTER(mark(z0), mark(z1)), MARK(z0)) MARK(filter(z0, z1)) -> c23(A__FILTER(mark(z0), mark(z1)), MARK(z1)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__SIEVE(cons(mark(s(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__primes), MARK(primes)) MARK(sieve(sieve(z0))) -> c17(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(from(z0))) -> c17(A__SIEVE(a__from(mark(z0))), MARK(from(z0))) MARK(sieve(head(z0))) -> c17(A__SIEVE(a__head(mark(z0))), MARK(head(z0))) MARK(sieve(tail(z0))) -> c17(A__SIEVE(a__tail(mark(z0))), MARK(tail(z0))) MARK(sieve(if(z0, z1, z2))) -> c17(A__SIEVE(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(sieve(filter(z0, z1))) -> c17(A__SIEVE(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(primes)) -> c19(A__HEAD(a__primes), MARK(primes)) MARK(head(sieve(z0))) -> c19(A__HEAD(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(head(from(z0))) -> c19(A__HEAD(a__from(mark(z0))), MARK(from(z0))) MARK(head(head(z0))) -> c19(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c19(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(if(z0, z1, z2))) -> c19(A__HEAD(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(head(filter(z0, z1))) -> c19(A__HEAD(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(head(s(z0))) -> c19(A__HEAD(s(mark(z0))), MARK(s(z0))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(divides(z0, z1))) -> c19(A__HEAD(divides(mark(z0), mark(z1))), MARK(divides(z0, z1))) S tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(tail(z0)) -> c20(A__TAIL(mark(z0)), MARK(z0)) MARK(if(z0, z1, z2)) -> c21(A__IF(mark(z0), z1, z2), MARK(z0)) MARK(filter(z0, z1)) -> c22(A__FILTER(mark(z0), mark(z1)), MARK(z0)) MARK(filter(z0, z1)) -> c23(A__FILTER(mark(z0), mark(z1)), MARK(z1)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__SIEVE(cons(mark(s(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__primes), MARK(primes)) MARK(sieve(sieve(z0))) -> c17(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(from(z0))) -> c17(A__SIEVE(a__from(mark(z0))), MARK(from(z0))) MARK(sieve(head(z0))) -> c17(A__SIEVE(a__head(mark(z0))), MARK(head(z0))) MARK(sieve(tail(z0))) -> c17(A__SIEVE(a__tail(mark(z0))), MARK(tail(z0))) MARK(sieve(if(z0, z1, z2))) -> c17(A__SIEVE(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(sieve(filter(z0, z1))) -> c17(A__SIEVE(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(primes)) -> c19(A__HEAD(a__primes), MARK(primes)) MARK(head(sieve(z0))) -> c19(A__HEAD(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(head(from(z0))) -> c19(A__HEAD(a__from(mark(z0))), MARK(from(z0))) MARK(head(head(z0))) -> c19(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c19(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(if(z0, z1, z2))) -> c19(A__HEAD(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(head(filter(z0, z1))) -> c19(A__HEAD(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(head(s(z0))) -> c19(A__HEAD(s(mark(z0))), MARK(s(z0))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(divides(z0, z1))) -> c19(A__HEAD(divides(mark(z0), mark(z1))), MARK(divides(z0, z1))) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c18_2, c20_2, c21_2, c22_2, c23_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_2, c_1, c17_2, c17_1, c19_2 ---------------------------------------- (71) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(tail(z0)) -> c20(A__TAIL(mark(z0)), MARK(z0)) MARK(if(z0, z1, z2)) -> c21(A__IF(mark(z0), z1, z2), MARK(z0)) MARK(filter(z0, z1)) -> c22(A__FILTER(mark(z0), mark(z1)), MARK(z0)) MARK(filter(z0, z1)) -> c23(A__FILTER(mark(z0), mark(z1)), MARK(z1)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__SIEVE(cons(mark(s(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__primes), MARK(primes)) MARK(sieve(sieve(z0))) -> c17(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(from(z0))) -> c17(A__SIEVE(a__from(mark(z0))), MARK(from(z0))) MARK(sieve(head(z0))) -> c17(A__SIEVE(a__head(mark(z0))), MARK(head(z0))) MARK(sieve(tail(z0))) -> c17(A__SIEVE(a__tail(mark(z0))), MARK(tail(z0))) MARK(sieve(if(z0, z1, z2))) -> c17(A__SIEVE(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(sieve(filter(z0, z1))) -> c17(A__SIEVE(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(primes)) -> c19(A__HEAD(a__primes), MARK(primes)) MARK(head(sieve(z0))) -> c19(A__HEAD(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(head(from(z0))) -> c19(A__HEAD(a__from(mark(z0))), MARK(from(z0))) MARK(head(head(z0))) -> c19(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c19(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(if(z0, z1, z2))) -> c19(A__HEAD(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(head(filter(z0, z1))) -> c19(A__HEAD(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) S tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(tail(z0)) -> c20(A__TAIL(mark(z0)), MARK(z0)) MARK(if(z0, z1, z2)) -> c21(A__IF(mark(z0), z1, z2), MARK(z0)) MARK(filter(z0, z1)) -> c22(A__FILTER(mark(z0), mark(z1)), MARK(z0)) MARK(filter(z0, z1)) -> c23(A__FILTER(mark(z0), mark(z1)), MARK(z1)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__SIEVE(cons(mark(s(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__primes), MARK(primes)) MARK(sieve(sieve(z0))) -> c17(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(from(z0))) -> c17(A__SIEVE(a__from(mark(z0))), MARK(from(z0))) MARK(sieve(head(z0))) -> c17(A__SIEVE(a__head(mark(z0))), MARK(head(z0))) MARK(sieve(tail(z0))) -> c17(A__SIEVE(a__tail(mark(z0))), MARK(tail(z0))) MARK(sieve(if(z0, z1, z2))) -> c17(A__SIEVE(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(sieve(filter(z0, z1))) -> c17(A__SIEVE(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(primes)) -> c19(A__HEAD(a__primes), MARK(primes)) MARK(head(sieve(z0))) -> c19(A__HEAD(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(head(from(z0))) -> c19(A__HEAD(a__from(mark(z0))), MARK(from(z0))) MARK(head(head(z0))) -> c19(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c19(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(if(z0, z1, z2))) -> c19(A__HEAD(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(head(filter(z0, z1))) -> c19(A__HEAD(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c18_2, c20_2, c21_2, c22_2, c23_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_2, c_1, c17_2, c17_1, c19_2, c19_1 ---------------------------------------- (73) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(tail(z0)) -> c20(A__TAIL(mark(z0)), MARK(z0)) by MARK(tail(primes)) -> c20(A__TAIL(a__primes), MARK(primes)) MARK(tail(sieve(z0))) -> c20(A__TAIL(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(tail(from(z0))) -> c20(A__TAIL(a__from(mark(z0))), MARK(from(z0))) MARK(tail(head(z0))) -> c20(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c20(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(if(z0, z1, z2))) -> c20(A__TAIL(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(tail(filter(z0, z1))) -> c20(A__TAIL(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(tail(s(z0))) -> c20(A__TAIL(s(mark(z0))), MARK(s(z0))) MARK(tail(0)) -> c20(A__TAIL(0), MARK(0)) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(true)) -> c20(A__TAIL(true), MARK(true)) MARK(tail(false)) -> c20(A__TAIL(false), MARK(false)) MARK(tail(divides(z0, z1))) -> c20(A__TAIL(divides(mark(z0), mark(z1))), MARK(divides(z0, z1))) ---------------------------------------- (74) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(if(z0, z1, z2)) -> c21(A__IF(mark(z0), z1, z2), MARK(z0)) MARK(filter(z0, z1)) -> c22(A__FILTER(mark(z0), mark(z1)), MARK(z0)) MARK(filter(z0, z1)) -> c23(A__FILTER(mark(z0), mark(z1)), MARK(z1)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__SIEVE(cons(mark(s(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__primes), MARK(primes)) MARK(sieve(sieve(z0))) -> c17(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(from(z0))) -> c17(A__SIEVE(a__from(mark(z0))), MARK(from(z0))) MARK(sieve(head(z0))) -> c17(A__SIEVE(a__head(mark(z0))), MARK(head(z0))) MARK(sieve(tail(z0))) -> c17(A__SIEVE(a__tail(mark(z0))), MARK(tail(z0))) MARK(sieve(if(z0, z1, z2))) -> c17(A__SIEVE(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(sieve(filter(z0, z1))) -> c17(A__SIEVE(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(primes)) -> c19(A__HEAD(a__primes), MARK(primes)) MARK(head(sieve(z0))) -> c19(A__HEAD(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(head(from(z0))) -> c19(A__HEAD(a__from(mark(z0))), MARK(from(z0))) MARK(head(head(z0))) -> c19(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c19(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(if(z0, z1, z2))) -> c19(A__HEAD(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(head(filter(z0, z1))) -> c19(A__HEAD(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(primes)) -> c20(A__TAIL(a__primes), MARK(primes)) MARK(tail(sieve(z0))) -> c20(A__TAIL(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(tail(from(z0))) -> c20(A__TAIL(a__from(mark(z0))), MARK(from(z0))) MARK(tail(head(z0))) -> c20(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c20(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(if(z0, z1, z2))) -> c20(A__TAIL(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(tail(filter(z0, z1))) -> c20(A__TAIL(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(tail(s(z0))) -> c20(A__TAIL(s(mark(z0))), MARK(s(z0))) MARK(tail(0)) -> c20(A__TAIL(0), MARK(0)) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(true)) -> c20(A__TAIL(true), MARK(true)) MARK(tail(false)) -> c20(A__TAIL(false), MARK(false)) MARK(tail(divides(z0, z1))) -> c20(A__TAIL(divides(mark(z0), mark(z1))), MARK(divides(z0, z1))) S tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(if(z0, z1, z2)) -> c21(A__IF(mark(z0), z1, z2), MARK(z0)) MARK(filter(z0, z1)) -> c22(A__FILTER(mark(z0), mark(z1)), MARK(z0)) MARK(filter(z0, z1)) -> c23(A__FILTER(mark(z0), mark(z1)), MARK(z1)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__SIEVE(cons(mark(s(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__primes), MARK(primes)) MARK(sieve(sieve(z0))) -> c17(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(from(z0))) -> c17(A__SIEVE(a__from(mark(z0))), MARK(from(z0))) MARK(sieve(head(z0))) -> c17(A__SIEVE(a__head(mark(z0))), MARK(head(z0))) MARK(sieve(tail(z0))) -> c17(A__SIEVE(a__tail(mark(z0))), MARK(tail(z0))) MARK(sieve(if(z0, z1, z2))) -> c17(A__SIEVE(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(sieve(filter(z0, z1))) -> c17(A__SIEVE(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(primes)) -> c19(A__HEAD(a__primes), MARK(primes)) MARK(head(sieve(z0))) -> c19(A__HEAD(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(head(from(z0))) -> c19(A__HEAD(a__from(mark(z0))), MARK(from(z0))) MARK(head(head(z0))) -> c19(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c19(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(if(z0, z1, z2))) -> c19(A__HEAD(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(head(filter(z0, z1))) -> c19(A__HEAD(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(primes)) -> c20(A__TAIL(a__primes), MARK(primes)) MARK(tail(sieve(z0))) -> c20(A__TAIL(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(tail(from(z0))) -> c20(A__TAIL(a__from(mark(z0))), MARK(from(z0))) MARK(tail(head(z0))) -> c20(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c20(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(if(z0, z1, z2))) -> c20(A__TAIL(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(tail(filter(z0, z1))) -> c20(A__TAIL(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(tail(s(z0))) -> c20(A__TAIL(s(mark(z0))), MARK(s(z0))) MARK(tail(0)) -> c20(A__TAIL(0), MARK(0)) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(true)) -> c20(A__TAIL(true), MARK(true)) MARK(tail(false)) -> c20(A__TAIL(false), MARK(false)) MARK(tail(divides(z0, z1))) -> c20(A__TAIL(divides(mark(z0), mark(z1))), MARK(divides(z0, z1))) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c18_2, c21_2, c22_2, c23_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_2, c_1, c17_2, c17_1, c19_2, c19_1, c20_2 ---------------------------------------- (75) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing nodes: MARK(tail(false)) -> c20(A__TAIL(false), MARK(false)) MARK(tail(true)) -> c20(A__TAIL(true), MARK(true)) MARK(tail(0)) -> c20(A__TAIL(0), MARK(0)) ---------------------------------------- (76) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(if(z0, z1, z2)) -> c21(A__IF(mark(z0), z1, z2), MARK(z0)) MARK(filter(z0, z1)) -> c22(A__FILTER(mark(z0), mark(z1)), MARK(z0)) MARK(filter(z0, z1)) -> c23(A__FILTER(mark(z0), mark(z1)), MARK(z1)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__SIEVE(cons(mark(s(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__primes), MARK(primes)) MARK(sieve(sieve(z0))) -> c17(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(from(z0))) -> c17(A__SIEVE(a__from(mark(z0))), MARK(from(z0))) MARK(sieve(head(z0))) -> c17(A__SIEVE(a__head(mark(z0))), MARK(head(z0))) MARK(sieve(tail(z0))) -> c17(A__SIEVE(a__tail(mark(z0))), MARK(tail(z0))) MARK(sieve(if(z0, z1, z2))) -> c17(A__SIEVE(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(sieve(filter(z0, z1))) -> c17(A__SIEVE(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(primes)) -> c19(A__HEAD(a__primes), MARK(primes)) MARK(head(sieve(z0))) -> c19(A__HEAD(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(head(from(z0))) -> c19(A__HEAD(a__from(mark(z0))), MARK(from(z0))) MARK(head(head(z0))) -> c19(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c19(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(if(z0, z1, z2))) -> c19(A__HEAD(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(head(filter(z0, z1))) -> c19(A__HEAD(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(primes)) -> c20(A__TAIL(a__primes), MARK(primes)) MARK(tail(sieve(z0))) -> c20(A__TAIL(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(tail(from(z0))) -> c20(A__TAIL(a__from(mark(z0))), MARK(from(z0))) MARK(tail(head(z0))) -> c20(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c20(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(if(z0, z1, z2))) -> c20(A__TAIL(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(tail(filter(z0, z1))) -> c20(A__TAIL(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(tail(s(z0))) -> c20(A__TAIL(s(mark(z0))), MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(divides(z0, z1))) -> c20(A__TAIL(divides(mark(z0), mark(z1))), MARK(divides(z0, z1))) S tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(if(z0, z1, z2)) -> c21(A__IF(mark(z0), z1, z2), MARK(z0)) MARK(filter(z0, z1)) -> c22(A__FILTER(mark(z0), mark(z1)), MARK(z0)) MARK(filter(z0, z1)) -> c23(A__FILTER(mark(z0), mark(z1)), MARK(z1)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__SIEVE(cons(mark(s(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__primes), MARK(primes)) MARK(sieve(sieve(z0))) -> c17(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(from(z0))) -> c17(A__SIEVE(a__from(mark(z0))), MARK(from(z0))) MARK(sieve(head(z0))) -> c17(A__SIEVE(a__head(mark(z0))), MARK(head(z0))) MARK(sieve(tail(z0))) -> c17(A__SIEVE(a__tail(mark(z0))), MARK(tail(z0))) MARK(sieve(if(z0, z1, z2))) -> c17(A__SIEVE(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(sieve(filter(z0, z1))) -> c17(A__SIEVE(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(primes)) -> c19(A__HEAD(a__primes), MARK(primes)) MARK(head(sieve(z0))) -> c19(A__HEAD(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(head(from(z0))) -> c19(A__HEAD(a__from(mark(z0))), MARK(from(z0))) MARK(head(head(z0))) -> c19(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c19(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(if(z0, z1, z2))) -> c19(A__HEAD(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(head(filter(z0, z1))) -> c19(A__HEAD(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(primes)) -> c20(A__TAIL(a__primes), MARK(primes)) MARK(tail(sieve(z0))) -> c20(A__TAIL(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(tail(from(z0))) -> c20(A__TAIL(a__from(mark(z0))), MARK(from(z0))) MARK(tail(head(z0))) -> c20(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c20(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(if(z0, z1, z2))) -> c20(A__TAIL(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(tail(filter(z0, z1))) -> c20(A__TAIL(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(tail(s(z0))) -> c20(A__TAIL(s(mark(z0))), MARK(s(z0))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(divides(z0, z1))) -> c20(A__TAIL(divides(mark(z0), mark(z1))), MARK(divides(z0, z1))) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c18_2, c21_2, c22_2, c23_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_2, c_1, c17_2, c17_1, c19_2, c19_1, c20_2 ---------------------------------------- (77) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (78) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(if(z0, z1, z2)) -> c21(A__IF(mark(z0), z1, z2), MARK(z0)) MARK(filter(z0, z1)) -> c22(A__FILTER(mark(z0), mark(z1)), MARK(z0)) MARK(filter(z0, z1)) -> c23(A__FILTER(mark(z0), mark(z1)), MARK(z1)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__SIEVE(cons(mark(s(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__primes), MARK(primes)) MARK(sieve(sieve(z0))) -> c17(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(from(z0))) -> c17(A__SIEVE(a__from(mark(z0))), MARK(from(z0))) MARK(sieve(head(z0))) -> c17(A__SIEVE(a__head(mark(z0))), MARK(head(z0))) MARK(sieve(tail(z0))) -> c17(A__SIEVE(a__tail(mark(z0))), MARK(tail(z0))) MARK(sieve(if(z0, z1, z2))) -> c17(A__SIEVE(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(sieve(filter(z0, z1))) -> c17(A__SIEVE(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(primes)) -> c19(A__HEAD(a__primes), MARK(primes)) MARK(head(sieve(z0))) -> c19(A__HEAD(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(head(from(z0))) -> c19(A__HEAD(a__from(mark(z0))), MARK(from(z0))) MARK(head(head(z0))) -> c19(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c19(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(if(z0, z1, z2))) -> c19(A__HEAD(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(head(filter(z0, z1))) -> c19(A__HEAD(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(primes)) -> c20(A__TAIL(a__primes), MARK(primes)) MARK(tail(sieve(z0))) -> c20(A__TAIL(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(tail(from(z0))) -> c20(A__TAIL(a__from(mark(z0))), MARK(from(z0))) MARK(tail(head(z0))) -> c20(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c20(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(if(z0, z1, z2))) -> c20(A__TAIL(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(tail(filter(z0, z1))) -> c20(A__TAIL(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) S tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(if(z0, z1, z2)) -> c21(A__IF(mark(z0), z1, z2), MARK(z0)) MARK(filter(z0, z1)) -> c22(A__FILTER(mark(z0), mark(z1)), MARK(z0)) MARK(filter(z0, z1)) -> c23(A__FILTER(mark(z0), mark(z1)), MARK(z1)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__SIEVE(cons(mark(s(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__primes), MARK(primes)) MARK(sieve(sieve(z0))) -> c17(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(from(z0))) -> c17(A__SIEVE(a__from(mark(z0))), MARK(from(z0))) MARK(sieve(head(z0))) -> c17(A__SIEVE(a__head(mark(z0))), MARK(head(z0))) MARK(sieve(tail(z0))) -> c17(A__SIEVE(a__tail(mark(z0))), MARK(tail(z0))) MARK(sieve(if(z0, z1, z2))) -> c17(A__SIEVE(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(sieve(filter(z0, z1))) -> c17(A__SIEVE(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(primes)) -> c19(A__HEAD(a__primes), MARK(primes)) MARK(head(sieve(z0))) -> c19(A__HEAD(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(head(from(z0))) -> c19(A__HEAD(a__from(mark(z0))), MARK(from(z0))) MARK(head(head(z0))) -> c19(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c19(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(if(z0, z1, z2))) -> c19(A__HEAD(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(head(filter(z0, z1))) -> c19(A__HEAD(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(primes)) -> c20(A__TAIL(a__primes), MARK(primes)) MARK(tail(sieve(z0))) -> c20(A__TAIL(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(tail(from(z0))) -> c20(A__TAIL(a__from(mark(z0))), MARK(from(z0))) MARK(tail(head(z0))) -> c20(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c20(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(if(z0, z1, z2))) -> c20(A__TAIL(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(tail(filter(z0, z1))) -> c20(A__TAIL(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c18_2, c21_2, c22_2, c23_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_2, c_1, c17_2, c17_1, c19_2, c19_1, c20_2, c20_1 ---------------------------------------- (79) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(if(z0, z1, z2)) -> c21(A__IF(mark(z0), z1, z2), MARK(z0)) by MARK(if(primes, x1, x2)) -> c21(A__IF(a__primes, x1, x2), MARK(primes)) MARK(if(sieve(z0), x1, x2)) -> c21(A__IF(a__sieve(mark(z0)), x1, x2), MARK(sieve(z0))) MARK(if(from(z0), x1, x2)) -> c21(A__IF(a__from(mark(z0)), x1, x2), MARK(from(z0))) MARK(if(head(z0), x1, x2)) -> c21(A__IF(a__head(mark(z0)), x1, x2), MARK(head(z0))) MARK(if(tail(z0), x1, x2)) -> c21(A__IF(a__tail(mark(z0)), x1, x2), MARK(tail(z0))) MARK(if(if(z0, z1, z2), x1, x2)) -> c21(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(filter(z0, z1), x1, x2)) -> c21(A__IF(a__filter(mark(z0), mark(z1)), x1, x2), MARK(filter(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(A__IF(s(mark(z0)), x1, x2), MARK(s(z0))) MARK(if(0, x1, x2)) -> c21(A__IF(0, x1, x2), MARK(0)) MARK(if(cons(z0, z1), x1, x2)) -> c21(A__IF(cons(mark(z0), z1), x1, x2), MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2), MARK(true)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2), MARK(false)) MARK(if(divides(z0, z1), x1, x2)) -> c21(A__IF(divides(mark(z0), mark(z1)), x1, x2), MARK(divides(z0, z1))) ---------------------------------------- (80) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(filter(z0, z1)) -> c22(A__FILTER(mark(z0), mark(z1)), MARK(z0)) MARK(filter(z0, z1)) -> c23(A__FILTER(mark(z0), mark(z1)), MARK(z1)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__SIEVE(cons(mark(s(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__primes), MARK(primes)) MARK(sieve(sieve(z0))) -> c17(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(from(z0))) -> c17(A__SIEVE(a__from(mark(z0))), MARK(from(z0))) MARK(sieve(head(z0))) -> c17(A__SIEVE(a__head(mark(z0))), MARK(head(z0))) MARK(sieve(tail(z0))) -> c17(A__SIEVE(a__tail(mark(z0))), MARK(tail(z0))) MARK(sieve(if(z0, z1, z2))) -> c17(A__SIEVE(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(sieve(filter(z0, z1))) -> c17(A__SIEVE(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(primes)) -> c19(A__HEAD(a__primes), MARK(primes)) MARK(head(sieve(z0))) -> c19(A__HEAD(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(head(from(z0))) -> c19(A__HEAD(a__from(mark(z0))), MARK(from(z0))) MARK(head(head(z0))) -> c19(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c19(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(if(z0, z1, z2))) -> c19(A__HEAD(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(head(filter(z0, z1))) -> c19(A__HEAD(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(primes)) -> c20(A__TAIL(a__primes), MARK(primes)) MARK(tail(sieve(z0))) -> c20(A__TAIL(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(tail(from(z0))) -> c20(A__TAIL(a__from(mark(z0))), MARK(from(z0))) MARK(tail(head(z0))) -> c20(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c20(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(if(z0, z1, z2))) -> c20(A__TAIL(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(tail(filter(z0, z1))) -> c20(A__TAIL(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(primes, x1, x2)) -> c21(A__IF(a__primes, x1, x2), MARK(primes)) MARK(if(sieve(z0), x1, x2)) -> c21(A__IF(a__sieve(mark(z0)), x1, x2), MARK(sieve(z0))) MARK(if(from(z0), x1, x2)) -> c21(A__IF(a__from(mark(z0)), x1, x2), MARK(from(z0))) MARK(if(head(z0), x1, x2)) -> c21(A__IF(a__head(mark(z0)), x1, x2), MARK(head(z0))) MARK(if(tail(z0), x1, x2)) -> c21(A__IF(a__tail(mark(z0)), x1, x2), MARK(tail(z0))) MARK(if(if(z0, z1, z2), x1, x2)) -> c21(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(filter(z0, z1), x1, x2)) -> c21(A__IF(a__filter(mark(z0), mark(z1)), x1, x2), MARK(filter(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(A__IF(s(mark(z0)), x1, x2), MARK(s(z0))) MARK(if(0, x1, x2)) -> c21(A__IF(0, x1, x2), MARK(0)) MARK(if(cons(z0, z1), x1, x2)) -> c21(A__IF(cons(mark(z0), z1), x1, x2), MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2), MARK(true)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2), MARK(false)) MARK(if(divides(z0, z1), x1, x2)) -> c21(A__IF(divides(mark(z0), mark(z1)), x1, x2), MARK(divides(z0, z1))) S tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(filter(z0, z1)) -> c22(A__FILTER(mark(z0), mark(z1)), MARK(z0)) MARK(filter(z0, z1)) -> c23(A__FILTER(mark(z0), mark(z1)), MARK(z1)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__SIEVE(cons(mark(s(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__primes), MARK(primes)) MARK(sieve(sieve(z0))) -> c17(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(from(z0))) -> c17(A__SIEVE(a__from(mark(z0))), MARK(from(z0))) MARK(sieve(head(z0))) -> c17(A__SIEVE(a__head(mark(z0))), MARK(head(z0))) MARK(sieve(tail(z0))) -> c17(A__SIEVE(a__tail(mark(z0))), MARK(tail(z0))) MARK(sieve(if(z0, z1, z2))) -> c17(A__SIEVE(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(sieve(filter(z0, z1))) -> c17(A__SIEVE(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(primes)) -> c19(A__HEAD(a__primes), MARK(primes)) MARK(head(sieve(z0))) -> c19(A__HEAD(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(head(from(z0))) -> c19(A__HEAD(a__from(mark(z0))), MARK(from(z0))) MARK(head(head(z0))) -> c19(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c19(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(if(z0, z1, z2))) -> c19(A__HEAD(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(head(filter(z0, z1))) -> c19(A__HEAD(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(primes)) -> c20(A__TAIL(a__primes), MARK(primes)) MARK(tail(sieve(z0))) -> c20(A__TAIL(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(tail(from(z0))) -> c20(A__TAIL(a__from(mark(z0))), MARK(from(z0))) MARK(tail(head(z0))) -> c20(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c20(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(if(z0, z1, z2))) -> c20(A__TAIL(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(tail(filter(z0, z1))) -> c20(A__TAIL(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(primes, x1, x2)) -> c21(A__IF(a__primes, x1, x2), MARK(primes)) MARK(if(sieve(z0), x1, x2)) -> c21(A__IF(a__sieve(mark(z0)), x1, x2), MARK(sieve(z0))) MARK(if(from(z0), x1, x2)) -> c21(A__IF(a__from(mark(z0)), x1, x2), MARK(from(z0))) MARK(if(head(z0), x1, x2)) -> c21(A__IF(a__head(mark(z0)), x1, x2), MARK(head(z0))) MARK(if(tail(z0), x1, x2)) -> c21(A__IF(a__tail(mark(z0)), x1, x2), MARK(tail(z0))) MARK(if(if(z0, z1, z2), x1, x2)) -> c21(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(filter(z0, z1), x1, x2)) -> c21(A__IF(a__filter(mark(z0), mark(z1)), x1, x2), MARK(filter(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(A__IF(s(mark(z0)), x1, x2), MARK(s(z0))) MARK(if(0, x1, x2)) -> c21(A__IF(0, x1, x2), MARK(0)) MARK(if(cons(z0, z1), x1, x2)) -> c21(A__IF(cons(mark(z0), z1), x1, x2), MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2), MARK(true)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2), MARK(false)) MARK(if(divides(z0, z1), x1, x2)) -> c21(A__IF(divides(mark(z0), mark(z1)), x1, x2), MARK(divides(z0, z1))) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c18_2, c22_2, c23_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_2, c_1, c17_2, c17_1, c19_2, c19_1, c20_2, c20_1, c21_2 ---------------------------------------- (81) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(if(0, x1, x2)) -> c21(A__IF(0, x1, x2), MARK(0)) ---------------------------------------- (82) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(filter(z0, z1)) -> c22(A__FILTER(mark(z0), mark(z1)), MARK(z0)) MARK(filter(z0, z1)) -> c23(A__FILTER(mark(z0), mark(z1)), MARK(z1)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__SIEVE(cons(mark(s(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__primes), MARK(primes)) MARK(sieve(sieve(z0))) -> c17(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(from(z0))) -> c17(A__SIEVE(a__from(mark(z0))), MARK(from(z0))) MARK(sieve(head(z0))) -> c17(A__SIEVE(a__head(mark(z0))), MARK(head(z0))) MARK(sieve(tail(z0))) -> c17(A__SIEVE(a__tail(mark(z0))), MARK(tail(z0))) MARK(sieve(if(z0, z1, z2))) -> c17(A__SIEVE(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(sieve(filter(z0, z1))) -> c17(A__SIEVE(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(primes)) -> c19(A__HEAD(a__primes), MARK(primes)) MARK(head(sieve(z0))) -> c19(A__HEAD(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(head(from(z0))) -> c19(A__HEAD(a__from(mark(z0))), MARK(from(z0))) MARK(head(head(z0))) -> c19(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c19(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(if(z0, z1, z2))) -> c19(A__HEAD(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(head(filter(z0, z1))) -> c19(A__HEAD(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(primes)) -> c20(A__TAIL(a__primes), MARK(primes)) MARK(tail(sieve(z0))) -> c20(A__TAIL(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(tail(from(z0))) -> c20(A__TAIL(a__from(mark(z0))), MARK(from(z0))) MARK(tail(head(z0))) -> c20(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c20(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(if(z0, z1, z2))) -> c20(A__TAIL(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(tail(filter(z0, z1))) -> c20(A__TAIL(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(primes, x1, x2)) -> c21(A__IF(a__primes, x1, x2), MARK(primes)) MARK(if(sieve(z0), x1, x2)) -> c21(A__IF(a__sieve(mark(z0)), x1, x2), MARK(sieve(z0))) MARK(if(from(z0), x1, x2)) -> c21(A__IF(a__from(mark(z0)), x1, x2), MARK(from(z0))) MARK(if(head(z0), x1, x2)) -> c21(A__IF(a__head(mark(z0)), x1, x2), MARK(head(z0))) MARK(if(tail(z0), x1, x2)) -> c21(A__IF(a__tail(mark(z0)), x1, x2), MARK(tail(z0))) MARK(if(if(z0, z1, z2), x1, x2)) -> c21(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(filter(z0, z1), x1, x2)) -> c21(A__IF(a__filter(mark(z0), mark(z1)), x1, x2), MARK(filter(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(A__IF(s(mark(z0)), x1, x2), MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(A__IF(cons(mark(z0), z1), x1, x2), MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2), MARK(true)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2), MARK(false)) MARK(if(divides(z0, z1), x1, x2)) -> c21(A__IF(divides(mark(z0), mark(z1)), x1, x2), MARK(divides(z0, z1))) S tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(filter(z0, z1)) -> c22(A__FILTER(mark(z0), mark(z1)), MARK(z0)) MARK(filter(z0, z1)) -> c23(A__FILTER(mark(z0), mark(z1)), MARK(z1)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__SIEVE(cons(mark(s(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__primes), MARK(primes)) MARK(sieve(sieve(z0))) -> c17(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(from(z0))) -> c17(A__SIEVE(a__from(mark(z0))), MARK(from(z0))) MARK(sieve(head(z0))) -> c17(A__SIEVE(a__head(mark(z0))), MARK(head(z0))) MARK(sieve(tail(z0))) -> c17(A__SIEVE(a__tail(mark(z0))), MARK(tail(z0))) MARK(sieve(if(z0, z1, z2))) -> c17(A__SIEVE(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(sieve(filter(z0, z1))) -> c17(A__SIEVE(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(primes)) -> c19(A__HEAD(a__primes), MARK(primes)) MARK(head(sieve(z0))) -> c19(A__HEAD(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(head(from(z0))) -> c19(A__HEAD(a__from(mark(z0))), MARK(from(z0))) MARK(head(head(z0))) -> c19(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c19(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(if(z0, z1, z2))) -> c19(A__HEAD(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(head(filter(z0, z1))) -> c19(A__HEAD(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(primes)) -> c20(A__TAIL(a__primes), MARK(primes)) MARK(tail(sieve(z0))) -> c20(A__TAIL(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(tail(from(z0))) -> c20(A__TAIL(a__from(mark(z0))), MARK(from(z0))) MARK(tail(head(z0))) -> c20(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c20(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(if(z0, z1, z2))) -> c20(A__TAIL(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(tail(filter(z0, z1))) -> c20(A__TAIL(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(primes, x1, x2)) -> c21(A__IF(a__primes, x1, x2), MARK(primes)) MARK(if(sieve(z0), x1, x2)) -> c21(A__IF(a__sieve(mark(z0)), x1, x2), MARK(sieve(z0))) MARK(if(from(z0), x1, x2)) -> c21(A__IF(a__from(mark(z0)), x1, x2), MARK(from(z0))) MARK(if(head(z0), x1, x2)) -> c21(A__IF(a__head(mark(z0)), x1, x2), MARK(head(z0))) MARK(if(tail(z0), x1, x2)) -> c21(A__IF(a__tail(mark(z0)), x1, x2), MARK(tail(z0))) MARK(if(if(z0, z1, z2), x1, x2)) -> c21(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(filter(z0, z1), x1, x2)) -> c21(A__IF(a__filter(mark(z0), mark(z1)), x1, x2), MARK(filter(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(A__IF(s(mark(z0)), x1, x2), MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(A__IF(cons(mark(z0), z1), x1, x2), MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2), MARK(true)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2), MARK(false)) MARK(if(divides(z0, z1), x1, x2)) -> c21(A__IF(divides(mark(z0), mark(z1)), x1, x2), MARK(divides(z0, z1))) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c18_2, c22_2, c23_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_2, c_1, c17_2, c17_1, c19_2, c19_1, c20_2, c20_1, c21_2 ---------------------------------------- (83) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 5 trailing tuple parts ---------------------------------------- (84) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(filter(z0, z1)) -> c22(A__FILTER(mark(z0), mark(z1)), MARK(z0)) MARK(filter(z0, z1)) -> c23(A__FILTER(mark(z0), mark(z1)), MARK(z1)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__SIEVE(cons(mark(s(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__primes), MARK(primes)) MARK(sieve(sieve(z0))) -> c17(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(from(z0))) -> c17(A__SIEVE(a__from(mark(z0))), MARK(from(z0))) MARK(sieve(head(z0))) -> c17(A__SIEVE(a__head(mark(z0))), MARK(head(z0))) MARK(sieve(tail(z0))) -> c17(A__SIEVE(a__tail(mark(z0))), MARK(tail(z0))) MARK(sieve(if(z0, z1, z2))) -> c17(A__SIEVE(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(sieve(filter(z0, z1))) -> c17(A__SIEVE(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(primes)) -> c19(A__HEAD(a__primes), MARK(primes)) MARK(head(sieve(z0))) -> c19(A__HEAD(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(head(from(z0))) -> c19(A__HEAD(a__from(mark(z0))), MARK(from(z0))) MARK(head(head(z0))) -> c19(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c19(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(if(z0, z1, z2))) -> c19(A__HEAD(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(head(filter(z0, z1))) -> c19(A__HEAD(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(primes)) -> c20(A__TAIL(a__primes), MARK(primes)) MARK(tail(sieve(z0))) -> c20(A__TAIL(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(tail(from(z0))) -> c20(A__TAIL(a__from(mark(z0))), MARK(from(z0))) MARK(tail(head(z0))) -> c20(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c20(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(if(z0, z1, z2))) -> c20(A__TAIL(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(tail(filter(z0, z1))) -> c20(A__TAIL(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(primes, x1, x2)) -> c21(A__IF(a__primes, x1, x2), MARK(primes)) MARK(if(sieve(z0), x1, x2)) -> c21(A__IF(a__sieve(mark(z0)), x1, x2), MARK(sieve(z0))) MARK(if(from(z0), x1, x2)) -> c21(A__IF(a__from(mark(z0)), x1, x2), MARK(from(z0))) MARK(if(head(z0), x1, x2)) -> c21(A__IF(a__head(mark(z0)), x1, x2), MARK(head(z0))) MARK(if(tail(z0), x1, x2)) -> c21(A__IF(a__tail(mark(z0)), x1, x2), MARK(tail(z0))) MARK(if(if(z0, z1, z2), x1, x2)) -> c21(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(filter(z0, z1), x1, x2)) -> c21(A__IF(a__filter(mark(z0), mark(z1)), x1, x2), MARK(filter(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) S tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(filter(z0, z1)) -> c22(A__FILTER(mark(z0), mark(z1)), MARK(z0)) MARK(filter(z0, z1)) -> c23(A__FILTER(mark(z0), mark(z1)), MARK(z1)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__SIEVE(cons(mark(s(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__primes), MARK(primes)) MARK(sieve(sieve(z0))) -> c17(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(from(z0))) -> c17(A__SIEVE(a__from(mark(z0))), MARK(from(z0))) MARK(sieve(head(z0))) -> c17(A__SIEVE(a__head(mark(z0))), MARK(head(z0))) MARK(sieve(tail(z0))) -> c17(A__SIEVE(a__tail(mark(z0))), MARK(tail(z0))) MARK(sieve(if(z0, z1, z2))) -> c17(A__SIEVE(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(sieve(filter(z0, z1))) -> c17(A__SIEVE(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(primes)) -> c19(A__HEAD(a__primes), MARK(primes)) MARK(head(sieve(z0))) -> c19(A__HEAD(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(head(from(z0))) -> c19(A__HEAD(a__from(mark(z0))), MARK(from(z0))) MARK(head(head(z0))) -> c19(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c19(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(if(z0, z1, z2))) -> c19(A__HEAD(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(head(filter(z0, z1))) -> c19(A__HEAD(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(primes)) -> c20(A__TAIL(a__primes), MARK(primes)) MARK(tail(sieve(z0))) -> c20(A__TAIL(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(tail(from(z0))) -> c20(A__TAIL(a__from(mark(z0))), MARK(from(z0))) MARK(tail(head(z0))) -> c20(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c20(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(if(z0, z1, z2))) -> c20(A__TAIL(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(tail(filter(z0, z1))) -> c20(A__TAIL(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(primes, x1, x2)) -> c21(A__IF(a__primes, x1, x2), MARK(primes)) MARK(if(sieve(z0), x1, x2)) -> c21(A__IF(a__sieve(mark(z0)), x1, x2), MARK(sieve(z0))) MARK(if(from(z0), x1, x2)) -> c21(A__IF(a__from(mark(z0)), x1, x2), MARK(from(z0))) MARK(if(head(z0), x1, x2)) -> c21(A__IF(a__head(mark(z0)), x1, x2), MARK(head(z0))) MARK(if(tail(z0), x1, x2)) -> c21(A__IF(a__tail(mark(z0)), x1, x2), MARK(tail(z0))) MARK(if(if(z0, z1, z2), x1, x2)) -> c21(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(filter(z0, z1), x1, x2)) -> c21(A__IF(a__filter(mark(z0), mark(z1)), x1, x2), MARK(filter(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c18_2, c22_2, c23_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_2, c_1, c17_2, c17_1, c19_2, c19_1, c20_2, c20_1, c21_2, c21_1 ---------------------------------------- (85) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(filter(z0, z1)) -> c22(A__FILTER(mark(z0), mark(z1)), MARK(z0)) by MARK(filter(x0, primes)) -> c22(A__FILTER(mark(x0), a__primes), MARK(x0)) MARK(filter(x0, sieve(z0))) -> c22(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, from(z0))) -> c22(A__FILTER(mark(x0), a__from(mark(z0))), MARK(x0)) MARK(filter(x0, head(z0))) -> c22(A__FILTER(mark(x0), a__head(mark(z0))), MARK(x0)) MARK(filter(x0, tail(z0))) -> c22(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(x0)) MARK(filter(x0, if(z0, z1, z2))) -> c22(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0)) MARK(filter(x0, filter(z0, z1))) -> c22(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(x0)) MARK(filter(x0, s(z0))) -> c22(A__FILTER(mark(x0), s(mark(z0))), MARK(x0)) MARK(filter(x0, 0)) -> c22(A__FILTER(mark(x0), 0), MARK(x0)) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(x0, true)) -> c22(A__FILTER(mark(x0), true), MARK(x0)) MARK(filter(x0, false)) -> c22(A__FILTER(mark(x0), false), MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(A__FILTER(mark(x0), divides(mark(z0), mark(z1))), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(0, x1)) -> c22(A__FILTER(0, mark(x1)), MARK(0)) MARK(filter(cons(z0, z1), x1)) -> c22(A__FILTER(cons(mark(z0), z1), mark(x1)), MARK(cons(z0, z1))) MARK(filter(true, x1)) -> c22(A__FILTER(true, mark(x1)), MARK(true)) MARK(filter(false, x1)) -> c22(A__FILTER(false, mark(x1)), MARK(false)) MARK(filter(divides(z0, z1), x1)) -> c22(A__FILTER(divides(mark(z0), mark(z1)), mark(x1)), MARK(divides(z0, z1))) ---------------------------------------- (86) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(filter(z0, z1)) -> c23(A__FILTER(mark(z0), mark(z1)), MARK(z1)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__SIEVE(cons(mark(s(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__primes), MARK(primes)) MARK(sieve(sieve(z0))) -> c17(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(from(z0))) -> c17(A__SIEVE(a__from(mark(z0))), MARK(from(z0))) MARK(sieve(head(z0))) -> c17(A__SIEVE(a__head(mark(z0))), MARK(head(z0))) MARK(sieve(tail(z0))) -> c17(A__SIEVE(a__tail(mark(z0))), MARK(tail(z0))) MARK(sieve(if(z0, z1, z2))) -> c17(A__SIEVE(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(sieve(filter(z0, z1))) -> c17(A__SIEVE(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(primes)) -> c19(A__HEAD(a__primes), MARK(primes)) MARK(head(sieve(z0))) -> c19(A__HEAD(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(head(from(z0))) -> c19(A__HEAD(a__from(mark(z0))), MARK(from(z0))) MARK(head(head(z0))) -> c19(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c19(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(if(z0, z1, z2))) -> c19(A__HEAD(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(head(filter(z0, z1))) -> c19(A__HEAD(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(primes)) -> c20(A__TAIL(a__primes), MARK(primes)) MARK(tail(sieve(z0))) -> c20(A__TAIL(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(tail(from(z0))) -> c20(A__TAIL(a__from(mark(z0))), MARK(from(z0))) MARK(tail(head(z0))) -> c20(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c20(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(if(z0, z1, z2))) -> c20(A__TAIL(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(tail(filter(z0, z1))) -> c20(A__TAIL(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(primes, x1, x2)) -> c21(A__IF(a__primes, x1, x2), MARK(primes)) MARK(if(sieve(z0), x1, x2)) -> c21(A__IF(a__sieve(mark(z0)), x1, x2), MARK(sieve(z0))) MARK(if(from(z0), x1, x2)) -> c21(A__IF(a__from(mark(z0)), x1, x2), MARK(from(z0))) MARK(if(head(z0), x1, x2)) -> c21(A__IF(a__head(mark(z0)), x1, x2), MARK(head(z0))) MARK(if(tail(z0), x1, x2)) -> c21(A__IF(a__tail(mark(z0)), x1, x2), MARK(tail(z0))) MARK(if(if(z0, z1, z2), x1, x2)) -> c21(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(filter(z0, z1), x1, x2)) -> c21(A__IF(a__filter(mark(z0), mark(z1)), x1, x2), MARK(filter(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c22(A__FILTER(mark(x0), a__primes), MARK(x0)) MARK(filter(x0, sieve(z0))) -> c22(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, from(z0))) -> c22(A__FILTER(mark(x0), a__from(mark(z0))), MARK(x0)) MARK(filter(x0, head(z0))) -> c22(A__FILTER(mark(x0), a__head(mark(z0))), MARK(x0)) MARK(filter(x0, tail(z0))) -> c22(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(x0)) MARK(filter(x0, if(z0, z1, z2))) -> c22(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0)) MARK(filter(x0, filter(z0, z1))) -> c22(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(x0)) MARK(filter(x0, s(z0))) -> c22(A__FILTER(mark(x0), s(mark(z0))), MARK(x0)) MARK(filter(x0, 0)) -> c22(A__FILTER(mark(x0), 0), MARK(x0)) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(x0, true)) -> c22(A__FILTER(mark(x0), true), MARK(x0)) MARK(filter(x0, false)) -> c22(A__FILTER(mark(x0), false), MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(A__FILTER(mark(x0), divides(mark(z0), mark(z1))), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(0, x1)) -> c22(A__FILTER(0, mark(x1)), MARK(0)) MARK(filter(cons(z0, z1), x1)) -> c22(A__FILTER(cons(mark(z0), z1), mark(x1)), MARK(cons(z0, z1))) MARK(filter(true, x1)) -> c22(A__FILTER(true, mark(x1)), MARK(true)) MARK(filter(false, x1)) -> c22(A__FILTER(false, mark(x1)), MARK(false)) MARK(filter(divides(z0, z1), x1)) -> c22(A__FILTER(divides(mark(z0), mark(z1)), mark(x1)), MARK(divides(z0, z1))) S tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(filter(z0, z1)) -> c23(A__FILTER(mark(z0), mark(z1)), MARK(z1)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__SIEVE(cons(mark(s(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__primes), MARK(primes)) MARK(sieve(sieve(z0))) -> c17(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(from(z0))) -> c17(A__SIEVE(a__from(mark(z0))), MARK(from(z0))) MARK(sieve(head(z0))) -> c17(A__SIEVE(a__head(mark(z0))), MARK(head(z0))) MARK(sieve(tail(z0))) -> c17(A__SIEVE(a__tail(mark(z0))), MARK(tail(z0))) MARK(sieve(if(z0, z1, z2))) -> c17(A__SIEVE(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(sieve(filter(z0, z1))) -> c17(A__SIEVE(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(primes)) -> c19(A__HEAD(a__primes), MARK(primes)) MARK(head(sieve(z0))) -> c19(A__HEAD(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(head(from(z0))) -> c19(A__HEAD(a__from(mark(z0))), MARK(from(z0))) MARK(head(head(z0))) -> c19(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c19(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(if(z0, z1, z2))) -> c19(A__HEAD(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(head(filter(z0, z1))) -> c19(A__HEAD(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(primes)) -> c20(A__TAIL(a__primes), MARK(primes)) MARK(tail(sieve(z0))) -> c20(A__TAIL(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(tail(from(z0))) -> c20(A__TAIL(a__from(mark(z0))), MARK(from(z0))) MARK(tail(head(z0))) -> c20(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c20(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(if(z0, z1, z2))) -> c20(A__TAIL(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(tail(filter(z0, z1))) -> c20(A__TAIL(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(primes, x1, x2)) -> c21(A__IF(a__primes, x1, x2), MARK(primes)) MARK(if(sieve(z0), x1, x2)) -> c21(A__IF(a__sieve(mark(z0)), x1, x2), MARK(sieve(z0))) MARK(if(from(z0), x1, x2)) -> c21(A__IF(a__from(mark(z0)), x1, x2), MARK(from(z0))) MARK(if(head(z0), x1, x2)) -> c21(A__IF(a__head(mark(z0)), x1, x2), MARK(head(z0))) MARK(if(tail(z0), x1, x2)) -> c21(A__IF(a__tail(mark(z0)), x1, x2), MARK(tail(z0))) MARK(if(if(z0, z1, z2), x1, x2)) -> c21(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(filter(z0, z1), x1, x2)) -> c21(A__IF(a__filter(mark(z0), mark(z1)), x1, x2), MARK(filter(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c22(A__FILTER(mark(x0), a__primes), MARK(x0)) MARK(filter(x0, sieve(z0))) -> c22(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, from(z0))) -> c22(A__FILTER(mark(x0), a__from(mark(z0))), MARK(x0)) MARK(filter(x0, head(z0))) -> c22(A__FILTER(mark(x0), a__head(mark(z0))), MARK(x0)) MARK(filter(x0, tail(z0))) -> c22(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(x0)) MARK(filter(x0, if(z0, z1, z2))) -> c22(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0)) MARK(filter(x0, filter(z0, z1))) -> c22(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(x0)) MARK(filter(x0, s(z0))) -> c22(A__FILTER(mark(x0), s(mark(z0))), MARK(x0)) MARK(filter(x0, 0)) -> c22(A__FILTER(mark(x0), 0), MARK(x0)) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(x0, true)) -> c22(A__FILTER(mark(x0), true), MARK(x0)) MARK(filter(x0, false)) -> c22(A__FILTER(mark(x0), false), MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(A__FILTER(mark(x0), divides(mark(z0), mark(z1))), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(0, x1)) -> c22(A__FILTER(0, mark(x1)), MARK(0)) MARK(filter(cons(z0, z1), x1)) -> c22(A__FILTER(cons(mark(z0), z1), mark(x1)), MARK(cons(z0, z1))) MARK(filter(true, x1)) -> c22(A__FILTER(true, mark(x1)), MARK(true)) MARK(filter(false, x1)) -> c22(A__FILTER(false, mark(x1)), MARK(false)) MARK(filter(divides(z0, z1), x1)) -> c22(A__FILTER(divides(mark(z0), mark(z1)), mark(x1)), MARK(divides(z0, z1))) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c18_2, c23_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_2, c_1, c17_2, c17_1, c19_2, c19_1, c20_2, c20_1, c21_2, c21_1, c22_2 ---------------------------------------- (87) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing nodes: MARK(filter(0, x1)) -> c22(A__FILTER(0, mark(x1)), MARK(0)) MARK(filter(false, x1)) -> c22(A__FILTER(false, mark(x1)), MARK(false)) MARK(filter(true, x1)) -> c22(A__FILTER(true, mark(x1)), MARK(true)) ---------------------------------------- (88) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(filter(z0, z1)) -> c23(A__FILTER(mark(z0), mark(z1)), MARK(z1)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__SIEVE(cons(mark(s(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__primes), MARK(primes)) MARK(sieve(sieve(z0))) -> c17(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(from(z0))) -> c17(A__SIEVE(a__from(mark(z0))), MARK(from(z0))) MARK(sieve(head(z0))) -> c17(A__SIEVE(a__head(mark(z0))), MARK(head(z0))) MARK(sieve(tail(z0))) -> c17(A__SIEVE(a__tail(mark(z0))), MARK(tail(z0))) MARK(sieve(if(z0, z1, z2))) -> c17(A__SIEVE(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(sieve(filter(z0, z1))) -> c17(A__SIEVE(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(primes)) -> c19(A__HEAD(a__primes), MARK(primes)) MARK(head(sieve(z0))) -> c19(A__HEAD(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(head(from(z0))) -> c19(A__HEAD(a__from(mark(z0))), MARK(from(z0))) MARK(head(head(z0))) -> c19(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c19(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(if(z0, z1, z2))) -> c19(A__HEAD(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(head(filter(z0, z1))) -> c19(A__HEAD(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(primes)) -> c20(A__TAIL(a__primes), MARK(primes)) MARK(tail(sieve(z0))) -> c20(A__TAIL(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(tail(from(z0))) -> c20(A__TAIL(a__from(mark(z0))), MARK(from(z0))) MARK(tail(head(z0))) -> c20(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c20(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(if(z0, z1, z2))) -> c20(A__TAIL(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(tail(filter(z0, z1))) -> c20(A__TAIL(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(primes, x1, x2)) -> c21(A__IF(a__primes, x1, x2), MARK(primes)) MARK(if(sieve(z0), x1, x2)) -> c21(A__IF(a__sieve(mark(z0)), x1, x2), MARK(sieve(z0))) MARK(if(from(z0), x1, x2)) -> c21(A__IF(a__from(mark(z0)), x1, x2), MARK(from(z0))) MARK(if(head(z0), x1, x2)) -> c21(A__IF(a__head(mark(z0)), x1, x2), MARK(head(z0))) MARK(if(tail(z0), x1, x2)) -> c21(A__IF(a__tail(mark(z0)), x1, x2), MARK(tail(z0))) MARK(if(if(z0, z1, z2), x1, x2)) -> c21(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(filter(z0, z1), x1, x2)) -> c21(A__IF(a__filter(mark(z0), mark(z1)), x1, x2), MARK(filter(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c22(A__FILTER(mark(x0), a__primes), MARK(x0)) MARK(filter(x0, sieve(z0))) -> c22(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, from(z0))) -> c22(A__FILTER(mark(x0), a__from(mark(z0))), MARK(x0)) MARK(filter(x0, head(z0))) -> c22(A__FILTER(mark(x0), a__head(mark(z0))), MARK(x0)) MARK(filter(x0, tail(z0))) -> c22(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(x0)) MARK(filter(x0, if(z0, z1, z2))) -> c22(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0)) MARK(filter(x0, filter(z0, z1))) -> c22(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(x0)) MARK(filter(x0, s(z0))) -> c22(A__FILTER(mark(x0), s(mark(z0))), MARK(x0)) MARK(filter(x0, 0)) -> c22(A__FILTER(mark(x0), 0), MARK(x0)) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(x0, true)) -> c22(A__FILTER(mark(x0), true), MARK(x0)) MARK(filter(x0, false)) -> c22(A__FILTER(mark(x0), false), MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(A__FILTER(mark(x0), divides(mark(z0), mark(z1))), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(cons(z0, z1), x1)) -> c22(A__FILTER(cons(mark(z0), z1), mark(x1)), MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(A__FILTER(divides(mark(z0), mark(z1)), mark(x1)), MARK(divides(z0, z1))) S tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(filter(z0, z1)) -> c23(A__FILTER(mark(z0), mark(z1)), MARK(z1)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__SIEVE(cons(mark(s(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__primes), MARK(primes)) MARK(sieve(sieve(z0))) -> c17(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(from(z0))) -> c17(A__SIEVE(a__from(mark(z0))), MARK(from(z0))) MARK(sieve(head(z0))) -> c17(A__SIEVE(a__head(mark(z0))), MARK(head(z0))) MARK(sieve(tail(z0))) -> c17(A__SIEVE(a__tail(mark(z0))), MARK(tail(z0))) MARK(sieve(if(z0, z1, z2))) -> c17(A__SIEVE(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(sieve(filter(z0, z1))) -> c17(A__SIEVE(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(primes)) -> c19(A__HEAD(a__primes), MARK(primes)) MARK(head(sieve(z0))) -> c19(A__HEAD(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(head(from(z0))) -> c19(A__HEAD(a__from(mark(z0))), MARK(from(z0))) MARK(head(head(z0))) -> c19(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c19(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(if(z0, z1, z2))) -> c19(A__HEAD(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(head(filter(z0, z1))) -> c19(A__HEAD(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(primes)) -> c20(A__TAIL(a__primes), MARK(primes)) MARK(tail(sieve(z0))) -> c20(A__TAIL(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(tail(from(z0))) -> c20(A__TAIL(a__from(mark(z0))), MARK(from(z0))) MARK(tail(head(z0))) -> c20(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c20(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(if(z0, z1, z2))) -> c20(A__TAIL(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(tail(filter(z0, z1))) -> c20(A__TAIL(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(primes, x1, x2)) -> c21(A__IF(a__primes, x1, x2), MARK(primes)) MARK(if(sieve(z0), x1, x2)) -> c21(A__IF(a__sieve(mark(z0)), x1, x2), MARK(sieve(z0))) MARK(if(from(z0), x1, x2)) -> c21(A__IF(a__from(mark(z0)), x1, x2), MARK(from(z0))) MARK(if(head(z0), x1, x2)) -> c21(A__IF(a__head(mark(z0)), x1, x2), MARK(head(z0))) MARK(if(tail(z0), x1, x2)) -> c21(A__IF(a__tail(mark(z0)), x1, x2), MARK(tail(z0))) MARK(if(if(z0, z1, z2), x1, x2)) -> c21(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(filter(z0, z1), x1, x2)) -> c21(A__IF(a__filter(mark(z0), mark(z1)), x1, x2), MARK(filter(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c22(A__FILTER(mark(x0), a__primes), MARK(x0)) MARK(filter(x0, sieve(z0))) -> c22(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, from(z0))) -> c22(A__FILTER(mark(x0), a__from(mark(z0))), MARK(x0)) MARK(filter(x0, head(z0))) -> c22(A__FILTER(mark(x0), a__head(mark(z0))), MARK(x0)) MARK(filter(x0, tail(z0))) -> c22(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(x0)) MARK(filter(x0, if(z0, z1, z2))) -> c22(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0)) MARK(filter(x0, filter(z0, z1))) -> c22(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(x0)) MARK(filter(x0, s(z0))) -> c22(A__FILTER(mark(x0), s(mark(z0))), MARK(x0)) MARK(filter(x0, 0)) -> c22(A__FILTER(mark(x0), 0), MARK(x0)) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(x0, true)) -> c22(A__FILTER(mark(x0), true), MARK(x0)) MARK(filter(x0, false)) -> c22(A__FILTER(mark(x0), false), MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(A__FILTER(mark(x0), divides(mark(z0), mark(z1))), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(cons(z0, z1), x1)) -> c22(A__FILTER(cons(mark(z0), z1), mark(x1)), MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(A__FILTER(divides(mark(z0), mark(z1)), mark(x1)), MARK(divides(z0, z1))) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c18_2, c23_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_2, c_1, c17_2, c17_1, c19_2, c19_1, c20_2, c20_1, c21_2, c21_1, c22_2 ---------------------------------------- (89) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 7 trailing tuple parts ---------------------------------------- (90) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(filter(z0, z1)) -> c23(A__FILTER(mark(z0), mark(z1)), MARK(z1)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__SIEVE(cons(mark(s(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__primes), MARK(primes)) MARK(sieve(sieve(z0))) -> c17(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(from(z0))) -> c17(A__SIEVE(a__from(mark(z0))), MARK(from(z0))) MARK(sieve(head(z0))) -> c17(A__SIEVE(a__head(mark(z0))), MARK(head(z0))) MARK(sieve(tail(z0))) -> c17(A__SIEVE(a__tail(mark(z0))), MARK(tail(z0))) MARK(sieve(if(z0, z1, z2))) -> c17(A__SIEVE(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(sieve(filter(z0, z1))) -> c17(A__SIEVE(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(primes)) -> c19(A__HEAD(a__primes), MARK(primes)) MARK(head(sieve(z0))) -> c19(A__HEAD(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(head(from(z0))) -> c19(A__HEAD(a__from(mark(z0))), MARK(from(z0))) MARK(head(head(z0))) -> c19(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c19(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(if(z0, z1, z2))) -> c19(A__HEAD(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(head(filter(z0, z1))) -> c19(A__HEAD(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(primes)) -> c20(A__TAIL(a__primes), MARK(primes)) MARK(tail(sieve(z0))) -> c20(A__TAIL(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(tail(from(z0))) -> c20(A__TAIL(a__from(mark(z0))), MARK(from(z0))) MARK(tail(head(z0))) -> c20(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c20(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(if(z0, z1, z2))) -> c20(A__TAIL(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(tail(filter(z0, z1))) -> c20(A__TAIL(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(primes, x1, x2)) -> c21(A__IF(a__primes, x1, x2), MARK(primes)) MARK(if(sieve(z0), x1, x2)) -> c21(A__IF(a__sieve(mark(z0)), x1, x2), MARK(sieve(z0))) MARK(if(from(z0), x1, x2)) -> c21(A__IF(a__from(mark(z0)), x1, x2), MARK(from(z0))) MARK(if(head(z0), x1, x2)) -> c21(A__IF(a__head(mark(z0)), x1, x2), MARK(head(z0))) MARK(if(tail(z0), x1, x2)) -> c21(A__IF(a__tail(mark(z0)), x1, x2), MARK(tail(z0))) MARK(if(if(z0, z1, z2), x1, x2)) -> c21(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(filter(z0, z1), x1, x2)) -> c21(A__IF(a__filter(mark(z0), mark(z1)), x1, x2), MARK(filter(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c22(A__FILTER(mark(x0), a__primes), MARK(x0)) MARK(filter(x0, sieve(z0))) -> c22(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, from(z0))) -> c22(A__FILTER(mark(x0), a__from(mark(z0))), MARK(x0)) MARK(filter(x0, head(z0))) -> c22(A__FILTER(mark(x0), a__head(mark(z0))), MARK(x0)) MARK(filter(x0, tail(z0))) -> c22(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(x0)) MARK(filter(x0, if(z0, z1, z2))) -> c22(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0)) MARK(filter(x0, filter(z0, z1))) -> c22(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(x0)) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(x0, s(z0))) -> c22(MARK(x0)) MARK(filter(x0, 0)) -> c22(MARK(x0)) MARK(filter(x0, true)) -> c22(MARK(x0)) MARK(filter(x0, false)) -> c22(MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), x1)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(MARK(divides(z0, z1))) S tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(filter(z0, z1)) -> c23(A__FILTER(mark(z0), mark(z1)), MARK(z1)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__SIEVE(cons(mark(s(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__primes), MARK(primes)) MARK(sieve(sieve(z0))) -> c17(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(from(z0))) -> c17(A__SIEVE(a__from(mark(z0))), MARK(from(z0))) MARK(sieve(head(z0))) -> c17(A__SIEVE(a__head(mark(z0))), MARK(head(z0))) MARK(sieve(tail(z0))) -> c17(A__SIEVE(a__tail(mark(z0))), MARK(tail(z0))) MARK(sieve(if(z0, z1, z2))) -> c17(A__SIEVE(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(sieve(filter(z0, z1))) -> c17(A__SIEVE(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(primes)) -> c19(A__HEAD(a__primes), MARK(primes)) MARK(head(sieve(z0))) -> c19(A__HEAD(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(head(from(z0))) -> c19(A__HEAD(a__from(mark(z0))), MARK(from(z0))) MARK(head(head(z0))) -> c19(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c19(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(if(z0, z1, z2))) -> c19(A__HEAD(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(head(filter(z0, z1))) -> c19(A__HEAD(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(primes)) -> c20(A__TAIL(a__primes), MARK(primes)) MARK(tail(sieve(z0))) -> c20(A__TAIL(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(tail(from(z0))) -> c20(A__TAIL(a__from(mark(z0))), MARK(from(z0))) MARK(tail(head(z0))) -> c20(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c20(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(if(z0, z1, z2))) -> c20(A__TAIL(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(tail(filter(z0, z1))) -> c20(A__TAIL(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(primes, x1, x2)) -> c21(A__IF(a__primes, x1, x2), MARK(primes)) MARK(if(sieve(z0), x1, x2)) -> c21(A__IF(a__sieve(mark(z0)), x1, x2), MARK(sieve(z0))) MARK(if(from(z0), x1, x2)) -> c21(A__IF(a__from(mark(z0)), x1, x2), MARK(from(z0))) MARK(if(head(z0), x1, x2)) -> c21(A__IF(a__head(mark(z0)), x1, x2), MARK(head(z0))) MARK(if(tail(z0), x1, x2)) -> c21(A__IF(a__tail(mark(z0)), x1, x2), MARK(tail(z0))) MARK(if(if(z0, z1, z2), x1, x2)) -> c21(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(filter(z0, z1), x1, x2)) -> c21(A__IF(a__filter(mark(z0), mark(z1)), x1, x2), MARK(filter(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c22(A__FILTER(mark(x0), a__primes), MARK(x0)) MARK(filter(x0, sieve(z0))) -> c22(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, from(z0))) -> c22(A__FILTER(mark(x0), a__from(mark(z0))), MARK(x0)) MARK(filter(x0, head(z0))) -> c22(A__FILTER(mark(x0), a__head(mark(z0))), MARK(x0)) MARK(filter(x0, tail(z0))) -> c22(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(x0)) MARK(filter(x0, if(z0, z1, z2))) -> c22(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0)) MARK(filter(x0, filter(z0, z1))) -> c22(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(x0)) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(x0, s(z0))) -> c22(MARK(x0)) MARK(filter(x0, 0)) -> c22(MARK(x0)) MARK(filter(x0, true)) -> c22(MARK(x0)) MARK(filter(x0, false)) -> c22(MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), x1)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(MARK(divides(z0, z1))) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c18_2, c23_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_2, c_1, c17_2, c17_1, c19_2, c19_1, c20_2, c20_1, c21_2, c21_1, c22_2, c22_1 ---------------------------------------- (91) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(filter(z0, z1)) -> c23(A__FILTER(mark(z0), mark(z1)), MARK(z1)) by MARK(filter(x0, primes)) -> c23(A__FILTER(mark(x0), a__primes), MARK(primes)) MARK(filter(x0, sieve(z0))) -> c23(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, from(z0))) -> c23(A__FILTER(mark(x0), a__from(mark(z0))), MARK(from(z0))) MARK(filter(x0, head(z0))) -> c23(A__FILTER(mark(x0), a__head(mark(z0))), MARK(head(z0))) MARK(filter(x0, tail(z0))) -> c23(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(tail(z0))) MARK(filter(x0, if(z0, z1, z2))) -> c23(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(filter(x0, filter(z0, z1))) -> c23(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(filter(x0, s(z0))) -> c23(A__FILTER(mark(x0), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, 0)) -> c23(A__FILTER(mark(x0), 0), MARK(0)) MARK(filter(x0, cons(z0, z1))) -> c23(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, true)) -> c23(A__FILTER(mark(x0), true), MARK(true)) MARK(filter(x0, false)) -> c23(A__FILTER(mark(x0), false), MARK(false)) MARK(filter(x0, divides(z0, z1))) -> c23(A__FILTER(mark(x0), divides(mark(z0), mark(z1))), MARK(divides(z0, z1))) MARK(filter(primes, x1)) -> c23(A__FILTER(a__primes, mark(x1)), MARK(x1)) MARK(filter(sieve(z0), x1)) -> c23(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(from(z0), x1)) -> c23(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(head(z0), x1)) -> c23(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(tail(z0), x1)) -> c23(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(if(z0, z1, z2), x1)) -> c23(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1)) MARK(filter(filter(z0, z1), x1)) -> c23(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(x1)) MARK(filter(s(z0), x1)) -> c23(A__FILTER(s(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(0, x1)) -> c23(A__FILTER(0, mark(x1)), MARK(x1)) MARK(filter(cons(z0, z1), x1)) -> c23(A__FILTER(cons(mark(z0), z1), mark(x1)), MARK(x1)) MARK(filter(true, x1)) -> c23(A__FILTER(true, mark(x1)), MARK(x1)) MARK(filter(false, x1)) -> c23(A__FILTER(false, mark(x1)), MARK(x1)) MARK(filter(divides(z0, z1), x1)) -> c23(A__FILTER(divides(mark(z0), mark(z1)), mark(x1)), MARK(x1)) ---------------------------------------- (92) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__SIEVE(cons(mark(s(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__primes), MARK(primes)) MARK(sieve(sieve(z0))) -> c17(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(from(z0))) -> c17(A__SIEVE(a__from(mark(z0))), MARK(from(z0))) MARK(sieve(head(z0))) -> c17(A__SIEVE(a__head(mark(z0))), MARK(head(z0))) MARK(sieve(tail(z0))) -> c17(A__SIEVE(a__tail(mark(z0))), MARK(tail(z0))) MARK(sieve(if(z0, z1, z2))) -> c17(A__SIEVE(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(sieve(filter(z0, z1))) -> c17(A__SIEVE(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(primes)) -> c19(A__HEAD(a__primes), MARK(primes)) MARK(head(sieve(z0))) -> c19(A__HEAD(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(head(from(z0))) -> c19(A__HEAD(a__from(mark(z0))), MARK(from(z0))) MARK(head(head(z0))) -> c19(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c19(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(if(z0, z1, z2))) -> c19(A__HEAD(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(head(filter(z0, z1))) -> c19(A__HEAD(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(primes)) -> c20(A__TAIL(a__primes), MARK(primes)) MARK(tail(sieve(z0))) -> c20(A__TAIL(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(tail(from(z0))) -> c20(A__TAIL(a__from(mark(z0))), MARK(from(z0))) MARK(tail(head(z0))) -> c20(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c20(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(if(z0, z1, z2))) -> c20(A__TAIL(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(tail(filter(z0, z1))) -> c20(A__TAIL(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(primes, x1, x2)) -> c21(A__IF(a__primes, x1, x2), MARK(primes)) MARK(if(sieve(z0), x1, x2)) -> c21(A__IF(a__sieve(mark(z0)), x1, x2), MARK(sieve(z0))) MARK(if(from(z0), x1, x2)) -> c21(A__IF(a__from(mark(z0)), x1, x2), MARK(from(z0))) MARK(if(head(z0), x1, x2)) -> c21(A__IF(a__head(mark(z0)), x1, x2), MARK(head(z0))) MARK(if(tail(z0), x1, x2)) -> c21(A__IF(a__tail(mark(z0)), x1, x2), MARK(tail(z0))) MARK(if(if(z0, z1, z2), x1, x2)) -> c21(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(filter(z0, z1), x1, x2)) -> c21(A__IF(a__filter(mark(z0), mark(z1)), x1, x2), MARK(filter(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c22(A__FILTER(mark(x0), a__primes), MARK(x0)) MARK(filter(x0, sieve(z0))) -> c22(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, from(z0))) -> c22(A__FILTER(mark(x0), a__from(mark(z0))), MARK(x0)) MARK(filter(x0, head(z0))) -> c22(A__FILTER(mark(x0), a__head(mark(z0))), MARK(x0)) MARK(filter(x0, tail(z0))) -> c22(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(x0)) MARK(filter(x0, if(z0, z1, z2))) -> c22(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0)) MARK(filter(x0, filter(z0, z1))) -> c22(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(x0)) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(x0, s(z0))) -> c22(MARK(x0)) MARK(filter(x0, 0)) -> c22(MARK(x0)) MARK(filter(x0, true)) -> c22(MARK(x0)) MARK(filter(x0, false)) -> c22(MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), x1)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c23(A__FILTER(mark(x0), a__primes), MARK(primes)) MARK(filter(x0, sieve(z0))) -> c23(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, from(z0))) -> c23(A__FILTER(mark(x0), a__from(mark(z0))), MARK(from(z0))) MARK(filter(x0, head(z0))) -> c23(A__FILTER(mark(x0), a__head(mark(z0))), MARK(head(z0))) MARK(filter(x0, tail(z0))) -> c23(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(tail(z0))) MARK(filter(x0, if(z0, z1, z2))) -> c23(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(filter(x0, filter(z0, z1))) -> c23(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(filter(x0, s(z0))) -> c23(A__FILTER(mark(x0), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, 0)) -> c23(A__FILTER(mark(x0), 0), MARK(0)) MARK(filter(x0, cons(z0, z1))) -> c23(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, true)) -> c23(A__FILTER(mark(x0), true), MARK(true)) MARK(filter(x0, false)) -> c23(A__FILTER(mark(x0), false), MARK(false)) MARK(filter(x0, divides(z0, z1))) -> c23(A__FILTER(mark(x0), divides(mark(z0), mark(z1))), MARK(divides(z0, z1))) MARK(filter(primes, x1)) -> c23(A__FILTER(a__primes, mark(x1)), MARK(x1)) MARK(filter(sieve(z0), x1)) -> c23(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(from(z0), x1)) -> c23(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(head(z0), x1)) -> c23(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(tail(z0), x1)) -> c23(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(if(z0, z1, z2), x1)) -> c23(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1)) MARK(filter(filter(z0, z1), x1)) -> c23(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(x1)) MARK(filter(s(z0), x1)) -> c23(A__FILTER(s(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(0, x1)) -> c23(A__FILTER(0, mark(x1)), MARK(x1)) MARK(filter(cons(z0, z1), x1)) -> c23(A__FILTER(cons(mark(z0), z1), mark(x1)), MARK(x1)) MARK(filter(true, x1)) -> c23(A__FILTER(true, mark(x1)), MARK(x1)) MARK(filter(false, x1)) -> c23(A__FILTER(false, mark(x1)), MARK(x1)) MARK(filter(divides(z0, z1), x1)) -> c23(A__FILTER(divides(mark(z0), mark(z1)), mark(x1)), MARK(x1)) S tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__SIEVE(cons(mark(s(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__primes), MARK(primes)) MARK(sieve(sieve(z0))) -> c17(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(from(z0))) -> c17(A__SIEVE(a__from(mark(z0))), MARK(from(z0))) MARK(sieve(head(z0))) -> c17(A__SIEVE(a__head(mark(z0))), MARK(head(z0))) MARK(sieve(tail(z0))) -> c17(A__SIEVE(a__tail(mark(z0))), MARK(tail(z0))) MARK(sieve(if(z0, z1, z2))) -> c17(A__SIEVE(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(sieve(filter(z0, z1))) -> c17(A__SIEVE(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(primes)) -> c19(A__HEAD(a__primes), MARK(primes)) MARK(head(sieve(z0))) -> c19(A__HEAD(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(head(from(z0))) -> c19(A__HEAD(a__from(mark(z0))), MARK(from(z0))) MARK(head(head(z0))) -> c19(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c19(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(if(z0, z1, z2))) -> c19(A__HEAD(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(head(filter(z0, z1))) -> c19(A__HEAD(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(primes)) -> c20(A__TAIL(a__primes), MARK(primes)) MARK(tail(sieve(z0))) -> c20(A__TAIL(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(tail(from(z0))) -> c20(A__TAIL(a__from(mark(z0))), MARK(from(z0))) MARK(tail(head(z0))) -> c20(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c20(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(if(z0, z1, z2))) -> c20(A__TAIL(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(tail(filter(z0, z1))) -> c20(A__TAIL(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(primes, x1, x2)) -> c21(A__IF(a__primes, x1, x2), MARK(primes)) MARK(if(sieve(z0), x1, x2)) -> c21(A__IF(a__sieve(mark(z0)), x1, x2), MARK(sieve(z0))) MARK(if(from(z0), x1, x2)) -> c21(A__IF(a__from(mark(z0)), x1, x2), MARK(from(z0))) MARK(if(head(z0), x1, x2)) -> c21(A__IF(a__head(mark(z0)), x1, x2), MARK(head(z0))) MARK(if(tail(z0), x1, x2)) -> c21(A__IF(a__tail(mark(z0)), x1, x2), MARK(tail(z0))) MARK(if(if(z0, z1, z2), x1, x2)) -> c21(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(filter(z0, z1), x1, x2)) -> c21(A__IF(a__filter(mark(z0), mark(z1)), x1, x2), MARK(filter(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c22(A__FILTER(mark(x0), a__primes), MARK(x0)) MARK(filter(x0, sieve(z0))) -> c22(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, from(z0))) -> c22(A__FILTER(mark(x0), a__from(mark(z0))), MARK(x0)) MARK(filter(x0, head(z0))) -> c22(A__FILTER(mark(x0), a__head(mark(z0))), MARK(x0)) MARK(filter(x0, tail(z0))) -> c22(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(x0)) MARK(filter(x0, if(z0, z1, z2))) -> c22(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0)) MARK(filter(x0, filter(z0, z1))) -> c22(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(x0)) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(x0, s(z0))) -> c22(MARK(x0)) MARK(filter(x0, 0)) -> c22(MARK(x0)) MARK(filter(x0, true)) -> c22(MARK(x0)) MARK(filter(x0, false)) -> c22(MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), x1)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c23(A__FILTER(mark(x0), a__primes), MARK(primes)) MARK(filter(x0, sieve(z0))) -> c23(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, from(z0))) -> c23(A__FILTER(mark(x0), a__from(mark(z0))), MARK(from(z0))) MARK(filter(x0, head(z0))) -> c23(A__FILTER(mark(x0), a__head(mark(z0))), MARK(head(z0))) MARK(filter(x0, tail(z0))) -> c23(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(tail(z0))) MARK(filter(x0, if(z0, z1, z2))) -> c23(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(filter(x0, filter(z0, z1))) -> c23(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(filter(x0, s(z0))) -> c23(A__FILTER(mark(x0), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, 0)) -> c23(A__FILTER(mark(x0), 0), MARK(0)) MARK(filter(x0, cons(z0, z1))) -> c23(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, true)) -> c23(A__FILTER(mark(x0), true), MARK(true)) MARK(filter(x0, false)) -> c23(A__FILTER(mark(x0), false), MARK(false)) MARK(filter(x0, divides(z0, z1))) -> c23(A__FILTER(mark(x0), divides(mark(z0), mark(z1))), MARK(divides(z0, z1))) MARK(filter(primes, x1)) -> c23(A__FILTER(a__primes, mark(x1)), MARK(x1)) MARK(filter(sieve(z0), x1)) -> c23(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(from(z0), x1)) -> c23(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(head(z0), x1)) -> c23(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(tail(z0), x1)) -> c23(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(if(z0, z1, z2), x1)) -> c23(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1)) MARK(filter(filter(z0, z1), x1)) -> c23(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(x1)) MARK(filter(s(z0), x1)) -> c23(A__FILTER(s(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(0, x1)) -> c23(A__FILTER(0, mark(x1)), MARK(x1)) MARK(filter(cons(z0, z1), x1)) -> c23(A__FILTER(cons(mark(z0), z1), mark(x1)), MARK(x1)) MARK(filter(true, x1)) -> c23(A__FILTER(true, mark(x1)), MARK(x1)) MARK(filter(false, x1)) -> c23(A__FILTER(false, mark(x1)), MARK(x1)) MARK(filter(divides(z0, z1), x1)) -> c23(A__FILTER(divides(mark(z0), mark(z1)), mark(x1)), MARK(x1)) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c18_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_2, c_1, c17_2, c17_1, c19_2, c19_1, c20_2, c20_1, c21_2, c21_1, c22_2, c22_1, c23_2 ---------------------------------------- (93) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing nodes: MARK(filter(x0, true)) -> c23(A__FILTER(mark(x0), true), MARK(true)) MARK(filter(x0, false)) -> c23(A__FILTER(mark(x0), false), MARK(false)) MARK(filter(x0, 0)) -> c23(A__FILTER(mark(x0), 0), MARK(0)) ---------------------------------------- (94) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__SIEVE(cons(mark(s(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__primes), MARK(primes)) MARK(sieve(sieve(z0))) -> c17(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(from(z0))) -> c17(A__SIEVE(a__from(mark(z0))), MARK(from(z0))) MARK(sieve(head(z0))) -> c17(A__SIEVE(a__head(mark(z0))), MARK(head(z0))) MARK(sieve(tail(z0))) -> c17(A__SIEVE(a__tail(mark(z0))), MARK(tail(z0))) MARK(sieve(if(z0, z1, z2))) -> c17(A__SIEVE(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(sieve(filter(z0, z1))) -> c17(A__SIEVE(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(primes)) -> c19(A__HEAD(a__primes), MARK(primes)) MARK(head(sieve(z0))) -> c19(A__HEAD(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(head(from(z0))) -> c19(A__HEAD(a__from(mark(z0))), MARK(from(z0))) MARK(head(head(z0))) -> c19(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c19(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(if(z0, z1, z2))) -> c19(A__HEAD(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(head(filter(z0, z1))) -> c19(A__HEAD(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(primes)) -> c20(A__TAIL(a__primes), MARK(primes)) MARK(tail(sieve(z0))) -> c20(A__TAIL(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(tail(from(z0))) -> c20(A__TAIL(a__from(mark(z0))), MARK(from(z0))) MARK(tail(head(z0))) -> c20(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c20(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(if(z0, z1, z2))) -> c20(A__TAIL(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(tail(filter(z0, z1))) -> c20(A__TAIL(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(primes, x1, x2)) -> c21(A__IF(a__primes, x1, x2), MARK(primes)) MARK(if(sieve(z0), x1, x2)) -> c21(A__IF(a__sieve(mark(z0)), x1, x2), MARK(sieve(z0))) MARK(if(from(z0), x1, x2)) -> c21(A__IF(a__from(mark(z0)), x1, x2), MARK(from(z0))) MARK(if(head(z0), x1, x2)) -> c21(A__IF(a__head(mark(z0)), x1, x2), MARK(head(z0))) MARK(if(tail(z0), x1, x2)) -> c21(A__IF(a__tail(mark(z0)), x1, x2), MARK(tail(z0))) MARK(if(if(z0, z1, z2), x1, x2)) -> c21(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(filter(z0, z1), x1, x2)) -> c21(A__IF(a__filter(mark(z0), mark(z1)), x1, x2), MARK(filter(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c22(A__FILTER(mark(x0), a__primes), MARK(x0)) MARK(filter(x0, sieve(z0))) -> c22(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, from(z0))) -> c22(A__FILTER(mark(x0), a__from(mark(z0))), MARK(x0)) MARK(filter(x0, head(z0))) -> c22(A__FILTER(mark(x0), a__head(mark(z0))), MARK(x0)) MARK(filter(x0, tail(z0))) -> c22(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(x0)) MARK(filter(x0, if(z0, z1, z2))) -> c22(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0)) MARK(filter(x0, filter(z0, z1))) -> c22(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(x0)) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(x0, s(z0))) -> c22(MARK(x0)) MARK(filter(x0, 0)) -> c22(MARK(x0)) MARK(filter(x0, true)) -> c22(MARK(x0)) MARK(filter(x0, false)) -> c22(MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), x1)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c23(A__FILTER(mark(x0), a__primes), MARK(primes)) MARK(filter(x0, sieve(z0))) -> c23(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, from(z0))) -> c23(A__FILTER(mark(x0), a__from(mark(z0))), MARK(from(z0))) MARK(filter(x0, head(z0))) -> c23(A__FILTER(mark(x0), a__head(mark(z0))), MARK(head(z0))) MARK(filter(x0, tail(z0))) -> c23(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(tail(z0))) MARK(filter(x0, if(z0, z1, z2))) -> c23(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(filter(x0, filter(z0, z1))) -> c23(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(filter(x0, s(z0))) -> c23(A__FILTER(mark(x0), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, cons(z0, z1))) -> c23(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, divides(z0, z1))) -> c23(A__FILTER(mark(x0), divides(mark(z0), mark(z1))), MARK(divides(z0, z1))) MARK(filter(primes, x1)) -> c23(A__FILTER(a__primes, mark(x1)), MARK(x1)) MARK(filter(sieve(z0), x1)) -> c23(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(from(z0), x1)) -> c23(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(head(z0), x1)) -> c23(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(tail(z0), x1)) -> c23(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(if(z0, z1, z2), x1)) -> c23(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1)) MARK(filter(filter(z0, z1), x1)) -> c23(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(x1)) MARK(filter(s(z0), x1)) -> c23(A__FILTER(s(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(0, x1)) -> c23(A__FILTER(0, mark(x1)), MARK(x1)) MARK(filter(cons(z0, z1), x1)) -> c23(A__FILTER(cons(mark(z0), z1), mark(x1)), MARK(x1)) MARK(filter(true, x1)) -> c23(A__FILTER(true, mark(x1)), MARK(x1)) MARK(filter(false, x1)) -> c23(A__FILTER(false, mark(x1)), MARK(x1)) MARK(filter(divides(z0, z1), x1)) -> c23(A__FILTER(divides(mark(z0), mark(z1)), mark(x1)), MARK(x1)) S tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__SIEVE(cons(mark(s(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__primes), MARK(primes)) MARK(sieve(sieve(z0))) -> c17(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(from(z0))) -> c17(A__SIEVE(a__from(mark(z0))), MARK(from(z0))) MARK(sieve(head(z0))) -> c17(A__SIEVE(a__head(mark(z0))), MARK(head(z0))) MARK(sieve(tail(z0))) -> c17(A__SIEVE(a__tail(mark(z0))), MARK(tail(z0))) MARK(sieve(if(z0, z1, z2))) -> c17(A__SIEVE(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(sieve(filter(z0, z1))) -> c17(A__SIEVE(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(primes)) -> c19(A__HEAD(a__primes), MARK(primes)) MARK(head(sieve(z0))) -> c19(A__HEAD(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(head(from(z0))) -> c19(A__HEAD(a__from(mark(z0))), MARK(from(z0))) MARK(head(head(z0))) -> c19(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c19(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(if(z0, z1, z2))) -> c19(A__HEAD(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(head(filter(z0, z1))) -> c19(A__HEAD(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(primes)) -> c20(A__TAIL(a__primes), MARK(primes)) MARK(tail(sieve(z0))) -> c20(A__TAIL(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(tail(from(z0))) -> c20(A__TAIL(a__from(mark(z0))), MARK(from(z0))) MARK(tail(head(z0))) -> c20(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c20(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(if(z0, z1, z2))) -> c20(A__TAIL(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(tail(filter(z0, z1))) -> c20(A__TAIL(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(primes, x1, x2)) -> c21(A__IF(a__primes, x1, x2), MARK(primes)) MARK(if(sieve(z0), x1, x2)) -> c21(A__IF(a__sieve(mark(z0)), x1, x2), MARK(sieve(z0))) MARK(if(from(z0), x1, x2)) -> c21(A__IF(a__from(mark(z0)), x1, x2), MARK(from(z0))) MARK(if(head(z0), x1, x2)) -> c21(A__IF(a__head(mark(z0)), x1, x2), MARK(head(z0))) MARK(if(tail(z0), x1, x2)) -> c21(A__IF(a__tail(mark(z0)), x1, x2), MARK(tail(z0))) MARK(if(if(z0, z1, z2), x1, x2)) -> c21(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(filter(z0, z1), x1, x2)) -> c21(A__IF(a__filter(mark(z0), mark(z1)), x1, x2), MARK(filter(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c22(A__FILTER(mark(x0), a__primes), MARK(x0)) MARK(filter(x0, sieve(z0))) -> c22(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, from(z0))) -> c22(A__FILTER(mark(x0), a__from(mark(z0))), MARK(x0)) MARK(filter(x0, head(z0))) -> c22(A__FILTER(mark(x0), a__head(mark(z0))), MARK(x0)) MARK(filter(x0, tail(z0))) -> c22(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(x0)) MARK(filter(x0, if(z0, z1, z2))) -> c22(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0)) MARK(filter(x0, filter(z0, z1))) -> c22(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(x0)) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(x0, s(z0))) -> c22(MARK(x0)) MARK(filter(x0, 0)) -> c22(MARK(x0)) MARK(filter(x0, true)) -> c22(MARK(x0)) MARK(filter(x0, false)) -> c22(MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), x1)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c23(A__FILTER(mark(x0), a__primes), MARK(primes)) MARK(filter(x0, sieve(z0))) -> c23(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, from(z0))) -> c23(A__FILTER(mark(x0), a__from(mark(z0))), MARK(from(z0))) MARK(filter(x0, head(z0))) -> c23(A__FILTER(mark(x0), a__head(mark(z0))), MARK(head(z0))) MARK(filter(x0, tail(z0))) -> c23(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(tail(z0))) MARK(filter(x0, if(z0, z1, z2))) -> c23(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(filter(x0, filter(z0, z1))) -> c23(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(filter(x0, s(z0))) -> c23(A__FILTER(mark(x0), s(mark(z0))), MARK(s(z0))) MARK(filter(x0, cons(z0, z1))) -> c23(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(x0, divides(z0, z1))) -> c23(A__FILTER(mark(x0), divides(mark(z0), mark(z1))), MARK(divides(z0, z1))) MARK(filter(primes, x1)) -> c23(A__FILTER(a__primes, mark(x1)), MARK(x1)) MARK(filter(sieve(z0), x1)) -> c23(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(from(z0), x1)) -> c23(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(head(z0), x1)) -> c23(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(tail(z0), x1)) -> c23(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(if(z0, z1, z2), x1)) -> c23(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1)) MARK(filter(filter(z0, z1), x1)) -> c23(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(x1)) MARK(filter(s(z0), x1)) -> c23(A__FILTER(s(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(0, x1)) -> c23(A__FILTER(0, mark(x1)), MARK(x1)) MARK(filter(cons(z0, z1), x1)) -> c23(A__FILTER(cons(mark(z0), z1), mark(x1)), MARK(x1)) MARK(filter(true, x1)) -> c23(A__FILTER(true, mark(x1)), MARK(x1)) MARK(filter(false, x1)) -> c23(A__FILTER(false, mark(x1)), MARK(x1)) MARK(filter(divides(z0, z1), x1)) -> c23(A__FILTER(divides(mark(z0), mark(z1)), mark(x1)), MARK(x1)) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c18_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_2, c_1, c17_2, c17_1, c19_2, c19_1, c20_2, c20_1, c21_2, c21_1, c22_2, c22_1, c23_2 ---------------------------------------- (95) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 7 trailing tuple parts ---------------------------------------- (96) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__SIEVE(cons(mark(s(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__primes), MARK(primes)) MARK(sieve(sieve(z0))) -> c17(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(from(z0))) -> c17(A__SIEVE(a__from(mark(z0))), MARK(from(z0))) MARK(sieve(head(z0))) -> c17(A__SIEVE(a__head(mark(z0))), MARK(head(z0))) MARK(sieve(tail(z0))) -> c17(A__SIEVE(a__tail(mark(z0))), MARK(tail(z0))) MARK(sieve(if(z0, z1, z2))) -> c17(A__SIEVE(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(sieve(filter(z0, z1))) -> c17(A__SIEVE(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(primes)) -> c19(A__HEAD(a__primes), MARK(primes)) MARK(head(sieve(z0))) -> c19(A__HEAD(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(head(from(z0))) -> c19(A__HEAD(a__from(mark(z0))), MARK(from(z0))) MARK(head(head(z0))) -> c19(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c19(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(if(z0, z1, z2))) -> c19(A__HEAD(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(head(filter(z0, z1))) -> c19(A__HEAD(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(primes)) -> c20(A__TAIL(a__primes), MARK(primes)) MARK(tail(sieve(z0))) -> c20(A__TAIL(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(tail(from(z0))) -> c20(A__TAIL(a__from(mark(z0))), MARK(from(z0))) MARK(tail(head(z0))) -> c20(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c20(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(if(z0, z1, z2))) -> c20(A__TAIL(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(tail(filter(z0, z1))) -> c20(A__TAIL(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(primes, x1, x2)) -> c21(A__IF(a__primes, x1, x2), MARK(primes)) MARK(if(sieve(z0), x1, x2)) -> c21(A__IF(a__sieve(mark(z0)), x1, x2), MARK(sieve(z0))) MARK(if(from(z0), x1, x2)) -> c21(A__IF(a__from(mark(z0)), x1, x2), MARK(from(z0))) MARK(if(head(z0), x1, x2)) -> c21(A__IF(a__head(mark(z0)), x1, x2), MARK(head(z0))) MARK(if(tail(z0), x1, x2)) -> c21(A__IF(a__tail(mark(z0)), x1, x2), MARK(tail(z0))) MARK(if(if(z0, z1, z2), x1, x2)) -> c21(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(filter(z0, z1), x1, x2)) -> c21(A__IF(a__filter(mark(z0), mark(z1)), x1, x2), MARK(filter(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c22(A__FILTER(mark(x0), a__primes), MARK(x0)) MARK(filter(x0, sieve(z0))) -> c22(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, from(z0))) -> c22(A__FILTER(mark(x0), a__from(mark(z0))), MARK(x0)) MARK(filter(x0, head(z0))) -> c22(A__FILTER(mark(x0), a__head(mark(z0))), MARK(x0)) MARK(filter(x0, tail(z0))) -> c22(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(x0)) MARK(filter(x0, if(z0, z1, z2))) -> c22(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0)) MARK(filter(x0, filter(z0, z1))) -> c22(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(x0)) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(x0, s(z0))) -> c22(MARK(x0)) MARK(filter(x0, 0)) -> c22(MARK(x0)) MARK(filter(x0, true)) -> c22(MARK(x0)) MARK(filter(x0, false)) -> c22(MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), x1)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c23(A__FILTER(mark(x0), a__primes), MARK(primes)) MARK(filter(x0, sieve(z0))) -> c23(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, from(z0))) -> c23(A__FILTER(mark(x0), a__from(mark(z0))), MARK(from(z0))) MARK(filter(x0, head(z0))) -> c23(A__FILTER(mark(x0), a__head(mark(z0))), MARK(head(z0))) MARK(filter(x0, tail(z0))) -> c23(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(tail(z0))) MARK(filter(x0, if(z0, z1, z2))) -> c23(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(filter(x0, filter(z0, z1))) -> c23(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(filter(x0, cons(z0, z1))) -> c23(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(primes, x1)) -> c23(A__FILTER(a__primes, mark(x1)), MARK(x1)) MARK(filter(sieve(z0), x1)) -> c23(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(from(z0), x1)) -> c23(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(head(z0), x1)) -> c23(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(tail(z0), x1)) -> c23(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(if(z0, z1, z2), x1)) -> c23(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1)) MARK(filter(filter(z0, z1), x1)) -> c23(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(x1)) MARK(filter(s(z0), x1)) -> c23(A__FILTER(s(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(x0, s(z0))) -> c23(MARK(s(z0))) MARK(filter(x0, divides(z0, z1))) -> c23(MARK(divides(z0, z1))) MARK(filter(0, x1)) -> c23(MARK(x1)) MARK(filter(cons(z0, z1), x1)) -> c23(MARK(x1)) MARK(filter(true, x1)) -> c23(MARK(x1)) MARK(filter(false, x1)) -> c23(MARK(x1)) MARK(filter(divides(z0, z1), x1)) -> c23(MARK(x1)) S tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__SIEVE(cons(mark(s(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__primes), MARK(primes)) MARK(sieve(sieve(z0))) -> c17(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(from(z0))) -> c17(A__SIEVE(a__from(mark(z0))), MARK(from(z0))) MARK(sieve(head(z0))) -> c17(A__SIEVE(a__head(mark(z0))), MARK(head(z0))) MARK(sieve(tail(z0))) -> c17(A__SIEVE(a__tail(mark(z0))), MARK(tail(z0))) MARK(sieve(if(z0, z1, z2))) -> c17(A__SIEVE(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(sieve(filter(z0, z1))) -> c17(A__SIEVE(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(primes)) -> c19(A__HEAD(a__primes), MARK(primes)) MARK(head(sieve(z0))) -> c19(A__HEAD(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(head(from(z0))) -> c19(A__HEAD(a__from(mark(z0))), MARK(from(z0))) MARK(head(head(z0))) -> c19(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c19(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(if(z0, z1, z2))) -> c19(A__HEAD(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(head(filter(z0, z1))) -> c19(A__HEAD(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(primes)) -> c20(A__TAIL(a__primes), MARK(primes)) MARK(tail(sieve(z0))) -> c20(A__TAIL(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(tail(from(z0))) -> c20(A__TAIL(a__from(mark(z0))), MARK(from(z0))) MARK(tail(head(z0))) -> c20(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c20(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(if(z0, z1, z2))) -> c20(A__TAIL(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(tail(filter(z0, z1))) -> c20(A__TAIL(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(primes, x1, x2)) -> c21(A__IF(a__primes, x1, x2), MARK(primes)) MARK(if(sieve(z0), x1, x2)) -> c21(A__IF(a__sieve(mark(z0)), x1, x2), MARK(sieve(z0))) MARK(if(from(z0), x1, x2)) -> c21(A__IF(a__from(mark(z0)), x1, x2), MARK(from(z0))) MARK(if(head(z0), x1, x2)) -> c21(A__IF(a__head(mark(z0)), x1, x2), MARK(head(z0))) MARK(if(tail(z0), x1, x2)) -> c21(A__IF(a__tail(mark(z0)), x1, x2), MARK(tail(z0))) MARK(if(if(z0, z1, z2), x1, x2)) -> c21(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(filter(z0, z1), x1, x2)) -> c21(A__IF(a__filter(mark(z0), mark(z1)), x1, x2), MARK(filter(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c22(A__FILTER(mark(x0), a__primes), MARK(x0)) MARK(filter(x0, sieve(z0))) -> c22(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, from(z0))) -> c22(A__FILTER(mark(x0), a__from(mark(z0))), MARK(x0)) MARK(filter(x0, head(z0))) -> c22(A__FILTER(mark(x0), a__head(mark(z0))), MARK(x0)) MARK(filter(x0, tail(z0))) -> c22(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(x0)) MARK(filter(x0, if(z0, z1, z2))) -> c22(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0)) MARK(filter(x0, filter(z0, z1))) -> c22(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(x0)) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(x0, s(z0))) -> c22(MARK(x0)) MARK(filter(x0, 0)) -> c22(MARK(x0)) MARK(filter(x0, true)) -> c22(MARK(x0)) MARK(filter(x0, false)) -> c22(MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), x1)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c23(A__FILTER(mark(x0), a__primes), MARK(primes)) MARK(filter(x0, sieve(z0))) -> c23(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, from(z0))) -> c23(A__FILTER(mark(x0), a__from(mark(z0))), MARK(from(z0))) MARK(filter(x0, head(z0))) -> c23(A__FILTER(mark(x0), a__head(mark(z0))), MARK(head(z0))) MARK(filter(x0, tail(z0))) -> c23(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(tail(z0))) MARK(filter(x0, if(z0, z1, z2))) -> c23(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(filter(x0, filter(z0, z1))) -> c23(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(filter(x0, cons(z0, z1))) -> c23(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(primes, x1)) -> c23(A__FILTER(a__primes, mark(x1)), MARK(x1)) MARK(filter(sieve(z0), x1)) -> c23(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(from(z0), x1)) -> c23(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(head(z0), x1)) -> c23(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(tail(z0), x1)) -> c23(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(if(z0, z1, z2), x1)) -> c23(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1)) MARK(filter(filter(z0, z1), x1)) -> c23(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(x1)) MARK(filter(s(z0), x1)) -> c23(A__FILTER(s(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(x0, s(z0))) -> c23(MARK(s(z0))) MARK(filter(x0, divides(z0, z1))) -> c23(MARK(divides(z0, z1))) MARK(filter(0, x1)) -> c23(MARK(x1)) MARK(filter(cons(z0, z1), x1)) -> c23(MARK(x1)) MARK(filter(true, x1)) -> c23(MARK(x1)) MARK(filter(false, x1)) -> c23(MARK(x1)) MARK(filter(divides(z0, z1), x1)) -> c23(MARK(x1)) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c18_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_2, c_1, c17_2, c17_1, c19_2, c19_1, c20_2, c20_1, c21_2, c21_1, c22_2, c22_1, c23_2, c23_1 ---------------------------------------- (97) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace A__PRIMES -> c(A__SIEVE(cons(mark(s(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) by A__PRIMES -> c(A__SIEVE(cons(s(mark(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) A__PRIMES -> c(A__FROM(s(s(0)))) ---------------------------------------- (98) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__primes), MARK(primes)) MARK(sieve(sieve(z0))) -> c17(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(from(z0))) -> c17(A__SIEVE(a__from(mark(z0))), MARK(from(z0))) MARK(sieve(head(z0))) -> c17(A__SIEVE(a__head(mark(z0))), MARK(head(z0))) MARK(sieve(tail(z0))) -> c17(A__SIEVE(a__tail(mark(z0))), MARK(tail(z0))) MARK(sieve(if(z0, z1, z2))) -> c17(A__SIEVE(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(sieve(filter(z0, z1))) -> c17(A__SIEVE(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(primes)) -> c19(A__HEAD(a__primes), MARK(primes)) MARK(head(sieve(z0))) -> c19(A__HEAD(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(head(from(z0))) -> c19(A__HEAD(a__from(mark(z0))), MARK(from(z0))) MARK(head(head(z0))) -> c19(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c19(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(if(z0, z1, z2))) -> c19(A__HEAD(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(head(filter(z0, z1))) -> c19(A__HEAD(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(primes)) -> c20(A__TAIL(a__primes), MARK(primes)) MARK(tail(sieve(z0))) -> c20(A__TAIL(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(tail(from(z0))) -> c20(A__TAIL(a__from(mark(z0))), MARK(from(z0))) MARK(tail(head(z0))) -> c20(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c20(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(if(z0, z1, z2))) -> c20(A__TAIL(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(tail(filter(z0, z1))) -> c20(A__TAIL(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(primes, x1, x2)) -> c21(A__IF(a__primes, x1, x2), MARK(primes)) MARK(if(sieve(z0), x1, x2)) -> c21(A__IF(a__sieve(mark(z0)), x1, x2), MARK(sieve(z0))) MARK(if(from(z0), x1, x2)) -> c21(A__IF(a__from(mark(z0)), x1, x2), MARK(from(z0))) MARK(if(head(z0), x1, x2)) -> c21(A__IF(a__head(mark(z0)), x1, x2), MARK(head(z0))) MARK(if(tail(z0), x1, x2)) -> c21(A__IF(a__tail(mark(z0)), x1, x2), MARK(tail(z0))) MARK(if(if(z0, z1, z2), x1, x2)) -> c21(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(filter(z0, z1), x1, x2)) -> c21(A__IF(a__filter(mark(z0), mark(z1)), x1, x2), MARK(filter(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c22(A__FILTER(mark(x0), a__primes), MARK(x0)) MARK(filter(x0, sieve(z0))) -> c22(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, from(z0))) -> c22(A__FILTER(mark(x0), a__from(mark(z0))), MARK(x0)) MARK(filter(x0, head(z0))) -> c22(A__FILTER(mark(x0), a__head(mark(z0))), MARK(x0)) MARK(filter(x0, tail(z0))) -> c22(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(x0)) MARK(filter(x0, if(z0, z1, z2))) -> c22(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0)) MARK(filter(x0, filter(z0, z1))) -> c22(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(x0)) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(x0, s(z0))) -> c22(MARK(x0)) MARK(filter(x0, 0)) -> c22(MARK(x0)) MARK(filter(x0, true)) -> c22(MARK(x0)) MARK(filter(x0, false)) -> c22(MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), x1)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c23(A__FILTER(mark(x0), a__primes), MARK(primes)) MARK(filter(x0, sieve(z0))) -> c23(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, from(z0))) -> c23(A__FILTER(mark(x0), a__from(mark(z0))), MARK(from(z0))) MARK(filter(x0, head(z0))) -> c23(A__FILTER(mark(x0), a__head(mark(z0))), MARK(head(z0))) MARK(filter(x0, tail(z0))) -> c23(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(tail(z0))) MARK(filter(x0, if(z0, z1, z2))) -> c23(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(filter(x0, filter(z0, z1))) -> c23(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(filter(x0, cons(z0, z1))) -> c23(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(primes, x1)) -> c23(A__FILTER(a__primes, mark(x1)), MARK(x1)) MARK(filter(sieve(z0), x1)) -> c23(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(from(z0), x1)) -> c23(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(head(z0), x1)) -> c23(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(tail(z0), x1)) -> c23(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(if(z0, z1, z2), x1)) -> c23(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1)) MARK(filter(filter(z0, z1), x1)) -> c23(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(x1)) MARK(filter(s(z0), x1)) -> c23(A__FILTER(s(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(x0, s(z0))) -> c23(MARK(s(z0))) MARK(filter(x0, divides(z0, z1))) -> c23(MARK(divides(z0, z1))) MARK(filter(0, x1)) -> c23(MARK(x1)) MARK(filter(cons(z0, z1), x1)) -> c23(MARK(x1)) MARK(filter(true, x1)) -> c23(MARK(x1)) MARK(filter(false, x1)) -> c23(MARK(x1)) MARK(filter(divides(z0, z1), x1)) -> c23(MARK(x1)) A__PRIMES -> c(A__SIEVE(cons(s(mark(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) S tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__primes), MARK(primes)) MARK(sieve(sieve(z0))) -> c17(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(from(z0))) -> c17(A__SIEVE(a__from(mark(z0))), MARK(from(z0))) MARK(sieve(head(z0))) -> c17(A__SIEVE(a__head(mark(z0))), MARK(head(z0))) MARK(sieve(tail(z0))) -> c17(A__SIEVE(a__tail(mark(z0))), MARK(tail(z0))) MARK(sieve(if(z0, z1, z2))) -> c17(A__SIEVE(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(sieve(filter(z0, z1))) -> c17(A__SIEVE(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(primes)) -> c19(A__HEAD(a__primes), MARK(primes)) MARK(head(sieve(z0))) -> c19(A__HEAD(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(head(from(z0))) -> c19(A__HEAD(a__from(mark(z0))), MARK(from(z0))) MARK(head(head(z0))) -> c19(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c19(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(if(z0, z1, z2))) -> c19(A__HEAD(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(head(filter(z0, z1))) -> c19(A__HEAD(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(primes)) -> c20(A__TAIL(a__primes), MARK(primes)) MARK(tail(sieve(z0))) -> c20(A__TAIL(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(tail(from(z0))) -> c20(A__TAIL(a__from(mark(z0))), MARK(from(z0))) MARK(tail(head(z0))) -> c20(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c20(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(if(z0, z1, z2))) -> c20(A__TAIL(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(tail(filter(z0, z1))) -> c20(A__TAIL(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(primes, x1, x2)) -> c21(A__IF(a__primes, x1, x2), MARK(primes)) MARK(if(sieve(z0), x1, x2)) -> c21(A__IF(a__sieve(mark(z0)), x1, x2), MARK(sieve(z0))) MARK(if(from(z0), x1, x2)) -> c21(A__IF(a__from(mark(z0)), x1, x2), MARK(from(z0))) MARK(if(head(z0), x1, x2)) -> c21(A__IF(a__head(mark(z0)), x1, x2), MARK(head(z0))) MARK(if(tail(z0), x1, x2)) -> c21(A__IF(a__tail(mark(z0)), x1, x2), MARK(tail(z0))) MARK(if(if(z0, z1, z2), x1, x2)) -> c21(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(filter(z0, z1), x1, x2)) -> c21(A__IF(a__filter(mark(z0), mark(z1)), x1, x2), MARK(filter(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c22(A__FILTER(mark(x0), a__primes), MARK(x0)) MARK(filter(x0, sieve(z0))) -> c22(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, from(z0))) -> c22(A__FILTER(mark(x0), a__from(mark(z0))), MARK(x0)) MARK(filter(x0, head(z0))) -> c22(A__FILTER(mark(x0), a__head(mark(z0))), MARK(x0)) MARK(filter(x0, tail(z0))) -> c22(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(x0)) MARK(filter(x0, if(z0, z1, z2))) -> c22(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0)) MARK(filter(x0, filter(z0, z1))) -> c22(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(x0)) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(x0, s(z0))) -> c22(MARK(x0)) MARK(filter(x0, 0)) -> c22(MARK(x0)) MARK(filter(x0, true)) -> c22(MARK(x0)) MARK(filter(x0, false)) -> c22(MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), x1)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c23(A__FILTER(mark(x0), a__primes), MARK(primes)) MARK(filter(x0, sieve(z0))) -> c23(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, from(z0))) -> c23(A__FILTER(mark(x0), a__from(mark(z0))), MARK(from(z0))) MARK(filter(x0, head(z0))) -> c23(A__FILTER(mark(x0), a__head(mark(z0))), MARK(head(z0))) MARK(filter(x0, tail(z0))) -> c23(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(tail(z0))) MARK(filter(x0, if(z0, z1, z2))) -> c23(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(filter(x0, filter(z0, z1))) -> c23(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(filter(x0, cons(z0, z1))) -> c23(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(primes, x1)) -> c23(A__FILTER(a__primes, mark(x1)), MARK(x1)) MARK(filter(sieve(z0), x1)) -> c23(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(from(z0), x1)) -> c23(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(head(z0), x1)) -> c23(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(tail(z0), x1)) -> c23(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(if(z0, z1, z2), x1)) -> c23(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1)) MARK(filter(filter(z0, z1), x1)) -> c23(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(x1)) MARK(filter(s(z0), x1)) -> c23(A__FILTER(s(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(x0, s(z0))) -> c23(MARK(s(z0))) MARK(filter(x0, divides(z0, z1))) -> c23(MARK(divides(z0, z1))) MARK(filter(0, x1)) -> c23(MARK(x1)) MARK(filter(cons(z0, z1), x1)) -> c23(MARK(x1)) MARK(filter(true, x1)) -> c23(MARK(x1)) MARK(filter(false, x1)) -> c23(MARK(x1)) MARK(filter(divides(z0, z1), x1)) -> c23(MARK(x1)) A__PRIMES -> c(A__SIEVE(cons(s(mark(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c18_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_1, c17_2, c17_1, c19_2, c19_1, c20_2, c20_1, c21_2, c21_1, c22_2, c22_1, c23_2, c23_1, c_2 ---------------------------------------- (99) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(sieve(primes)) -> c17(A__SIEVE(a__primes), MARK(primes)) by MARK(sieve(primes)) -> c17(A__SIEVE(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(sieve(primes)) -> c17(A__SIEVE(primes), MARK(primes)) ---------------------------------------- (100) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(sieve(z0))) -> c17(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(from(z0))) -> c17(A__SIEVE(a__from(mark(z0))), MARK(from(z0))) MARK(sieve(head(z0))) -> c17(A__SIEVE(a__head(mark(z0))), MARK(head(z0))) MARK(sieve(tail(z0))) -> c17(A__SIEVE(a__tail(mark(z0))), MARK(tail(z0))) MARK(sieve(if(z0, z1, z2))) -> c17(A__SIEVE(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(sieve(filter(z0, z1))) -> c17(A__SIEVE(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(primes)) -> c19(A__HEAD(a__primes), MARK(primes)) MARK(head(sieve(z0))) -> c19(A__HEAD(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(head(from(z0))) -> c19(A__HEAD(a__from(mark(z0))), MARK(from(z0))) MARK(head(head(z0))) -> c19(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c19(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(if(z0, z1, z2))) -> c19(A__HEAD(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(head(filter(z0, z1))) -> c19(A__HEAD(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(primes)) -> c20(A__TAIL(a__primes), MARK(primes)) MARK(tail(sieve(z0))) -> c20(A__TAIL(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(tail(from(z0))) -> c20(A__TAIL(a__from(mark(z0))), MARK(from(z0))) MARK(tail(head(z0))) -> c20(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c20(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(if(z0, z1, z2))) -> c20(A__TAIL(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(tail(filter(z0, z1))) -> c20(A__TAIL(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(primes, x1, x2)) -> c21(A__IF(a__primes, x1, x2), MARK(primes)) MARK(if(sieve(z0), x1, x2)) -> c21(A__IF(a__sieve(mark(z0)), x1, x2), MARK(sieve(z0))) MARK(if(from(z0), x1, x2)) -> c21(A__IF(a__from(mark(z0)), x1, x2), MARK(from(z0))) MARK(if(head(z0), x1, x2)) -> c21(A__IF(a__head(mark(z0)), x1, x2), MARK(head(z0))) MARK(if(tail(z0), x1, x2)) -> c21(A__IF(a__tail(mark(z0)), x1, x2), MARK(tail(z0))) MARK(if(if(z0, z1, z2), x1, x2)) -> c21(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(filter(z0, z1), x1, x2)) -> c21(A__IF(a__filter(mark(z0), mark(z1)), x1, x2), MARK(filter(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c22(A__FILTER(mark(x0), a__primes), MARK(x0)) MARK(filter(x0, sieve(z0))) -> c22(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, from(z0))) -> c22(A__FILTER(mark(x0), a__from(mark(z0))), MARK(x0)) MARK(filter(x0, head(z0))) -> c22(A__FILTER(mark(x0), a__head(mark(z0))), MARK(x0)) MARK(filter(x0, tail(z0))) -> c22(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(x0)) MARK(filter(x0, if(z0, z1, z2))) -> c22(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0)) MARK(filter(x0, filter(z0, z1))) -> c22(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(x0)) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(x0, s(z0))) -> c22(MARK(x0)) MARK(filter(x0, 0)) -> c22(MARK(x0)) MARK(filter(x0, true)) -> c22(MARK(x0)) MARK(filter(x0, false)) -> c22(MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), x1)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c23(A__FILTER(mark(x0), a__primes), MARK(primes)) MARK(filter(x0, sieve(z0))) -> c23(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, from(z0))) -> c23(A__FILTER(mark(x0), a__from(mark(z0))), MARK(from(z0))) MARK(filter(x0, head(z0))) -> c23(A__FILTER(mark(x0), a__head(mark(z0))), MARK(head(z0))) MARK(filter(x0, tail(z0))) -> c23(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(tail(z0))) MARK(filter(x0, if(z0, z1, z2))) -> c23(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(filter(x0, filter(z0, z1))) -> c23(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(filter(x0, cons(z0, z1))) -> c23(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(primes, x1)) -> c23(A__FILTER(a__primes, mark(x1)), MARK(x1)) MARK(filter(sieve(z0), x1)) -> c23(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(from(z0), x1)) -> c23(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(head(z0), x1)) -> c23(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(tail(z0), x1)) -> c23(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(if(z0, z1, z2), x1)) -> c23(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1)) MARK(filter(filter(z0, z1), x1)) -> c23(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(x1)) MARK(filter(s(z0), x1)) -> c23(A__FILTER(s(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(x0, s(z0))) -> c23(MARK(s(z0))) MARK(filter(x0, divides(z0, z1))) -> c23(MARK(divides(z0, z1))) MARK(filter(0, x1)) -> c23(MARK(x1)) MARK(filter(cons(z0, z1), x1)) -> c23(MARK(x1)) MARK(filter(true, x1)) -> c23(MARK(x1)) MARK(filter(false, x1)) -> c23(MARK(x1)) MARK(filter(divides(z0, z1), x1)) -> c23(MARK(x1)) A__PRIMES -> c(A__SIEVE(cons(s(mark(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(sieve(primes)) -> c17(A__SIEVE(primes), MARK(primes)) S tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(sieve(z0))) -> c17(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(from(z0))) -> c17(A__SIEVE(a__from(mark(z0))), MARK(from(z0))) MARK(sieve(head(z0))) -> c17(A__SIEVE(a__head(mark(z0))), MARK(head(z0))) MARK(sieve(tail(z0))) -> c17(A__SIEVE(a__tail(mark(z0))), MARK(tail(z0))) MARK(sieve(if(z0, z1, z2))) -> c17(A__SIEVE(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(sieve(filter(z0, z1))) -> c17(A__SIEVE(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(primes)) -> c19(A__HEAD(a__primes), MARK(primes)) MARK(head(sieve(z0))) -> c19(A__HEAD(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(head(from(z0))) -> c19(A__HEAD(a__from(mark(z0))), MARK(from(z0))) MARK(head(head(z0))) -> c19(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c19(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(if(z0, z1, z2))) -> c19(A__HEAD(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(head(filter(z0, z1))) -> c19(A__HEAD(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(primes)) -> c20(A__TAIL(a__primes), MARK(primes)) MARK(tail(sieve(z0))) -> c20(A__TAIL(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(tail(from(z0))) -> c20(A__TAIL(a__from(mark(z0))), MARK(from(z0))) MARK(tail(head(z0))) -> c20(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c20(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(if(z0, z1, z2))) -> c20(A__TAIL(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(tail(filter(z0, z1))) -> c20(A__TAIL(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(primes, x1, x2)) -> c21(A__IF(a__primes, x1, x2), MARK(primes)) MARK(if(sieve(z0), x1, x2)) -> c21(A__IF(a__sieve(mark(z0)), x1, x2), MARK(sieve(z0))) MARK(if(from(z0), x1, x2)) -> c21(A__IF(a__from(mark(z0)), x1, x2), MARK(from(z0))) MARK(if(head(z0), x1, x2)) -> c21(A__IF(a__head(mark(z0)), x1, x2), MARK(head(z0))) MARK(if(tail(z0), x1, x2)) -> c21(A__IF(a__tail(mark(z0)), x1, x2), MARK(tail(z0))) MARK(if(if(z0, z1, z2), x1, x2)) -> c21(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(filter(z0, z1), x1, x2)) -> c21(A__IF(a__filter(mark(z0), mark(z1)), x1, x2), MARK(filter(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c22(A__FILTER(mark(x0), a__primes), MARK(x0)) MARK(filter(x0, sieve(z0))) -> c22(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, from(z0))) -> c22(A__FILTER(mark(x0), a__from(mark(z0))), MARK(x0)) MARK(filter(x0, head(z0))) -> c22(A__FILTER(mark(x0), a__head(mark(z0))), MARK(x0)) MARK(filter(x0, tail(z0))) -> c22(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(x0)) MARK(filter(x0, if(z0, z1, z2))) -> c22(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0)) MARK(filter(x0, filter(z0, z1))) -> c22(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(x0)) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(x0, s(z0))) -> c22(MARK(x0)) MARK(filter(x0, 0)) -> c22(MARK(x0)) MARK(filter(x0, true)) -> c22(MARK(x0)) MARK(filter(x0, false)) -> c22(MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), x1)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c23(A__FILTER(mark(x0), a__primes), MARK(primes)) MARK(filter(x0, sieve(z0))) -> c23(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, from(z0))) -> c23(A__FILTER(mark(x0), a__from(mark(z0))), MARK(from(z0))) MARK(filter(x0, head(z0))) -> c23(A__FILTER(mark(x0), a__head(mark(z0))), MARK(head(z0))) MARK(filter(x0, tail(z0))) -> c23(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(tail(z0))) MARK(filter(x0, if(z0, z1, z2))) -> c23(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(filter(x0, filter(z0, z1))) -> c23(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(filter(x0, cons(z0, z1))) -> c23(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(primes, x1)) -> c23(A__FILTER(a__primes, mark(x1)), MARK(x1)) MARK(filter(sieve(z0), x1)) -> c23(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(from(z0), x1)) -> c23(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(head(z0), x1)) -> c23(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(tail(z0), x1)) -> c23(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(if(z0, z1, z2), x1)) -> c23(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1)) MARK(filter(filter(z0, z1), x1)) -> c23(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(x1)) MARK(filter(s(z0), x1)) -> c23(A__FILTER(s(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(x0, s(z0))) -> c23(MARK(s(z0))) MARK(filter(x0, divides(z0, z1))) -> c23(MARK(divides(z0, z1))) MARK(filter(0, x1)) -> c23(MARK(x1)) MARK(filter(cons(z0, z1), x1)) -> c23(MARK(x1)) MARK(filter(true, x1)) -> c23(MARK(x1)) MARK(filter(false, x1)) -> c23(MARK(x1)) MARK(filter(divides(z0, z1), x1)) -> c23(MARK(x1)) A__PRIMES -> c(A__SIEVE(cons(s(mark(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(sieve(primes)) -> c17(A__SIEVE(primes), MARK(primes)) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c18_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_1, c17_2, c17_1, c19_2, c19_1, c20_2, c20_1, c21_2, c21_1, c22_2, c22_1, c23_2, c23_1, c_2 ---------------------------------------- (101) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (102) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(sieve(z0))) -> c17(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(from(z0))) -> c17(A__SIEVE(a__from(mark(z0))), MARK(from(z0))) MARK(sieve(head(z0))) -> c17(A__SIEVE(a__head(mark(z0))), MARK(head(z0))) MARK(sieve(tail(z0))) -> c17(A__SIEVE(a__tail(mark(z0))), MARK(tail(z0))) MARK(sieve(if(z0, z1, z2))) -> c17(A__SIEVE(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(sieve(filter(z0, z1))) -> c17(A__SIEVE(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(primes)) -> c19(A__HEAD(a__primes), MARK(primes)) MARK(head(sieve(z0))) -> c19(A__HEAD(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(head(from(z0))) -> c19(A__HEAD(a__from(mark(z0))), MARK(from(z0))) MARK(head(head(z0))) -> c19(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c19(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(if(z0, z1, z2))) -> c19(A__HEAD(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(head(filter(z0, z1))) -> c19(A__HEAD(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(primes)) -> c20(A__TAIL(a__primes), MARK(primes)) MARK(tail(sieve(z0))) -> c20(A__TAIL(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(tail(from(z0))) -> c20(A__TAIL(a__from(mark(z0))), MARK(from(z0))) MARK(tail(head(z0))) -> c20(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c20(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(if(z0, z1, z2))) -> c20(A__TAIL(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(tail(filter(z0, z1))) -> c20(A__TAIL(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(primes, x1, x2)) -> c21(A__IF(a__primes, x1, x2), MARK(primes)) MARK(if(sieve(z0), x1, x2)) -> c21(A__IF(a__sieve(mark(z0)), x1, x2), MARK(sieve(z0))) MARK(if(from(z0), x1, x2)) -> c21(A__IF(a__from(mark(z0)), x1, x2), MARK(from(z0))) MARK(if(head(z0), x1, x2)) -> c21(A__IF(a__head(mark(z0)), x1, x2), MARK(head(z0))) MARK(if(tail(z0), x1, x2)) -> c21(A__IF(a__tail(mark(z0)), x1, x2), MARK(tail(z0))) MARK(if(if(z0, z1, z2), x1, x2)) -> c21(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(filter(z0, z1), x1, x2)) -> c21(A__IF(a__filter(mark(z0), mark(z1)), x1, x2), MARK(filter(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c22(A__FILTER(mark(x0), a__primes), MARK(x0)) MARK(filter(x0, sieve(z0))) -> c22(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, from(z0))) -> c22(A__FILTER(mark(x0), a__from(mark(z0))), MARK(x0)) MARK(filter(x0, head(z0))) -> c22(A__FILTER(mark(x0), a__head(mark(z0))), MARK(x0)) MARK(filter(x0, tail(z0))) -> c22(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(x0)) MARK(filter(x0, if(z0, z1, z2))) -> c22(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0)) MARK(filter(x0, filter(z0, z1))) -> c22(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(x0)) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(x0, s(z0))) -> c22(MARK(x0)) MARK(filter(x0, 0)) -> c22(MARK(x0)) MARK(filter(x0, true)) -> c22(MARK(x0)) MARK(filter(x0, false)) -> c22(MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), x1)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c23(A__FILTER(mark(x0), a__primes), MARK(primes)) MARK(filter(x0, sieve(z0))) -> c23(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, from(z0))) -> c23(A__FILTER(mark(x0), a__from(mark(z0))), MARK(from(z0))) MARK(filter(x0, head(z0))) -> c23(A__FILTER(mark(x0), a__head(mark(z0))), MARK(head(z0))) MARK(filter(x0, tail(z0))) -> c23(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(tail(z0))) MARK(filter(x0, if(z0, z1, z2))) -> c23(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(filter(x0, filter(z0, z1))) -> c23(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(filter(x0, cons(z0, z1))) -> c23(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(primes, x1)) -> c23(A__FILTER(a__primes, mark(x1)), MARK(x1)) MARK(filter(sieve(z0), x1)) -> c23(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(from(z0), x1)) -> c23(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(head(z0), x1)) -> c23(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(tail(z0), x1)) -> c23(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(if(z0, z1, z2), x1)) -> c23(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1)) MARK(filter(filter(z0, z1), x1)) -> c23(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(x1)) MARK(filter(s(z0), x1)) -> c23(A__FILTER(s(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(x0, s(z0))) -> c23(MARK(s(z0))) MARK(filter(x0, divides(z0, z1))) -> c23(MARK(divides(z0, z1))) MARK(filter(0, x1)) -> c23(MARK(x1)) MARK(filter(cons(z0, z1), x1)) -> c23(MARK(x1)) MARK(filter(true, x1)) -> c23(MARK(x1)) MARK(filter(false, x1)) -> c23(MARK(x1)) MARK(filter(divides(z0, z1), x1)) -> c23(MARK(x1)) A__PRIMES -> c(A__SIEVE(cons(s(mark(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(sieve(primes)) -> c17(MARK(primes)) S tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(sieve(z0))) -> c17(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(sieve(from(z0))) -> c17(A__SIEVE(a__from(mark(z0))), MARK(from(z0))) MARK(sieve(head(z0))) -> c17(A__SIEVE(a__head(mark(z0))), MARK(head(z0))) MARK(sieve(tail(z0))) -> c17(A__SIEVE(a__tail(mark(z0))), MARK(tail(z0))) MARK(sieve(if(z0, z1, z2))) -> c17(A__SIEVE(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(sieve(filter(z0, z1))) -> c17(A__SIEVE(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(primes)) -> c19(A__HEAD(a__primes), MARK(primes)) MARK(head(sieve(z0))) -> c19(A__HEAD(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(head(from(z0))) -> c19(A__HEAD(a__from(mark(z0))), MARK(from(z0))) MARK(head(head(z0))) -> c19(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c19(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(if(z0, z1, z2))) -> c19(A__HEAD(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(head(filter(z0, z1))) -> c19(A__HEAD(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(primes)) -> c20(A__TAIL(a__primes), MARK(primes)) MARK(tail(sieve(z0))) -> c20(A__TAIL(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(tail(from(z0))) -> c20(A__TAIL(a__from(mark(z0))), MARK(from(z0))) MARK(tail(head(z0))) -> c20(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c20(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(if(z0, z1, z2))) -> c20(A__TAIL(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(tail(filter(z0, z1))) -> c20(A__TAIL(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(primes, x1, x2)) -> c21(A__IF(a__primes, x1, x2), MARK(primes)) MARK(if(sieve(z0), x1, x2)) -> c21(A__IF(a__sieve(mark(z0)), x1, x2), MARK(sieve(z0))) MARK(if(from(z0), x1, x2)) -> c21(A__IF(a__from(mark(z0)), x1, x2), MARK(from(z0))) MARK(if(head(z0), x1, x2)) -> c21(A__IF(a__head(mark(z0)), x1, x2), MARK(head(z0))) MARK(if(tail(z0), x1, x2)) -> c21(A__IF(a__tail(mark(z0)), x1, x2), MARK(tail(z0))) MARK(if(if(z0, z1, z2), x1, x2)) -> c21(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(filter(z0, z1), x1, x2)) -> c21(A__IF(a__filter(mark(z0), mark(z1)), x1, x2), MARK(filter(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c22(A__FILTER(mark(x0), a__primes), MARK(x0)) MARK(filter(x0, sieve(z0))) -> c22(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, from(z0))) -> c22(A__FILTER(mark(x0), a__from(mark(z0))), MARK(x0)) MARK(filter(x0, head(z0))) -> c22(A__FILTER(mark(x0), a__head(mark(z0))), MARK(x0)) MARK(filter(x0, tail(z0))) -> c22(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(x0)) MARK(filter(x0, if(z0, z1, z2))) -> c22(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0)) MARK(filter(x0, filter(z0, z1))) -> c22(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(x0)) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(x0, s(z0))) -> c22(MARK(x0)) MARK(filter(x0, 0)) -> c22(MARK(x0)) MARK(filter(x0, true)) -> c22(MARK(x0)) MARK(filter(x0, false)) -> c22(MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), x1)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c23(A__FILTER(mark(x0), a__primes), MARK(primes)) MARK(filter(x0, sieve(z0))) -> c23(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, from(z0))) -> c23(A__FILTER(mark(x0), a__from(mark(z0))), MARK(from(z0))) MARK(filter(x0, head(z0))) -> c23(A__FILTER(mark(x0), a__head(mark(z0))), MARK(head(z0))) MARK(filter(x0, tail(z0))) -> c23(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(tail(z0))) MARK(filter(x0, if(z0, z1, z2))) -> c23(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(filter(x0, filter(z0, z1))) -> c23(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(filter(x0, cons(z0, z1))) -> c23(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(primes, x1)) -> c23(A__FILTER(a__primes, mark(x1)), MARK(x1)) MARK(filter(sieve(z0), x1)) -> c23(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(from(z0), x1)) -> c23(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(head(z0), x1)) -> c23(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(tail(z0), x1)) -> c23(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(if(z0, z1, z2), x1)) -> c23(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1)) MARK(filter(filter(z0, z1), x1)) -> c23(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(x1)) MARK(filter(s(z0), x1)) -> c23(A__FILTER(s(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(x0, s(z0))) -> c23(MARK(s(z0))) MARK(filter(x0, divides(z0, z1))) -> c23(MARK(divides(z0, z1))) MARK(filter(0, x1)) -> c23(MARK(x1)) MARK(filter(cons(z0, z1), x1)) -> c23(MARK(x1)) MARK(filter(true, x1)) -> c23(MARK(x1)) MARK(filter(false, x1)) -> c23(MARK(x1)) MARK(filter(divides(z0, z1), x1)) -> c23(MARK(x1)) A__PRIMES -> c(A__SIEVE(cons(s(mark(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(sieve(primes)) -> c17(MARK(primes)) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c18_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_1, c17_2, c17_1, c19_2, c19_1, c20_2, c20_1, c21_2, c21_1, c22_2, c22_1, c23_2, c23_1, c_2 ---------------------------------------- (103) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(sieve(sieve(z0))) -> c17(A__SIEVE(a__sieve(mark(z0))), MARK(sieve(z0))) by MARK(sieve(sieve(x0))) -> c17(A__SIEVE(sieve(mark(x0))), MARK(sieve(x0))) MARK(sieve(sieve(primes))) -> c17(A__SIEVE(a__sieve(a__primes)), MARK(sieve(primes))) MARK(sieve(sieve(sieve(z0)))) -> c17(A__SIEVE(a__sieve(a__sieve(mark(z0)))), MARK(sieve(sieve(z0)))) MARK(sieve(sieve(from(z0)))) -> c17(A__SIEVE(a__sieve(a__from(mark(z0)))), MARK(sieve(from(z0)))) MARK(sieve(sieve(head(z0)))) -> c17(A__SIEVE(a__sieve(a__head(mark(z0)))), MARK(sieve(head(z0)))) MARK(sieve(sieve(tail(z0)))) -> c17(A__SIEVE(a__sieve(a__tail(mark(z0)))), MARK(sieve(tail(z0)))) MARK(sieve(sieve(if(z0, z1, z2)))) -> c17(A__SIEVE(a__sieve(a__if(mark(z0), z1, z2))), MARK(sieve(if(z0, z1, z2)))) MARK(sieve(sieve(filter(z0, z1)))) -> c17(A__SIEVE(a__sieve(a__filter(mark(z0), mark(z1)))), MARK(sieve(filter(z0, z1)))) MARK(sieve(sieve(s(z0)))) -> c17(A__SIEVE(a__sieve(s(mark(z0)))), MARK(sieve(s(z0)))) MARK(sieve(sieve(0))) -> c17(A__SIEVE(a__sieve(0)), MARK(sieve(0))) MARK(sieve(sieve(cons(z0, z1)))) -> c17(A__SIEVE(a__sieve(cons(mark(z0), z1))), MARK(sieve(cons(z0, z1)))) MARK(sieve(sieve(true))) -> c17(A__SIEVE(a__sieve(true)), MARK(sieve(true))) MARK(sieve(sieve(false))) -> c17(A__SIEVE(a__sieve(false)), MARK(sieve(false))) MARK(sieve(sieve(divides(z0, z1)))) -> c17(A__SIEVE(a__sieve(divides(mark(z0), mark(z1)))), MARK(sieve(divides(z0, z1)))) ---------------------------------------- (104) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(from(z0))) -> c17(A__SIEVE(a__from(mark(z0))), MARK(from(z0))) MARK(sieve(head(z0))) -> c17(A__SIEVE(a__head(mark(z0))), MARK(head(z0))) MARK(sieve(tail(z0))) -> c17(A__SIEVE(a__tail(mark(z0))), MARK(tail(z0))) MARK(sieve(if(z0, z1, z2))) -> c17(A__SIEVE(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(sieve(filter(z0, z1))) -> c17(A__SIEVE(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(primes)) -> c19(A__HEAD(a__primes), MARK(primes)) MARK(head(sieve(z0))) -> c19(A__HEAD(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(head(from(z0))) -> c19(A__HEAD(a__from(mark(z0))), MARK(from(z0))) MARK(head(head(z0))) -> c19(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c19(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(if(z0, z1, z2))) -> c19(A__HEAD(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(head(filter(z0, z1))) -> c19(A__HEAD(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(primes)) -> c20(A__TAIL(a__primes), MARK(primes)) MARK(tail(sieve(z0))) -> c20(A__TAIL(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(tail(from(z0))) -> c20(A__TAIL(a__from(mark(z0))), MARK(from(z0))) MARK(tail(head(z0))) -> c20(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c20(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(if(z0, z1, z2))) -> c20(A__TAIL(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(tail(filter(z0, z1))) -> c20(A__TAIL(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(primes, x1, x2)) -> c21(A__IF(a__primes, x1, x2), MARK(primes)) MARK(if(sieve(z0), x1, x2)) -> c21(A__IF(a__sieve(mark(z0)), x1, x2), MARK(sieve(z0))) MARK(if(from(z0), x1, x2)) -> c21(A__IF(a__from(mark(z0)), x1, x2), MARK(from(z0))) MARK(if(head(z0), x1, x2)) -> c21(A__IF(a__head(mark(z0)), x1, x2), MARK(head(z0))) MARK(if(tail(z0), x1, x2)) -> c21(A__IF(a__tail(mark(z0)), x1, x2), MARK(tail(z0))) MARK(if(if(z0, z1, z2), x1, x2)) -> c21(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(filter(z0, z1), x1, x2)) -> c21(A__IF(a__filter(mark(z0), mark(z1)), x1, x2), MARK(filter(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c22(A__FILTER(mark(x0), a__primes), MARK(x0)) MARK(filter(x0, sieve(z0))) -> c22(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, from(z0))) -> c22(A__FILTER(mark(x0), a__from(mark(z0))), MARK(x0)) MARK(filter(x0, head(z0))) -> c22(A__FILTER(mark(x0), a__head(mark(z0))), MARK(x0)) MARK(filter(x0, tail(z0))) -> c22(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(x0)) MARK(filter(x0, if(z0, z1, z2))) -> c22(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0)) MARK(filter(x0, filter(z0, z1))) -> c22(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(x0)) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(x0, s(z0))) -> c22(MARK(x0)) MARK(filter(x0, 0)) -> c22(MARK(x0)) MARK(filter(x0, true)) -> c22(MARK(x0)) MARK(filter(x0, false)) -> c22(MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), x1)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c23(A__FILTER(mark(x0), a__primes), MARK(primes)) MARK(filter(x0, sieve(z0))) -> c23(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, from(z0))) -> c23(A__FILTER(mark(x0), a__from(mark(z0))), MARK(from(z0))) MARK(filter(x0, head(z0))) -> c23(A__FILTER(mark(x0), a__head(mark(z0))), MARK(head(z0))) MARK(filter(x0, tail(z0))) -> c23(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(tail(z0))) MARK(filter(x0, if(z0, z1, z2))) -> c23(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(filter(x0, filter(z0, z1))) -> c23(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(filter(x0, cons(z0, z1))) -> c23(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(primes, x1)) -> c23(A__FILTER(a__primes, mark(x1)), MARK(x1)) MARK(filter(sieve(z0), x1)) -> c23(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(from(z0), x1)) -> c23(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(head(z0), x1)) -> c23(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(tail(z0), x1)) -> c23(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(if(z0, z1, z2), x1)) -> c23(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1)) MARK(filter(filter(z0, z1), x1)) -> c23(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(x1)) MARK(filter(s(z0), x1)) -> c23(A__FILTER(s(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(x0, s(z0))) -> c23(MARK(s(z0))) MARK(filter(x0, divides(z0, z1))) -> c23(MARK(divides(z0, z1))) MARK(filter(0, x1)) -> c23(MARK(x1)) MARK(filter(cons(z0, z1), x1)) -> c23(MARK(x1)) MARK(filter(true, x1)) -> c23(MARK(x1)) MARK(filter(false, x1)) -> c23(MARK(x1)) MARK(filter(divides(z0, z1), x1)) -> c23(MARK(x1)) A__PRIMES -> c(A__SIEVE(cons(s(mark(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(sieve(primes)) -> c17(MARK(primes)) MARK(sieve(sieve(x0))) -> c17(A__SIEVE(sieve(mark(x0))), MARK(sieve(x0))) MARK(sieve(sieve(primes))) -> c17(A__SIEVE(a__sieve(a__primes)), MARK(sieve(primes))) MARK(sieve(sieve(sieve(z0)))) -> c17(A__SIEVE(a__sieve(a__sieve(mark(z0)))), MARK(sieve(sieve(z0)))) MARK(sieve(sieve(from(z0)))) -> c17(A__SIEVE(a__sieve(a__from(mark(z0)))), MARK(sieve(from(z0)))) MARK(sieve(sieve(head(z0)))) -> c17(A__SIEVE(a__sieve(a__head(mark(z0)))), MARK(sieve(head(z0)))) MARK(sieve(sieve(tail(z0)))) -> c17(A__SIEVE(a__sieve(a__tail(mark(z0)))), MARK(sieve(tail(z0)))) MARK(sieve(sieve(if(z0, z1, z2)))) -> c17(A__SIEVE(a__sieve(a__if(mark(z0), z1, z2))), MARK(sieve(if(z0, z1, z2)))) MARK(sieve(sieve(filter(z0, z1)))) -> c17(A__SIEVE(a__sieve(a__filter(mark(z0), mark(z1)))), MARK(sieve(filter(z0, z1)))) MARK(sieve(sieve(s(z0)))) -> c17(A__SIEVE(a__sieve(s(mark(z0)))), MARK(sieve(s(z0)))) MARK(sieve(sieve(0))) -> c17(A__SIEVE(a__sieve(0)), MARK(sieve(0))) MARK(sieve(sieve(cons(z0, z1)))) -> c17(A__SIEVE(a__sieve(cons(mark(z0), z1))), MARK(sieve(cons(z0, z1)))) MARK(sieve(sieve(true))) -> c17(A__SIEVE(a__sieve(true)), MARK(sieve(true))) MARK(sieve(sieve(false))) -> c17(A__SIEVE(a__sieve(false)), MARK(sieve(false))) MARK(sieve(sieve(divides(z0, z1)))) -> c17(A__SIEVE(a__sieve(divides(mark(z0), mark(z1)))), MARK(sieve(divides(z0, z1)))) S tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(from(z0))) -> c17(A__SIEVE(a__from(mark(z0))), MARK(from(z0))) MARK(sieve(head(z0))) -> c17(A__SIEVE(a__head(mark(z0))), MARK(head(z0))) MARK(sieve(tail(z0))) -> c17(A__SIEVE(a__tail(mark(z0))), MARK(tail(z0))) MARK(sieve(if(z0, z1, z2))) -> c17(A__SIEVE(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(sieve(filter(z0, z1))) -> c17(A__SIEVE(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(primes)) -> c19(A__HEAD(a__primes), MARK(primes)) MARK(head(sieve(z0))) -> c19(A__HEAD(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(head(from(z0))) -> c19(A__HEAD(a__from(mark(z0))), MARK(from(z0))) MARK(head(head(z0))) -> c19(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c19(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(if(z0, z1, z2))) -> c19(A__HEAD(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(head(filter(z0, z1))) -> c19(A__HEAD(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(primes)) -> c20(A__TAIL(a__primes), MARK(primes)) MARK(tail(sieve(z0))) -> c20(A__TAIL(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(tail(from(z0))) -> c20(A__TAIL(a__from(mark(z0))), MARK(from(z0))) MARK(tail(head(z0))) -> c20(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c20(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(if(z0, z1, z2))) -> c20(A__TAIL(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(tail(filter(z0, z1))) -> c20(A__TAIL(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(primes, x1, x2)) -> c21(A__IF(a__primes, x1, x2), MARK(primes)) MARK(if(sieve(z0), x1, x2)) -> c21(A__IF(a__sieve(mark(z0)), x1, x2), MARK(sieve(z0))) MARK(if(from(z0), x1, x2)) -> c21(A__IF(a__from(mark(z0)), x1, x2), MARK(from(z0))) MARK(if(head(z0), x1, x2)) -> c21(A__IF(a__head(mark(z0)), x1, x2), MARK(head(z0))) MARK(if(tail(z0), x1, x2)) -> c21(A__IF(a__tail(mark(z0)), x1, x2), MARK(tail(z0))) MARK(if(if(z0, z1, z2), x1, x2)) -> c21(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(filter(z0, z1), x1, x2)) -> c21(A__IF(a__filter(mark(z0), mark(z1)), x1, x2), MARK(filter(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c22(A__FILTER(mark(x0), a__primes), MARK(x0)) MARK(filter(x0, sieve(z0))) -> c22(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, from(z0))) -> c22(A__FILTER(mark(x0), a__from(mark(z0))), MARK(x0)) MARK(filter(x0, head(z0))) -> c22(A__FILTER(mark(x0), a__head(mark(z0))), MARK(x0)) MARK(filter(x0, tail(z0))) -> c22(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(x0)) MARK(filter(x0, if(z0, z1, z2))) -> c22(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0)) MARK(filter(x0, filter(z0, z1))) -> c22(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(x0)) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(x0, s(z0))) -> c22(MARK(x0)) MARK(filter(x0, 0)) -> c22(MARK(x0)) MARK(filter(x0, true)) -> c22(MARK(x0)) MARK(filter(x0, false)) -> c22(MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), x1)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c23(A__FILTER(mark(x0), a__primes), MARK(primes)) MARK(filter(x0, sieve(z0))) -> c23(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, from(z0))) -> c23(A__FILTER(mark(x0), a__from(mark(z0))), MARK(from(z0))) MARK(filter(x0, head(z0))) -> c23(A__FILTER(mark(x0), a__head(mark(z0))), MARK(head(z0))) MARK(filter(x0, tail(z0))) -> c23(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(tail(z0))) MARK(filter(x0, if(z0, z1, z2))) -> c23(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(filter(x0, filter(z0, z1))) -> c23(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(filter(x0, cons(z0, z1))) -> c23(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(primes, x1)) -> c23(A__FILTER(a__primes, mark(x1)), MARK(x1)) MARK(filter(sieve(z0), x1)) -> c23(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(from(z0), x1)) -> c23(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(head(z0), x1)) -> c23(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(tail(z0), x1)) -> c23(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(if(z0, z1, z2), x1)) -> c23(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1)) MARK(filter(filter(z0, z1), x1)) -> c23(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(x1)) MARK(filter(s(z0), x1)) -> c23(A__FILTER(s(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(x0, s(z0))) -> c23(MARK(s(z0))) MARK(filter(x0, divides(z0, z1))) -> c23(MARK(divides(z0, z1))) MARK(filter(0, x1)) -> c23(MARK(x1)) MARK(filter(cons(z0, z1), x1)) -> c23(MARK(x1)) MARK(filter(true, x1)) -> c23(MARK(x1)) MARK(filter(false, x1)) -> c23(MARK(x1)) MARK(filter(divides(z0, z1), x1)) -> c23(MARK(x1)) A__PRIMES -> c(A__SIEVE(cons(s(mark(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(sieve(primes)) -> c17(MARK(primes)) MARK(sieve(sieve(x0))) -> c17(A__SIEVE(sieve(mark(x0))), MARK(sieve(x0))) MARK(sieve(sieve(primes))) -> c17(A__SIEVE(a__sieve(a__primes)), MARK(sieve(primes))) MARK(sieve(sieve(sieve(z0)))) -> c17(A__SIEVE(a__sieve(a__sieve(mark(z0)))), MARK(sieve(sieve(z0)))) MARK(sieve(sieve(from(z0)))) -> c17(A__SIEVE(a__sieve(a__from(mark(z0)))), MARK(sieve(from(z0)))) MARK(sieve(sieve(head(z0)))) -> c17(A__SIEVE(a__sieve(a__head(mark(z0)))), MARK(sieve(head(z0)))) MARK(sieve(sieve(tail(z0)))) -> c17(A__SIEVE(a__sieve(a__tail(mark(z0)))), MARK(sieve(tail(z0)))) MARK(sieve(sieve(if(z0, z1, z2)))) -> c17(A__SIEVE(a__sieve(a__if(mark(z0), z1, z2))), MARK(sieve(if(z0, z1, z2)))) MARK(sieve(sieve(filter(z0, z1)))) -> c17(A__SIEVE(a__sieve(a__filter(mark(z0), mark(z1)))), MARK(sieve(filter(z0, z1)))) MARK(sieve(sieve(s(z0)))) -> c17(A__SIEVE(a__sieve(s(mark(z0)))), MARK(sieve(s(z0)))) MARK(sieve(sieve(0))) -> c17(A__SIEVE(a__sieve(0)), MARK(sieve(0))) MARK(sieve(sieve(cons(z0, z1)))) -> c17(A__SIEVE(a__sieve(cons(mark(z0), z1))), MARK(sieve(cons(z0, z1)))) MARK(sieve(sieve(true))) -> c17(A__SIEVE(a__sieve(true)), MARK(sieve(true))) MARK(sieve(sieve(false))) -> c17(A__SIEVE(a__sieve(false)), MARK(sieve(false))) MARK(sieve(sieve(divides(z0, z1)))) -> c17(A__SIEVE(a__sieve(divides(mark(z0), mark(z1)))), MARK(sieve(divides(z0, z1)))) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c18_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_1, c17_2, c17_1, c19_2, c19_1, c20_2, c20_1, c21_2, c21_1, c22_2, c22_1, c23_2, c23_1, c_2 ---------------------------------------- (105) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing tuple parts ---------------------------------------- (106) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(from(z0))) -> c17(A__SIEVE(a__from(mark(z0))), MARK(from(z0))) MARK(sieve(head(z0))) -> c17(A__SIEVE(a__head(mark(z0))), MARK(head(z0))) MARK(sieve(tail(z0))) -> c17(A__SIEVE(a__tail(mark(z0))), MARK(tail(z0))) MARK(sieve(if(z0, z1, z2))) -> c17(A__SIEVE(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(sieve(filter(z0, z1))) -> c17(A__SIEVE(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(primes)) -> c19(A__HEAD(a__primes), MARK(primes)) MARK(head(sieve(z0))) -> c19(A__HEAD(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(head(from(z0))) -> c19(A__HEAD(a__from(mark(z0))), MARK(from(z0))) MARK(head(head(z0))) -> c19(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c19(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(if(z0, z1, z2))) -> c19(A__HEAD(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(head(filter(z0, z1))) -> c19(A__HEAD(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(primes)) -> c20(A__TAIL(a__primes), MARK(primes)) MARK(tail(sieve(z0))) -> c20(A__TAIL(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(tail(from(z0))) -> c20(A__TAIL(a__from(mark(z0))), MARK(from(z0))) MARK(tail(head(z0))) -> c20(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c20(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(if(z0, z1, z2))) -> c20(A__TAIL(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(tail(filter(z0, z1))) -> c20(A__TAIL(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(primes, x1, x2)) -> c21(A__IF(a__primes, x1, x2), MARK(primes)) MARK(if(sieve(z0), x1, x2)) -> c21(A__IF(a__sieve(mark(z0)), x1, x2), MARK(sieve(z0))) MARK(if(from(z0), x1, x2)) -> c21(A__IF(a__from(mark(z0)), x1, x2), MARK(from(z0))) MARK(if(head(z0), x1, x2)) -> c21(A__IF(a__head(mark(z0)), x1, x2), MARK(head(z0))) MARK(if(tail(z0), x1, x2)) -> c21(A__IF(a__tail(mark(z0)), x1, x2), MARK(tail(z0))) MARK(if(if(z0, z1, z2), x1, x2)) -> c21(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(filter(z0, z1), x1, x2)) -> c21(A__IF(a__filter(mark(z0), mark(z1)), x1, x2), MARK(filter(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c22(A__FILTER(mark(x0), a__primes), MARK(x0)) MARK(filter(x0, sieve(z0))) -> c22(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, from(z0))) -> c22(A__FILTER(mark(x0), a__from(mark(z0))), MARK(x0)) MARK(filter(x0, head(z0))) -> c22(A__FILTER(mark(x0), a__head(mark(z0))), MARK(x0)) MARK(filter(x0, tail(z0))) -> c22(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(x0)) MARK(filter(x0, if(z0, z1, z2))) -> c22(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0)) MARK(filter(x0, filter(z0, z1))) -> c22(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(x0)) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(x0, s(z0))) -> c22(MARK(x0)) MARK(filter(x0, 0)) -> c22(MARK(x0)) MARK(filter(x0, true)) -> c22(MARK(x0)) MARK(filter(x0, false)) -> c22(MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), x1)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c23(A__FILTER(mark(x0), a__primes), MARK(primes)) MARK(filter(x0, sieve(z0))) -> c23(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, from(z0))) -> c23(A__FILTER(mark(x0), a__from(mark(z0))), MARK(from(z0))) MARK(filter(x0, head(z0))) -> c23(A__FILTER(mark(x0), a__head(mark(z0))), MARK(head(z0))) MARK(filter(x0, tail(z0))) -> c23(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(tail(z0))) MARK(filter(x0, if(z0, z1, z2))) -> c23(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(filter(x0, filter(z0, z1))) -> c23(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(filter(x0, cons(z0, z1))) -> c23(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(primes, x1)) -> c23(A__FILTER(a__primes, mark(x1)), MARK(x1)) MARK(filter(sieve(z0), x1)) -> c23(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(from(z0), x1)) -> c23(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(head(z0), x1)) -> c23(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(tail(z0), x1)) -> c23(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(if(z0, z1, z2), x1)) -> c23(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1)) MARK(filter(filter(z0, z1), x1)) -> c23(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(x1)) MARK(filter(s(z0), x1)) -> c23(A__FILTER(s(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(x0, s(z0))) -> c23(MARK(s(z0))) MARK(filter(x0, divides(z0, z1))) -> c23(MARK(divides(z0, z1))) MARK(filter(0, x1)) -> c23(MARK(x1)) MARK(filter(cons(z0, z1), x1)) -> c23(MARK(x1)) MARK(filter(true, x1)) -> c23(MARK(x1)) MARK(filter(false, x1)) -> c23(MARK(x1)) MARK(filter(divides(z0, z1), x1)) -> c23(MARK(x1)) A__PRIMES -> c(A__SIEVE(cons(s(mark(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(sieve(primes)) -> c17(MARK(primes)) MARK(sieve(sieve(primes))) -> c17(A__SIEVE(a__sieve(a__primes)), MARK(sieve(primes))) MARK(sieve(sieve(sieve(z0)))) -> c17(A__SIEVE(a__sieve(a__sieve(mark(z0)))), MARK(sieve(sieve(z0)))) MARK(sieve(sieve(from(z0)))) -> c17(A__SIEVE(a__sieve(a__from(mark(z0)))), MARK(sieve(from(z0)))) MARK(sieve(sieve(head(z0)))) -> c17(A__SIEVE(a__sieve(a__head(mark(z0)))), MARK(sieve(head(z0)))) MARK(sieve(sieve(tail(z0)))) -> c17(A__SIEVE(a__sieve(a__tail(mark(z0)))), MARK(sieve(tail(z0)))) MARK(sieve(sieve(if(z0, z1, z2)))) -> c17(A__SIEVE(a__sieve(a__if(mark(z0), z1, z2))), MARK(sieve(if(z0, z1, z2)))) MARK(sieve(sieve(filter(z0, z1)))) -> c17(A__SIEVE(a__sieve(a__filter(mark(z0), mark(z1)))), MARK(sieve(filter(z0, z1)))) MARK(sieve(sieve(s(z0)))) -> c17(A__SIEVE(a__sieve(s(mark(z0)))), MARK(sieve(s(z0)))) MARK(sieve(sieve(cons(z0, z1)))) -> c17(A__SIEVE(a__sieve(cons(mark(z0), z1))), MARK(sieve(cons(z0, z1)))) MARK(sieve(sieve(divides(z0, z1)))) -> c17(A__SIEVE(a__sieve(divides(mark(z0), mark(z1)))), MARK(sieve(divides(z0, z1)))) MARK(sieve(sieve(x0))) -> c17(MARK(sieve(x0))) MARK(sieve(sieve(0))) -> c17(A__SIEVE(a__sieve(0))) MARK(sieve(sieve(true))) -> c17(A__SIEVE(a__sieve(true))) MARK(sieve(sieve(false))) -> c17(A__SIEVE(a__sieve(false))) S tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(from(z0))) -> c17(A__SIEVE(a__from(mark(z0))), MARK(from(z0))) MARK(sieve(head(z0))) -> c17(A__SIEVE(a__head(mark(z0))), MARK(head(z0))) MARK(sieve(tail(z0))) -> c17(A__SIEVE(a__tail(mark(z0))), MARK(tail(z0))) MARK(sieve(if(z0, z1, z2))) -> c17(A__SIEVE(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(sieve(filter(z0, z1))) -> c17(A__SIEVE(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(primes)) -> c19(A__HEAD(a__primes), MARK(primes)) MARK(head(sieve(z0))) -> c19(A__HEAD(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(head(from(z0))) -> c19(A__HEAD(a__from(mark(z0))), MARK(from(z0))) MARK(head(head(z0))) -> c19(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c19(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(if(z0, z1, z2))) -> c19(A__HEAD(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(head(filter(z0, z1))) -> c19(A__HEAD(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(primes)) -> c20(A__TAIL(a__primes), MARK(primes)) MARK(tail(sieve(z0))) -> c20(A__TAIL(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(tail(from(z0))) -> c20(A__TAIL(a__from(mark(z0))), MARK(from(z0))) MARK(tail(head(z0))) -> c20(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c20(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(if(z0, z1, z2))) -> c20(A__TAIL(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(tail(filter(z0, z1))) -> c20(A__TAIL(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(primes, x1, x2)) -> c21(A__IF(a__primes, x1, x2), MARK(primes)) MARK(if(sieve(z0), x1, x2)) -> c21(A__IF(a__sieve(mark(z0)), x1, x2), MARK(sieve(z0))) MARK(if(from(z0), x1, x2)) -> c21(A__IF(a__from(mark(z0)), x1, x2), MARK(from(z0))) MARK(if(head(z0), x1, x2)) -> c21(A__IF(a__head(mark(z0)), x1, x2), MARK(head(z0))) MARK(if(tail(z0), x1, x2)) -> c21(A__IF(a__tail(mark(z0)), x1, x2), MARK(tail(z0))) MARK(if(if(z0, z1, z2), x1, x2)) -> c21(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(filter(z0, z1), x1, x2)) -> c21(A__IF(a__filter(mark(z0), mark(z1)), x1, x2), MARK(filter(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c22(A__FILTER(mark(x0), a__primes), MARK(x0)) MARK(filter(x0, sieve(z0))) -> c22(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, from(z0))) -> c22(A__FILTER(mark(x0), a__from(mark(z0))), MARK(x0)) MARK(filter(x0, head(z0))) -> c22(A__FILTER(mark(x0), a__head(mark(z0))), MARK(x0)) MARK(filter(x0, tail(z0))) -> c22(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(x0)) MARK(filter(x0, if(z0, z1, z2))) -> c22(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0)) MARK(filter(x0, filter(z0, z1))) -> c22(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(x0)) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(x0, s(z0))) -> c22(MARK(x0)) MARK(filter(x0, 0)) -> c22(MARK(x0)) MARK(filter(x0, true)) -> c22(MARK(x0)) MARK(filter(x0, false)) -> c22(MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), x1)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c23(A__FILTER(mark(x0), a__primes), MARK(primes)) MARK(filter(x0, sieve(z0))) -> c23(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, from(z0))) -> c23(A__FILTER(mark(x0), a__from(mark(z0))), MARK(from(z0))) MARK(filter(x0, head(z0))) -> c23(A__FILTER(mark(x0), a__head(mark(z0))), MARK(head(z0))) MARK(filter(x0, tail(z0))) -> c23(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(tail(z0))) MARK(filter(x0, if(z0, z1, z2))) -> c23(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(filter(x0, filter(z0, z1))) -> c23(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(filter(x0, cons(z0, z1))) -> c23(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(primes, x1)) -> c23(A__FILTER(a__primes, mark(x1)), MARK(x1)) MARK(filter(sieve(z0), x1)) -> c23(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(from(z0), x1)) -> c23(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(head(z0), x1)) -> c23(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(tail(z0), x1)) -> c23(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(if(z0, z1, z2), x1)) -> c23(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1)) MARK(filter(filter(z0, z1), x1)) -> c23(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(x1)) MARK(filter(s(z0), x1)) -> c23(A__FILTER(s(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(x0, s(z0))) -> c23(MARK(s(z0))) MARK(filter(x0, divides(z0, z1))) -> c23(MARK(divides(z0, z1))) MARK(filter(0, x1)) -> c23(MARK(x1)) MARK(filter(cons(z0, z1), x1)) -> c23(MARK(x1)) MARK(filter(true, x1)) -> c23(MARK(x1)) MARK(filter(false, x1)) -> c23(MARK(x1)) MARK(filter(divides(z0, z1), x1)) -> c23(MARK(x1)) A__PRIMES -> c(A__SIEVE(cons(s(mark(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(sieve(primes)) -> c17(MARK(primes)) MARK(sieve(sieve(primes))) -> c17(A__SIEVE(a__sieve(a__primes)), MARK(sieve(primes))) MARK(sieve(sieve(sieve(z0)))) -> c17(A__SIEVE(a__sieve(a__sieve(mark(z0)))), MARK(sieve(sieve(z0)))) MARK(sieve(sieve(from(z0)))) -> c17(A__SIEVE(a__sieve(a__from(mark(z0)))), MARK(sieve(from(z0)))) MARK(sieve(sieve(head(z0)))) -> c17(A__SIEVE(a__sieve(a__head(mark(z0)))), MARK(sieve(head(z0)))) MARK(sieve(sieve(tail(z0)))) -> c17(A__SIEVE(a__sieve(a__tail(mark(z0)))), MARK(sieve(tail(z0)))) MARK(sieve(sieve(if(z0, z1, z2)))) -> c17(A__SIEVE(a__sieve(a__if(mark(z0), z1, z2))), MARK(sieve(if(z0, z1, z2)))) MARK(sieve(sieve(filter(z0, z1)))) -> c17(A__SIEVE(a__sieve(a__filter(mark(z0), mark(z1)))), MARK(sieve(filter(z0, z1)))) MARK(sieve(sieve(s(z0)))) -> c17(A__SIEVE(a__sieve(s(mark(z0)))), MARK(sieve(s(z0)))) MARK(sieve(sieve(cons(z0, z1)))) -> c17(A__SIEVE(a__sieve(cons(mark(z0), z1))), MARK(sieve(cons(z0, z1)))) MARK(sieve(sieve(divides(z0, z1)))) -> c17(A__SIEVE(a__sieve(divides(mark(z0), mark(z1)))), MARK(sieve(divides(z0, z1)))) MARK(sieve(sieve(x0))) -> c17(MARK(sieve(x0))) MARK(sieve(sieve(0))) -> c17(A__SIEVE(a__sieve(0))) MARK(sieve(sieve(true))) -> c17(A__SIEVE(a__sieve(true))) MARK(sieve(sieve(false))) -> c17(A__SIEVE(a__sieve(false))) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c18_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_1, c17_2, c17_1, c19_2, c19_1, c20_2, c20_1, c21_2, c21_1, c22_2, c22_1, c23_2, c23_1, c_2 ---------------------------------------- (107) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(sieve(from(z0))) -> c17(A__SIEVE(a__from(mark(z0))), MARK(from(z0))) by MARK(sieve(from(x0))) -> c17(A__SIEVE(cons(mark(mark(x0)), from(s(mark(x0))))), MARK(from(x0))) MARK(sieve(from(x0))) -> c17(A__SIEVE(from(mark(x0))), MARK(from(x0))) MARK(sieve(from(primes))) -> c17(A__SIEVE(a__from(a__primes)), MARK(from(primes))) MARK(sieve(from(sieve(z0)))) -> c17(A__SIEVE(a__from(a__sieve(mark(z0)))), MARK(from(sieve(z0)))) MARK(sieve(from(from(z0)))) -> c17(A__SIEVE(a__from(a__from(mark(z0)))), MARK(from(from(z0)))) MARK(sieve(from(head(z0)))) -> c17(A__SIEVE(a__from(a__head(mark(z0)))), MARK(from(head(z0)))) MARK(sieve(from(tail(z0)))) -> c17(A__SIEVE(a__from(a__tail(mark(z0)))), MARK(from(tail(z0)))) MARK(sieve(from(if(z0, z1, z2)))) -> c17(A__SIEVE(a__from(a__if(mark(z0), z1, z2))), MARK(from(if(z0, z1, z2)))) MARK(sieve(from(filter(z0, z1)))) -> c17(A__SIEVE(a__from(a__filter(mark(z0), mark(z1)))), MARK(from(filter(z0, z1)))) MARK(sieve(from(s(z0)))) -> c17(A__SIEVE(a__from(s(mark(z0)))), MARK(from(s(z0)))) MARK(sieve(from(0))) -> c17(A__SIEVE(a__from(0)), MARK(from(0))) MARK(sieve(from(cons(z0, z1)))) -> c17(A__SIEVE(a__from(cons(mark(z0), z1))), MARK(from(cons(z0, z1)))) MARK(sieve(from(true))) -> c17(A__SIEVE(a__from(true)), MARK(from(true))) MARK(sieve(from(false))) -> c17(A__SIEVE(a__from(false)), MARK(from(false))) MARK(sieve(from(divides(z0, z1)))) -> c17(A__SIEVE(a__from(divides(mark(z0), mark(z1)))), MARK(from(divides(z0, z1)))) MARK(sieve(from(x0))) -> c17(MARK(from(x0))) ---------------------------------------- (108) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(head(z0))) -> c17(A__SIEVE(a__head(mark(z0))), MARK(head(z0))) MARK(sieve(tail(z0))) -> c17(A__SIEVE(a__tail(mark(z0))), MARK(tail(z0))) MARK(sieve(if(z0, z1, z2))) -> c17(A__SIEVE(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(sieve(filter(z0, z1))) -> c17(A__SIEVE(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(primes)) -> c19(A__HEAD(a__primes), MARK(primes)) MARK(head(sieve(z0))) -> c19(A__HEAD(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(head(from(z0))) -> c19(A__HEAD(a__from(mark(z0))), MARK(from(z0))) MARK(head(head(z0))) -> c19(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c19(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(if(z0, z1, z2))) -> c19(A__HEAD(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(head(filter(z0, z1))) -> c19(A__HEAD(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(primes)) -> c20(A__TAIL(a__primes), MARK(primes)) MARK(tail(sieve(z0))) -> c20(A__TAIL(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(tail(from(z0))) -> c20(A__TAIL(a__from(mark(z0))), MARK(from(z0))) MARK(tail(head(z0))) -> c20(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c20(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(if(z0, z1, z2))) -> c20(A__TAIL(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(tail(filter(z0, z1))) -> c20(A__TAIL(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(primes, x1, x2)) -> c21(A__IF(a__primes, x1, x2), MARK(primes)) MARK(if(sieve(z0), x1, x2)) -> c21(A__IF(a__sieve(mark(z0)), x1, x2), MARK(sieve(z0))) MARK(if(from(z0), x1, x2)) -> c21(A__IF(a__from(mark(z0)), x1, x2), MARK(from(z0))) MARK(if(head(z0), x1, x2)) -> c21(A__IF(a__head(mark(z0)), x1, x2), MARK(head(z0))) MARK(if(tail(z0), x1, x2)) -> c21(A__IF(a__tail(mark(z0)), x1, x2), MARK(tail(z0))) MARK(if(if(z0, z1, z2), x1, x2)) -> c21(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(filter(z0, z1), x1, x2)) -> c21(A__IF(a__filter(mark(z0), mark(z1)), x1, x2), MARK(filter(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c22(A__FILTER(mark(x0), a__primes), MARK(x0)) MARK(filter(x0, sieve(z0))) -> c22(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, from(z0))) -> c22(A__FILTER(mark(x0), a__from(mark(z0))), MARK(x0)) MARK(filter(x0, head(z0))) -> c22(A__FILTER(mark(x0), a__head(mark(z0))), MARK(x0)) MARK(filter(x0, tail(z0))) -> c22(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(x0)) MARK(filter(x0, if(z0, z1, z2))) -> c22(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0)) MARK(filter(x0, filter(z0, z1))) -> c22(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(x0)) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(x0, s(z0))) -> c22(MARK(x0)) MARK(filter(x0, 0)) -> c22(MARK(x0)) MARK(filter(x0, true)) -> c22(MARK(x0)) MARK(filter(x0, false)) -> c22(MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), x1)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c23(A__FILTER(mark(x0), a__primes), MARK(primes)) MARK(filter(x0, sieve(z0))) -> c23(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, from(z0))) -> c23(A__FILTER(mark(x0), a__from(mark(z0))), MARK(from(z0))) MARK(filter(x0, head(z0))) -> c23(A__FILTER(mark(x0), a__head(mark(z0))), MARK(head(z0))) MARK(filter(x0, tail(z0))) -> c23(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(tail(z0))) MARK(filter(x0, if(z0, z1, z2))) -> c23(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(filter(x0, filter(z0, z1))) -> c23(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(filter(x0, cons(z0, z1))) -> c23(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(primes, x1)) -> c23(A__FILTER(a__primes, mark(x1)), MARK(x1)) MARK(filter(sieve(z0), x1)) -> c23(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(from(z0), x1)) -> c23(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(head(z0), x1)) -> c23(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(tail(z0), x1)) -> c23(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(if(z0, z1, z2), x1)) -> c23(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1)) MARK(filter(filter(z0, z1), x1)) -> c23(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(x1)) MARK(filter(s(z0), x1)) -> c23(A__FILTER(s(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(x0, s(z0))) -> c23(MARK(s(z0))) MARK(filter(x0, divides(z0, z1))) -> c23(MARK(divides(z0, z1))) MARK(filter(0, x1)) -> c23(MARK(x1)) MARK(filter(cons(z0, z1), x1)) -> c23(MARK(x1)) MARK(filter(true, x1)) -> c23(MARK(x1)) MARK(filter(false, x1)) -> c23(MARK(x1)) MARK(filter(divides(z0, z1), x1)) -> c23(MARK(x1)) A__PRIMES -> c(A__SIEVE(cons(s(mark(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(sieve(primes)) -> c17(MARK(primes)) MARK(sieve(sieve(primes))) -> c17(A__SIEVE(a__sieve(a__primes)), MARK(sieve(primes))) MARK(sieve(sieve(sieve(z0)))) -> c17(A__SIEVE(a__sieve(a__sieve(mark(z0)))), MARK(sieve(sieve(z0)))) MARK(sieve(sieve(from(z0)))) -> c17(A__SIEVE(a__sieve(a__from(mark(z0)))), MARK(sieve(from(z0)))) MARK(sieve(sieve(head(z0)))) -> c17(A__SIEVE(a__sieve(a__head(mark(z0)))), MARK(sieve(head(z0)))) MARK(sieve(sieve(tail(z0)))) -> c17(A__SIEVE(a__sieve(a__tail(mark(z0)))), MARK(sieve(tail(z0)))) MARK(sieve(sieve(if(z0, z1, z2)))) -> c17(A__SIEVE(a__sieve(a__if(mark(z0), z1, z2))), MARK(sieve(if(z0, z1, z2)))) MARK(sieve(sieve(filter(z0, z1)))) -> c17(A__SIEVE(a__sieve(a__filter(mark(z0), mark(z1)))), MARK(sieve(filter(z0, z1)))) MARK(sieve(sieve(s(z0)))) -> c17(A__SIEVE(a__sieve(s(mark(z0)))), MARK(sieve(s(z0)))) MARK(sieve(sieve(cons(z0, z1)))) -> c17(A__SIEVE(a__sieve(cons(mark(z0), z1))), MARK(sieve(cons(z0, z1)))) MARK(sieve(sieve(divides(z0, z1)))) -> c17(A__SIEVE(a__sieve(divides(mark(z0), mark(z1)))), MARK(sieve(divides(z0, z1)))) MARK(sieve(sieve(x0))) -> c17(MARK(sieve(x0))) MARK(sieve(sieve(0))) -> c17(A__SIEVE(a__sieve(0))) MARK(sieve(sieve(true))) -> c17(A__SIEVE(a__sieve(true))) MARK(sieve(sieve(false))) -> c17(A__SIEVE(a__sieve(false))) MARK(sieve(from(x0))) -> c17(A__SIEVE(cons(mark(mark(x0)), from(s(mark(x0))))), MARK(from(x0))) MARK(sieve(from(x0))) -> c17(A__SIEVE(from(mark(x0))), MARK(from(x0))) MARK(sieve(from(primes))) -> c17(A__SIEVE(a__from(a__primes)), MARK(from(primes))) MARK(sieve(from(sieve(z0)))) -> c17(A__SIEVE(a__from(a__sieve(mark(z0)))), MARK(from(sieve(z0)))) MARK(sieve(from(from(z0)))) -> c17(A__SIEVE(a__from(a__from(mark(z0)))), MARK(from(from(z0)))) MARK(sieve(from(head(z0)))) -> c17(A__SIEVE(a__from(a__head(mark(z0)))), MARK(from(head(z0)))) MARK(sieve(from(tail(z0)))) -> c17(A__SIEVE(a__from(a__tail(mark(z0)))), MARK(from(tail(z0)))) MARK(sieve(from(if(z0, z1, z2)))) -> c17(A__SIEVE(a__from(a__if(mark(z0), z1, z2))), MARK(from(if(z0, z1, z2)))) MARK(sieve(from(filter(z0, z1)))) -> c17(A__SIEVE(a__from(a__filter(mark(z0), mark(z1)))), MARK(from(filter(z0, z1)))) MARK(sieve(from(s(z0)))) -> c17(A__SIEVE(a__from(s(mark(z0)))), MARK(from(s(z0)))) MARK(sieve(from(0))) -> c17(A__SIEVE(a__from(0)), MARK(from(0))) MARK(sieve(from(cons(z0, z1)))) -> c17(A__SIEVE(a__from(cons(mark(z0), z1))), MARK(from(cons(z0, z1)))) MARK(sieve(from(true))) -> c17(A__SIEVE(a__from(true)), MARK(from(true))) MARK(sieve(from(false))) -> c17(A__SIEVE(a__from(false)), MARK(from(false))) MARK(sieve(from(divides(z0, z1)))) -> c17(A__SIEVE(a__from(divides(mark(z0), mark(z1)))), MARK(from(divides(z0, z1)))) MARK(sieve(from(x0))) -> c17(MARK(from(x0))) S tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(head(z0))) -> c17(A__SIEVE(a__head(mark(z0))), MARK(head(z0))) MARK(sieve(tail(z0))) -> c17(A__SIEVE(a__tail(mark(z0))), MARK(tail(z0))) MARK(sieve(if(z0, z1, z2))) -> c17(A__SIEVE(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(sieve(filter(z0, z1))) -> c17(A__SIEVE(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(primes)) -> c19(A__HEAD(a__primes), MARK(primes)) MARK(head(sieve(z0))) -> c19(A__HEAD(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(head(from(z0))) -> c19(A__HEAD(a__from(mark(z0))), MARK(from(z0))) MARK(head(head(z0))) -> c19(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c19(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(if(z0, z1, z2))) -> c19(A__HEAD(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(head(filter(z0, z1))) -> c19(A__HEAD(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(primes)) -> c20(A__TAIL(a__primes), MARK(primes)) MARK(tail(sieve(z0))) -> c20(A__TAIL(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(tail(from(z0))) -> c20(A__TAIL(a__from(mark(z0))), MARK(from(z0))) MARK(tail(head(z0))) -> c20(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c20(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(if(z0, z1, z2))) -> c20(A__TAIL(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(tail(filter(z0, z1))) -> c20(A__TAIL(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(primes, x1, x2)) -> c21(A__IF(a__primes, x1, x2), MARK(primes)) MARK(if(sieve(z0), x1, x2)) -> c21(A__IF(a__sieve(mark(z0)), x1, x2), MARK(sieve(z0))) MARK(if(from(z0), x1, x2)) -> c21(A__IF(a__from(mark(z0)), x1, x2), MARK(from(z0))) MARK(if(head(z0), x1, x2)) -> c21(A__IF(a__head(mark(z0)), x1, x2), MARK(head(z0))) MARK(if(tail(z0), x1, x2)) -> c21(A__IF(a__tail(mark(z0)), x1, x2), MARK(tail(z0))) MARK(if(if(z0, z1, z2), x1, x2)) -> c21(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(filter(z0, z1), x1, x2)) -> c21(A__IF(a__filter(mark(z0), mark(z1)), x1, x2), MARK(filter(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c22(A__FILTER(mark(x0), a__primes), MARK(x0)) MARK(filter(x0, sieve(z0))) -> c22(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, from(z0))) -> c22(A__FILTER(mark(x0), a__from(mark(z0))), MARK(x0)) MARK(filter(x0, head(z0))) -> c22(A__FILTER(mark(x0), a__head(mark(z0))), MARK(x0)) MARK(filter(x0, tail(z0))) -> c22(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(x0)) MARK(filter(x0, if(z0, z1, z2))) -> c22(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0)) MARK(filter(x0, filter(z0, z1))) -> c22(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(x0)) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(x0, s(z0))) -> c22(MARK(x0)) MARK(filter(x0, 0)) -> c22(MARK(x0)) MARK(filter(x0, true)) -> c22(MARK(x0)) MARK(filter(x0, false)) -> c22(MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), x1)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c23(A__FILTER(mark(x0), a__primes), MARK(primes)) MARK(filter(x0, sieve(z0))) -> c23(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, from(z0))) -> c23(A__FILTER(mark(x0), a__from(mark(z0))), MARK(from(z0))) MARK(filter(x0, head(z0))) -> c23(A__FILTER(mark(x0), a__head(mark(z0))), MARK(head(z0))) MARK(filter(x0, tail(z0))) -> c23(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(tail(z0))) MARK(filter(x0, if(z0, z1, z2))) -> c23(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(filter(x0, filter(z0, z1))) -> c23(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(filter(x0, cons(z0, z1))) -> c23(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(primes, x1)) -> c23(A__FILTER(a__primes, mark(x1)), MARK(x1)) MARK(filter(sieve(z0), x1)) -> c23(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(from(z0), x1)) -> c23(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(head(z0), x1)) -> c23(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(tail(z0), x1)) -> c23(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(if(z0, z1, z2), x1)) -> c23(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1)) MARK(filter(filter(z0, z1), x1)) -> c23(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(x1)) MARK(filter(s(z0), x1)) -> c23(A__FILTER(s(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(x0, s(z0))) -> c23(MARK(s(z0))) MARK(filter(x0, divides(z0, z1))) -> c23(MARK(divides(z0, z1))) MARK(filter(0, x1)) -> c23(MARK(x1)) MARK(filter(cons(z0, z1), x1)) -> c23(MARK(x1)) MARK(filter(true, x1)) -> c23(MARK(x1)) MARK(filter(false, x1)) -> c23(MARK(x1)) MARK(filter(divides(z0, z1), x1)) -> c23(MARK(x1)) A__PRIMES -> c(A__SIEVE(cons(s(mark(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(sieve(primes)) -> c17(MARK(primes)) MARK(sieve(sieve(primes))) -> c17(A__SIEVE(a__sieve(a__primes)), MARK(sieve(primes))) MARK(sieve(sieve(sieve(z0)))) -> c17(A__SIEVE(a__sieve(a__sieve(mark(z0)))), MARK(sieve(sieve(z0)))) MARK(sieve(sieve(from(z0)))) -> c17(A__SIEVE(a__sieve(a__from(mark(z0)))), MARK(sieve(from(z0)))) MARK(sieve(sieve(head(z0)))) -> c17(A__SIEVE(a__sieve(a__head(mark(z0)))), MARK(sieve(head(z0)))) MARK(sieve(sieve(tail(z0)))) -> c17(A__SIEVE(a__sieve(a__tail(mark(z0)))), MARK(sieve(tail(z0)))) MARK(sieve(sieve(if(z0, z1, z2)))) -> c17(A__SIEVE(a__sieve(a__if(mark(z0), z1, z2))), MARK(sieve(if(z0, z1, z2)))) MARK(sieve(sieve(filter(z0, z1)))) -> c17(A__SIEVE(a__sieve(a__filter(mark(z0), mark(z1)))), MARK(sieve(filter(z0, z1)))) MARK(sieve(sieve(s(z0)))) -> c17(A__SIEVE(a__sieve(s(mark(z0)))), MARK(sieve(s(z0)))) MARK(sieve(sieve(cons(z0, z1)))) -> c17(A__SIEVE(a__sieve(cons(mark(z0), z1))), MARK(sieve(cons(z0, z1)))) MARK(sieve(sieve(divides(z0, z1)))) -> c17(A__SIEVE(a__sieve(divides(mark(z0), mark(z1)))), MARK(sieve(divides(z0, z1)))) MARK(sieve(sieve(x0))) -> c17(MARK(sieve(x0))) MARK(sieve(sieve(0))) -> c17(A__SIEVE(a__sieve(0))) MARK(sieve(sieve(true))) -> c17(A__SIEVE(a__sieve(true))) MARK(sieve(sieve(false))) -> c17(A__SIEVE(a__sieve(false))) MARK(sieve(from(x0))) -> c17(A__SIEVE(cons(mark(mark(x0)), from(s(mark(x0))))), MARK(from(x0))) MARK(sieve(from(x0))) -> c17(A__SIEVE(from(mark(x0))), MARK(from(x0))) MARK(sieve(from(primes))) -> c17(A__SIEVE(a__from(a__primes)), MARK(from(primes))) MARK(sieve(from(sieve(z0)))) -> c17(A__SIEVE(a__from(a__sieve(mark(z0)))), MARK(from(sieve(z0)))) MARK(sieve(from(from(z0)))) -> c17(A__SIEVE(a__from(a__from(mark(z0)))), MARK(from(from(z0)))) MARK(sieve(from(head(z0)))) -> c17(A__SIEVE(a__from(a__head(mark(z0)))), MARK(from(head(z0)))) MARK(sieve(from(tail(z0)))) -> c17(A__SIEVE(a__from(a__tail(mark(z0)))), MARK(from(tail(z0)))) MARK(sieve(from(if(z0, z1, z2)))) -> c17(A__SIEVE(a__from(a__if(mark(z0), z1, z2))), MARK(from(if(z0, z1, z2)))) MARK(sieve(from(filter(z0, z1)))) -> c17(A__SIEVE(a__from(a__filter(mark(z0), mark(z1)))), MARK(from(filter(z0, z1)))) MARK(sieve(from(s(z0)))) -> c17(A__SIEVE(a__from(s(mark(z0)))), MARK(from(s(z0)))) MARK(sieve(from(0))) -> c17(A__SIEVE(a__from(0)), MARK(from(0))) MARK(sieve(from(cons(z0, z1)))) -> c17(A__SIEVE(a__from(cons(mark(z0), z1))), MARK(from(cons(z0, z1)))) MARK(sieve(from(true))) -> c17(A__SIEVE(a__from(true)), MARK(from(true))) MARK(sieve(from(false))) -> c17(A__SIEVE(a__from(false)), MARK(from(false))) MARK(sieve(from(divides(z0, z1)))) -> c17(A__SIEVE(a__from(divides(mark(z0), mark(z1)))), MARK(from(divides(z0, z1)))) MARK(sieve(from(x0))) -> c17(MARK(from(x0))) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c18_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_1, c17_2, c17_1, c19_2, c19_1, c20_2, c20_1, c21_2, c21_1, c22_2, c22_1, c23_2, c23_1, c_2 ---------------------------------------- (109) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (110) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(head(z0))) -> c17(A__SIEVE(a__head(mark(z0))), MARK(head(z0))) MARK(sieve(tail(z0))) -> c17(A__SIEVE(a__tail(mark(z0))), MARK(tail(z0))) MARK(sieve(if(z0, z1, z2))) -> c17(A__SIEVE(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(sieve(filter(z0, z1))) -> c17(A__SIEVE(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(primes)) -> c19(A__HEAD(a__primes), MARK(primes)) MARK(head(sieve(z0))) -> c19(A__HEAD(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(head(from(z0))) -> c19(A__HEAD(a__from(mark(z0))), MARK(from(z0))) MARK(head(head(z0))) -> c19(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c19(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(if(z0, z1, z2))) -> c19(A__HEAD(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(head(filter(z0, z1))) -> c19(A__HEAD(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(primes)) -> c20(A__TAIL(a__primes), MARK(primes)) MARK(tail(sieve(z0))) -> c20(A__TAIL(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(tail(from(z0))) -> c20(A__TAIL(a__from(mark(z0))), MARK(from(z0))) MARK(tail(head(z0))) -> c20(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c20(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(if(z0, z1, z2))) -> c20(A__TAIL(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(tail(filter(z0, z1))) -> c20(A__TAIL(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(primes, x1, x2)) -> c21(A__IF(a__primes, x1, x2), MARK(primes)) MARK(if(sieve(z0), x1, x2)) -> c21(A__IF(a__sieve(mark(z0)), x1, x2), MARK(sieve(z0))) MARK(if(from(z0), x1, x2)) -> c21(A__IF(a__from(mark(z0)), x1, x2), MARK(from(z0))) MARK(if(head(z0), x1, x2)) -> c21(A__IF(a__head(mark(z0)), x1, x2), MARK(head(z0))) MARK(if(tail(z0), x1, x2)) -> c21(A__IF(a__tail(mark(z0)), x1, x2), MARK(tail(z0))) MARK(if(if(z0, z1, z2), x1, x2)) -> c21(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(filter(z0, z1), x1, x2)) -> c21(A__IF(a__filter(mark(z0), mark(z1)), x1, x2), MARK(filter(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c22(A__FILTER(mark(x0), a__primes), MARK(x0)) MARK(filter(x0, sieve(z0))) -> c22(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, from(z0))) -> c22(A__FILTER(mark(x0), a__from(mark(z0))), MARK(x0)) MARK(filter(x0, head(z0))) -> c22(A__FILTER(mark(x0), a__head(mark(z0))), MARK(x0)) MARK(filter(x0, tail(z0))) -> c22(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(x0)) MARK(filter(x0, if(z0, z1, z2))) -> c22(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0)) MARK(filter(x0, filter(z0, z1))) -> c22(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(x0)) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(x0, s(z0))) -> c22(MARK(x0)) MARK(filter(x0, 0)) -> c22(MARK(x0)) MARK(filter(x0, true)) -> c22(MARK(x0)) MARK(filter(x0, false)) -> c22(MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), x1)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c23(A__FILTER(mark(x0), a__primes), MARK(primes)) MARK(filter(x0, sieve(z0))) -> c23(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, from(z0))) -> c23(A__FILTER(mark(x0), a__from(mark(z0))), MARK(from(z0))) MARK(filter(x0, head(z0))) -> c23(A__FILTER(mark(x0), a__head(mark(z0))), MARK(head(z0))) MARK(filter(x0, tail(z0))) -> c23(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(tail(z0))) MARK(filter(x0, if(z0, z1, z2))) -> c23(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(filter(x0, filter(z0, z1))) -> c23(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(filter(x0, cons(z0, z1))) -> c23(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(primes, x1)) -> c23(A__FILTER(a__primes, mark(x1)), MARK(x1)) MARK(filter(sieve(z0), x1)) -> c23(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(from(z0), x1)) -> c23(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(head(z0), x1)) -> c23(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(tail(z0), x1)) -> c23(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(if(z0, z1, z2), x1)) -> c23(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1)) MARK(filter(filter(z0, z1), x1)) -> c23(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(x1)) MARK(filter(s(z0), x1)) -> c23(A__FILTER(s(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(x0, s(z0))) -> c23(MARK(s(z0))) MARK(filter(x0, divides(z0, z1))) -> c23(MARK(divides(z0, z1))) MARK(filter(0, x1)) -> c23(MARK(x1)) MARK(filter(cons(z0, z1), x1)) -> c23(MARK(x1)) MARK(filter(true, x1)) -> c23(MARK(x1)) MARK(filter(false, x1)) -> c23(MARK(x1)) MARK(filter(divides(z0, z1), x1)) -> c23(MARK(x1)) A__PRIMES -> c(A__SIEVE(cons(s(mark(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(sieve(primes)) -> c17(MARK(primes)) MARK(sieve(sieve(primes))) -> c17(A__SIEVE(a__sieve(a__primes)), MARK(sieve(primes))) MARK(sieve(sieve(sieve(z0)))) -> c17(A__SIEVE(a__sieve(a__sieve(mark(z0)))), MARK(sieve(sieve(z0)))) MARK(sieve(sieve(from(z0)))) -> c17(A__SIEVE(a__sieve(a__from(mark(z0)))), MARK(sieve(from(z0)))) MARK(sieve(sieve(head(z0)))) -> c17(A__SIEVE(a__sieve(a__head(mark(z0)))), MARK(sieve(head(z0)))) MARK(sieve(sieve(tail(z0)))) -> c17(A__SIEVE(a__sieve(a__tail(mark(z0)))), MARK(sieve(tail(z0)))) MARK(sieve(sieve(if(z0, z1, z2)))) -> c17(A__SIEVE(a__sieve(a__if(mark(z0), z1, z2))), MARK(sieve(if(z0, z1, z2)))) MARK(sieve(sieve(filter(z0, z1)))) -> c17(A__SIEVE(a__sieve(a__filter(mark(z0), mark(z1)))), MARK(sieve(filter(z0, z1)))) MARK(sieve(sieve(s(z0)))) -> c17(A__SIEVE(a__sieve(s(mark(z0)))), MARK(sieve(s(z0)))) MARK(sieve(sieve(cons(z0, z1)))) -> c17(A__SIEVE(a__sieve(cons(mark(z0), z1))), MARK(sieve(cons(z0, z1)))) MARK(sieve(sieve(divides(z0, z1)))) -> c17(A__SIEVE(a__sieve(divides(mark(z0), mark(z1)))), MARK(sieve(divides(z0, z1)))) MARK(sieve(sieve(x0))) -> c17(MARK(sieve(x0))) MARK(sieve(sieve(0))) -> c17(A__SIEVE(a__sieve(0))) MARK(sieve(sieve(true))) -> c17(A__SIEVE(a__sieve(true))) MARK(sieve(sieve(false))) -> c17(A__SIEVE(a__sieve(false))) MARK(sieve(from(x0))) -> c17(A__SIEVE(cons(mark(mark(x0)), from(s(mark(x0))))), MARK(from(x0))) MARK(sieve(from(primes))) -> c17(A__SIEVE(a__from(a__primes)), MARK(from(primes))) MARK(sieve(from(sieve(z0)))) -> c17(A__SIEVE(a__from(a__sieve(mark(z0)))), MARK(from(sieve(z0)))) MARK(sieve(from(from(z0)))) -> c17(A__SIEVE(a__from(a__from(mark(z0)))), MARK(from(from(z0)))) MARK(sieve(from(head(z0)))) -> c17(A__SIEVE(a__from(a__head(mark(z0)))), MARK(from(head(z0)))) MARK(sieve(from(tail(z0)))) -> c17(A__SIEVE(a__from(a__tail(mark(z0)))), MARK(from(tail(z0)))) MARK(sieve(from(if(z0, z1, z2)))) -> c17(A__SIEVE(a__from(a__if(mark(z0), z1, z2))), MARK(from(if(z0, z1, z2)))) MARK(sieve(from(filter(z0, z1)))) -> c17(A__SIEVE(a__from(a__filter(mark(z0), mark(z1)))), MARK(from(filter(z0, z1)))) MARK(sieve(from(s(z0)))) -> c17(A__SIEVE(a__from(s(mark(z0)))), MARK(from(s(z0)))) MARK(sieve(from(0))) -> c17(A__SIEVE(a__from(0)), MARK(from(0))) MARK(sieve(from(cons(z0, z1)))) -> c17(A__SIEVE(a__from(cons(mark(z0), z1))), MARK(from(cons(z0, z1)))) MARK(sieve(from(true))) -> c17(A__SIEVE(a__from(true)), MARK(from(true))) MARK(sieve(from(false))) -> c17(A__SIEVE(a__from(false)), MARK(from(false))) MARK(sieve(from(divides(z0, z1)))) -> c17(A__SIEVE(a__from(divides(mark(z0), mark(z1)))), MARK(from(divides(z0, z1)))) MARK(sieve(from(x0))) -> c17(MARK(from(x0))) S tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(head(z0))) -> c17(A__SIEVE(a__head(mark(z0))), MARK(head(z0))) MARK(sieve(tail(z0))) -> c17(A__SIEVE(a__tail(mark(z0))), MARK(tail(z0))) MARK(sieve(if(z0, z1, z2))) -> c17(A__SIEVE(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(sieve(filter(z0, z1))) -> c17(A__SIEVE(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(primes)) -> c19(A__HEAD(a__primes), MARK(primes)) MARK(head(sieve(z0))) -> c19(A__HEAD(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(head(from(z0))) -> c19(A__HEAD(a__from(mark(z0))), MARK(from(z0))) MARK(head(head(z0))) -> c19(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c19(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(if(z0, z1, z2))) -> c19(A__HEAD(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(head(filter(z0, z1))) -> c19(A__HEAD(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(primes)) -> c20(A__TAIL(a__primes), MARK(primes)) MARK(tail(sieve(z0))) -> c20(A__TAIL(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(tail(from(z0))) -> c20(A__TAIL(a__from(mark(z0))), MARK(from(z0))) MARK(tail(head(z0))) -> c20(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c20(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(if(z0, z1, z2))) -> c20(A__TAIL(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(tail(filter(z0, z1))) -> c20(A__TAIL(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(primes, x1, x2)) -> c21(A__IF(a__primes, x1, x2), MARK(primes)) MARK(if(sieve(z0), x1, x2)) -> c21(A__IF(a__sieve(mark(z0)), x1, x2), MARK(sieve(z0))) MARK(if(from(z0), x1, x2)) -> c21(A__IF(a__from(mark(z0)), x1, x2), MARK(from(z0))) MARK(if(head(z0), x1, x2)) -> c21(A__IF(a__head(mark(z0)), x1, x2), MARK(head(z0))) MARK(if(tail(z0), x1, x2)) -> c21(A__IF(a__tail(mark(z0)), x1, x2), MARK(tail(z0))) MARK(if(if(z0, z1, z2), x1, x2)) -> c21(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(filter(z0, z1), x1, x2)) -> c21(A__IF(a__filter(mark(z0), mark(z1)), x1, x2), MARK(filter(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c22(A__FILTER(mark(x0), a__primes), MARK(x0)) MARK(filter(x0, sieve(z0))) -> c22(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, from(z0))) -> c22(A__FILTER(mark(x0), a__from(mark(z0))), MARK(x0)) MARK(filter(x0, head(z0))) -> c22(A__FILTER(mark(x0), a__head(mark(z0))), MARK(x0)) MARK(filter(x0, tail(z0))) -> c22(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(x0)) MARK(filter(x0, if(z0, z1, z2))) -> c22(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0)) MARK(filter(x0, filter(z0, z1))) -> c22(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(x0)) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(x0, s(z0))) -> c22(MARK(x0)) MARK(filter(x0, 0)) -> c22(MARK(x0)) MARK(filter(x0, true)) -> c22(MARK(x0)) MARK(filter(x0, false)) -> c22(MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), x1)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c23(A__FILTER(mark(x0), a__primes), MARK(primes)) MARK(filter(x0, sieve(z0))) -> c23(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, from(z0))) -> c23(A__FILTER(mark(x0), a__from(mark(z0))), MARK(from(z0))) MARK(filter(x0, head(z0))) -> c23(A__FILTER(mark(x0), a__head(mark(z0))), MARK(head(z0))) MARK(filter(x0, tail(z0))) -> c23(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(tail(z0))) MARK(filter(x0, if(z0, z1, z2))) -> c23(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(filter(x0, filter(z0, z1))) -> c23(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(filter(x0, cons(z0, z1))) -> c23(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(primes, x1)) -> c23(A__FILTER(a__primes, mark(x1)), MARK(x1)) MARK(filter(sieve(z0), x1)) -> c23(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(from(z0), x1)) -> c23(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(head(z0), x1)) -> c23(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(tail(z0), x1)) -> c23(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(if(z0, z1, z2), x1)) -> c23(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1)) MARK(filter(filter(z0, z1), x1)) -> c23(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(x1)) MARK(filter(s(z0), x1)) -> c23(A__FILTER(s(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(x0, s(z0))) -> c23(MARK(s(z0))) MARK(filter(x0, divides(z0, z1))) -> c23(MARK(divides(z0, z1))) MARK(filter(0, x1)) -> c23(MARK(x1)) MARK(filter(cons(z0, z1), x1)) -> c23(MARK(x1)) MARK(filter(true, x1)) -> c23(MARK(x1)) MARK(filter(false, x1)) -> c23(MARK(x1)) MARK(filter(divides(z0, z1), x1)) -> c23(MARK(x1)) A__PRIMES -> c(A__SIEVE(cons(s(mark(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(sieve(primes)) -> c17(MARK(primes)) MARK(sieve(sieve(primes))) -> c17(A__SIEVE(a__sieve(a__primes)), MARK(sieve(primes))) MARK(sieve(sieve(sieve(z0)))) -> c17(A__SIEVE(a__sieve(a__sieve(mark(z0)))), MARK(sieve(sieve(z0)))) MARK(sieve(sieve(from(z0)))) -> c17(A__SIEVE(a__sieve(a__from(mark(z0)))), MARK(sieve(from(z0)))) MARK(sieve(sieve(head(z0)))) -> c17(A__SIEVE(a__sieve(a__head(mark(z0)))), MARK(sieve(head(z0)))) MARK(sieve(sieve(tail(z0)))) -> c17(A__SIEVE(a__sieve(a__tail(mark(z0)))), MARK(sieve(tail(z0)))) MARK(sieve(sieve(if(z0, z1, z2)))) -> c17(A__SIEVE(a__sieve(a__if(mark(z0), z1, z2))), MARK(sieve(if(z0, z1, z2)))) MARK(sieve(sieve(filter(z0, z1)))) -> c17(A__SIEVE(a__sieve(a__filter(mark(z0), mark(z1)))), MARK(sieve(filter(z0, z1)))) MARK(sieve(sieve(s(z0)))) -> c17(A__SIEVE(a__sieve(s(mark(z0)))), MARK(sieve(s(z0)))) MARK(sieve(sieve(cons(z0, z1)))) -> c17(A__SIEVE(a__sieve(cons(mark(z0), z1))), MARK(sieve(cons(z0, z1)))) MARK(sieve(sieve(divides(z0, z1)))) -> c17(A__SIEVE(a__sieve(divides(mark(z0), mark(z1)))), MARK(sieve(divides(z0, z1)))) MARK(sieve(sieve(x0))) -> c17(MARK(sieve(x0))) MARK(sieve(sieve(0))) -> c17(A__SIEVE(a__sieve(0))) MARK(sieve(sieve(true))) -> c17(A__SIEVE(a__sieve(true))) MARK(sieve(sieve(false))) -> c17(A__SIEVE(a__sieve(false))) MARK(sieve(from(x0))) -> c17(A__SIEVE(cons(mark(mark(x0)), from(s(mark(x0))))), MARK(from(x0))) MARK(sieve(from(primes))) -> c17(A__SIEVE(a__from(a__primes)), MARK(from(primes))) MARK(sieve(from(sieve(z0)))) -> c17(A__SIEVE(a__from(a__sieve(mark(z0)))), MARK(from(sieve(z0)))) MARK(sieve(from(from(z0)))) -> c17(A__SIEVE(a__from(a__from(mark(z0)))), MARK(from(from(z0)))) MARK(sieve(from(head(z0)))) -> c17(A__SIEVE(a__from(a__head(mark(z0)))), MARK(from(head(z0)))) MARK(sieve(from(tail(z0)))) -> c17(A__SIEVE(a__from(a__tail(mark(z0)))), MARK(from(tail(z0)))) MARK(sieve(from(if(z0, z1, z2)))) -> c17(A__SIEVE(a__from(a__if(mark(z0), z1, z2))), MARK(from(if(z0, z1, z2)))) MARK(sieve(from(filter(z0, z1)))) -> c17(A__SIEVE(a__from(a__filter(mark(z0), mark(z1)))), MARK(from(filter(z0, z1)))) MARK(sieve(from(s(z0)))) -> c17(A__SIEVE(a__from(s(mark(z0)))), MARK(from(s(z0)))) MARK(sieve(from(0))) -> c17(A__SIEVE(a__from(0)), MARK(from(0))) MARK(sieve(from(cons(z0, z1)))) -> c17(A__SIEVE(a__from(cons(mark(z0), z1))), MARK(from(cons(z0, z1)))) MARK(sieve(from(true))) -> c17(A__SIEVE(a__from(true)), MARK(from(true))) MARK(sieve(from(false))) -> c17(A__SIEVE(a__from(false)), MARK(from(false))) MARK(sieve(from(divides(z0, z1)))) -> c17(A__SIEVE(a__from(divides(mark(z0), mark(z1)))), MARK(from(divides(z0, z1)))) MARK(sieve(from(x0))) -> c17(MARK(from(x0))) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c18_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_1, c17_2, c17_1, c19_2, c19_1, c20_2, c20_1, c21_2, c21_1, c22_2, c22_1, c23_2, c23_1, c_2 ---------------------------------------- (111) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(sieve(head(z0))) -> c17(A__SIEVE(a__head(mark(z0))), MARK(head(z0))) by MARK(sieve(head(x0))) -> c17(A__SIEVE(head(mark(x0))), MARK(head(x0))) MARK(sieve(head(primes))) -> c17(A__SIEVE(a__head(a__primes)), MARK(head(primes))) MARK(sieve(head(sieve(z0)))) -> c17(A__SIEVE(a__head(a__sieve(mark(z0)))), MARK(head(sieve(z0)))) MARK(sieve(head(from(z0)))) -> c17(A__SIEVE(a__head(a__from(mark(z0)))), MARK(head(from(z0)))) MARK(sieve(head(head(z0)))) -> c17(A__SIEVE(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(sieve(head(tail(z0)))) -> c17(A__SIEVE(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(sieve(head(if(z0, z1, z2)))) -> c17(A__SIEVE(a__head(a__if(mark(z0), z1, z2))), MARK(head(if(z0, z1, z2)))) MARK(sieve(head(filter(z0, z1)))) -> c17(A__SIEVE(a__head(a__filter(mark(z0), mark(z1)))), MARK(head(filter(z0, z1)))) MARK(sieve(head(s(z0)))) -> c17(A__SIEVE(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(sieve(head(0))) -> c17(A__SIEVE(a__head(0)), MARK(head(0))) MARK(sieve(head(cons(z0, z1)))) -> c17(A__SIEVE(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(sieve(head(true))) -> c17(A__SIEVE(a__head(true)), MARK(head(true))) MARK(sieve(head(false))) -> c17(A__SIEVE(a__head(false)), MARK(head(false))) MARK(sieve(head(divides(z0, z1)))) -> c17(A__SIEVE(a__head(divides(mark(z0), mark(z1)))), MARK(head(divides(z0, z1)))) ---------------------------------------- (112) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(tail(z0))) -> c17(A__SIEVE(a__tail(mark(z0))), MARK(tail(z0))) MARK(sieve(if(z0, z1, z2))) -> c17(A__SIEVE(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(sieve(filter(z0, z1))) -> c17(A__SIEVE(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(primes)) -> c19(A__HEAD(a__primes), MARK(primes)) MARK(head(sieve(z0))) -> c19(A__HEAD(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(head(from(z0))) -> c19(A__HEAD(a__from(mark(z0))), MARK(from(z0))) MARK(head(head(z0))) -> c19(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c19(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(if(z0, z1, z2))) -> c19(A__HEAD(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(head(filter(z0, z1))) -> c19(A__HEAD(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(primes)) -> c20(A__TAIL(a__primes), MARK(primes)) MARK(tail(sieve(z0))) -> c20(A__TAIL(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(tail(from(z0))) -> c20(A__TAIL(a__from(mark(z0))), MARK(from(z0))) MARK(tail(head(z0))) -> c20(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c20(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(if(z0, z1, z2))) -> c20(A__TAIL(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(tail(filter(z0, z1))) -> c20(A__TAIL(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(primes, x1, x2)) -> c21(A__IF(a__primes, x1, x2), MARK(primes)) MARK(if(sieve(z0), x1, x2)) -> c21(A__IF(a__sieve(mark(z0)), x1, x2), MARK(sieve(z0))) MARK(if(from(z0), x1, x2)) -> c21(A__IF(a__from(mark(z0)), x1, x2), MARK(from(z0))) MARK(if(head(z0), x1, x2)) -> c21(A__IF(a__head(mark(z0)), x1, x2), MARK(head(z0))) MARK(if(tail(z0), x1, x2)) -> c21(A__IF(a__tail(mark(z0)), x1, x2), MARK(tail(z0))) MARK(if(if(z0, z1, z2), x1, x2)) -> c21(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(filter(z0, z1), x1, x2)) -> c21(A__IF(a__filter(mark(z0), mark(z1)), x1, x2), MARK(filter(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c22(A__FILTER(mark(x0), a__primes), MARK(x0)) MARK(filter(x0, sieve(z0))) -> c22(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, from(z0))) -> c22(A__FILTER(mark(x0), a__from(mark(z0))), MARK(x0)) MARK(filter(x0, head(z0))) -> c22(A__FILTER(mark(x0), a__head(mark(z0))), MARK(x0)) MARK(filter(x0, tail(z0))) -> c22(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(x0)) MARK(filter(x0, if(z0, z1, z2))) -> c22(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0)) MARK(filter(x0, filter(z0, z1))) -> c22(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(x0)) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(x0, s(z0))) -> c22(MARK(x0)) MARK(filter(x0, 0)) -> c22(MARK(x0)) MARK(filter(x0, true)) -> c22(MARK(x0)) MARK(filter(x0, false)) -> c22(MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), x1)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c23(A__FILTER(mark(x0), a__primes), MARK(primes)) MARK(filter(x0, sieve(z0))) -> c23(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, from(z0))) -> c23(A__FILTER(mark(x0), a__from(mark(z0))), MARK(from(z0))) MARK(filter(x0, head(z0))) -> c23(A__FILTER(mark(x0), a__head(mark(z0))), MARK(head(z0))) MARK(filter(x0, tail(z0))) -> c23(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(tail(z0))) MARK(filter(x0, if(z0, z1, z2))) -> c23(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(filter(x0, filter(z0, z1))) -> c23(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(filter(x0, cons(z0, z1))) -> c23(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(primes, x1)) -> c23(A__FILTER(a__primes, mark(x1)), MARK(x1)) MARK(filter(sieve(z0), x1)) -> c23(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(from(z0), x1)) -> c23(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(head(z0), x1)) -> c23(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(tail(z0), x1)) -> c23(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(if(z0, z1, z2), x1)) -> c23(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1)) MARK(filter(filter(z0, z1), x1)) -> c23(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(x1)) MARK(filter(s(z0), x1)) -> c23(A__FILTER(s(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(x0, s(z0))) -> c23(MARK(s(z0))) MARK(filter(x0, divides(z0, z1))) -> c23(MARK(divides(z0, z1))) MARK(filter(0, x1)) -> c23(MARK(x1)) MARK(filter(cons(z0, z1), x1)) -> c23(MARK(x1)) MARK(filter(true, x1)) -> c23(MARK(x1)) MARK(filter(false, x1)) -> c23(MARK(x1)) MARK(filter(divides(z0, z1), x1)) -> c23(MARK(x1)) A__PRIMES -> c(A__SIEVE(cons(s(mark(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(sieve(primes)) -> c17(MARK(primes)) MARK(sieve(sieve(primes))) -> c17(A__SIEVE(a__sieve(a__primes)), MARK(sieve(primes))) MARK(sieve(sieve(sieve(z0)))) -> c17(A__SIEVE(a__sieve(a__sieve(mark(z0)))), MARK(sieve(sieve(z0)))) MARK(sieve(sieve(from(z0)))) -> c17(A__SIEVE(a__sieve(a__from(mark(z0)))), MARK(sieve(from(z0)))) MARK(sieve(sieve(head(z0)))) -> c17(A__SIEVE(a__sieve(a__head(mark(z0)))), MARK(sieve(head(z0)))) MARK(sieve(sieve(tail(z0)))) -> c17(A__SIEVE(a__sieve(a__tail(mark(z0)))), MARK(sieve(tail(z0)))) MARK(sieve(sieve(if(z0, z1, z2)))) -> c17(A__SIEVE(a__sieve(a__if(mark(z0), z1, z2))), MARK(sieve(if(z0, z1, z2)))) MARK(sieve(sieve(filter(z0, z1)))) -> c17(A__SIEVE(a__sieve(a__filter(mark(z0), mark(z1)))), MARK(sieve(filter(z0, z1)))) MARK(sieve(sieve(s(z0)))) -> c17(A__SIEVE(a__sieve(s(mark(z0)))), MARK(sieve(s(z0)))) MARK(sieve(sieve(cons(z0, z1)))) -> c17(A__SIEVE(a__sieve(cons(mark(z0), z1))), MARK(sieve(cons(z0, z1)))) MARK(sieve(sieve(divides(z0, z1)))) -> c17(A__SIEVE(a__sieve(divides(mark(z0), mark(z1)))), MARK(sieve(divides(z0, z1)))) MARK(sieve(sieve(x0))) -> c17(MARK(sieve(x0))) MARK(sieve(sieve(0))) -> c17(A__SIEVE(a__sieve(0))) MARK(sieve(sieve(true))) -> c17(A__SIEVE(a__sieve(true))) MARK(sieve(sieve(false))) -> c17(A__SIEVE(a__sieve(false))) MARK(sieve(from(x0))) -> c17(A__SIEVE(cons(mark(mark(x0)), from(s(mark(x0))))), MARK(from(x0))) MARK(sieve(from(primes))) -> c17(A__SIEVE(a__from(a__primes)), MARK(from(primes))) MARK(sieve(from(sieve(z0)))) -> c17(A__SIEVE(a__from(a__sieve(mark(z0)))), MARK(from(sieve(z0)))) MARK(sieve(from(from(z0)))) -> c17(A__SIEVE(a__from(a__from(mark(z0)))), MARK(from(from(z0)))) MARK(sieve(from(head(z0)))) -> c17(A__SIEVE(a__from(a__head(mark(z0)))), MARK(from(head(z0)))) MARK(sieve(from(tail(z0)))) -> c17(A__SIEVE(a__from(a__tail(mark(z0)))), MARK(from(tail(z0)))) MARK(sieve(from(if(z0, z1, z2)))) -> c17(A__SIEVE(a__from(a__if(mark(z0), z1, z2))), MARK(from(if(z0, z1, z2)))) MARK(sieve(from(filter(z0, z1)))) -> c17(A__SIEVE(a__from(a__filter(mark(z0), mark(z1)))), MARK(from(filter(z0, z1)))) MARK(sieve(from(s(z0)))) -> c17(A__SIEVE(a__from(s(mark(z0)))), MARK(from(s(z0)))) MARK(sieve(from(0))) -> c17(A__SIEVE(a__from(0)), MARK(from(0))) MARK(sieve(from(cons(z0, z1)))) -> c17(A__SIEVE(a__from(cons(mark(z0), z1))), MARK(from(cons(z0, z1)))) MARK(sieve(from(true))) -> c17(A__SIEVE(a__from(true)), MARK(from(true))) MARK(sieve(from(false))) -> c17(A__SIEVE(a__from(false)), MARK(from(false))) MARK(sieve(from(divides(z0, z1)))) -> c17(A__SIEVE(a__from(divides(mark(z0), mark(z1)))), MARK(from(divides(z0, z1)))) MARK(sieve(from(x0))) -> c17(MARK(from(x0))) MARK(sieve(head(x0))) -> c17(A__SIEVE(head(mark(x0))), MARK(head(x0))) MARK(sieve(head(primes))) -> c17(A__SIEVE(a__head(a__primes)), MARK(head(primes))) MARK(sieve(head(sieve(z0)))) -> c17(A__SIEVE(a__head(a__sieve(mark(z0)))), MARK(head(sieve(z0)))) MARK(sieve(head(from(z0)))) -> c17(A__SIEVE(a__head(a__from(mark(z0)))), MARK(head(from(z0)))) MARK(sieve(head(head(z0)))) -> c17(A__SIEVE(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(sieve(head(tail(z0)))) -> c17(A__SIEVE(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(sieve(head(if(z0, z1, z2)))) -> c17(A__SIEVE(a__head(a__if(mark(z0), z1, z2))), MARK(head(if(z0, z1, z2)))) MARK(sieve(head(filter(z0, z1)))) -> c17(A__SIEVE(a__head(a__filter(mark(z0), mark(z1)))), MARK(head(filter(z0, z1)))) MARK(sieve(head(s(z0)))) -> c17(A__SIEVE(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(sieve(head(0))) -> c17(A__SIEVE(a__head(0)), MARK(head(0))) MARK(sieve(head(cons(z0, z1)))) -> c17(A__SIEVE(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(sieve(head(true))) -> c17(A__SIEVE(a__head(true)), MARK(head(true))) MARK(sieve(head(false))) -> c17(A__SIEVE(a__head(false)), MARK(head(false))) MARK(sieve(head(divides(z0, z1)))) -> c17(A__SIEVE(a__head(divides(mark(z0), mark(z1)))), MARK(head(divides(z0, z1)))) S tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(tail(z0))) -> c17(A__SIEVE(a__tail(mark(z0))), MARK(tail(z0))) MARK(sieve(if(z0, z1, z2))) -> c17(A__SIEVE(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(sieve(filter(z0, z1))) -> c17(A__SIEVE(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(primes)) -> c19(A__HEAD(a__primes), MARK(primes)) MARK(head(sieve(z0))) -> c19(A__HEAD(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(head(from(z0))) -> c19(A__HEAD(a__from(mark(z0))), MARK(from(z0))) MARK(head(head(z0))) -> c19(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c19(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(if(z0, z1, z2))) -> c19(A__HEAD(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(head(filter(z0, z1))) -> c19(A__HEAD(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(primes)) -> c20(A__TAIL(a__primes), MARK(primes)) MARK(tail(sieve(z0))) -> c20(A__TAIL(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(tail(from(z0))) -> c20(A__TAIL(a__from(mark(z0))), MARK(from(z0))) MARK(tail(head(z0))) -> c20(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c20(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(if(z0, z1, z2))) -> c20(A__TAIL(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(tail(filter(z0, z1))) -> c20(A__TAIL(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(primes, x1, x2)) -> c21(A__IF(a__primes, x1, x2), MARK(primes)) MARK(if(sieve(z0), x1, x2)) -> c21(A__IF(a__sieve(mark(z0)), x1, x2), MARK(sieve(z0))) MARK(if(from(z0), x1, x2)) -> c21(A__IF(a__from(mark(z0)), x1, x2), MARK(from(z0))) MARK(if(head(z0), x1, x2)) -> c21(A__IF(a__head(mark(z0)), x1, x2), MARK(head(z0))) MARK(if(tail(z0), x1, x2)) -> c21(A__IF(a__tail(mark(z0)), x1, x2), MARK(tail(z0))) MARK(if(if(z0, z1, z2), x1, x2)) -> c21(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(filter(z0, z1), x1, x2)) -> c21(A__IF(a__filter(mark(z0), mark(z1)), x1, x2), MARK(filter(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c22(A__FILTER(mark(x0), a__primes), MARK(x0)) MARK(filter(x0, sieve(z0))) -> c22(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, from(z0))) -> c22(A__FILTER(mark(x0), a__from(mark(z0))), MARK(x0)) MARK(filter(x0, head(z0))) -> c22(A__FILTER(mark(x0), a__head(mark(z0))), MARK(x0)) MARK(filter(x0, tail(z0))) -> c22(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(x0)) MARK(filter(x0, if(z0, z1, z2))) -> c22(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0)) MARK(filter(x0, filter(z0, z1))) -> c22(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(x0)) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(x0, s(z0))) -> c22(MARK(x0)) MARK(filter(x0, 0)) -> c22(MARK(x0)) MARK(filter(x0, true)) -> c22(MARK(x0)) MARK(filter(x0, false)) -> c22(MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), x1)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c23(A__FILTER(mark(x0), a__primes), MARK(primes)) MARK(filter(x0, sieve(z0))) -> c23(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, from(z0))) -> c23(A__FILTER(mark(x0), a__from(mark(z0))), MARK(from(z0))) MARK(filter(x0, head(z0))) -> c23(A__FILTER(mark(x0), a__head(mark(z0))), MARK(head(z0))) MARK(filter(x0, tail(z0))) -> c23(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(tail(z0))) MARK(filter(x0, if(z0, z1, z2))) -> c23(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(filter(x0, filter(z0, z1))) -> c23(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(filter(x0, cons(z0, z1))) -> c23(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(primes, x1)) -> c23(A__FILTER(a__primes, mark(x1)), MARK(x1)) MARK(filter(sieve(z0), x1)) -> c23(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(from(z0), x1)) -> c23(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(head(z0), x1)) -> c23(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(tail(z0), x1)) -> c23(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(if(z0, z1, z2), x1)) -> c23(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1)) MARK(filter(filter(z0, z1), x1)) -> c23(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(x1)) MARK(filter(s(z0), x1)) -> c23(A__FILTER(s(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(x0, s(z0))) -> c23(MARK(s(z0))) MARK(filter(x0, divides(z0, z1))) -> c23(MARK(divides(z0, z1))) MARK(filter(0, x1)) -> c23(MARK(x1)) MARK(filter(cons(z0, z1), x1)) -> c23(MARK(x1)) MARK(filter(true, x1)) -> c23(MARK(x1)) MARK(filter(false, x1)) -> c23(MARK(x1)) MARK(filter(divides(z0, z1), x1)) -> c23(MARK(x1)) A__PRIMES -> c(A__SIEVE(cons(s(mark(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(sieve(primes)) -> c17(MARK(primes)) MARK(sieve(sieve(primes))) -> c17(A__SIEVE(a__sieve(a__primes)), MARK(sieve(primes))) MARK(sieve(sieve(sieve(z0)))) -> c17(A__SIEVE(a__sieve(a__sieve(mark(z0)))), MARK(sieve(sieve(z0)))) MARK(sieve(sieve(from(z0)))) -> c17(A__SIEVE(a__sieve(a__from(mark(z0)))), MARK(sieve(from(z0)))) MARK(sieve(sieve(head(z0)))) -> c17(A__SIEVE(a__sieve(a__head(mark(z0)))), MARK(sieve(head(z0)))) MARK(sieve(sieve(tail(z0)))) -> c17(A__SIEVE(a__sieve(a__tail(mark(z0)))), MARK(sieve(tail(z0)))) MARK(sieve(sieve(if(z0, z1, z2)))) -> c17(A__SIEVE(a__sieve(a__if(mark(z0), z1, z2))), MARK(sieve(if(z0, z1, z2)))) MARK(sieve(sieve(filter(z0, z1)))) -> c17(A__SIEVE(a__sieve(a__filter(mark(z0), mark(z1)))), MARK(sieve(filter(z0, z1)))) MARK(sieve(sieve(s(z0)))) -> c17(A__SIEVE(a__sieve(s(mark(z0)))), MARK(sieve(s(z0)))) MARK(sieve(sieve(cons(z0, z1)))) -> c17(A__SIEVE(a__sieve(cons(mark(z0), z1))), MARK(sieve(cons(z0, z1)))) MARK(sieve(sieve(divides(z0, z1)))) -> c17(A__SIEVE(a__sieve(divides(mark(z0), mark(z1)))), MARK(sieve(divides(z0, z1)))) MARK(sieve(sieve(x0))) -> c17(MARK(sieve(x0))) MARK(sieve(sieve(0))) -> c17(A__SIEVE(a__sieve(0))) MARK(sieve(sieve(true))) -> c17(A__SIEVE(a__sieve(true))) MARK(sieve(sieve(false))) -> c17(A__SIEVE(a__sieve(false))) MARK(sieve(from(x0))) -> c17(A__SIEVE(cons(mark(mark(x0)), from(s(mark(x0))))), MARK(from(x0))) MARK(sieve(from(primes))) -> c17(A__SIEVE(a__from(a__primes)), MARK(from(primes))) MARK(sieve(from(sieve(z0)))) -> c17(A__SIEVE(a__from(a__sieve(mark(z0)))), MARK(from(sieve(z0)))) MARK(sieve(from(from(z0)))) -> c17(A__SIEVE(a__from(a__from(mark(z0)))), MARK(from(from(z0)))) MARK(sieve(from(head(z0)))) -> c17(A__SIEVE(a__from(a__head(mark(z0)))), MARK(from(head(z0)))) MARK(sieve(from(tail(z0)))) -> c17(A__SIEVE(a__from(a__tail(mark(z0)))), MARK(from(tail(z0)))) MARK(sieve(from(if(z0, z1, z2)))) -> c17(A__SIEVE(a__from(a__if(mark(z0), z1, z2))), MARK(from(if(z0, z1, z2)))) MARK(sieve(from(filter(z0, z1)))) -> c17(A__SIEVE(a__from(a__filter(mark(z0), mark(z1)))), MARK(from(filter(z0, z1)))) MARK(sieve(from(s(z0)))) -> c17(A__SIEVE(a__from(s(mark(z0)))), MARK(from(s(z0)))) MARK(sieve(from(0))) -> c17(A__SIEVE(a__from(0)), MARK(from(0))) MARK(sieve(from(cons(z0, z1)))) -> c17(A__SIEVE(a__from(cons(mark(z0), z1))), MARK(from(cons(z0, z1)))) MARK(sieve(from(true))) -> c17(A__SIEVE(a__from(true)), MARK(from(true))) MARK(sieve(from(false))) -> c17(A__SIEVE(a__from(false)), MARK(from(false))) MARK(sieve(from(divides(z0, z1)))) -> c17(A__SIEVE(a__from(divides(mark(z0), mark(z1)))), MARK(from(divides(z0, z1)))) MARK(sieve(from(x0))) -> c17(MARK(from(x0))) MARK(sieve(head(x0))) -> c17(A__SIEVE(head(mark(x0))), MARK(head(x0))) MARK(sieve(head(primes))) -> c17(A__SIEVE(a__head(a__primes)), MARK(head(primes))) MARK(sieve(head(sieve(z0)))) -> c17(A__SIEVE(a__head(a__sieve(mark(z0)))), MARK(head(sieve(z0)))) MARK(sieve(head(from(z0)))) -> c17(A__SIEVE(a__head(a__from(mark(z0)))), MARK(head(from(z0)))) MARK(sieve(head(head(z0)))) -> c17(A__SIEVE(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(sieve(head(tail(z0)))) -> c17(A__SIEVE(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(sieve(head(if(z0, z1, z2)))) -> c17(A__SIEVE(a__head(a__if(mark(z0), z1, z2))), MARK(head(if(z0, z1, z2)))) MARK(sieve(head(filter(z0, z1)))) -> c17(A__SIEVE(a__head(a__filter(mark(z0), mark(z1)))), MARK(head(filter(z0, z1)))) MARK(sieve(head(s(z0)))) -> c17(A__SIEVE(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(sieve(head(0))) -> c17(A__SIEVE(a__head(0)), MARK(head(0))) MARK(sieve(head(cons(z0, z1)))) -> c17(A__SIEVE(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(sieve(head(true))) -> c17(A__SIEVE(a__head(true)), MARK(head(true))) MARK(sieve(head(false))) -> c17(A__SIEVE(a__head(false)), MARK(head(false))) MARK(sieve(head(divides(z0, z1)))) -> c17(A__SIEVE(a__head(divides(mark(z0), mark(z1)))), MARK(head(divides(z0, z1)))) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c18_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_1, c17_2, c17_1, c19_2, c19_1, c20_2, c20_1, c21_2, c21_1, c22_2, c22_1, c23_2, c23_1, c_2 ---------------------------------------- (113) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing tuple parts ---------------------------------------- (114) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(tail(z0))) -> c17(A__SIEVE(a__tail(mark(z0))), MARK(tail(z0))) MARK(sieve(if(z0, z1, z2))) -> c17(A__SIEVE(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(sieve(filter(z0, z1))) -> c17(A__SIEVE(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(primes)) -> c19(A__HEAD(a__primes), MARK(primes)) MARK(head(sieve(z0))) -> c19(A__HEAD(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(head(from(z0))) -> c19(A__HEAD(a__from(mark(z0))), MARK(from(z0))) MARK(head(head(z0))) -> c19(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c19(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(if(z0, z1, z2))) -> c19(A__HEAD(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(head(filter(z0, z1))) -> c19(A__HEAD(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(primes)) -> c20(A__TAIL(a__primes), MARK(primes)) MARK(tail(sieve(z0))) -> c20(A__TAIL(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(tail(from(z0))) -> c20(A__TAIL(a__from(mark(z0))), MARK(from(z0))) MARK(tail(head(z0))) -> c20(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c20(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(if(z0, z1, z2))) -> c20(A__TAIL(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) 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c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(x0, s(z0))) -> c22(MARK(x0)) MARK(filter(x0, 0)) -> c22(MARK(x0)) MARK(filter(x0, true)) -> c22(MARK(x0)) MARK(filter(x0, false)) -> c22(MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), x1)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> 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z1)))) MARK(sieve(sieve(x0))) -> c17(MARK(sieve(x0))) MARK(sieve(sieve(0))) -> c17(A__SIEVE(a__sieve(0))) MARK(sieve(sieve(true))) -> c17(A__SIEVE(a__sieve(true))) MARK(sieve(sieve(false))) -> c17(A__SIEVE(a__sieve(false))) MARK(sieve(from(x0))) -> c17(A__SIEVE(cons(mark(mark(x0)), from(s(mark(x0))))), MARK(from(x0))) MARK(sieve(from(primes))) -> c17(A__SIEVE(a__from(a__primes)), MARK(from(primes))) MARK(sieve(from(sieve(z0)))) -> c17(A__SIEVE(a__from(a__sieve(mark(z0)))), MARK(from(sieve(z0)))) MARK(sieve(from(from(z0)))) -> c17(A__SIEVE(a__from(a__from(mark(z0)))), MARK(from(from(z0)))) MARK(sieve(from(head(z0)))) -> c17(A__SIEVE(a__from(a__head(mark(z0)))), MARK(from(head(z0)))) MARK(sieve(from(tail(z0)))) -> c17(A__SIEVE(a__from(a__tail(mark(z0)))), MARK(from(tail(z0)))) MARK(sieve(from(if(z0, z1, z2)))) -> c17(A__SIEVE(a__from(a__if(mark(z0), z1, z2))), MARK(from(if(z0, z1, z2)))) MARK(sieve(from(filter(z0, z1)))) -> c17(A__SIEVE(a__from(a__filter(mark(z0), mark(z1)))), MARK(from(filter(z0, z1)))) MARK(sieve(from(s(z0)))) -> c17(A__SIEVE(a__from(s(mark(z0)))), MARK(from(s(z0)))) MARK(sieve(from(0))) -> c17(A__SIEVE(a__from(0)), MARK(from(0))) MARK(sieve(from(cons(z0, z1)))) -> c17(A__SIEVE(a__from(cons(mark(z0), z1))), MARK(from(cons(z0, z1)))) MARK(sieve(from(true))) -> c17(A__SIEVE(a__from(true)), MARK(from(true))) MARK(sieve(from(false))) -> c17(A__SIEVE(a__from(false)), MARK(from(false))) MARK(sieve(from(divides(z0, z1)))) -> c17(A__SIEVE(a__from(divides(mark(z0), mark(z1)))), MARK(from(divides(z0, z1)))) MARK(sieve(from(x0))) -> c17(MARK(from(x0))) MARK(sieve(head(primes))) -> c17(A__SIEVE(a__head(a__primes)), MARK(head(primes))) MARK(sieve(head(sieve(z0)))) -> c17(A__SIEVE(a__head(a__sieve(mark(z0)))), MARK(head(sieve(z0)))) MARK(sieve(head(from(z0)))) -> c17(A__SIEVE(a__head(a__from(mark(z0)))), MARK(head(from(z0)))) MARK(sieve(head(head(z0)))) -> c17(A__SIEVE(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(sieve(head(tail(z0)))) -> c17(A__SIEVE(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(sieve(head(if(z0, z1, z2)))) -> c17(A__SIEVE(a__head(a__if(mark(z0), z1, z2))), MARK(head(if(z0, z1, z2)))) MARK(sieve(head(filter(z0, z1)))) -> c17(A__SIEVE(a__head(a__filter(mark(z0), mark(z1)))), MARK(head(filter(z0, z1)))) MARK(sieve(head(s(z0)))) -> c17(A__SIEVE(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(sieve(head(cons(z0, z1)))) -> c17(A__SIEVE(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(sieve(head(divides(z0, z1)))) -> c17(A__SIEVE(a__head(divides(mark(z0), mark(z1)))), MARK(head(divides(z0, z1)))) MARK(sieve(head(x0))) -> c17(MARK(head(x0))) MARK(sieve(head(0))) -> c17(A__SIEVE(a__head(0))) MARK(sieve(head(true))) -> c17(A__SIEVE(a__head(true))) MARK(sieve(head(false))) -> c17(A__SIEVE(a__head(false))) S tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(tail(z0))) -> c17(A__SIEVE(a__tail(mark(z0))), MARK(tail(z0))) MARK(sieve(if(z0, z1, z2))) -> c17(A__SIEVE(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(sieve(filter(z0, z1))) -> c17(A__SIEVE(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(primes)) -> c19(A__HEAD(a__primes), MARK(primes)) MARK(head(sieve(z0))) -> c19(A__HEAD(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(head(from(z0))) -> c19(A__HEAD(a__from(mark(z0))), MARK(from(z0))) MARK(head(head(z0))) -> c19(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c19(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(if(z0, z1, z2))) -> c19(A__HEAD(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(head(filter(z0, z1))) -> c19(A__HEAD(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(primes)) -> c20(A__TAIL(a__primes), MARK(primes)) MARK(tail(sieve(z0))) -> c20(A__TAIL(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(tail(from(z0))) -> c20(A__TAIL(a__from(mark(z0))), MARK(from(z0))) MARK(tail(head(z0))) -> c20(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c20(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(if(z0, z1, z2))) -> c20(A__TAIL(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(tail(filter(z0, z1))) -> c20(A__TAIL(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(primes, x1, x2)) -> c21(A__IF(a__primes, x1, x2), MARK(primes)) MARK(if(sieve(z0), x1, x2)) -> c21(A__IF(a__sieve(mark(z0)), x1, x2), MARK(sieve(z0))) MARK(if(from(z0), x1, x2)) -> c21(A__IF(a__from(mark(z0)), x1, x2), MARK(from(z0))) MARK(if(head(z0), x1, x2)) -> c21(A__IF(a__head(mark(z0)), x1, x2), MARK(head(z0))) MARK(if(tail(z0), x1, x2)) -> c21(A__IF(a__tail(mark(z0)), x1, x2), MARK(tail(z0))) MARK(if(if(z0, z1, z2), x1, x2)) -> c21(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(filter(z0, z1), x1, x2)) -> c21(A__IF(a__filter(mark(z0), mark(z1)), x1, x2), MARK(filter(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c22(A__FILTER(mark(x0), a__primes), MARK(x0)) MARK(filter(x0, sieve(z0))) -> c22(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, from(z0))) -> c22(A__FILTER(mark(x0), a__from(mark(z0))), MARK(x0)) MARK(filter(x0, head(z0))) -> c22(A__FILTER(mark(x0), a__head(mark(z0))), MARK(x0)) MARK(filter(x0, tail(z0))) -> c22(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(x0)) MARK(filter(x0, if(z0, z1, z2))) -> c22(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0)) MARK(filter(x0, filter(z0, z1))) -> c22(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(x0)) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(x0, s(z0))) -> c22(MARK(x0)) MARK(filter(x0, 0)) -> c22(MARK(x0)) MARK(filter(x0, true)) -> c22(MARK(x0)) MARK(filter(x0, false)) -> c22(MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), x1)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c23(A__FILTER(mark(x0), a__primes), MARK(primes)) MARK(filter(x0, sieve(z0))) -> c23(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, from(z0))) -> c23(A__FILTER(mark(x0), a__from(mark(z0))), MARK(from(z0))) MARK(filter(x0, head(z0))) -> c23(A__FILTER(mark(x0), a__head(mark(z0))), MARK(head(z0))) MARK(filter(x0, tail(z0))) -> c23(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(tail(z0))) MARK(filter(x0, if(z0, z1, z2))) -> c23(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(filter(x0, filter(z0, z1))) -> c23(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(filter(x0, cons(z0, z1))) -> c23(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(primes, x1)) -> c23(A__FILTER(a__primes, mark(x1)), MARK(x1)) MARK(filter(sieve(z0), x1)) -> c23(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(from(z0), x1)) -> c23(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(head(z0), x1)) -> c23(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(tail(z0), x1)) -> c23(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(if(z0, z1, z2), x1)) -> c23(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1)) MARK(filter(filter(z0, z1), x1)) -> c23(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(x1)) MARK(filter(s(z0), x1)) -> c23(A__FILTER(s(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(x0, s(z0))) -> c23(MARK(s(z0))) MARK(filter(x0, divides(z0, z1))) -> c23(MARK(divides(z0, z1))) MARK(filter(0, x1)) -> c23(MARK(x1)) MARK(filter(cons(z0, z1), x1)) -> c23(MARK(x1)) MARK(filter(true, x1)) -> c23(MARK(x1)) MARK(filter(false, x1)) -> c23(MARK(x1)) MARK(filter(divides(z0, z1), x1)) -> c23(MARK(x1)) A__PRIMES -> c(A__SIEVE(cons(s(mark(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(sieve(primes)) -> c17(MARK(primes)) MARK(sieve(sieve(primes))) -> c17(A__SIEVE(a__sieve(a__primes)), MARK(sieve(primes))) MARK(sieve(sieve(sieve(z0)))) -> c17(A__SIEVE(a__sieve(a__sieve(mark(z0)))), MARK(sieve(sieve(z0)))) MARK(sieve(sieve(from(z0)))) -> c17(A__SIEVE(a__sieve(a__from(mark(z0)))), MARK(sieve(from(z0)))) MARK(sieve(sieve(head(z0)))) -> c17(A__SIEVE(a__sieve(a__head(mark(z0)))), MARK(sieve(head(z0)))) MARK(sieve(sieve(tail(z0)))) -> c17(A__SIEVE(a__sieve(a__tail(mark(z0)))), MARK(sieve(tail(z0)))) MARK(sieve(sieve(if(z0, z1, z2)))) -> c17(A__SIEVE(a__sieve(a__if(mark(z0), z1, z2))), MARK(sieve(if(z0, z1, z2)))) MARK(sieve(sieve(filter(z0, z1)))) -> c17(A__SIEVE(a__sieve(a__filter(mark(z0), mark(z1)))), MARK(sieve(filter(z0, z1)))) MARK(sieve(sieve(s(z0)))) -> c17(A__SIEVE(a__sieve(s(mark(z0)))), MARK(sieve(s(z0)))) MARK(sieve(sieve(cons(z0, z1)))) -> c17(A__SIEVE(a__sieve(cons(mark(z0), z1))), MARK(sieve(cons(z0, z1)))) MARK(sieve(sieve(divides(z0, z1)))) -> c17(A__SIEVE(a__sieve(divides(mark(z0), mark(z1)))), MARK(sieve(divides(z0, z1)))) MARK(sieve(sieve(x0))) -> c17(MARK(sieve(x0))) MARK(sieve(sieve(0))) -> c17(A__SIEVE(a__sieve(0))) MARK(sieve(sieve(true))) -> c17(A__SIEVE(a__sieve(true))) MARK(sieve(sieve(false))) -> c17(A__SIEVE(a__sieve(false))) MARK(sieve(from(x0))) -> c17(A__SIEVE(cons(mark(mark(x0)), from(s(mark(x0))))), MARK(from(x0))) MARK(sieve(from(primes))) -> c17(A__SIEVE(a__from(a__primes)), MARK(from(primes))) MARK(sieve(from(sieve(z0)))) -> c17(A__SIEVE(a__from(a__sieve(mark(z0)))), MARK(from(sieve(z0)))) MARK(sieve(from(from(z0)))) -> c17(A__SIEVE(a__from(a__from(mark(z0)))), MARK(from(from(z0)))) MARK(sieve(from(head(z0)))) -> c17(A__SIEVE(a__from(a__head(mark(z0)))), MARK(from(head(z0)))) MARK(sieve(from(tail(z0)))) -> c17(A__SIEVE(a__from(a__tail(mark(z0)))), MARK(from(tail(z0)))) MARK(sieve(from(if(z0, z1, z2)))) -> c17(A__SIEVE(a__from(a__if(mark(z0), z1, z2))), MARK(from(if(z0, z1, z2)))) MARK(sieve(from(filter(z0, z1)))) -> c17(A__SIEVE(a__from(a__filter(mark(z0), mark(z1)))), MARK(from(filter(z0, z1)))) MARK(sieve(from(s(z0)))) -> c17(A__SIEVE(a__from(s(mark(z0)))), MARK(from(s(z0)))) MARK(sieve(from(0))) -> c17(A__SIEVE(a__from(0)), MARK(from(0))) MARK(sieve(from(cons(z0, z1)))) -> c17(A__SIEVE(a__from(cons(mark(z0), z1))), MARK(from(cons(z0, z1)))) MARK(sieve(from(true))) -> c17(A__SIEVE(a__from(true)), MARK(from(true))) MARK(sieve(from(false))) -> c17(A__SIEVE(a__from(false)), MARK(from(false))) MARK(sieve(from(divides(z0, z1)))) -> c17(A__SIEVE(a__from(divides(mark(z0), mark(z1)))), MARK(from(divides(z0, z1)))) MARK(sieve(from(x0))) -> c17(MARK(from(x0))) MARK(sieve(head(primes))) -> c17(A__SIEVE(a__head(a__primes)), MARK(head(primes))) MARK(sieve(head(sieve(z0)))) -> c17(A__SIEVE(a__head(a__sieve(mark(z0)))), MARK(head(sieve(z0)))) MARK(sieve(head(from(z0)))) -> c17(A__SIEVE(a__head(a__from(mark(z0)))), MARK(head(from(z0)))) MARK(sieve(head(head(z0)))) -> c17(A__SIEVE(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(sieve(head(tail(z0)))) -> c17(A__SIEVE(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(sieve(head(if(z0, z1, z2)))) -> c17(A__SIEVE(a__head(a__if(mark(z0), z1, z2))), MARK(head(if(z0, z1, z2)))) MARK(sieve(head(filter(z0, z1)))) -> c17(A__SIEVE(a__head(a__filter(mark(z0), mark(z1)))), MARK(head(filter(z0, z1)))) MARK(sieve(head(s(z0)))) -> c17(A__SIEVE(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(sieve(head(cons(z0, z1)))) -> c17(A__SIEVE(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(sieve(head(divides(z0, z1)))) -> c17(A__SIEVE(a__head(divides(mark(z0), mark(z1)))), MARK(head(divides(z0, z1)))) MARK(sieve(head(x0))) -> c17(MARK(head(x0))) MARK(sieve(head(0))) -> c17(A__SIEVE(a__head(0))) MARK(sieve(head(true))) -> c17(A__SIEVE(a__head(true))) MARK(sieve(head(false))) -> c17(A__SIEVE(a__head(false))) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c18_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_1, c17_2, c17_1, c19_2, c19_1, c20_2, c20_1, c21_2, c21_1, c22_2, c22_1, c23_2, c23_1, c_2 ---------------------------------------- (115) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(sieve(tail(z0))) -> c17(A__SIEVE(a__tail(mark(z0))), MARK(tail(z0))) by MARK(sieve(tail(x0))) -> c17(A__SIEVE(tail(mark(x0))), MARK(tail(x0))) MARK(sieve(tail(primes))) -> c17(A__SIEVE(a__tail(a__primes)), MARK(tail(primes))) MARK(sieve(tail(sieve(z0)))) -> c17(A__SIEVE(a__tail(a__sieve(mark(z0)))), MARK(tail(sieve(z0)))) MARK(sieve(tail(from(z0)))) -> c17(A__SIEVE(a__tail(a__from(mark(z0)))), MARK(tail(from(z0)))) MARK(sieve(tail(head(z0)))) -> c17(A__SIEVE(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(sieve(tail(tail(z0)))) -> c17(A__SIEVE(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(sieve(tail(if(z0, z1, z2)))) -> c17(A__SIEVE(a__tail(a__if(mark(z0), z1, z2))), MARK(tail(if(z0, z1, z2)))) MARK(sieve(tail(filter(z0, z1)))) -> c17(A__SIEVE(a__tail(a__filter(mark(z0), mark(z1)))), MARK(tail(filter(z0, z1)))) MARK(sieve(tail(s(z0)))) -> c17(A__SIEVE(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(sieve(tail(0))) -> c17(A__SIEVE(a__tail(0)), MARK(tail(0))) MARK(sieve(tail(cons(z0, z1)))) -> c17(A__SIEVE(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(sieve(tail(true))) -> c17(A__SIEVE(a__tail(true)), MARK(tail(true))) MARK(sieve(tail(false))) -> c17(A__SIEVE(a__tail(false)), MARK(tail(false))) MARK(sieve(tail(divides(z0, z1)))) -> c17(A__SIEVE(a__tail(divides(mark(z0), mark(z1)))), MARK(tail(divides(z0, z1)))) ---------------------------------------- (116) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(if(z0, z1, z2))) -> c17(A__SIEVE(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(sieve(filter(z0, z1))) -> c17(A__SIEVE(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(primes)) -> c19(A__HEAD(a__primes), MARK(primes)) MARK(head(sieve(z0))) -> c19(A__HEAD(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(head(from(z0))) -> c19(A__HEAD(a__from(mark(z0))), MARK(from(z0))) MARK(head(head(z0))) -> c19(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c19(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(if(z0, z1, z2))) -> c19(A__HEAD(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(head(filter(z0, z1))) -> c19(A__HEAD(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(primes)) -> c20(A__TAIL(a__primes), MARK(primes)) MARK(tail(sieve(z0))) -> c20(A__TAIL(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(tail(from(z0))) -> c20(A__TAIL(a__from(mark(z0))), MARK(from(z0))) MARK(tail(head(z0))) -> c20(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c20(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(if(z0, z1, z2))) -> c20(A__TAIL(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(tail(filter(z0, z1))) -> c20(A__TAIL(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(primes, x1, x2)) -> c21(A__IF(a__primes, x1, x2), MARK(primes)) MARK(if(sieve(z0), x1, x2)) -> c21(A__IF(a__sieve(mark(z0)), x1, x2), MARK(sieve(z0))) MARK(if(from(z0), x1, x2)) -> c21(A__IF(a__from(mark(z0)), x1, x2), MARK(from(z0))) MARK(if(head(z0), x1, x2)) -> c21(A__IF(a__head(mark(z0)), x1, x2), MARK(head(z0))) MARK(if(tail(z0), x1, x2)) -> c21(A__IF(a__tail(mark(z0)), x1, x2), MARK(tail(z0))) MARK(if(if(z0, z1, z2), x1, x2)) -> c21(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(filter(z0, z1), x1, x2)) -> c21(A__IF(a__filter(mark(z0), mark(z1)), x1, x2), MARK(filter(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c22(A__FILTER(mark(x0), a__primes), MARK(x0)) MARK(filter(x0, sieve(z0))) -> c22(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, from(z0))) -> c22(A__FILTER(mark(x0), a__from(mark(z0))), MARK(x0)) MARK(filter(x0, head(z0))) -> c22(A__FILTER(mark(x0), a__head(mark(z0))), MARK(x0)) MARK(filter(x0, tail(z0))) -> c22(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(x0)) MARK(filter(x0, if(z0, z1, z2))) -> c22(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0)) MARK(filter(x0, filter(z0, z1))) -> c22(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(x0)) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(x0, s(z0))) -> c22(MARK(x0)) MARK(filter(x0, 0)) -> c22(MARK(x0)) MARK(filter(x0, true)) -> c22(MARK(x0)) MARK(filter(x0, false)) -> c22(MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), x1)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> 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MARK(filter(x0, divides(z0, z1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), x1)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c23(A__FILTER(mark(x0), a__primes), MARK(primes)) MARK(filter(x0, sieve(z0))) -> c23(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, from(z0))) -> c23(A__FILTER(mark(x0), a__from(mark(z0))), MARK(from(z0))) MARK(filter(x0, head(z0))) -> c23(A__FILTER(mark(x0), a__head(mark(z0))), MARK(head(z0))) MARK(filter(x0, tail(z0))) -> c23(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(tail(z0))) MARK(filter(x0, if(z0, z1, z2))) -> c23(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(filter(x0, filter(z0, z1))) -> c23(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(filter(x0, cons(z0, z1))) -> c23(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(primes, x1)) -> c23(A__FILTER(a__primes, mark(x1)), MARK(x1)) MARK(filter(sieve(z0), x1)) -> c23(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(from(z0), x1)) -> c23(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(head(z0), x1)) -> c23(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(tail(z0), x1)) -> c23(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(if(z0, z1, z2), x1)) -> c23(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1)) MARK(filter(filter(z0, z1), x1)) -> c23(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(x1)) MARK(filter(s(z0), x1)) -> c23(A__FILTER(s(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(x0, s(z0))) -> c23(MARK(s(z0))) MARK(filter(x0, divides(z0, z1))) -> c23(MARK(divides(z0, z1))) MARK(filter(0, x1)) -> c23(MARK(x1)) MARK(filter(cons(z0, z1), x1)) -> c23(MARK(x1)) MARK(filter(true, x1)) -> c23(MARK(x1)) MARK(filter(false, x1)) -> c23(MARK(x1)) MARK(filter(divides(z0, z1), x1)) -> c23(MARK(x1)) A__PRIMES -> c(A__SIEVE(cons(s(mark(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(sieve(primes)) -> c17(MARK(primes)) MARK(sieve(sieve(primes))) -> c17(A__SIEVE(a__sieve(a__primes)), MARK(sieve(primes))) MARK(sieve(sieve(sieve(z0)))) -> c17(A__SIEVE(a__sieve(a__sieve(mark(z0)))), MARK(sieve(sieve(z0)))) MARK(sieve(sieve(from(z0)))) -> c17(A__SIEVE(a__sieve(a__from(mark(z0)))), MARK(sieve(from(z0)))) MARK(sieve(sieve(head(z0)))) -> c17(A__SIEVE(a__sieve(a__head(mark(z0)))), MARK(sieve(head(z0)))) MARK(sieve(sieve(tail(z0)))) -> c17(A__SIEVE(a__sieve(a__tail(mark(z0)))), MARK(sieve(tail(z0)))) MARK(sieve(sieve(if(z0, z1, z2)))) -> c17(A__SIEVE(a__sieve(a__if(mark(z0), z1, z2))), MARK(sieve(if(z0, z1, z2)))) MARK(sieve(sieve(filter(z0, z1)))) -> c17(A__SIEVE(a__sieve(a__filter(mark(z0), mark(z1)))), MARK(sieve(filter(z0, z1)))) MARK(sieve(sieve(s(z0)))) -> c17(A__SIEVE(a__sieve(s(mark(z0)))), MARK(sieve(s(z0)))) MARK(sieve(sieve(cons(z0, z1)))) -> c17(A__SIEVE(a__sieve(cons(mark(z0), z1))), MARK(sieve(cons(z0, z1)))) MARK(sieve(sieve(divides(z0, z1)))) -> c17(A__SIEVE(a__sieve(divides(mark(z0), mark(z1)))), MARK(sieve(divides(z0, z1)))) MARK(sieve(sieve(x0))) -> c17(MARK(sieve(x0))) MARK(sieve(sieve(0))) -> c17(A__SIEVE(a__sieve(0))) MARK(sieve(sieve(true))) -> c17(A__SIEVE(a__sieve(true))) MARK(sieve(sieve(false))) -> c17(A__SIEVE(a__sieve(false))) MARK(sieve(from(x0))) -> c17(A__SIEVE(cons(mark(mark(x0)), from(s(mark(x0))))), MARK(from(x0))) MARK(sieve(from(primes))) -> c17(A__SIEVE(a__from(a__primes)), MARK(from(primes))) MARK(sieve(from(sieve(z0)))) -> c17(A__SIEVE(a__from(a__sieve(mark(z0)))), MARK(from(sieve(z0)))) MARK(sieve(from(from(z0)))) -> c17(A__SIEVE(a__from(a__from(mark(z0)))), MARK(from(from(z0)))) MARK(sieve(from(head(z0)))) -> c17(A__SIEVE(a__from(a__head(mark(z0)))), MARK(from(head(z0)))) MARK(sieve(from(tail(z0)))) -> c17(A__SIEVE(a__from(a__tail(mark(z0)))), MARK(from(tail(z0)))) MARK(sieve(from(if(z0, z1, z2)))) -> c17(A__SIEVE(a__from(a__if(mark(z0), z1, z2))), MARK(from(if(z0, z1, z2)))) MARK(sieve(from(filter(z0, z1)))) -> c17(A__SIEVE(a__from(a__filter(mark(z0), mark(z1)))), MARK(from(filter(z0, z1)))) MARK(sieve(from(s(z0)))) -> c17(A__SIEVE(a__from(s(mark(z0)))), MARK(from(s(z0)))) MARK(sieve(from(0))) -> c17(A__SIEVE(a__from(0)), MARK(from(0))) MARK(sieve(from(cons(z0, z1)))) -> c17(A__SIEVE(a__from(cons(mark(z0), z1))), MARK(from(cons(z0, z1)))) MARK(sieve(from(true))) -> c17(A__SIEVE(a__from(true)), MARK(from(true))) MARK(sieve(from(false))) -> c17(A__SIEVE(a__from(false)), MARK(from(false))) MARK(sieve(from(divides(z0, z1)))) -> c17(A__SIEVE(a__from(divides(mark(z0), mark(z1)))), MARK(from(divides(z0, z1)))) MARK(sieve(from(x0))) -> c17(MARK(from(x0))) MARK(sieve(head(primes))) -> c17(A__SIEVE(a__head(a__primes)), MARK(head(primes))) MARK(sieve(head(sieve(z0)))) -> c17(A__SIEVE(a__head(a__sieve(mark(z0)))), MARK(head(sieve(z0)))) MARK(sieve(head(from(z0)))) -> c17(A__SIEVE(a__head(a__from(mark(z0)))), MARK(head(from(z0)))) MARK(sieve(head(head(z0)))) -> c17(A__SIEVE(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(sieve(head(tail(z0)))) -> c17(A__SIEVE(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(sieve(head(if(z0, z1, z2)))) -> c17(A__SIEVE(a__head(a__if(mark(z0), z1, z2))), MARK(head(if(z0, z1, z2)))) MARK(sieve(head(filter(z0, z1)))) -> c17(A__SIEVE(a__head(a__filter(mark(z0), mark(z1)))), MARK(head(filter(z0, z1)))) MARK(sieve(head(s(z0)))) -> c17(A__SIEVE(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(sieve(head(cons(z0, z1)))) -> c17(A__SIEVE(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(sieve(head(divides(z0, z1)))) -> c17(A__SIEVE(a__head(divides(mark(z0), mark(z1)))), MARK(head(divides(z0, z1)))) MARK(sieve(head(x0))) -> c17(MARK(head(x0))) MARK(sieve(head(0))) -> c17(A__SIEVE(a__head(0))) MARK(sieve(head(true))) -> c17(A__SIEVE(a__head(true))) MARK(sieve(head(false))) -> c17(A__SIEVE(a__head(false))) MARK(sieve(tail(x0))) -> c17(A__SIEVE(tail(mark(x0))), MARK(tail(x0))) MARK(sieve(tail(primes))) -> c17(A__SIEVE(a__tail(a__primes)), MARK(tail(primes))) MARK(sieve(tail(sieve(z0)))) -> c17(A__SIEVE(a__tail(a__sieve(mark(z0)))), MARK(tail(sieve(z0)))) MARK(sieve(tail(from(z0)))) -> c17(A__SIEVE(a__tail(a__from(mark(z0)))), MARK(tail(from(z0)))) MARK(sieve(tail(head(z0)))) -> c17(A__SIEVE(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(sieve(tail(tail(z0)))) -> c17(A__SIEVE(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(sieve(tail(if(z0, z1, z2)))) -> c17(A__SIEVE(a__tail(a__if(mark(z0), z1, z2))), MARK(tail(if(z0, z1, z2)))) MARK(sieve(tail(filter(z0, z1)))) -> c17(A__SIEVE(a__tail(a__filter(mark(z0), mark(z1)))), MARK(tail(filter(z0, z1)))) MARK(sieve(tail(s(z0)))) -> c17(A__SIEVE(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(sieve(tail(0))) -> c17(A__SIEVE(a__tail(0)), MARK(tail(0))) MARK(sieve(tail(cons(z0, z1)))) -> c17(A__SIEVE(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(sieve(tail(true))) -> c17(A__SIEVE(a__tail(true)), MARK(tail(true))) MARK(sieve(tail(false))) -> c17(A__SIEVE(a__tail(false)), MARK(tail(false))) MARK(sieve(tail(divides(z0, z1)))) -> c17(A__SIEVE(a__tail(divides(mark(z0), mark(z1)))), MARK(tail(divides(z0, z1)))) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c18_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_1, c17_2, c17_1, c19_2, c19_1, c20_2, c20_1, c21_2, c21_1, c22_2, c22_1, c23_2, c23_1, c_2 ---------------------------------------- (117) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing tuple parts ---------------------------------------- (118) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(if(z0, z1, z2))) -> c17(A__SIEVE(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(sieve(filter(z0, z1))) -> c17(A__SIEVE(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(primes)) -> c19(A__HEAD(a__primes), MARK(primes)) MARK(head(sieve(z0))) -> c19(A__HEAD(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(head(from(z0))) -> c19(A__HEAD(a__from(mark(z0))), MARK(from(z0))) MARK(head(head(z0))) -> c19(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c19(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(if(z0, z1, z2))) -> c19(A__HEAD(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(head(filter(z0, z1))) -> c19(A__HEAD(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(primes)) -> c20(A__TAIL(a__primes), MARK(primes)) MARK(tail(sieve(z0))) -> c20(A__TAIL(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(tail(from(z0))) -> c20(A__TAIL(a__from(mark(z0))), MARK(from(z0))) MARK(tail(head(z0))) -> c20(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c20(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(if(z0, z1, z2))) -> c20(A__TAIL(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(tail(filter(z0, z1))) -> c20(A__TAIL(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(primes, x1, x2)) -> c21(A__IF(a__primes, x1, x2), MARK(primes)) MARK(if(sieve(z0), x1, x2)) -> c21(A__IF(a__sieve(mark(z0)), x1, x2), MARK(sieve(z0))) MARK(if(from(z0), x1, x2)) -> c21(A__IF(a__from(mark(z0)), x1, x2), MARK(from(z0))) MARK(if(head(z0), x1, x2)) -> c21(A__IF(a__head(mark(z0)), x1, x2), MARK(head(z0))) MARK(if(tail(z0), x1, x2)) -> c21(A__IF(a__tail(mark(z0)), x1, x2), MARK(tail(z0))) MARK(if(if(z0, z1, z2), x1, x2)) -> c21(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(filter(z0, z1), x1, x2)) -> c21(A__IF(a__filter(mark(z0), mark(z1)), x1, x2), MARK(filter(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c22(A__FILTER(mark(x0), a__primes), MARK(x0)) MARK(filter(x0, sieve(z0))) -> c22(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, from(z0))) -> c22(A__FILTER(mark(x0), a__from(mark(z0))), MARK(x0)) MARK(filter(x0, head(z0))) -> c22(A__FILTER(mark(x0), a__head(mark(z0))), MARK(x0)) MARK(filter(x0, tail(z0))) -> c22(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(x0)) MARK(filter(x0, if(z0, z1, z2))) -> c22(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0)) MARK(filter(x0, filter(z0, z1))) -> c22(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(x0)) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(x0, s(z0))) -> c22(MARK(x0)) MARK(filter(x0, 0)) -> c22(MARK(x0)) MARK(filter(x0, true)) -> c22(MARK(x0)) MARK(filter(x0, false)) -> c22(MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), x1)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c23(A__FILTER(mark(x0), a__primes), MARK(primes)) MARK(filter(x0, sieve(z0))) -> c23(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, from(z0))) -> c23(A__FILTER(mark(x0), a__from(mark(z0))), MARK(from(z0))) MARK(filter(x0, head(z0))) -> c23(A__FILTER(mark(x0), a__head(mark(z0))), MARK(head(z0))) MARK(filter(x0, tail(z0))) -> c23(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(tail(z0))) MARK(filter(x0, if(z0, z1, z2))) -> c23(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(filter(x0, filter(z0, z1))) -> c23(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(filter(x0, cons(z0, z1))) -> c23(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(primes, x1)) -> c23(A__FILTER(a__primes, mark(x1)), MARK(x1)) MARK(filter(sieve(z0), x1)) -> c23(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(from(z0), x1)) -> c23(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(head(z0), x1)) -> c23(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(tail(z0), x1)) -> c23(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(if(z0, z1, z2), x1)) -> c23(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1)) MARK(filter(filter(z0, z1), x1)) -> c23(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(x1)) MARK(filter(s(z0), x1)) -> c23(A__FILTER(s(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(x0, s(z0))) -> c23(MARK(s(z0))) MARK(filter(x0, divides(z0, z1))) -> c23(MARK(divides(z0, z1))) MARK(filter(0, x1)) -> c23(MARK(x1)) MARK(filter(cons(z0, z1), x1)) -> c23(MARK(x1)) MARK(filter(true, x1)) -> c23(MARK(x1)) MARK(filter(false, x1)) -> c23(MARK(x1)) MARK(filter(divides(z0, z1), x1)) -> c23(MARK(x1)) A__PRIMES -> c(A__SIEVE(cons(s(mark(s(0))), 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MARK(filter(x0, s(z0))) -> c22(MARK(x0)) MARK(filter(x0, 0)) -> c22(MARK(x0)) MARK(filter(x0, true)) -> c22(MARK(x0)) MARK(filter(x0, false)) -> c22(MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), x1)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c23(A__FILTER(mark(x0), a__primes), MARK(primes)) MARK(filter(x0, sieve(z0))) -> c23(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, from(z0))) -> c23(A__FILTER(mark(x0), a__from(mark(z0))), MARK(from(z0))) MARK(filter(x0, head(z0))) -> c23(A__FILTER(mark(x0), a__head(mark(z0))), MARK(head(z0))) MARK(filter(x0, tail(z0))) -> c23(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(tail(z0))) MARK(filter(x0, if(z0, z1, z2))) -> c23(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(filter(x0, filter(z0, z1))) -> c23(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) 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-> c23(MARK(x1)) MARK(filter(false, x1)) -> c23(MARK(x1)) MARK(filter(divides(z0, z1), x1)) -> c23(MARK(x1)) A__PRIMES -> c(A__SIEVE(cons(s(mark(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(sieve(primes)) -> c17(MARK(primes)) MARK(sieve(sieve(primes))) -> c17(A__SIEVE(a__sieve(a__primes)), MARK(sieve(primes))) MARK(sieve(sieve(sieve(z0)))) -> c17(A__SIEVE(a__sieve(a__sieve(mark(z0)))), MARK(sieve(sieve(z0)))) MARK(sieve(sieve(from(z0)))) -> c17(A__SIEVE(a__sieve(a__from(mark(z0)))), MARK(sieve(from(z0)))) MARK(sieve(sieve(head(z0)))) -> c17(A__SIEVE(a__sieve(a__head(mark(z0)))), MARK(sieve(head(z0)))) MARK(sieve(sieve(tail(z0)))) -> c17(A__SIEVE(a__sieve(a__tail(mark(z0)))), MARK(sieve(tail(z0)))) MARK(sieve(sieve(if(z0, z1, z2)))) -> c17(A__SIEVE(a__sieve(a__if(mark(z0), z1, z2))), MARK(sieve(if(z0, z1, z2)))) MARK(sieve(sieve(filter(z0, z1)))) -> c17(A__SIEVE(a__sieve(a__filter(mark(z0), mark(z1)))), MARK(sieve(filter(z0, z1)))) MARK(sieve(sieve(s(z0)))) -> c17(A__SIEVE(a__sieve(s(mark(z0)))), MARK(sieve(s(z0)))) MARK(sieve(sieve(cons(z0, z1)))) -> c17(A__SIEVE(a__sieve(cons(mark(z0), z1))), MARK(sieve(cons(z0, z1)))) MARK(sieve(sieve(divides(z0, z1)))) -> c17(A__SIEVE(a__sieve(divides(mark(z0), mark(z1)))), MARK(sieve(divides(z0, z1)))) MARK(sieve(sieve(x0))) -> c17(MARK(sieve(x0))) MARK(sieve(sieve(0))) -> c17(A__SIEVE(a__sieve(0))) MARK(sieve(sieve(true))) -> c17(A__SIEVE(a__sieve(true))) MARK(sieve(sieve(false))) -> c17(A__SIEVE(a__sieve(false))) MARK(sieve(from(x0))) -> c17(A__SIEVE(cons(mark(mark(x0)), from(s(mark(x0))))), MARK(from(x0))) MARK(sieve(from(primes))) -> c17(A__SIEVE(a__from(a__primes)), MARK(from(primes))) MARK(sieve(from(sieve(z0)))) -> c17(A__SIEVE(a__from(a__sieve(mark(z0)))), MARK(from(sieve(z0)))) MARK(sieve(from(from(z0)))) -> c17(A__SIEVE(a__from(a__from(mark(z0)))), MARK(from(from(z0)))) MARK(sieve(from(head(z0)))) -> c17(A__SIEVE(a__from(a__head(mark(z0)))), MARK(from(head(z0)))) MARK(sieve(from(tail(z0)))) -> c17(A__SIEVE(a__from(a__tail(mark(z0)))), MARK(from(tail(z0)))) MARK(sieve(from(if(z0, z1, z2)))) -> c17(A__SIEVE(a__from(a__if(mark(z0), z1, z2))), MARK(from(if(z0, z1, z2)))) MARK(sieve(from(filter(z0, z1)))) -> c17(A__SIEVE(a__from(a__filter(mark(z0), mark(z1)))), MARK(from(filter(z0, z1)))) MARK(sieve(from(s(z0)))) -> c17(A__SIEVE(a__from(s(mark(z0)))), MARK(from(s(z0)))) MARK(sieve(from(0))) -> c17(A__SIEVE(a__from(0)), MARK(from(0))) MARK(sieve(from(cons(z0, z1)))) -> c17(A__SIEVE(a__from(cons(mark(z0), z1))), MARK(from(cons(z0, z1)))) MARK(sieve(from(true))) -> c17(A__SIEVE(a__from(true)), MARK(from(true))) MARK(sieve(from(false))) -> c17(A__SIEVE(a__from(false)), MARK(from(false))) MARK(sieve(from(divides(z0, z1)))) -> c17(A__SIEVE(a__from(divides(mark(z0), mark(z1)))), MARK(from(divides(z0, z1)))) MARK(sieve(from(x0))) -> c17(MARK(from(x0))) MARK(sieve(head(primes))) -> c17(A__SIEVE(a__head(a__primes)), MARK(head(primes))) MARK(sieve(head(sieve(z0)))) -> c17(A__SIEVE(a__head(a__sieve(mark(z0)))), MARK(head(sieve(z0)))) MARK(sieve(head(from(z0)))) -> c17(A__SIEVE(a__head(a__from(mark(z0)))), MARK(head(from(z0)))) MARK(sieve(head(head(z0)))) -> c17(A__SIEVE(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(sieve(head(tail(z0)))) -> c17(A__SIEVE(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(sieve(head(if(z0, z1, z2)))) -> c17(A__SIEVE(a__head(a__if(mark(z0), z1, z2))), MARK(head(if(z0, z1, z2)))) MARK(sieve(head(filter(z0, z1)))) -> c17(A__SIEVE(a__head(a__filter(mark(z0), mark(z1)))), MARK(head(filter(z0, z1)))) MARK(sieve(head(s(z0)))) -> c17(A__SIEVE(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(sieve(head(cons(z0, z1)))) -> c17(A__SIEVE(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(sieve(head(divides(z0, z1)))) -> c17(A__SIEVE(a__head(divides(mark(z0), mark(z1)))), MARK(head(divides(z0, z1)))) MARK(sieve(head(x0))) -> c17(MARK(head(x0))) MARK(sieve(head(0))) -> c17(A__SIEVE(a__head(0))) MARK(sieve(head(true))) -> c17(A__SIEVE(a__head(true))) MARK(sieve(head(false))) -> c17(A__SIEVE(a__head(false))) MARK(sieve(tail(primes))) -> c17(A__SIEVE(a__tail(a__primes)), MARK(tail(primes))) MARK(sieve(tail(sieve(z0)))) -> c17(A__SIEVE(a__tail(a__sieve(mark(z0)))), MARK(tail(sieve(z0)))) MARK(sieve(tail(from(z0)))) -> c17(A__SIEVE(a__tail(a__from(mark(z0)))), MARK(tail(from(z0)))) MARK(sieve(tail(head(z0)))) -> c17(A__SIEVE(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(sieve(tail(tail(z0)))) -> c17(A__SIEVE(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(sieve(tail(if(z0, z1, z2)))) -> c17(A__SIEVE(a__tail(a__if(mark(z0), z1, z2))), MARK(tail(if(z0, z1, z2)))) MARK(sieve(tail(filter(z0, z1)))) -> c17(A__SIEVE(a__tail(a__filter(mark(z0), mark(z1)))), MARK(tail(filter(z0, z1)))) MARK(sieve(tail(s(z0)))) -> c17(A__SIEVE(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(sieve(tail(cons(z0, z1)))) -> c17(A__SIEVE(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(sieve(tail(divides(z0, z1)))) -> c17(A__SIEVE(a__tail(divides(mark(z0), mark(z1)))), MARK(tail(divides(z0, z1)))) MARK(sieve(tail(x0))) -> c17(MARK(tail(x0))) MARK(sieve(tail(0))) -> c17(A__SIEVE(a__tail(0))) MARK(sieve(tail(true))) -> c17(A__SIEVE(a__tail(true))) MARK(sieve(tail(false))) -> c17(A__SIEVE(a__tail(false))) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c18_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_1, c17_2, c17_1, c19_2, c19_1, c20_2, c20_1, c21_2, c21_1, c22_2, c22_1, c23_2, c23_1, c_2 ---------------------------------------- (119) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(sieve(if(z0, z1, z2))) -> c17(A__SIEVE(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) by MARK(sieve(if(x0, z1, z2))) -> c17(A__SIEVE(if(mark(x0), z1, z2)), MARK(if(x0, z1, z2))) MARK(sieve(if(primes, x1, x2))) -> c17(A__SIEVE(a__if(a__primes, x1, x2)), MARK(if(primes, x1, x2))) MARK(sieve(if(sieve(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__sieve(mark(z0)), x1, x2)), MARK(if(sieve(z0), x1, x2))) MARK(sieve(if(from(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__from(mark(z0)), x1, x2)), MARK(if(from(z0), x1, x2))) MARK(sieve(if(head(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__head(mark(z0)), x1, x2)), MARK(if(head(z0), x1, x2))) MARK(sieve(if(tail(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__tail(mark(z0)), x1, x2)), MARK(if(tail(z0), x1, x2))) MARK(sieve(if(if(z0, z1, z2), x1, x2))) -> c17(A__SIEVE(a__if(a__if(mark(z0), z1, z2), x1, x2)), MARK(if(if(z0, z1, z2), x1, x2))) MARK(sieve(if(filter(z0, z1), x1, x2))) -> c17(A__SIEVE(a__if(a__filter(mark(z0), mark(z1)), x1, x2)), MARK(if(filter(z0, z1), x1, x2))) MARK(sieve(if(s(z0), x1, x2))) -> c17(A__SIEVE(a__if(s(mark(z0)), x1, x2)), MARK(if(s(z0), x1, x2))) MARK(sieve(if(0, x1, x2))) -> c17(A__SIEVE(a__if(0, x1, x2)), MARK(if(0, x1, x2))) MARK(sieve(if(cons(z0, z1), x1, x2))) -> c17(A__SIEVE(a__if(cons(mark(z0), z1), x1, x2)), MARK(if(cons(z0, z1), x1, x2))) MARK(sieve(if(true, x1, x2))) -> c17(A__SIEVE(a__if(true, x1, x2)), MARK(if(true, x1, x2))) MARK(sieve(if(false, x1, x2))) -> c17(A__SIEVE(a__if(false, x1, x2)), MARK(if(false, x1, x2))) MARK(sieve(if(divides(z0, z1), x1, x2))) -> c17(A__SIEVE(a__if(divides(mark(z0), mark(z1)), x1, x2)), MARK(if(divides(z0, z1), x1, x2))) ---------------------------------------- (120) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(filter(z0, z1))) -> c17(A__SIEVE(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(primes)) -> c19(A__HEAD(a__primes), MARK(primes)) MARK(head(sieve(z0))) -> c19(A__HEAD(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(head(from(z0))) -> c19(A__HEAD(a__from(mark(z0))), MARK(from(z0))) MARK(head(head(z0))) -> c19(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c19(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(if(z0, z1, z2))) -> c19(A__HEAD(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(head(filter(z0, z1))) -> c19(A__HEAD(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(primes)) -> c20(A__TAIL(a__primes), MARK(primes)) MARK(tail(sieve(z0))) -> c20(A__TAIL(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(tail(from(z0))) -> c20(A__TAIL(a__from(mark(z0))), MARK(from(z0))) MARK(tail(head(z0))) -> c20(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c20(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(if(z0, z1, z2))) -> c20(A__TAIL(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(tail(filter(z0, z1))) -> c20(A__TAIL(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(primes, x1, x2)) -> c21(A__IF(a__primes, x1, x2), MARK(primes)) MARK(if(sieve(z0), x1, x2)) -> c21(A__IF(a__sieve(mark(z0)), x1, x2), MARK(sieve(z0))) MARK(if(from(z0), x1, x2)) -> c21(A__IF(a__from(mark(z0)), x1, x2), MARK(from(z0))) MARK(if(head(z0), x1, x2)) -> c21(A__IF(a__head(mark(z0)), x1, x2), MARK(head(z0))) MARK(if(tail(z0), x1, x2)) -> c21(A__IF(a__tail(mark(z0)), x1, x2), MARK(tail(z0))) MARK(if(if(z0, z1, z2), x1, x2)) -> c21(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(filter(z0, z1), x1, x2)) -> c21(A__IF(a__filter(mark(z0), mark(z1)), x1, x2), MARK(filter(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c22(A__FILTER(mark(x0), a__primes), MARK(x0)) MARK(filter(x0, sieve(z0))) -> c22(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, from(z0))) -> c22(A__FILTER(mark(x0), a__from(mark(z0))), MARK(x0)) MARK(filter(x0, head(z0))) -> c22(A__FILTER(mark(x0), a__head(mark(z0))), MARK(x0)) MARK(filter(x0, tail(z0))) -> c22(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(x0)) MARK(filter(x0, if(z0, z1, z2))) -> c22(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0)) MARK(filter(x0, filter(z0, z1))) -> c22(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(x0)) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(x0, s(z0))) -> c22(MARK(x0)) MARK(filter(x0, 0)) -> c22(MARK(x0)) MARK(filter(x0, true)) -> c22(MARK(x0)) MARK(filter(x0, false)) -> c22(MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), x1)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c23(A__FILTER(mark(x0), a__primes), MARK(primes)) MARK(filter(x0, sieve(z0))) -> c23(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, from(z0))) -> c23(A__FILTER(mark(x0), a__from(mark(z0))), MARK(from(z0))) MARK(filter(x0, head(z0))) -> c23(A__FILTER(mark(x0), a__head(mark(z0))), MARK(head(z0))) MARK(filter(x0, tail(z0))) -> c23(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(tail(z0))) MARK(filter(x0, if(z0, z1, z2))) -> c23(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(filter(x0, filter(z0, z1))) -> c23(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(filter(x0, cons(z0, z1))) -> c23(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(primes, x1)) -> c23(A__FILTER(a__primes, mark(x1)), MARK(x1)) MARK(filter(sieve(z0), x1)) -> c23(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(from(z0), x1)) -> c23(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(head(z0), x1)) -> c23(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(tail(z0), x1)) -> c23(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(if(z0, z1, z2), x1)) -> c23(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1)) MARK(filter(filter(z0, z1), x1)) -> c23(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(x1)) MARK(filter(s(z0), x1)) -> c23(A__FILTER(s(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(x0, s(z0))) -> c23(MARK(s(z0))) MARK(filter(x0, divides(z0, z1))) -> c23(MARK(divides(z0, z1))) MARK(filter(0, x1)) -> c23(MARK(x1)) MARK(filter(cons(z0, z1), x1)) -> c23(MARK(x1)) MARK(filter(true, x1)) -> c23(MARK(x1)) MARK(filter(false, x1)) -> c23(MARK(x1)) MARK(filter(divides(z0, z1), x1)) -> c23(MARK(x1)) A__PRIMES -> c(A__SIEVE(cons(s(mark(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(sieve(primes)) -> c17(MARK(primes)) MARK(sieve(sieve(primes))) -> c17(A__SIEVE(a__sieve(a__primes)), MARK(sieve(primes))) MARK(sieve(sieve(sieve(z0)))) -> c17(A__SIEVE(a__sieve(a__sieve(mark(z0)))), MARK(sieve(sieve(z0)))) MARK(sieve(sieve(from(z0)))) -> c17(A__SIEVE(a__sieve(a__from(mark(z0)))), MARK(sieve(from(z0)))) 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MARK(tail(tail(z0)))) MARK(sieve(tail(if(z0, z1, z2)))) -> c17(A__SIEVE(a__tail(a__if(mark(z0), z1, z2))), MARK(tail(if(z0, z1, z2)))) MARK(sieve(tail(filter(z0, z1)))) -> c17(A__SIEVE(a__tail(a__filter(mark(z0), mark(z1)))), MARK(tail(filter(z0, z1)))) MARK(sieve(tail(s(z0)))) -> c17(A__SIEVE(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(sieve(tail(cons(z0, z1)))) -> c17(A__SIEVE(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(sieve(tail(divides(z0, z1)))) -> c17(A__SIEVE(a__tail(divides(mark(z0), mark(z1)))), MARK(tail(divides(z0, z1)))) MARK(sieve(tail(x0))) -> c17(MARK(tail(x0))) MARK(sieve(tail(0))) -> c17(A__SIEVE(a__tail(0))) MARK(sieve(tail(true))) -> c17(A__SIEVE(a__tail(true))) MARK(sieve(tail(false))) -> c17(A__SIEVE(a__tail(false))) MARK(sieve(if(x0, z1, z2))) -> c17(A__SIEVE(if(mark(x0), z1, z2)), MARK(if(x0, z1, z2))) MARK(sieve(if(primes, x1, x2))) -> c17(A__SIEVE(a__if(a__primes, x1, x2)), MARK(if(primes, x1, x2))) MARK(sieve(if(sieve(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__sieve(mark(z0)), x1, x2)), MARK(if(sieve(z0), x1, x2))) MARK(sieve(if(from(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__from(mark(z0)), x1, x2)), MARK(if(from(z0), x1, x2))) MARK(sieve(if(head(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__head(mark(z0)), x1, x2)), MARK(if(head(z0), x1, x2))) MARK(sieve(if(tail(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__tail(mark(z0)), x1, x2)), MARK(if(tail(z0), x1, x2))) MARK(sieve(if(if(z0, z1, z2), x1, x2))) -> c17(A__SIEVE(a__if(a__if(mark(z0), z1, z2), x1, x2)), MARK(if(if(z0, z1, z2), x1, x2))) MARK(sieve(if(filter(z0, z1), x1, x2))) -> c17(A__SIEVE(a__if(a__filter(mark(z0), mark(z1)), x1, x2)), MARK(if(filter(z0, z1), x1, x2))) MARK(sieve(if(s(z0), x1, x2))) -> c17(A__SIEVE(a__if(s(mark(z0)), x1, x2)), MARK(if(s(z0), x1, x2))) MARK(sieve(if(0, x1, x2))) -> c17(A__SIEVE(a__if(0, x1, x2)), MARK(if(0, x1, x2))) MARK(sieve(if(cons(z0, z1), x1, x2))) -> c17(A__SIEVE(a__if(cons(mark(z0), z1), x1, x2)), MARK(if(cons(z0, z1), x1, x2))) MARK(sieve(if(true, x1, x2))) -> c17(A__SIEVE(a__if(true, x1, x2)), MARK(if(true, x1, x2))) MARK(sieve(if(false, x1, x2))) -> c17(A__SIEVE(a__if(false, x1, x2)), MARK(if(false, x1, x2))) MARK(sieve(if(divides(z0, z1), x1, x2))) -> c17(A__SIEVE(a__if(divides(mark(z0), mark(z1)), x1, x2)), MARK(if(divides(z0, z1), x1, x2))) S tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(filter(z0, z1))) -> c17(A__SIEVE(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(primes)) -> c19(A__HEAD(a__primes), MARK(primes)) MARK(head(sieve(z0))) -> c19(A__HEAD(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(head(from(z0))) -> c19(A__HEAD(a__from(mark(z0))), MARK(from(z0))) MARK(head(head(z0))) -> c19(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c19(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(if(z0, z1, z2))) -> c19(A__HEAD(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(head(filter(z0, z1))) -> c19(A__HEAD(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(primes)) -> c20(A__TAIL(a__primes), MARK(primes)) MARK(tail(sieve(z0))) -> c20(A__TAIL(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(tail(from(z0))) -> c20(A__TAIL(a__from(mark(z0))), MARK(from(z0))) MARK(tail(head(z0))) -> c20(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c20(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(if(z0, z1, z2))) -> c20(A__TAIL(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(tail(filter(z0, z1))) -> c20(A__TAIL(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(primes, x1, x2)) -> c21(A__IF(a__primes, x1, x2), MARK(primes)) MARK(if(sieve(z0), x1, x2)) -> c21(A__IF(a__sieve(mark(z0)), x1, x2), MARK(sieve(z0))) MARK(if(from(z0), x1, x2)) -> c21(A__IF(a__from(mark(z0)), x1, x2), MARK(from(z0))) MARK(if(head(z0), x1, x2)) -> c21(A__IF(a__head(mark(z0)), x1, x2), MARK(head(z0))) MARK(if(tail(z0), x1, x2)) -> c21(A__IF(a__tail(mark(z0)), x1, x2), MARK(tail(z0))) MARK(if(if(z0, z1, z2), x1, x2)) -> c21(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(filter(z0, z1), x1, x2)) -> c21(A__IF(a__filter(mark(z0), mark(z1)), x1, x2), MARK(filter(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c22(A__FILTER(mark(x0), a__primes), MARK(x0)) MARK(filter(x0, sieve(z0))) -> c22(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, from(z0))) -> c22(A__FILTER(mark(x0), a__from(mark(z0))), MARK(x0)) MARK(filter(x0, head(z0))) -> c22(A__FILTER(mark(x0), a__head(mark(z0))), MARK(x0)) MARK(filter(x0, tail(z0))) -> c22(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(x0)) MARK(filter(x0, if(z0, z1, z2))) -> c22(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0)) MARK(filter(x0, filter(z0, z1))) -> c22(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(x0)) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(x0, s(z0))) -> c22(MARK(x0)) MARK(filter(x0, 0)) -> c22(MARK(x0)) MARK(filter(x0, true)) -> c22(MARK(x0)) MARK(filter(x0, false)) -> c22(MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), x1)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c23(A__FILTER(mark(x0), a__primes), MARK(primes)) MARK(filter(x0, sieve(z0))) -> c23(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, from(z0))) -> c23(A__FILTER(mark(x0), a__from(mark(z0))), MARK(from(z0))) MARK(filter(x0, head(z0))) -> c23(A__FILTER(mark(x0), a__head(mark(z0))), MARK(head(z0))) MARK(filter(x0, tail(z0))) -> c23(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(tail(z0))) MARK(filter(x0, if(z0, z1, z2))) -> c23(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(filter(x0, filter(z0, z1))) -> c23(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(filter(x0, cons(z0, z1))) -> c23(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(primes, x1)) -> c23(A__FILTER(a__primes, mark(x1)), MARK(x1)) MARK(filter(sieve(z0), x1)) -> c23(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(from(z0), x1)) -> c23(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(head(z0), x1)) -> c23(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(tail(z0), x1)) -> c23(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(if(z0, z1, z2), x1)) -> c23(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1)) MARK(filter(filter(z0, z1), x1)) -> c23(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(x1)) MARK(filter(s(z0), x1)) -> c23(A__FILTER(s(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(x0, s(z0))) -> c23(MARK(s(z0))) MARK(filter(x0, divides(z0, z1))) -> c23(MARK(divides(z0, z1))) MARK(filter(0, x1)) -> c23(MARK(x1)) MARK(filter(cons(z0, z1), x1)) -> c23(MARK(x1)) MARK(filter(true, x1)) -> c23(MARK(x1)) MARK(filter(false, x1)) -> c23(MARK(x1)) MARK(filter(divides(z0, z1), x1)) -> c23(MARK(x1)) A__PRIMES -> c(A__SIEVE(cons(s(mark(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(sieve(primes)) -> c17(MARK(primes)) MARK(sieve(sieve(primes))) -> c17(A__SIEVE(a__sieve(a__primes)), MARK(sieve(primes))) MARK(sieve(sieve(sieve(z0)))) -> c17(A__SIEVE(a__sieve(a__sieve(mark(z0)))), MARK(sieve(sieve(z0)))) MARK(sieve(sieve(from(z0)))) -> c17(A__SIEVE(a__sieve(a__from(mark(z0)))), MARK(sieve(from(z0)))) MARK(sieve(sieve(head(z0)))) -> c17(A__SIEVE(a__sieve(a__head(mark(z0)))), MARK(sieve(head(z0)))) MARK(sieve(sieve(tail(z0)))) -> c17(A__SIEVE(a__sieve(a__tail(mark(z0)))), MARK(sieve(tail(z0)))) MARK(sieve(sieve(if(z0, z1, z2)))) -> c17(A__SIEVE(a__sieve(a__if(mark(z0), z1, z2))), MARK(sieve(if(z0, z1, z2)))) MARK(sieve(sieve(filter(z0, z1)))) -> c17(A__SIEVE(a__sieve(a__filter(mark(z0), mark(z1)))), MARK(sieve(filter(z0, z1)))) MARK(sieve(sieve(s(z0)))) -> c17(A__SIEVE(a__sieve(s(mark(z0)))), MARK(sieve(s(z0)))) MARK(sieve(sieve(cons(z0, z1)))) -> c17(A__SIEVE(a__sieve(cons(mark(z0), z1))), MARK(sieve(cons(z0, z1)))) MARK(sieve(sieve(divides(z0, z1)))) -> c17(A__SIEVE(a__sieve(divides(mark(z0), mark(z1)))), MARK(sieve(divides(z0, z1)))) MARK(sieve(sieve(x0))) -> c17(MARK(sieve(x0))) MARK(sieve(sieve(0))) -> c17(A__SIEVE(a__sieve(0))) MARK(sieve(sieve(true))) -> c17(A__SIEVE(a__sieve(true))) MARK(sieve(sieve(false))) -> c17(A__SIEVE(a__sieve(false))) MARK(sieve(from(x0))) -> c17(A__SIEVE(cons(mark(mark(x0)), from(s(mark(x0))))), MARK(from(x0))) MARK(sieve(from(primes))) -> c17(A__SIEVE(a__from(a__primes)), MARK(from(primes))) MARK(sieve(from(sieve(z0)))) -> c17(A__SIEVE(a__from(a__sieve(mark(z0)))), MARK(from(sieve(z0)))) MARK(sieve(from(from(z0)))) -> c17(A__SIEVE(a__from(a__from(mark(z0)))), MARK(from(from(z0)))) MARK(sieve(from(head(z0)))) -> c17(A__SIEVE(a__from(a__head(mark(z0)))), MARK(from(head(z0)))) MARK(sieve(from(tail(z0)))) -> c17(A__SIEVE(a__from(a__tail(mark(z0)))), MARK(from(tail(z0)))) MARK(sieve(from(if(z0, z1, z2)))) -> c17(A__SIEVE(a__from(a__if(mark(z0), z1, z2))), MARK(from(if(z0, z1, z2)))) MARK(sieve(from(filter(z0, z1)))) -> c17(A__SIEVE(a__from(a__filter(mark(z0), mark(z1)))), MARK(from(filter(z0, z1)))) MARK(sieve(from(s(z0)))) -> c17(A__SIEVE(a__from(s(mark(z0)))), MARK(from(s(z0)))) MARK(sieve(from(0))) -> c17(A__SIEVE(a__from(0)), MARK(from(0))) MARK(sieve(from(cons(z0, z1)))) -> c17(A__SIEVE(a__from(cons(mark(z0), z1))), MARK(from(cons(z0, z1)))) MARK(sieve(from(true))) -> c17(A__SIEVE(a__from(true)), MARK(from(true))) MARK(sieve(from(false))) -> c17(A__SIEVE(a__from(false)), MARK(from(false))) MARK(sieve(from(divides(z0, z1)))) -> c17(A__SIEVE(a__from(divides(mark(z0), mark(z1)))), MARK(from(divides(z0, z1)))) MARK(sieve(from(x0))) -> c17(MARK(from(x0))) MARK(sieve(head(primes))) -> c17(A__SIEVE(a__head(a__primes)), MARK(head(primes))) MARK(sieve(head(sieve(z0)))) -> c17(A__SIEVE(a__head(a__sieve(mark(z0)))), MARK(head(sieve(z0)))) MARK(sieve(head(from(z0)))) -> c17(A__SIEVE(a__head(a__from(mark(z0)))), MARK(head(from(z0)))) MARK(sieve(head(head(z0)))) -> c17(A__SIEVE(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(sieve(head(tail(z0)))) -> c17(A__SIEVE(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(sieve(head(if(z0, z1, z2)))) -> c17(A__SIEVE(a__head(a__if(mark(z0), z1, z2))), MARK(head(if(z0, z1, z2)))) MARK(sieve(head(filter(z0, z1)))) -> c17(A__SIEVE(a__head(a__filter(mark(z0), mark(z1)))), MARK(head(filter(z0, z1)))) MARK(sieve(head(s(z0)))) -> c17(A__SIEVE(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(sieve(head(cons(z0, z1)))) -> c17(A__SIEVE(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(sieve(head(divides(z0, z1)))) -> c17(A__SIEVE(a__head(divides(mark(z0), mark(z1)))), MARK(head(divides(z0, z1)))) MARK(sieve(head(x0))) -> c17(MARK(head(x0))) MARK(sieve(head(0))) -> c17(A__SIEVE(a__head(0))) MARK(sieve(head(true))) -> c17(A__SIEVE(a__head(true))) MARK(sieve(head(false))) -> c17(A__SIEVE(a__head(false))) MARK(sieve(tail(primes))) -> c17(A__SIEVE(a__tail(a__primes)), MARK(tail(primes))) MARK(sieve(tail(sieve(z0)))) -> c17(A__SIEVE(a__tail(a__sieve(mark(z0)))), MARK(tail(sieve(z0)))) MARK(sieve(tail(from(z0)))) -> c17(A__SIEVE(a__tail(a__from(mark(z0)))), MARK(tail(from(z0)))) MARK(sieve(tail(head(z0)))) -> c17(A__SIEVE(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(sieve(tail(tail(z0)))) -> c17(A__SIEVE(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(sieve(tail(if(z0, z1, z2)))) -> c17(A__SIEVE(a__tail(a__if(mark(z0), z1, z2))), MARK(tail(if(z0, z1, z2)))) MARK(sieve(tail(filter(z0, z1)))) -> c17(A__SIEVE(a__tail(a__filter(mark(z0), mark(z1)))), MARK(tail(filter(z0, z1)))) MARK(sieve(tail(s(z0)))) -> c17(A__SIEVE(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(sieve(tail(cons(z0, z1)))) -> c17(A__SIEVE(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(sieve(tail(divides(z0, z1)))) -> c17(A__SIEVE(a__tail(divides(mark(z0), mark(z1)))), MARK(tail(divides(z0, z1)))) MARK(sieve(tail(x0))) -> c17(MARK(tail(x0))) MARK(sieve(tail(0))) -> c17(A__SIEVE(a__tail(0))) MARK(sieve(tail(true))) -> c17(A__SIEVE(a__tail(true))) MARK(sieve(tail(false))) -> c17(A__SIEVE(a__tail(false))) MARK(sieve(if(x0, z1, z2))) -> c17(A__SIEVE(if(mark(x0), z1, z2)), MARK(if(x0, z1, z2))) MARK(sieve(if(primes, x1, x2))) -> c17(A__SIEVE(a__if(a__primes, x1, x2)), MARK(if(primes, x1, x2))) MARK(sieve(if(sieve(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__sieve(mark(z0)), x1, x2)), MARK(if(sieve(z0), x1, x2))) MARK(sieve(if(from(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__from(mark(z0)), x1, x2)), MARK(if(from(z0), x1, x2))) MARK(sieve(if(head(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__head(mark(z0)), x1, x2)), MARK(if(head(z0), x1, x2))) MARK(sieve(if(tail(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__tail(mark(z0)), x1, x2)), MARK(if(tail(z0), x1, x2))) MARK(sieve(if(if(z0, z1, z2), x1, x2))) -> c17(A__SIEVE(a__if(a__if(mark(z0), z1, z2), x1, x2)), MARK(if(if(z0, z1, z2), x1, x2))) MARK(sieve(if(filter(z0, z1), x1, x2))) -> c17(A__SIEVE(a__if(a__filter(mark(z0), mark(z1)), x1, x2)), MARK(if(filter(z0, z1), x1, x2))) MARK(sieve(if(s(z0), x1, x2))) -> c17(A__SIEVE(a__if(s(mark(z0)), x1, x2)), MARK(if(s(z0), x1, x2))) MARK(sieve(if(0, x1, x2))) -> c17(A__SIEVE(a__if(0, x1, x2)), MARK(if(0, x1, x2))) MARK(sieve(if(cons(z0, z1), x1, x2))) -> c17(A__SIEVE(a__if(cons(mark(z0), z1), x1, x2)), MARK(if(cons(z0, z1), x1, x2))) MARK(sieve(if(true, x1, x2))) -> c17(A__SIEVE(a__if(true, x1, x2)), MARK(if(true, x1, x2))) MARK(sieve(if(false, x1, x2))) -> c17(A__SIEVE(a__if(false, x1, x2)), MARK(if(false, x1, x2))) MARK(sieve(if(divides(z0, z1), x1, x2))) -> c17(A__SIEVE(a__if(divides(mark(z0), mark(z1)), x1, x2)), MARK(if(divides(z0, z1), x1, x2))) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c18_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_1, c17_2, c17_1, c19_2, c19_1, c20_2, c20_1, c21_2, c21_1, c22_2, c22_1, c23_2, c23_1, c_2 ---------------------------------------- (121) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (122) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(filter(z0, z1))) -> c17(A__SIEVE(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(primes)) -> c19(A__HEAD(a__primes), MARK(primes)) MARK(head(sieve(z0))) -> c19(A__HEAD(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(head(from(z0))) -> c19(A__HEAD(a__from(mark(z0))), MARK(from(z0))) MARK(head(head(z0))) -> c19(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c19(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(if(z0, z1, z2))) -> c19(A__HEAD(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(head(filter(z0, z1))) -> c19(A__HEAD(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(primes)) -> c20(A__TAIL(a__primes), MARK(primes)) MARK(tail(sieve(z0))) -> c20(A__TAIL(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(tail(from(z0))) -> c20(A__TAIL(a__from(mark(z0))), MARK(from(z0))) MARK(tail(head(z0))) -> c20(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c20(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(if(z0, z1, z2))) -> c20(A__TAIL(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(tail(filter(z0, z1))) -> c20(A__TAIL(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(primes, x1, x2)) -> c21(A__IF(a__primes, x1, x2), MARK(primes)) MARK(if(sieve(z0), x1, x2)) -> c21(A__IF(a__sieve(mark(z0)), x1, x2), MARK(sieve(z0))) MARK(if(from(z0), x1, x2)) -> c21(A__IF(a__from(mark(z0)), x1, x2), MARK(from(z0))) MARK(if(head(z0), x1, x2)) -> c21(A__IF(a__head(mark(z0)), x1, x2), MARK(head(z0))) MARK(if(tail(z0), x1, x2)) -> c21(A__IF(a__tail(mark(z0)), x1, x2), MARK(tail(z0))) MARK(if(if(z0, z1, z2), x1, x2)) -> c21(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(filter(z0, z1), x1, x2)) -> c21(A__IF(a__filter(mark(z0), mark(z1)), x1, x2), MARK(filter(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c22(A__FILTER(mark(x0), a__primes), MARK(x0)) MARK(filter(x0, sieve(z0))) -> c22(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, from(z0))) -> c22(A__FILTER(mark(x0), a__from(mark(z0))), MARK(x0)) MARK(filter(x0, head(z0))) -> c22(A__FILTER(mark(x0), a__head(mark(z0))), MARK(x0)) MARK(filter(x0, tail(z0))) -> c22(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(x0)) MARK(filter(x0, if(z0, z1, z2))) -> c22(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0)) MARK(filter(x0, filter(z0, z1))) -> c22(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(x0)) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(x0, s(z0))) -> c22(MARK(x0)) MARK(filter(x0, 0)) -> c22(MARK(x0)) MARK(filter(x0, true)) -> c22(MARK(x0)) MARK(filter(x0, false)) -> c22(MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), x1)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c23(A__FILTER(mark(x0), a__primes), MARK(primes)) MARK(filter(x0, sieve(z0))) -> c23(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, from(z0))) -> c23(A__FILTER(mark(x0), a__from(mark(z0))), MARK(from(z0))) MARK(filter(x0, head(z0))) -> c23(A__FILTER(mark(x0), a__head(mark(z0))), MARK(head(z0))) MARK(filter(x0, tail(z0))) -> c23(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(tail(z0))) MARK(filter(x0, if(z0, z1, z2))) -> c23(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(filter(x0, filter(z0, z1))) -> c23(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(filter(x0, cons(z0, z1))) -> c23(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(primes, x1)) -> c23(A__FILTER(a__primes, mark(x1)), MARK(x1)) MARK(filter(sieve(z0), x1)) -> c23(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(from(z0), x1)) -> c23(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(head(z0), x1)) -> c23(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(tail(z0), x1)) -> c23(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(if(z0, z1, z2), x1)) -> c23(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1)) 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c22(MARK(x0)) MARK(filter(x0, true)) -> c22(MARK(x0)) MARK(filter(x0, false)) -> c22(MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), x1)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c23(A__FILTER(mark(x0), a__primes), MARK(primes)) MARK(filter(x0, sieve(z0))) -> c23(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, from(z0))) -> c23(A__FILTER(mark(x0), a__from(mark(z0))), MARK(from(z0))) MARK(filter(x0, head(z0))) -> c23(A__FILTER(mark(x0), a__head(mark(z0))), MARK(head(z0))) MARK(filter(x0, tail(z0))) -> c23(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(tail(z0))) MARK(filter(x0, if(z0, z1, z2))) -> c23(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(filter(x0, filter(z0, z1))) -> c23(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(filter(x0, cons(z0, z1))) -> c23(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(primes, x1)) -> c23(A__FILTER(a__primes, mark(x1)), MARK(x1)) MARK(filter(sieve(z0), x1)) -> c23(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(from(z0), x1)) -> c23(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(head(z0), x1)) -> c23(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(tail(z0), x1)) -> c23(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(if(z0, z1, z2), x1)) -> c23(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1)) MARK(filter(filter(z0, z1), x1)) -> c23(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(x1)) MARK(filter(s(z0), x1)) -> c23(A__FILTER(s(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(x0, s(z0))) -> c23(MARK(s(z0))) MARK(filter(x0, divides(z0, z1))) -> c23(MARK(divides(z0, z1))) MARK(filter(0, x1)) -> c23(MARK(x1)) MARK(filter(cons(z0, z1), x1)) -> c23(MARK(x1)) MARK(filter(true, x1)) -> c23(MARK(x1)) MARK(filter(false, x1)) -> c23(MARK(x1)) MARK(filter(divides(z0, z1), x1)) -> c23(MARK(x1)) A__PRIMES -> c(A__SIEVE(cons(s(mark(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(sieve(primes)) -> c17(MARK(primes)) MARK(sieve(sieve(primes))) -> c17(A__SIEVE(a__sieve(a__primes)), MARK(sieve(primes))) MARK(sieve(sieve(sieve(z0)))) -> c17(A__SIEVE(a__sieve(a__sieve(mark(z0)))), MARK(sieve(sieve(z0)))) MARK(sieve(sieve(from(z0)))) -> c17(A__SIEVE(a__sieve(a__from(mark(z0)))), MARK(sieve(from(z0)))) MARK(sieve(sieve(head(z0)))) -> c17(A__SIEVE(a__sieve(a__head(mark(z0)))), MARK(sieve(head(z0)))) MARK(sieve(sieve(tail(z0)))) -> c17(A__SIEVE(a__sieve(a__tail(mark(z0)))), MARK(sieve(tail(z0)))) MARK(sieve(sieve(if(z0, z1, z2)))) -> c17(A__SIEVE(a__sieve(a__if(mark(z0), z1, z2))), MARK(sieve(if(z0, z1, z2)))) MARK(sieve(sieve(filter(z0, z1)))) -> c17(A__SIEVE(a__sieve(a__filter(mark(z0), mark(z1)))), MARK(sieve(filter(z0, z1)))) MARK(sieve(sieve(s(z0)))) -> c17(A__SIEVE(a__sieve(s(mark(z0)))), MARK(sieve(s(z0)))) MARK(sieve(sieve(cons(z0, z1)))) -> c17(A__SIEVE(a__sieve(cons(mark(z0), z1))), MARK(sieve(cons(z0, z1)))) MARK(sieve(sieve(divides(z0, z1)))) -> c17(A__SIEVE(a__sieve(divides(mark(z0), mark(z1)))), MARK(sieve(divides(z0, z1)))) MARK(sieve(sieve(x0))) -> c17(MARK(sieve(x0))) MARK(sieve(sieve(0))) -> c17(A__SIEVE(a__sieve(0))) MARK(sieve(sieve(true))) -> c17(A__SIEVE(a__sieve(true))) MARK(sieve(sieve(false))) -> c17(A__SIEVE(a__sieve(false))) MARK(sieve(from(x0))) -> c17(A__SIEVE(cons(mark(mark(x0)), from(s(mark(x0))))), MARK(from(x0))) MARK(sieve(from(primes))) -> c17(A__SIEVE(a__from(a__primes)), MARK(from(primes))) MARK(sieve(from(sieve(z0)))) -> c17(A__SIEVE(a__from(a__sieve(mark(z0)))), MARK(from(sieve(z0)))) MARK(sieve(from(from(z0)))) -> c17(A__SIEVE(a__from(a__from(mark(z0)))), MARK(from(from(z0)))) MARK(sieve(from(head(z0)))) -> c17(A__SIEVE(a__from(a__head(mark(z0)))), MARK(from(head(z0)))) MARK(sieve(from(tail(z0)))) -> c17(A__SIEVE(a__from(a__tail(mark(z0)))), MARK(from(tail(z0)))) MARK(sieve(from(if(z0, z1, z2)))) -> c17(A__SIEVE(a__from(a__if(mark(z0), z1, z2))), MARK(from(if(z0, z1, z2)))) MARK(sieve(from(filter(z0, z1)))) -> c17(A__SIEVE(a__from(a__filter(mark(z0), mark(z1)))), MARK(from(filter(z0, z1)))) MARK(sieve(from(s(z0)))) -> c17(A__SIEVE(a__from(s(mark(z0)))), MARK(from(s(z0)))) MARK(sieve(from(0))) -> c17(A__SIEVE(a__from(0)), MARK(from(0))) MARK(sieve(from(cons(z0, z1)))) -> c17(A__SIEVE(a__from(cons(mark(z0), z1))), MARK(from(cons(z0, z1)))) MARK(sieve(from(true))) -> c17(A__SIEVE(a__from(true)), MARK(from(true))) MARK(sieve(from(false))) -> c17(A__SIEVE(a__from(false)), MARK(from(false))) MARK(sieve(from(divides(z0, z1)))) -> c17(A__SIEVE(a__from(divides(mark(z0), mark(z1)))), MARK(from(divides(z0, z1)))) MARK(sieve(from(x0))) -> c17(MARK(from(x0))) MARK(sieve(head(primes))) -> c17(A__SIEVE(a__head(a__primes)), MARK(head(primes))) MARK(sieve(head(sieve(z0)))) -> c17(A__SIEVE(a__head(a__sieve(mark(z0)))), MARK(head(sieve(z0)))) MARK(sieve(head(from(z0)))) -> c17(A__SIEVE(a__head(a__from(mark(z0)))), MARK(head(from(z0)))) MARK(sieve(head(head(z0)))) -> c17(A__SIEVE(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(sieve(head(tail(z0)))) -> c17(A__SIEVE(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(sieve(head(if(z0, z1, z2)))) -> c17(A__SIEVE(a__head(a__if(mark(z0), z1, z2))), MARK(head(if(z0, z1, z2)))) MARK(sieve(head(filter(z0, z1)))) -> c17(A__SIEVE(a__head(a__filter(mark(z0), mark(z1)))), MARK(head(filter(z0, z1)))) MARK(sieve(head(s(z0)))) -> c17(A__SIEVE(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(sieve(head(cons(z0, z1)))) -> c17(A__SIEVE(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(sieve(head(divides(z0, z1)))) -> c17(A__SIEVE(a__head(divides(mark(z0), mark(z1)))), MARK(head(divides(z0, z1)))) MARK(sieve(head(x0))) -> c17(MARK(head(x0))) MARK(sieve(head(0))) -> c17(A__SIEVE(a__head(0))) MARK(sieve(head(true))) -> c17(A__SIEVE(a__head(true))) MARK(sieve(head(false))) -> c17(A__SIEVE(a__head(false))) MARK(sieve(tail(primes))) -> c17(A__SIEVE(a__tail(a__primes)), MARK(tail(primes))) MARK(sieve(tail(sieve(z0)))) -> c17(A__SIEVE(a__tail(a__sieve(mark(z0)))), MARK(tail(sieve(z0)))) MARK(sieve(tail(from(z0)))) -> c17(A__SIEVE(a__tail(a__from(mark(z0)))), MARK(tail(from(z0)))) MARK(sieve(tail(head(z0)))) -> c17(A__SIEVE(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(sieve(tail(tail(z0)))) -> c17(A__SIEVE(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(sieve(tail(if(z0, z1, z2)))) -> c17(A__SIEVE(a__tail(a__if(mark(z0), z1, z2))), MARK(tail(if(z0, z1, z2)))) MARK(sieve(tail(filter(z0, z1)))) -> c17(A__SIEVE(a__tail(a__filter(mark(z0), mark(z1)))), MARK(tail(filter(z0, z1)))) MARK(sieve(tail(s(z0)))) -> c17(A__SIEVE(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(sieve(tail(cons(z0, z1)))) -> c17(A__SIEVE(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(sieve(tail(divides(z0, z1)))) -> c17(A__SIEVE(a__tail(divides(mark(z0), mark(z1)))), MARK(tail(divides(z0, z1)))) MARK(sieve(tail(x0))) -> c17(MARK(tail(x0))) MARK(sieve(tail(0))) -> c17(A__SIEVE(a__tail(0))) MARK(sieve(tail(true))) -> c17(A__SIEVE(a__tail(true))) MARK(sieve(tail(false))) -> c17(A__SIEVE(a__tail(false))) MARK(sieve(if(primes, x1, x2))) -> c17(A__SIEVE(a__if(a__primes, x1, x2)), MARK(if(primes, x1, x2))) MARK(sieve(if(sieve(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__sieve(mark(z0)), x1, x2)), MARK(if(sieve(z0), x1, x2))) MARK(sieve(if(from(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__from(mark(z0)), x1, x2)), MARK(if(from(z0), x1, x2))) MARK(sieve(if(head(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__head(mark(z0)), x1, x2)), MARK(if(head(z0), x1, x2))) MARK(sieve(if(tail(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__tail(mark(z0)), x1, x2)), MARK(if(tail(z0), x1, x2))) MARK(sieve(if(if(z0, z1, z2), x1, x2))) -> c17(A__SIEVE(a__if(a__if(mark(z0), z1, z2), x1, x2)), MARK(if(if(z0, z1, z2), x1, x2))) MARK(sieve(if(filter(z0, z1), x1, x2))) -> c17(A__SIEVE(a__if(a__filter(mark(z0), mark(z1)), x1, x2)), MARK(if(filter(z0, z1), x1, x2))) MARK(sieve(if(s(z0), x1, x2))) -> c17(A__SIEVE(a__if(s(mark(z0)), x1, x2)), MARK(if(s(z0), x1, x2))) MARK(sieve(if(cons(z0, z1), x1, x2))) -> c17(A__SIEVE(a__if(cons(mark(z0), z1), x1, x2)), MARK(if(cons(z0, z1), x1, x2))) MARK(sieve(if(true, x1, x2))) -> c17(A__SIEVE(a__if(true, x1, x2)), MARK(if(true, x1, x2))) MARK(sieve(if(false, x1, x2))) -> c17(A__SIEVE(a__if(false, x1, x2)), MARK(if(false, x1, x2))) MARK(sieve(if(divides(z0, z1), x1, x2))) -> c17(A__SIEVE(a__if(divides(mark(z0), mark(z1)), x1, x2)), MARK(if(divides(z0, z1), x1, x2))) MARK(sieve(if(x0, z1, z2))) -> c17(MARK(if(x0, z1, z2))) MARK(sieve(if(0, x1, x2))) -> c17(A__SIEVE(a__if(0, x1, x2))) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c18_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_1, c17_2, c17_1, c19_2, c19_1, c20_2, c20_1, c21_2, c21_1, c22_2, c22_1, c23_2, c23_1, c_2 ---------------------------------------- (123) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(sieve(filter(z0, z1))) -> c17(A__SIEVE(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) by MARK(sieve(filter(x0, x1))) -> c17(A__SIEVE(filter(mark(x0), mark(x1))), MARK(filter(x0, x1))) MARK(sieve(filter(x0, primes))) -> c17(A__SIEVE(a__filter(mark(x0), a__primes)), MARK(filter(x0, primes))) MARK(sieve(filter(x0, sieve(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), a__sieve(mark(z0)))), MARK(filter(x0, sieve(z0)))) MARK(sieve(filter(x0, from(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), a__from(mark(z0)))), MARK(filter(x0, from(z0)))) MARK(sieve(filter(x0, head(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), a__head(mark(z0)))), MARK(filter(x0, head(z0)))) MARK(sieve(filter(x0, tail(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), a__tail(mark(z0)))), MARK(filter(x0, tail(z0)))) MARK(sieve(filter(x0, if(z0, z1, z2)))) -> c17(A__SIEVE(a__filter(mark(x0), a__if(mark(z0), z1, z2))), MARK(filter(x0, if(z0, z1, z2)))) MARK(sieve(filter(x0, filter(z0, z1)))) -> c17(A__SIEVE(a__filter(mark(x0), a__filter(mark(z0), mark(z1)))), MARK(filter(x0, filter(z0, z1)))) MARK(sieve(filter(x0, s(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), s(mark(z0)))), MARK(filter(x0, s(z0)))) MARK(sieve(filter(x0, 0))) -> c17(A__SIEVE(a__filter(mark(x0), 0)), MARK(filter(x0, 0))) MARK(sieve(filter(x0, cons(z0, z1)))) -> c17(A__SIEVE(a__filter(mark(x0), cons(mark(z0), z1))), MARK(filter(x0, cons(z0, z1)))) MARK(sieve(filter(x0, true))) -> c17(A__SIEVE(a__filter(mark(x0), true)), MARK(filter(x0, true))) MARK(sieve(filter(x0, false))) -> c17(A__SIEVE(a__filter(mark(x0), false)), MARK(filter(x0, false))) MARK(sieve(filter(x0, divides(z0, z1)))) -> c17(A__SIEVE(a__filter(mark(x0), divides(mark(z0), mark(z1)))), MARK(filter(x0, divides(z0, z1)))) MARK(sieve(filter(primes, x1))) -> c17(A__SIEVE(a__filter(a__primes, mark(x1))), MARK(filter(primes, x1))) MARK(sieve(filter(sieve(z0), x1))) -> c17(A__SIEVE(a__filter(a__sieve(mark(z0)), mark(x1))), MARK(filter(sieve(z0), x1))) MARK(sieve(filter(from(z0), x1))) -> c17(A__SIEVE(a__filter(a__from(mark(z0)), mark(x1))), MARK(filter(from(z0), x1))) MARK(sieve(filter(head(z0), x1))) -> c17(A__SIEVE(a__filter(a__head(mark(z0)), mark(x1))), MARK(filter(head(z0), x1))) MARK(sieve(filter(tail(z0), x1))) -> c17(A__SIEVE(a__filter(a__tail(mark(z0)), mark(x1))), MARK(filter(tail(z0), x1))) MARK(sieve(filter(if(z0, z1, z2), x1))) -> c17(A__SIEVE(a__filter(a__if(mark(z0), z1, z2), mark(x1))), MARK(filter(if(z0, z1, z2), x1))) MARK(sieve(filter(filter(z0, z1), x1))) -> c17(A__SIEVE(a__filter(a__filter(mark(z0), mark(z1)), mark(x1))), MARK(filter(filter(z0, z1), x1))) MARK(sieve(filter(s(z0), x1))) -> c17(A__SIEVE(a__filter(s(mark(z0)), mark(x1))), MARK(filter(s(z0), x1))) MARK(sieve(filter(0, x1))) -> c17(A__SIEVE(a__filter(0, mark(x1))), MARK(filter(0, x1))) MARK(sieve(filter(cons(z0, z1), x1))) -> c17(A__SIEVE(a__filter(cons(mark(z0), z1), mark(x1))), MARK(filter(cons(z0, z1), x1))) MARK(sieve(filter(true, x1))) -> c17(A__SIEVE(a__filter(true, mark(x1))), MARK(filter(true, x1))) MARK(sieve(filter(false, x1))) -> c17(A__SIEVE(a__filter(false, mark(x1))), MARK(filter(false, x1))) MARK(sieve(filter(divides(z0, z1), x1))) -> c17(A__SIEVE(a__filter(divides(mark(z0), mark(z1)), mark(x1))), MARK(filter(divides(z0, z1), x1))) ---------------------------------------- (124) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(primes)) -> c19(A__HEAD(a__primes), MARK(primes)) MARK(head(sieve(z0))) -> c19(A__HEAD(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(head(from(z0))) -> c19(A__HEAD(a__from(mark(z0))), MARK(from(z0))) MARK(head(head(z0))) -> c19(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c19(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(if(z0, z1, z2))) -> c19(A__HEAD(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(head(filter(z0, z1))) -> c19(A__HEAD(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(primes)) -> c20(A__TAIL(a__primes), MARK(primes)) MARK(tail(sieve(z0))) -> c20(A__TAIL(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(tail(from(z0))) -> c20(A__TAIL(a__from(mark(z0))), MARK(from(z0))) MARK(tail(head(z0))) -> c20(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c20(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(if(z0, z1, z2))) -> c20(A__TAIL(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(tail(filter(z0, z1))) -> c20(A__TAIL(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) 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MARK(sieve(tail(s(z0)))) -> c17(A__SIEVE(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(sieve(tail(cons(z0, z1)))) -> c17(A__SIEVE(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(sieve(tail(divides(z0, z1)))) -> c17(A__SIEVE(a__tail(divides(mark(z0), mark(z1)))), MARK(tail(divides(z0, z1)))) MARK(sieve(tail(x0))) -> c17(MARK(tail(x0))) MARK(sieve(tail(0))) -> c17(A__SIEVE(a__tail(0))) MARK(sieve(tail(true))) -> c17(A__SIEVE(a__tail(true))) MARK(sieve(tail(false))) -> c17(A__SIEVE(a__tail(false))) MARK(sieve(if(primes, x1, x2))) -> c17(A__SIEVE(a__if(a__primes, x1, x2)), MARK(if(primes, x1, x2))) MARK(sieve(if(sieve(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__sieve(mark(z0)), x1, x2)), MARK(if(sieve(z0), x1, x2))) MARK(sieve(if(from(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__from(mark(z0)), x1, x2)), MARK(if(from(z0), x1, x2))) MARK(sieve(if(head(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__head(mark(z0)), x1, x2)), MARK(if(head(z0), x1, x2))) MARK(sieve(if(tail(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__tail(mark(z0)), x1, x2)), MARK(if(tail(z0), x1, x2))) MARK(sieve(if(if(z0, z1, z2), x1, x2))) -> c17(A__SIEVE(a__if(a__if(mark(z0), z1, z2), x1, x2)), MARK(if(if(z0, z1, z2), x1, x2))) MARK(sieve(if(filter(z0, z1), x1, x2))) -> c17(A__SIEVE(a__if(a__filter(mark(z0), mark(z1)), x1, x2)), MARK(if(filter(z0, z1), x1, x2))) MARK(sieve(if(s(z0), x1, x2))) -> c17(A__SIEVE(a__if(s(mark(z0)), x1, x2)), MARK(if(s(z0), x1, x2))) MARK(sieve(if(cons(z0, z1), x1, x2))) -> c17(A__SIEVE(a__if(cons(mark(z0), z1), x1, x2)), MARK(if(cons(z0, z1), x1, x2))) MARK(sieve(if(true, x1, x2))) -> c17(A__SIEVE(a__if(true, x1, x2)), MARK(if(true, x1, x2))) MARK(sieve(if(false, x1, x2))) -> c17(A__SIEVE(a__if(false, x1, x2)), MARK(if(false, x1, x2))) MARK(sieve(if(divides(z0, z1), x1, x2))) -> c17(A__SIEVE(a__if(divides(mark(z0), mark(z1)), x1, x2)), MARK(if(divides(z0, z1), x1, x2))) MARK(sieve(if(x0, z1, z2))) -> c17(MARK(if(x0, z1, z2))) MARK(sieve(if(0, x1, x2))) -> c17(A__SIEVE(a__if(0, x1, x2))) MARK(sieve(filter(x0, x1))) -> c17(A__SIEVE(filter(mark(x0), mark(x1))), MARK(filter(x0, x1))) MARK(sieve(filter(x0, primes))) -> c17(A__SIEVE(a__filter(mark(x0), a__primes)), MARK(filter(x0, primes))) MARK(sieve(filter(x0, sieve(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), a__sieve(mark(z0)))), MARK(filter(x0, sieve(z0)))) MARK(sieve(filter(x0, from(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), a__from(mark(z0)))), MARK(filter(x0, from(z0)))) MARK(sieve(filter(x0, head(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), a__head(mark(z0)))), MARK(filter(x0, head(z0)))) MARK(sieve(filter(x0, tail(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), a__tail(mark(z0)))), MARK(filter(x0, tail(z0)))) MARK(sieve(filter(x0, if(z0, z1, z2)))) -> c17(A__SIEVE(a__filter(mark(x0), a__if(mark(z0), z1, z2))), MARK(filter(x0, if(z0, z1, z2)))) MARK(sieve(filter(x0, filter(z0, z1)))) -> c17(A__SIEVE(a__filter(mark(x0), a__filter(mark(z0), mark(z1)))), MARK(filter(x0, filter(z0, z1)))) MARK(sieve(filter(x0, s(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), s(mark(z0)))), MARK(filter(x0, s(z0)))) MARK(sieve(filter(x0, 0))) -> c17(A__SIEVE(a__filter(mark(x0), 0)), MARK(filter(x0, 0))) MARK(sieve(filter(x0, cons(z0, z1)))) -> c17(A__SIEVE(a__filter(mark(x0), cons(mark(z0), z1))), MARK(filter(x0, cons(z0, z1)))) MARK(sieve(filter(x0, true))) -> c17(A__SIEVE(a__filter(mark(x0), true)), MARK(filter(x0, true))) MARK(sieve(filter(x0, false))) -> c17(A__SIEVE(a__filter(mark(x0), false)), MARK(filter(x0, false))) MARK(sieve(filter(x0, divides(z0, z1)))) -> c17(A__SIEVE(a__filter(mark(x0), divides(mark(z0), mark(z1)))), MARK(filter(x0, divides(z0, z1)))) MARK(sieve(filter(primes, x1))) -> c17(A__SIEVE(a__filter(a__primes, mark(x1))), MARK(filter(primes, x1))) MARK(sieve(filter(sieve(z0), x1))) -> c17(A__SIEVE(a__filter(a__sieve(mark(z0)), mark(x1))), MARK(filter(sieve(z0), x1))) MARK(sieve(filter(from(z0), x1))) -> c17(A__SIEVE(a__filter(a__from(mark(z0)), mark(x1))), MARK(filter(from(z0), x1))) MARK(sieve(filter(head(z0), x1))) -> c17(A__SIEVE(a__filter(a__head(mark(z0)), mark(x1))), MARK(filter(head(z0), x1))) MARK(sieve(filter(tail(z0), x1))) -> c17(A__SIEVE(a__filter(a__tail(mark(z0)), mark(x1))), MARK(filter(tail(z0), x1))) MARK(sieve(filter(if(z0, z1, z2), x1))) -> c17(A__SIEVE(a__filter(a__if(mark(z0), z1, z2), mark(x1))), MARK(filter(if(z0, z1, z2), x1))) MARK(sieve(filter(filter(z0, z1), x1))) -> c17(A__SIEVE(a__filter(a__filter(mark(z0), mark(z1)), mark(x1))), MARK(filter(filter(z0, z1), x1))) MARK(sieve(filter(s(z0), x1))) -> c17(A__SIEVE(a__filter(s(mark(z0)), mark(x1))), MARK(filter(s(z0), x1))) MARK(sieve(filter(0, x1))) -> c17(A__SIEVE(a__filter(0, mark(x1))), MARK(filter(0, x1))) MARK(sieve(filter(cons(z0, z1), x1))) -> c17(A__SIEVE(a__filter(cons(mark(z0), z1), mark(x1))), MARK(filter(cons(z0, z1), x1))) MARK(sieve(filter(true, x1))) -> c17(A__SIEVE(a__filter(true, mark(x1))), MARK(filter(true, x1))) MARK(sieve(filter(false, x1))) -> c17(A__SIEVE(a__filter(false, mark(x1))), MARK(filter(false, x1))) MARK(sieve(filter(divides(z0, z1), x1))) -> c17(A__SIEVE(a__filter(divides(mark(z0), mark(z1)), mark(x1))), MARK(filter(divides(z0, z1), x1))) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c18_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_1, c17_2, c17_1, c19_2, c19_1, c20_2, c20_1, c21_2, c21_1, c22_2, c22_1, c23_2, c23_1, c_2 ---------------------------------------- (125) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (126) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(primes)) -> c19(A__HEAD(a__primes), MARK(primes)) MARK(head(sieve(z0))) -> c19(A__HEAD(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(head(from(z0))) -> c19(A__HEAD(a__from(mark(z0))), MARK(from(z0))) MARK(head(head(z0))) -> c19(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c19(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(if(z0, z1, z2))) -> c19(A__HEAD(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(head(filter(z0, z1))) -> c19(A__HEAD(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(primes)) -> c20(A__TAIL(a__primes), MARK(primes)) MARK(tail(sieve(z0))) -> c20(A__TAIL(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(tail(from(z0))) -> c20(A__TAIL(a__from(mark(z0))), MARK(from(z0))) MARK(tail(head(z0))) -> c20(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c20(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(if(z0, z1, z2))) -> c20(A__TAIL(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(tail(filter(z0, z1))) -> c20(A__TAIL(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(primes, x1, x2)) -> c21(A__IF(a__primes, x1, x2), MARK(primes)) MARK(if(sieve(z0), x1, x2)) -> c21(A__IF(a__sieve(mark(z0)), x1, x2), MARK(sieve(z0))) MARK(if(from(z0), x1, x2)) -> c21(A__IF(a__from(mark(z0)), x1, x2), MARK(from(z0))) MARK(if(head(z0), x1, x2)) -> c21(A__IF(a__head(mark(z0)), x1, x2), MARK(head(z0))) MARK(if(tail(z0), x1, x2)) -> c21(A__IF(a__tail(mark(z0)), x1, x2), MARK(tail(z0))) MARK(if(if(z0, z1, z2), x1, x2)) -> c21(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(filter(z0, z1), x1, x2)) -> c21(A__IF(a__filter(mark(z0), mark(z1)), x1, x2), MARK(filter(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c22(A__FILTER(mark(x0), a__primes), MARK(x0)) MARK(filter(x0, sieve(z0))) -> c22(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, from(z0))) -> c22(A__FILTER(mark(x0), a__from(mark(z0))), MARK(x0)) MARK(filter(x0, head(z0))) -> c22(A__FILTER(mark(x0), a__head(mark(z0))), MARK(x0)) MARK(filter(x0, tail(z0))) -> c22(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(x0)) MARK(filter(x0, if(z0, z1, z2))) -> c22(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0)) MARK(filter(x0, filter(z0, z1))) -> c22(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(x0)) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(x0, s(z0))) -> c22(MARK(x0)) MARK(filter(x0, 0)) -> c22(MARK(x0)) MARK(filter(x0, true)) -> c22(MARK(x0)) MARK(filter(x0, false)) -> c22(MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), x1)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c23(A__FILTER(mark(x0), a__primes), MARK(primes)) MARK(filter(x0, sieve(z0))) -> c23(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, from(z0))) -> c23(A__FILTER(mark(x0), a__from(mark(z0))), MARK(from(z0))) MARK(filter(x0, head(z0))) -> c23(A__FILTER(mark(x0), a__head(mark(z0))), MARK(head(z0))) MARK(filter(x0, tail(z0))) -> c23(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(tail(z0))) MARK(filter(x0, if(z0, z1, z2))) -> c23(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(filter(x0, filter(z0, z1))) -> c23(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(filter(x0, cons(z0, z1))) -> c23(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(primes, x1)) -> c23(A__FILTER(a__primes, mark(x1)), MARK(x1)) MARK(filter(sieve(z0), x1)) -> c23(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(from(z0), x1)) -> c23(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(head(z0), x1)) -> c23(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(tail(z0), x1)) -> c23(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(if(z0, z1, z2), x1)) -> c23(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1)) MARK(filter(filter(z0, z1), x1)) -> c23(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(x1)) MARK(filter(s(z0), x1)) -> c23(A__FILTER(s(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(x0, s(z0))) -> c23(MARK(s(z0))) MARK(filter(x0, divides(z0, z1))) -> c23(MARK(divides(z0, z1))) MARK(filter(0, x1)) -> c23(MARK(x1)) MARK(filter(cons(z0, z1), x1)) -> c23(MARK(x1)) MARK(filter(true, x1)) -> c23(MARK(x1)) MARK(filter(false, x1)) -> c23(MARK(x1)) MARK(filter(divides(z0, z1), x1)) -> c23(MARK(x1)) A__PRIMES -> c(A__SIEVE(cons(s(mark(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(sieve(primes)) -> c17(MARK(primes)) MARK(sieve(sieve(primes))) -> c17(A__SIEVE(a__sieve(a__primes)), MARK(sieve(primes))) MARK(sieve(sieve(sieve(z0)))) -> c17(A__SIEVE(a__sieve(a__sieve(mark(z0)))), MARK(sieve(sieve(z0)))) MARK(sieve(sieve(from(z0)))) -> c17(A__SIEVE(a__sieve(a__from(mark(z0)))), MARK(sieve(from(z0)))) MARK(sieve(sieve(head(z0)))) -> c17(A__SIEVE(a__sieve(a__head(mark(z0)))), MARK(sieve(head(z0)))) MARK(sieve(sieve(tail(z0)))) -> c17(A__SIEVE(a__sieve(a__tail(mark(z0)))), MARK(sieve(tail(z0)))) MARK(sieve(sieve(if(z0, z1, z2)))) -> c17(A__SIEVE(a__sieve(a__if(mark(z0), z1, z2))), MARK(sieve(if(z0, z1, z2)))) MARK(sieve(sieve(filter(z0, z1)))) -> c17(A__SIEVE(a__sieve(a__filter(mark(z0), mark(z1)))), MARK(sieve(filter(z0, z1)))) MARK(sieve(sieve(s(z0)))) -> c17(A__SIEVE(a__sieve(s(mark(z0)))), MARK(sieve(s(z0)))) MARK(sieve(sieve(cons(z0, z1)))) -> c17(A__SIEVE(a__sieve(cons(mark(z0), z1))), MARK(sieve(cons(z0, z1)))) MARK(sieve(sieve(divides(z0, z1)))) -> c17(A__SIEVE(a__sieve(divides(mark(z0), mark(z1)))), MARK(sieve(divides(z0, z1)))) MARK(sieve(sieve(x0))) -> c17(MARK(sieve(x0))) MARK(sieve(sieve(0))) -> c17(A__SIEVE(a__sieve(0))) MARK(sieve(sieve(true))) -> c17(A__SIEVE(a__sieve(true))) MARK(sieve(sieve(false))) -> c17(A__SIEVE(a__sieve(false))) MARK(sieve(from(x0))) -> c17(A__SIEVE(cons(mark(mark(x0)), from(s(mark(x0))))), MARK(from(x0))) MARK(sieve(from(primes))) -> c17(A__SIEVE(a__from(a__primes)), MARK(from(primes))) MARK(sieve(from(sieve(z0)))) -> c17(A__SIEVE(a__from(a__sieve(mark(z0)))), MARK(from(sieve(z0)))) MARK(sieve(from(from(z0)))) -> c17(A__SIEVE(a__from(a__from(mark(z0)))), MARK(from(from(z0)))) MARK(sieve(from(head(z0)))) -> c17(A__SIEVE(a__from(a__head(mark(z0)))), MARK(from(head(z0)))) MARK(sieve(from(tail(z0)))) -> c17(A__SIEVE(a__from(a__tail(mark(z0)))), MARK(from(tail(z0)))) MARK(sieve(from(if(z0, z1, z2)))) -> c17(A__SIEVE(a__from(a__if(mark(z0), z1, z2))), MARK(from(if(z0, z1, z2)))) MARK(sieve(from(filter(z0, z1)))) -> c17(A__SIEVE(a__from(a__filter(mark(z0), mark(z1)))), MARK(from(filter(z0, z1)))) MARK(sieve(from(s(z0)))) -> c17(A__SIEVE(a__from(s(mark(z0)))), MARK(from(s(z0)))) MARK(sieve(from(0))) -> 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MARK(from(z0))) MARK(if(head(z0), x1, x2)) -> c21(A__IF(a__head(mark(z0)), x1, x2), MARK(head(z0))) MARK(if(tail(z0), x1, x2)) -> c21(A__IF(a__tail(mark(z0)), x1, x2), MARK(tail(z0))) MARK(if(if(z0, z1, z2), x1, x2)) -> c21(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(filter(z0, z1), x1, x2)) -> c21(A__IF(a__filter(mark(z0), mark(z1)), x1, x2), MARK(filter(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c22(A__FILTER(mark(x0), a__primes), MARK(x0)) MARK(filter(x0, sieve(z0))) -> c22(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, from(z0))) -> c22(A__FILTER(mark(x0), a__from(mark(z0))), MARK(x0)) MARK(filter(x0, head(z0))) -> c22(A__FILTER(mark(x0), a__head(mark(z0))), MARK(x0)) MARK(filter(x0, tail(z0))) -> c22(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(x0)) MARK(filter(x0, if(z0, z1, z2))) -> c22(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0)) MARK(filter(x0, filter(z0, z1))) -> c22(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(x0)) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(x0, s(z0))) -> c22(MARK(x0)) MARK(filter(x0, 0)) -> c22(MARK(x0)) MARK(filter(x0, true)) -> c22(MARK(x0)) MARK(filter(x0, false)) -> c22(MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), x1)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c23(A__FILTER(mark(x0), a__primes), MARK(primes)) MARK(filter(x0, sieve(z0))) -> c23(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, from(z0))) -> c23(A__FILTER(mark(x0), a__from(mark(z0))), MARK(from(z0))) MARK(filter(x0, head(z0))) -> c23(A__FILTER(mark(x0), a__head(mark(z0))), MARK(head(z0))) MARK(filter(x0, tail(z0))) -> c23(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(tail(z0))) MARK(filter(x0, if(z0, z1, z2))) -> c23(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(filter(x0, filter(z0, z1))) -> c23(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(filter(x0, cons(z0, z1))) -> c23(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(primes, x1)) -> c23(A__FILTER(a__primes, mark(x1)), MARK(x1)) MARK(filter(sieve(z0), x1)) -> c23(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(from(z0), x1)) -> c23(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(head(z0), x1)) -> c23(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(tail(z0), x1)) -> c23(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(if(z0, z1, z2), x1)) -> c23(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1)) MARK(filter(filter(z0, z1), x1)) -> c23(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(x1)) MARK(filter(s(z0), x1)) -> c23(A__FILTER(s(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(x0, s(z0))) -> c23(MARK(s(z0))) MARK(filter(x0, divides(z0, z1))) -> c23(MARK(divides(z0, z1))) MARK(filter(0, x1)) -> c23(MARK(x1)) MARK(filter(cons(z0, z1), x1)) -> c23(MARK(x1)) MARK(filter(true, x1)) -> c23(MARK(x1)) MARK(filter(false, x1)) -> c23(MARK(x1)) MARK(filter(divides(z0, z1), x1)) -> c23(MARK(x1)) A__PRIMES -> c(A__SIEVE(cons(s(mark(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(sieve(primes)) -> c17(MARK(primes)) MARK(sieve(sieve(primes))) -> c17(A__SIEVE(a__sieve(a__primes)), MARK(sieve(primes))) MARK(sieve(sieve(sieve(z0)))) -> c17(A__SIEVE(a__sieve(a__sieve(mark(z0)))), MARK(sieve(sieve(z0)))) MARK(sieve(sieve(from(z0)))) -> c17(A__SIEVE(a__sieve(a__from(mark(z0)))), MARK(sieve(from(z0)))) MARK(sieve(sieve(head(z0)))) -> c17(A__SIEVE(a__sieve(a__head(mark(z0)))), MARK(sieve(head(z0)))) MARK(sieve(sieve(tail(z0)))) -> c17(A__SIEVE(a__sieve(a__tail(mark(z0)))), MARK(sieve(tail(z0)))) MARK(sieve(sieve(if(z0, z1, z2)))) -> c17(A__SIEVE(a__sieve(a__if(mark(z0), z1, z2))), MARK(sieve(if(z0, z1, z2)))) MARK(sieve(sieve(filter(z0, z1)))) -> c17(A__SIEVE(a__sieve(a__filter(mark(z0), mark(z1)))), MARK(sieve(filter(z0, z1)))) MARK(sieve(sieve(s(z0)))) -> c17(A__SIEVE(a__sieve(s(mark(z0)))), MARK(sieve(s(z0)))) MARK(sieve(sieve(cons(z0, z1)))) -> c17(A__SIEVE(a__sieve(cons(mark(z0), z1))), MARK(sieve(cons(z0, z1)))) MARK(sieve(sieve(divides(z0, z1)))) -> c17(A__SIEVE(a__sieve(divides(mark(z0), mark(z1)))), MARK(sieve(divides(z0, z1)))) MARK(sieve(sieve(x0))) -> c17(MARK(sieve(x0))) MARK(sieve(sieve(0))) -> c17(A__SIEVE(a__sieve(0))) MARK(sieve(sieve(true))) -> c17(A__SIEVE(a__sieve(true))) MARK(sieve(sieve(false))) -> c17(A__SIEVE(a__sieve(false))) MARK(sieve(from(x0))) -> c17(A__SIEVE(cons(mark(mark(x0)), from(s(mark(x0))))), MARK(from(x0))) MARK(sieve(from(primes))) -> c17(A__SIEVE(a__from(a__primes)), MARK(from(primes))) MARK(sieve(from(sieve(z0)))) -> c17(A__SIEVE(a__from(a__sieve(mark(z0)))), MARK(from(sieve(z0)))) MARK(sieve(from(from(z0)))) -> c17(A__SIEVE(a__from(a__from(mark(z0)))), MARK(from(from(z0)))) MARK(sieve(from(head(z0)))) -> c17(A__SIEVE(a__from(a__head(mark(z0)))), MARK(from(head(z0)))) MARK(sieve(from(tail(z0)))) -> c17(A__SIEVE(a__from(a__tail(mark(z0)))), MARK(from(tail(z0)))) MARK(sieve(from(if(z0, z1, z2)))) -> c17(A__SIEVE(a__from(a__if(mark(z0), z1, z2))), MARK(from(if(z0, z1, z2)))) MARK(sieve(from(filter(z0, z1)))) -> c17(A__SIEVE(a__from(a__filter(mark(z0), mark(z1)))), MARK(from(filter(z0, z1)))) MARK(sieve(from(s(z0)))) -> c17(A__SIEVE(a__from(s(mark(z0)))), MARK(from(s(z0)))) MARK(sieve(from(0))) -> c17(A__SIEVE(a__from(0)), MARK(from(0))) MARK(sieve(from(cons(z0, z1)))) -> c17(A__SIEVE(a__from(cons(mark(z0), z1))), MARK(from(cons(z0, z1)))) MARK(sieve(from(true))) -> c17(A__SIEVE(a__from(true)), MARK(from(true))) MARK(sieve(from(false))) -> c17(A__SIEVE(a__from(false)), MARK(from(false))) MARK(sieve(from(divides(z0, z1)))) -> c17(A__SIEVE(a__from(divides(mark(z0), mark(z1)))), MARK(from(divides(z0, z1)))) MARK(sieve(from(x0))) -> c17(MARK(from(x0))) MARK(sieve(head(primes))) -> c17(A__SIEVE(a__head(a__primes)), MARK(head(primes))) MARK(sieve(head(sieve(z0)))) -> c17(A__SIEVE(a__head(a__sieve(mark(z0)))), MARK(head(sieve(z0)))) MARK(sieve(head(from(z0)))) -> c17(A__SIEVE(a__head(a__from(mark(z0)))), MARK(head(from(z0)))) MARK(sieve(head(head(z0)))) -> c17(A__SIEVE(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(sieve(head(tail(z0)))) -> c17(A__SIEVE(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(sieve(head(if(z0, z1, z2)))) -> c17(A__SIEVE(a__head(a__if(mark(z0), z1, z2))), MARK(head(if(z0, z1, z2)))) MARK(sieve(head(filter(z0, z1)))) -> c17(A__SIEVE(a__head(a__filter(mark(z0), mark(z1)))), MARK(head(filter(z0, z1)))) MARK(sieve(head(s(z0)))) -> c17(A__SIEVE(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(sieve(head(cons(z0, z1)))) -> c17(A__SIEVE(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(sieve(head(divides(z0, z1)))) -> c17(A__SIEVE(a__head(divides(mark(z0), mark(z1)))), MARK(head(divides(z0, z1)))) MARK(sieve(head(x0))) -> c17(MARK(head(x0))) MARK(sieve(head(0))) -> c17(A__SIEVE(a__head(0))) MARK(sieve(head(true))) -> c17(A__SIEVE(a__head(true))) MARK(sieve(head(false))) -> c17(A__SIEVE(a__head(false))) MARK(sieve(tail(primes))) -> c17(A__SIEVE(a__tail(a__primes)), MARK(tail(primes))) MARK(sieve(tail(sieve(z0)))) -> c17(A__SIEVE(a__tail(a__sieve(mark(z0)))), MARK(tail(sieve(z0)))) MARK(sieve(tail(from(z0)))) -> c17(A__SIEVE(a__tail(a__from(mark(z0)))), MARK(tail(from(z0)))) MARK(sieve(tail(head(z0)))) -> c17(A__SIEVE(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(sieve(tail(tail(z0)))) -> c17(A__SIEVE(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(sieve(tail(if(z0, z1, z2)))) -> c17(A__SIEVE(a__tail(a__if(mark(z0), z1, z2))), MARK(tail(if(z0, z1, z2)))) MARK(sieve(tail(filter(z0, z1)))) -> c17(A__SIEVE(a__tail(a__filter(mark(z0), mark(z1)))), MARK(tail(filter(z0, z1)))) MARK(sieve(tail(s(z0)))) -> c17(A__SIEVE(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(sieve(tail(cons(z0, z1)))) -> c17(A__SIEVE(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(sieve(tail(divides(z0, z1)))) -> c17(A__SIEVE(a__tail(divides(mark(z0), mark(z1)))), MARK(tail(divides(z0, z1)))) MARK(sieve(tail(x0))) -> c17(MARK(tail(x0))) MARK(sieve(tail(0))) -> c17(A__SIEVE(a__tail(0))) MARK(sieve(tail(true))) -> c17(A__SIEVE(a__tail(true))) MARK(sieve(tail(false))) -> c17(A__SIEVE(a__tail(false))) MARK(sieve(if(primes, x1, x2))) -> c17(A__SIEVE(a__if(a__primes, x1, x2)), MARK(if(primes, x1, x2))) MARK(sieve(if(sieve(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__sieve(mark(z0)), x1, x2)), MARK(if(sieve(z0), x1, x2))) MARK(sieve(if(from(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__from(mark(z0)), x1, x2)), MARK(if(from(z0), x1, x2))) MARK(sieve(if(head(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__head(mark(z0)), x1, x2)), MARK(if(head(z0), x1, x2))) MARK(sieve(if(tail(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__tail(mark(z0)), x1, x2)), MARK(if(tail(z0), x1, x2))) MARK(sieve(if(if(z0, z1, z2), x1, x2))) -> c17(A__SIEVE(a__if(a__if(mark(z0), z1, z2), x1, x2)), MARK(if(if(z0, z1, z2), x1, x2))) MARK(sieve(if(filter(z0, z1), x1, x2))) -> c17(A__SIEVE(a__if(a__filter(mark(z0), mark(z1)), x1, x2)), MARK(if(filter(z0, z1), x1, x2))) MARK(sieve(if(s(z0), x1, x2))) -> c17(A__SIEVE(a__if(s(mark(z0)), x1, x2)), MARK(if(s(z0), x1, x2))) MARK(sieve(if(cons(z0, z1), x1, x2))) -> c17(A__SIEVE(a__if(cons(mark(z0), z1), x1, x2)), MARK(if(cons(z0, z1), x1, x2))) MARK(sieve(if(true, x1, x2))) -> c17(A__SIEVE(a__if(true, x1, x2)), MARK(if(true, x1, x2))) MARK(sieve(if(false, x1, x2))) -> c17(A__SIEVE(a__if(false, x1, x2)), MARK(if(false, x1, x2))) MARK(sieve(if(divides(z0, z1), x1, x2))) -> c17(A__SIEVE(a__if(divides(mark(z0), mark(z1)), x1, x2)), MARK(if(divides(z0, z1), x1, x2))) MARK(sieve(if(x0, z1, z2))) -> c17(MARK(if(x0, z1, z2))) MARK(sieve(if(0, x1, x2))) -> c17(A__SIEVE(a__if(0, x1, x2))) MARK(sieve(filter(x0, primes))) -> c17(A__SIEVE(a__filter(mark(x0), a__primes)), MARK(filter(x0, primes))) MARK(sieve(filter(x0, sieve(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), a__sieve(mark(z0)))), MARK(filter(x0, sieve(z0)))) MARK(sieve(filter(x0, from(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), a__from(mark(z0)))), MARK(filter(x0, from(z0)))) MARK(sieve(filter(x0, head(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), a__head(mark(z0)))), MARK(filter(x0, head(z0)))) MARK(sieve(filter(x0, tail(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), a__tail(mark(z0)))), MARK(filter(x0, tail(z0)))) MARK(sieve(filter(x0, if(z0, z1, z2)))) -> c17(A__SIEVE(a__filter(mark(x0), a__if(mark(z0), z1, z2))), MARK(filter(x0, if(z0, z1, z2)))) MARK(sieve(filter(x0, filter(z0, z1)))) -> c17(A__SIEVE(a__filter(mark(x0), a__filter(mark(z0), mark(z1)))), MARK(filter(x0, filter(z0, z1)))) MARK(sieve(filter(x0, s(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), s(mark(z0)))), MARK(filter(x0, s(z0)))) MARK(sieve(filter(x0, 0))) -> c17(A__SIEVE(a__filter(mark(x0), 0)), MARK(filter(x0, 0))) MARK(sieve(filter(x0, cons(z0, z1)))) -> c17(A__SIEVE(a__filter(mark(x0), cons(mark(z0), z1))), MARK(filter(x0, cons(z0, z1)))) MARK(sieve(filter(x0, true))) -> c17(A__SIEVE(a__filter(mark(x0), true)), MARK(filter(x0, true))) MARK(sieve(filter(x0, false))) -> c17(A__SIEVE(a__filter(mark(x0), false)), MARK(filter(x0, false))) MARK(sieve(filter(x0, divides(z0, z1)))) -> c17(A__SIEVE(a__filter(mark(x0), divides(mark(z0), mark(z1)))), MARK(filter(x0, divides(z0, z1)))) MARK(sieve(filter(primes, x1))) -> c17(A__SIEVE(a__filter(a__primes, mark(x1))), MARK(filter(primes, x1))) MARK(sieve(filter(sieve(z0), x1))) -> c17(A__SIEVE(a__filter(a__sieve(mark(z0)), mark(x1))), MARK(filter(sieve(z0), x1))) MARK(sieve(filter(from(z0), x1))) -> c17(A__SIEVE(a__filter(a__from(mark(z0)), mark(x1))), MARK(filter(from(z0), x1))) MARK(sieve(filter(head(z0), x1))) -> c17(A__SIEVE(a__filter(a__head(mark(z0)), mark(x1))), MARK(filter(head(z0), x1))) MARK(sieve(filter(tail(z0), x1))) -> c17(A__SIEVE(a__filter(a__tail(mark(z0)), mark(x1))), MARK(filter(tail(z0), x1))) MARK(sieve(filter(if(z0, z1, z2), x1))) -> c17(A__SIEVE(a__filter(a__if(mark(z0), z1, z2), mark(x1))), MARK(filter(if(z0, z1, z2), x1))) MARK(sieve(filter(filter(z0, z1), x1))) -> c17(A__SIEVE(a__filter(a__filter(mark(z0), mark(z1)), mark(x1))), MARK(filter(filter(z0, z1), x1))) MARK(sieve(filter(s(z0), x1))) -> c17(A__SIEVE(a__filter(s(mark(z0)), mark(x1))), MARK(filter(s(z0), x1))) MARK(sieve(filter(0, x1))) -> c17(A__SIEVE(a__filter(0, mark(x1))), MARK(filter(0, x1))) MARK(sieve(filter(cons(z0, z1), x1))) -> c17(A__SIEVE(a__filter(cons(mark(z0), z1), mark(x1))), MARK(filter(cons(z0, z1), x1))) MARK(sieve(filter(true, x1))) -> c17(A__SIEVE(a__filter(true, mark(x1))), MARK(filter(true, x1))) MARK(sieve(filter(false, x1))) -> c17(A__SIEVE(a__filter(false, mark(x1))), MARK(filter(false, x1))) MARK(sieve(filter(divides(z0, z1), x1))) -> c17(A__SIEVE(a__filter(divides(mark(z0), mark(z1)), mark(x1))), MARK(filter(divides(z0, z1), x1))) MARK(sieve(filter(x0, x1))) -> c17(MARK(filter(x0, x1))) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c18_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_1, c17_2, c17_1, c19_2, c19_1, c20_2, c20_1, c21_2, c21_1, c22_2, c22_1, c23_2, c23_1, c_2 ---------------------------------------- (127) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(head(primes)) -> c19(A__HEAD(a__primes), MARK(primes)) by MARK(head(primes)) -> c19(A__HEAD(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(head(primes)) -> c19(A__HEAD(primes), MARK(primes)) ---------------------------------------- (128) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(sieve(z0))) -> c19(A__HEAD(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(head(from(z0))) -> c19(A__HEAD(a__from(mark(z0))), MARK(from(z0))) MARK(head(head(z0))) -> c19(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c19(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(if(z0, z1, z2))) -> c19(A__HEAD(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(head(filter(z0, z1))) -> c19(A__HEAD(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(primes)) -> c20(A__TAIL(a__primes), MARK(primes)) MARK(tail(sieve(z0))) -> c20(A__TAIL(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(tail(from(z0))) -> c20(A__TAIL(a__from(mark(z0))), MARK(from(z0))) MARK(tail(head(z0))) -> c20(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c20(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(if(z0, z1, z2))) -> c20(A__TAIL(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(tail(filter(z0, z1))) -> c20(A__TAIL(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(primes, x1, x2)) -> c21(A__IF(a__primes, x1, x2), MARK(primes)) MARK(if(sieve(z0), x1, x2)) -> c21(A__IF(a__sieve(mark(z0)), x1, x2), MARK(sieve(z0))) MARK(if(from(z0), x1, x2)) -> c21(A__IF(a__from(mark(z0)), x1, x2), MARK(from(z0))) MARK(if(head(z0), x1, x2)) -> c21(A__IF(a__head(mark(z0)), x1, x2), MARK(head(z0))) MARK(if(tail(z0), x1, x2)) -> c21(A__IF(a__tail(mark(z0)), x1, x2), MARK(tail(z0))) MARK(if(if(z0, z1, z2), x1, x2)) -> c21(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(filter(z0, z1), x1, x2)) -> c21(A__IF(a__filter(mark(z0), mark(z1)), x1, x2), MARK(filter(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c22(A__FILTER(mark(x0), a__primes), MARK(x0)) MARK(filter(x0, sieve(z0))) -> c22(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, from(z0))) -> c22(A__FILTER(mark(x0), a__from(mark(z0))), MARK(x0)) MARK(filter(x0, head(z0))) -> c22(A__FILTER(mark(x0), a__head(mark(z0))), MARK(x0)) MARK(filter(x0, tail(z0))) -> c22(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(x0)) MARK(filter(x0, if(z0, z1, z2))) -> c22(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0)) MARK(filter(x0, filter(z0, z1))) -> c22(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(x0)) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(x0, s(z0))) -> c22(MARK(x0)) MARK(filter(x0, 0)) -> c22(MARK(x0)) MARK(filter(x0, true)) -> c22(MARK(x0)) MARK(filter(x0, false)) -> c22(MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), x1)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c23(A__FILTER(mark(x0), a__primes), MARK(primes)) MARK(filter(x0, sieve(z0))) -> c23(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, from(z0))) -> c23(A__FILTER(mark(x0), a__from(mark(z0))), MARK(from(z0))) MARK(filter(x0, head(z0))) -> c23(A__FILTER(mark(x0), a__head(mark(z0))), MARK(head(z0))) MARK(filter(x0, tail(z0))) -> c23(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(tail(z0))) MARK(filter(x0, if(z0, z1, z2))) -> c23(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(filter(x0, filter(z0, z1))) -> c23(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(filter(x0, cons(z0, z1))) -> c23(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(primes, x1)) -> c23(A__FILTER(a__primes, mark(x1)), MARK(x1)) MARK(filter(sieve(z0), x1)) -> c23(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(from(z0), x1)) -> c23(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(head(z0), x1)) -> c23(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(tail(z0), x1)) -> c23(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(if(z0, z1, z2), x1)) -> c23(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1)) MARK(filter(filter(z0, z1), x1)) -> c23(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(x1)) MARK(filter(s(z0), x1)) -> c23(A__FILTER(s(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(x0, s(z0))) -> c23(MARK(s(z0))) MARK(filter(x0, divides(z0, z1))) -> c23(MARK(divides(z0, z1))) MARK(filter(0, x1)) -> c23(MARK(x1)) MARK(filter(cons(z0, z1), x1)) -> c23(MARK(x1)) MARK(filter(true, x1)) -> c23(MARK(x1)) MARK(filter(false, x1)) -> c23(MARK(x1)) MARK(filter(divides(z0, z1), x1)) -> c23(MARK(x1)) A__PRIMES -> c(A__SIEVE(cons(s(mark(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(sieve(primes)) -> c17(MARK(primes)) MARK(sieve(sieve(primes))) -> c17(A__SIEVE(a__sieve(a__primes)), MARK(sieve(primes))) MARK(sieve(sieve(sieve(z0)))) -> c17(A__SIEVE(a__sieve(a__sieve(mark(z0)))), MARK(sieve(sieve(z0)))) MARK(sieve(sieve(from(z0)))) -> c17(A__SIEVE(a__sieve(a__from(mark(z0)))), MARK(sieve(from(z0)))) MARK(sieve(sieve(head(z0)))) -> c17(A__SIEVE(a__sieve(a__head(mark(z0)))), MARK(sieve(head(z0)))) MARK(sieve(sieve(tail(z0)))) -> c17(A__SIEVE(a__sieve(a__tail(mark(z0)))), MARK(sieve(tail(z0)))) MARK(sieve(sieve(if(z0, z1, z2)))) -> c17(A__SIEVE(a__sieve(a__if(mark(z0), z1, z2))), MARK(sieve(if(z0, z1, z2)))) MARK(sieve(sieve(filter(z0, z1)))) -> c17(A__SIEVE(a__sieve(a__filter(mark(z0), mark(z1)))), MARK(sieve(filter(z0, z1)))) MARK(sieve(sieve(s(z0)))) -> c17(A__SIEVE(a__sieve(s(mark(z0)))), MARK(sieve(s(z0)))) MARK(sieve(sieve(cons(z0, z1)))) -> c17(A__SIEVE(a__sieve(cons(mark(z0), z1))), MARK(sieve(cons(z0, z1)))) MARK(sieve(sieve(divides(z0, z1)))) -> c17(A__SIEVE(a__sieve(divides(mark(z0), mark(z1)))), MARK(sieve(divides(z0, z1)))) MARK(sieve(sieve(x0))) -> c17(MARK(sieve(x0))) MARK(sieve(sieve(0))) -> c17(A__SIEVE(a__sieve(0))) MARK(sieve(sieve(true))) -> c17(A__SIEVE(a__sieve(true))) MARK(sieve(sieve(false))) -> c17(A__SIEVE(a__sieve(false))) MARK(sieve(from(x0))) -> c17(A__SIEVE(cons(mark(mark(x0)), from(s(mark(x0))))), MARK(from(x0))) MARK(sieve(from(primes))) -> c17(A__SIEVE(a__from(a__primes)), MARK(from(primes))) MARK(sieve(from(sieve(z0)))) -> c17(A__SIEVE(a__from(a__sieve(mark(z0)))), MARK(from(sieve(z0)))) MARK(sieve(from(from(z0)))) -> c17(A__SIEVE(a__from(a__from(mark(z0)))), MARK(from(from(z0)))) MARK(sieve(from(head(z0)))) -> c17(A__SIEVE(a__from(a__head(mark(z0)))), MARK(from(head(z0)))) MARK(sieve(from(tail(z0)))) -> c17(A__SIEVE(a__from(a__tail(mark(z0)))), MARK(from(tail(z0)))) MARK(sieve(from(if(z0, z1, z2)))) -> c17(A__SIEVE(a__from(a__if(mark(z0), z1, z2))), MARK(from(if(z0, z1, z2)))) MARK(sieve(from(filter(z0, z1)))) -> c17(A__SIEVE(a__from(a__filter(mark(z0), mark(z1)))), MARK(from(filter(z0, z1)))) MARK(sieve(from(s(z0)))) -> c17(A__SIEVE(a__from(s(mark(z0)))), MARK(from(s(z0)))) MARK(sieve(from(0))) -> c17(A__SIEVE(a__from(0)), MARK(from(0))) MARK(sieve(from(cons(z0, z1)))) -> c17(A__SIEVE(a__from(cons(mark(z0), z1))), MARK(from(cons(z0, z1)))) MARK(sieve(from(true))) -> c17(A__SIEVE(a__from(true)), MARK(from(true))) MARK(sieve(from(false))) -> c17(A__SIEVE(a__from(false)), MARK(from(false))) MARK(sieve(from(divides(z0, z1)))) -> c17(A__SIEVE(a__from(divides(mark(z0), mark(z1)))), MARK(from(divides(z0, z1)))) MARK(sieve(from(x0))) -> c17(MARK(from(x0))) MARK(sieve(head(primes))) -> c17(A__SIEVE(a__head(a__primes)), MARK(head(primes))) MARK(sieve(head(sieve(z0)))) -> c17(A__SIEVE(a__head(a__sieve(mark(z0)))), MARK(head(sieve(z0)))) MARK(sieve(head(from(z0)))) -> c17(A__SIEVE(a__head(a__from(mark(z0)))), MARK(head(from(z0)))) MARK(sieve(head(head(z0)))) -> c17(A__SIEVE(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(sieve(head(tail(z0)))) -> c17(A__SIEVE(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(sieve(head(if(z0, z1, z2)))) -> c17(A__SIEVE(a__head(a__if(mark(z0), z1, z2))), MARK(head(if(z0, z1, z2)))) MARK(sieve(head(filter(z0, z1)))) -> c17(A__SIEVE(a__head(a__filter(mark(z0), mark(z1)))), MARK(head(filter(z0, z1)))) MARK(sieve(head(s(z0)))) -> c17(A__SIEVE(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(sieve(head(cons(z0, z1)))) -> c17(A__SIEVE(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(sieve(head(divides(z0, z1)))) -> c17(A__SIEVE(a__head(divides(mark(z0), mark(z1)))), MARK(head(divides(z0, z1)))) MARK(sieve(head(x0))) -> c17(MARK(head(x0))) MARK(sieve(head(0))) -> c17(A__SIEVE(a__head(0))) MARK(sieve(head(true))) -> c17(A__SIEVE(a__head(true))) MARK(sieve(head(false))) -> c17(A__SIEVE(a__head(false))) MARK(sieve(tail(primes))) -> c17(A__SIEVE(a__tail(a__primes)), MARK(tail(primes))) MARK(sieve(tail(sieve(z0)))) -> c17(A__SIEVE(a__tail(a__sieve(mark(z0)))), MARK(tail(sieve(z0)))) MARK(sieve(tail(from(z0)))) -> c17(A__SIEVE(a__tail(a__from(mark(z0)))), MARK(tail(from(z0)))) MARK(sieve(tail(head(z0)))) -> c17(A__SIEVE(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(sieve(tail(tail(z0)))) -> c17(A__SIEVE(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(sieve(tail(if(z0, z1, z2)))) -> c17(A__SIEVE(a__tail(a__if(mark(z0), z1, z2))), MARK(tail(if(z0, z1, z2)))) MARK(sieve(tail(filter(z0, z1)))) -> c17(A__SIEVE(a__tail(a__filter(mark(z0), mark(z1)))), MARK(tail(filter(z0, z1)))) MARK(sieve(tail(s(z0)))) -> c17(A__SIEVE(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(sieve(tail(cons(z0, z1)))) -> c17(A__SIEVE(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(sieve(tail(divides(z0, z1)))) -> c17(A__SIEVE(a__tail(divides(mark(z0), mark(z1)))), MARK(tail(divides(z0, z1)))) MARK(sieve(tail(x0))) -> c17(MARK(tail(x0))) MARK(sieve(tail(0))) -> c17(A__SIEVE(a__tail(0))) MARK(sieve(tail(true))) -> c17(A__SIEVE(a__tail(true))) MARK(sieve(tail(false))) -> c17(A__SIEVE(a__tail(false))) MARK(sieve(if(primes, x1, x2))) -> c17(A__SIEVE(a__if(a__primes, x1, x2)), MARK(if(primes, x1, x2))) MARK(sieve(if(sieve(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__sieve(mark(z0)), x1, x2)), MARK(if(sieve(z0), x1, x2))) MARK(sieve(if(from(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__from(mark(z0)), x1, x2)), MARK(if(from(z0), x1, x2))) MARK(sieve(if(head(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__head(mark(z0)), x1, x2)), MARK(if(head(z0), x1, x2))) MARK(sieve(if(tail(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__tail(mark(z0)), x1, x2)), MARK(if(tail(z0), x1, x2))) MARK(sieve(if(if(z0, z1, z2), x1, x2))) -> c17(A__SIEVE(a__if(a__if(mark(z0), z1, z2), x1, x2)), MARK(if(if(z0, z1, z2), x1, x2))) MARK(sieve(if(filter(z0, z1), x1, x2))) -> c17(A__SIEVE(a__if(a__filter(mark(z0), mark(z1)), x1, x2)), MARK(if(filter(z0, z1), x1, x2))) MARK(sieve(if(s(z0), x1, x2))) -> c17(A__SIEVE(a__if(s(mark(z0)), x1, x2)), MARK(if(s(z0), x1, x2))) MARK(sieve(if(cons(z0, z1), x1, x2))) -> c17(A__SIEVE(a__if(cons(mark(z0), z1), x1, x2)), MARK(if(cons(z0, z1), x1, x2))) MARK(sieve(if(true, x1, x2))) -> c17(A__SIEVE(a__if(true, x1, x2)), MARK(if(true, x1, x2))) MARK(sieve(if(false, x1, x2))) -> c17(A__SIEVE(a__if(false, x1, x2)), MARK(if(false, x1, x2))) MARK(sieve(if(divides(z0, z1), x1, x2))) -> c17(A__SIEVE(a__if(divides(mark(z0), mark(z1)), x1, x2)), MARK(if(divides(z0, z1), x1, x2))) MARK(sieve(if(x0, z1, z2))) -> c17(MARK(if(x0, z1, z2))) MARK(sieve(if(0, x1, x2))) -> c17(A__SIEVE(a__if(0, x1, x2))) MARK(sieve(filter(x0, primes))) -> c17(A__SIEVE(a__filter(mark(x0), a__primes)), MARK(filter(x0, primes))) MARK(sieve(filter(x0, sieve(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), a__sieve(mark(z0)))), MARK(filter(x0, sieve(z0)))) MARK(sieve(filter(x0, from(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), a__from(mark(z0)))), MARK(filter(x0, from(z0)))) MARK(sieve(filter(x0, head(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), a__head(mark(z0)))), MARK(filter(x0, head(z0)))) MARK(sieve(filter(x0, tail(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), a__tail(mark(z0)))), MARK(filter(x0, tail(z0)))) MARK(sieve(filter(x0, if(z0, z1, z2)))) -> c17(A__SIEVE(a__filter(mark(x0), a__if(mark(z0), z1, z2))), MARK(filter(x0, if(z0, z1, z2)))) MARK(sieve(filter(x0, filter(z0, z1)))) -> c17(A__SIEVE(a__filter(mark(x0), a__filter(mark(z0), mark(z1)))), MARK(filter(x0, filter(z0, z1)))) MARK(sieve(filter(x0, s(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), s(mark(z0)))), MARK(filter(x0, s(z0)))) MARK(sieve(filter(x0, 0))) -> c17(A__SIEVE(a__filter(mark(x0), 0)), MARK(filter(x0, 0))) MARK(sieve(filter(x0, cons(z0, z1)))) -> c17(A__SIEVE(a__filter(mark(x0), cons(mark(z0), z1))), MARK(filter(x0, cons(z0, z1)))) MARK(sieve(filter(x0, true))) -> c17(A__SIEVE(a__filter(mark(x0), true)), MARK(filter(x0, true))) MARK(sieve(filter(x0, false))) -> c17(A__SIEVE(a__filter(mark(x0), false)), MARK(filter(x0, false))) MARK(sieve(filter(x0, divides(z0, z1)))) -> c17(A__SIEVE(a__filter(mark(x0), divides(mark(z0), mark(z1)))), MARK(filter(x0, divides(z0, z1)))) MARK(sieve(filter(primes, x1))) -> c17(A__SIEVE(a__filter(a__primes, mark(x1))), MARK(filter(primes, x1))) MARK(sieve(filter(sieve(z0), x1))) -> c17(A__SIEVE(a__filter(a__sieve(mark(z0)), mark(x1))), MARK(filter(sieve(z0), x1))) MARK(sieve(filter(from(z0), x1))) -> c17(A__SIEVE(a__filter(a__from(mark(z0)), mark(x1))), MARK(filter(from(z0), x1))) MARK(sieve(filter(head(z0), x1))) -> c17(A__SIEVE(a__filter(a__head(mark(z0)), mark(x1))), MARK(filter(head(z0), x1))) MARK(sieve(filter(tail(z0), x1))) -> c17(A__SIEVE(a__filter(a__tail(mark(z0)), mark(x1))), MARK(filter(tail(z0), x1))) MARK(sieve(filter(if(z0, z1, z2), x1))) -> c17(A__SIEVE(a__filter(a__if(mark(z0), z1, z2), mark(x1))), MARK(filter(if(z0, z1, z2), x1))) MARK(sieve(filter(filter(z0, z1), x1))) -> c17(A__SIEVE(a__filter(a__filter(mark(z0), mark(z1)), mark(x1))), MARK(filter(filter(z0, z1), x1))) MARK(sieve(filter(s(z0), x1))) -> c17(A__SIEVE(a__filter(s(mark(z0)), mark(x1))), 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z1, z2)))) MARK(sieve(tail(filter(z0, z1)))) -> c17(A__SIEVE(a__tail(a__filter(mark(z0), mark(z1)))), MARK(tail(filter(z0, z1)))) MARK(sieve(tail(s(z0)))) -> c17(A__SIEVE(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(sieve(tail(cons(z0, z1)))) -> c17(A__SIEVE(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(sieve(tail(divides(z0, z1)))) -> c17(A__SIEVE(a__tail(divides(mark(z0), mark(z1)))), MARK(tail(divides(z0, z1)))) MARK(sieve(tail(x0))) -> c17(MARK(tail(x0))) MARK(sieve(tail(0))) -> c17(A__SIEVE(a__tail(0))) MARK(sieve(tail(true))) -> c17(A__SIEVE(a__tail(true))) MARK(sieve(tail(false))) -> c17(A__SIEVE(a__tail(false))) MARK(sieve(if(primes, x1, x2))) -> c17(A__SIEVE(a__if(a__primes, x1, x2)), MARK(if(primes, x1, x2))) MARK(sieve(if(sieve(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__sieve(mark(z0)), x1, x2)), MARK(if(sieve(z0), x1, x2))) MARK(sieve(if(from(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__from(mark(z0)), x1, x2)), MARK(if(from(z0), x1, x2))) MARK(sieve(if(head(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__head(mark(z0)), x1, x2)), MARK(if(head(z0), x1, x2))) MARK(sieve(if(tail(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__tail(mark(z0)), x1, x2)), MARK(if(tail(z0), x1, x2))) MARK(sieve(if(if(z0, z1, z2), x1, x2))) -> c17(A__SIEVE(a__if(a__if(mark(z0), z1, z2), x1, x2)), MARK(if(if(z0, z1, z2), x1, x2))) MARK(sieve(if(filter(z0, z1), x1, x2))) -> c17(A__SIEVE(a__if(a__filter(mark(z0), mark(z1)), x1, x2)), MARK(if(filter(z0, z1), x1, x2))) MARK(sieve(if(s(z0), x1, x2))) -> c17(A__SIEVE(a__if(s(mark(z0)), x1, x2)), MARK(if(s(z0), x1, x2))) MARK(sieve(if(cons(z0, z1), x1, x2))) -> c17(A__SIEVE(a__if(cons(mark(z0), z1), x1, x2)), MARK(if(cons(z0, z1), x1, x2))) MARK(sieve(if(true, x1, x2))) -> c17(A__SIEVE(a__if(true, x1, x2)), MARK(if(true, x1, x2))) MARK(sieve(if(false, x1, x2))) -> c17(A__SIEVE(a__if(false, x1, x2)), MARK(if(false, x1, x2))) MARK(sieve(if(divides(z0, z1), x1, x2))) -> c17(A__SIEVE(a__if(divides(mark(z0), mark(z1)), x1, x2)), MARK(if(divides(z0, z1), x1, x2))) MARK(sieve(if(x0, z1, z2))) -> c17(MARK(if(x0, z1, z2))) MARK(sieve(if(0, x1, x2))) -> c17(A__SIEVE(a__if(0, x1, x2))) MARK(sieve(filter(x0, primes))) -> c17(A__SIEVE(a__filter(mark(x0), a__primes)), MARK(filter(x0, primes))) MARK(sieve(filter(x0, sieve(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), a__sieve(mark(z0)))), MARK(filter(x0, sieve(z0)))) MARK(sieve(filter(x0, from(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), a__from(mark(z0)))), MARK(filter(x0, from(z0)))) MARK(sieve(filter(x0, head(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), a__head(mark(z0)))), MARK(filter(x0, head(z0)))) MARK(sieve(filter(x0, tail(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), a__tail(mark(z0)))), MARK(filter(x0, tail(z0)))) MARK(sieve(filter(x0, if(z0, z1, z2)))) -> c17(A__SIEVE(a__filter(mark(x0), a__if(mark(z0), z1, z2))), MARK(filter(x0, if(z0, z1, z2)))) MARK(sieve(filter(x0, filter(z0, z1)))) -> c17(A__SIEVE(a__filter(mark(x0), a__filter(mark(z0), mark(z1)))), MARK(filter(x0, filter(z0, z1)))) MARK(sieve(filter(x0, s(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), s(mark(z0)))), MARK(filter(x0, s(z0)))) MARK(sieve(filter(x0, 0))) -> c17(A__SIEVE(a__filter(mark(x0), 0)), MARK(filter(x0, 0))) MARK(sieve(filter(x0, cons(z0, z1)))) -> c17(A__SIEVE(a__filter(mark(x0), cons(mark(z0), z1))), MARK(filter(x0, cons(z0, z1)))) MARK(sieve(filter(x0, true))) -> c17(A__SIEVE(a__filter(mark(x0), true)), MARK(filter(x0, true))) MARK(sieve(filter(x0, false))) -> c17(A__SIEVE(a__filter(mark(x0), false)), MARK(filter(x0, false))) MARK(sieve(filter(x0, divides(z0, z1)))) -> c17(A__SIEVE(a__filter(mark(x0), divides(mark(z0), mark(z1)))), MARK(filter(x0, divides(z0, z1)))) MARK(sieve(filter(primes, x1))) -> c17(A__SIEVE(a__filter(a__primes, mark(x1))), MARK(filter(primes, x1))) MARK(sieve(filter(sieve(z0), x1))) -> c17(A__SIEVE(a__filter(a__sieve(mark(z0)), mark(x1))), MARK(filter(sieve(z0), x1))) MARK(sieve(filter(from(z0), x1))) -> c17(A__SIEVE(a__filter(a__from(mark(z0)), mark(x1))), MARK(filter(from(z0), x1))) MARK(sieve(filter(head(z0), x1))) -> c17(A__SIEVE(a__filter(a__head(mark(z0)), mark(x1))), MARK(filter(head(z0), x1))) MARK(sieve(filter(tail(z0), x1))) -> c17(A__SIEVE(a__filter(a__tail(mark(z0)), mark(x1))), MARK(filter(tail(z0), x1))) MARK(sieve(filter(if(z0, z1, z2), x1))) -> c17(A__SIEVE(a__filter(a__if(mark(z0), z1, z2), mark(x1))), MARK(filter(if(z0, z1, z2), x1))) MARK(sieve(filter(filter(z0, z1), x1))) -> c17(A__SIEVE(a__filter(a__filter(mark(z0), mark(z1)), mark(x1))), MARK(filter(filter(z0, z1), x1))) MARK(sieve(filter(s(z0), x1))) -> c17(A__SIEVE(a__filter(s(mark(z0)), mark(x1))), MARK(filter(s(z0), x1))) MARK(sieve(filter(0, x1))) -> c17(A__SIEVE(a__filter(0, mark(x1))), MARK(filter(0, x1))) MARK(sieve(filter(cons(z0, z1), x1))) -> c17(A__SIEVE(a__filter(cons(mark(z0), z1), mark(x1))), MARK(filter(cons(z0, z1), x1))) MARK(sieve(filter(true, x1))) -> c17(A__SIEVE(a__filter(true, mark(x1))), MARK(filter(true, x1))) MARK(sieve(filter(false, x1))) -> c17(A__SIEVE(a__filter(false, mark(x1))), MARK(filter(false, x1))) MARK(sieve(filter(divides(z0, z1), x1))) -> c17(A__SIEVE(a__filter(divides(mark(z0), mark(z1)), mark(x1))), MARK(filter(divides(z0, z1), x1))) MARK(sieve(filter(x0, x1))) -> c17(MARK(filter(x0, x1))) MARK(head(primes)) -> c19(A__HEAD(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(head(primes)) -> c19(A__HEAD(primes), MARK(primes)) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c18_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_1, c17_2, c17_1, c19_2, c19_1, c20_2, c20_1, c21_2, c21_1, c22_2, c22_1, c23_2, c23_1, c_2 ---------------------------------------- (129) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (130) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(sieve(z0))) -> c19(A__HEAD(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(head(from(z0))) -> c19(A__HEAD(a__from(mark(z0))), MARK(from(z0))) MARK(head(head(z0))) -> c19(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c19(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(if(z0, z1, z2))) -> c19(A__HEAD(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(head(filter(z0, z1))) -> c19(A__HEAD(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(primes)) -> c20(A__TAIL(a__primes), MARK(primes)) MARK(tail(sieve(z0))) -> c20(A__TAIL(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(tail(from(z0))) -> c20(A__TAIL(a__from(mark(z0))), MARK(from(z0))) MARK(tail(head(z0))) -> c20(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c20(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(if(z0, z1, z2))) -> c20(A__TAIL(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(tail(filter(z0, z1))) -> c20(A__TAIL(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(primes, x1, x2)) -> c21(A__IF(a__primes, x1, x2), MARK(primes)) MARK(if(sieve(z0), x1, x2)) -> c21(A__IF(a__sieve(mark(z0)), x1, x2), MARK(sieve(z0))) MARK(if(from(z0), x1, x2)) -> c21(A__IF(a__from(mark(z0)), x1, x2), MARK(from(z0))) MARK(if(head(z0), x1, x2)) -> c21(A__IF(a__head(mark(z0)), x1, x2), MARK(head(z0))) MARK(if(tail(z0), x1, x2)) -> c21(A__IF(a__tail(mark(z0)), x1, x2), MARK(tail(z0))) MARK(if(if(z0, z1, z2), x1, x2)) -> c21(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(filter(z0, z1), x1, x2)) -> c21(A__IF(a__filter(mark(z0), mark(z1)), x1, x2), MARK(filter(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c22(A__FILTER(mark(x0), a__primes), MARK(x0)) MARK(filter(x0, sieve(z0))) -> c22(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, from(z0))) -> c22(A__FILTER(mark(x0), a__from(mark(z0))), MARK(x0)) MARK(filter(x0, head(z0))) -> c22(A__FILTER(mark(x0), a__head(mark(z0))), MARK(x0)) MARK(filter(x0, tail(z0))) -> c22(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(x0)) MARK(filter(x0, if(z0, z1, z2))) -> c22(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0)) MARK(filter(x0, filter(z0, z1))) -> c22(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(x0)) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(x0, s(z0))) -> c22(MARK(x0)) MARK(filter(x0, 0)) -> c22(MARK(x0)) MARK(filter(x0, true)) -> c22(MARK(x0)) MARK(filter(x0, false)) -> c22(MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), x1)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c23(A__FILTER(mark(x0), a__primes), MARK(primes)) MARK(filter(x0, sieve(z0))) -> c23(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, from(z0))) -> c23(A__FILTER(mark(x0), a__from(mark(z0))), MARK(from(z0))) MARK(filter(x0, head(z0))) -> c23(A__FILTER(mark(x0), a__head(mark(z0))), MARK(head(z0))) MARK(filter(x0, tail(z0))) -> c23(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(tail(z0))) MARK(filter(x0, if(z0, z1, z2))) -> c23(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(filter(x0, filter(z0, z1))) -> c23(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(filter(x0, cons(z0, z1))) -> c23(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(primes, x1)) -> c23(A__FILTER(a__primes, mark(x1)), MARK(x1)) MARK(filter(sieve(z0), x1)) -> c23(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(from(z0), x1)) -> c23(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(head(z0), x1)) -> c23(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(tail(z0), x1)) -> c23(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(if(z0, z1, z2), x1)) -> c23(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1)) MARK(filter(filter(z0, z1), x1)) -> c23(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(x1)) MARK(filter(s(z0), x1)) -> c23(A__FILTER(s(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(x0, s(z0))) -> c23(MARK(s(z0))) MARK(filter(x0, divides(z0, z1))) -> c23(MARK(divides(z0, z1))) MARK(filter(0, x1)) -> c23(MARK(x1)) MARK(filter(cons(z0, z1), x1)) -> c23(MARK(x1)) MARK(filter(true, x1)) -> c23(MARK(x1)) MARK(filter(false, x1)) -> c23(MARK(x1)) MARK(filter(divides(z0, z1), x1)) -> c23(MARK(x1)) A__PRIMES -> c(A__SIEVE(cons(s(mark(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(sieve(primes)) -> c17(MARK(primes)) MARK(sieve(sieve(primes))) -> c17(A__SIEVE(a__sieve(a__primes)), MARK(sieve(primes))) MARK(sieve(sieve(sieve(z0)))) -> c17(A__SIEVE(a__sieve(a__sieve(mark(z0)))), MARK(sieve(sieve(z0)))) MARK(sieve(sieve(from(z0)))) -> c17(A__SIEVE(a__sieve(a__from(mark(z0)))), MARK(sieve(from(z0)))) MARK(sieve(sieve(head(z0)))) -> c17(A__SIEVE(a__sieve(a__head(mark(z0)))), MARK(sieve(head(z0)))) MARK(sieve(sieve(tail(z0)))) -> c17(A__SIEVE(a__sieve(a__tail(mark(z0)))), MARK(sieve(tail(z0)))) MARK(sieve(sieve(if(z0, z1, z2)))) -> c17(A__SIEVE(a__sieve(a__if(mark(z0), z1, z2))), MARK(sieve(if(z0, z1, z2)))) MARK(sieve(sieve(filter(z0, z1)))) -> c17(A__SIEVE(a__sieve(a__filter(mark(z0), mark(z1)))), MARK(sieve(filter(z0, z1)))) MARK(sieve(sieve(s(z0)))) -> c17(A__SIEVE(a__sieve(s(mark(z0)))), MARK(sieve(s(z0)))) MARK(sieve(sieve(cons(z0, z1)))) -> c17(A__SIEVE(a__sieve(cons(mark(z0), z1))), MARK(sieve(cons(z0, z1)))) MARK(sieve(sieve(divides(z0, z1)))) -> c17(A__SIEVE(a__sieve(divides(mark(z0), mark(z1)))), MARK(sieve(divides(z0, z1)))) MARK(sieve(sieve(x0))) -> c17(MARK(sieve(x0))) MARK(sieve(sieve(0))) -> c17(A__SIEVE(a__sieve(0))) MARK(sieve(sieve(true))) -> c17(A__SIEVE(a__sieve(true))) MARK(sieve(sieve(false))) -> c17(A__SIEVE(a__sieve(false))) MARK(sieve(from(x0))) -> c17(A__SIEVE(cons(mark(mark(x0)), from(s(mark(x0))))), MARK(from(x0))) MARK(sieve(from(primes))) -> c17(A__SIEVE(a__from(a__primes)), MARK(from(primes))) MARK(sieve(from(sieve(z0)))) -> c17(A__SIEVE(a__from(a__sieve(mark(z0)))), MARK(from(sieve(z0)))) MARK(sieve(from(from(z0)))) -> c17(A__SIEVE(a__from(a__from(mark(z0)))), MARK(from(from(z0)))) MARK(sieve(from(head(z0)))) -> c17(A__SIEVE(a__from(a__head(mark(z0)))), MARK(from(head(z0)))) MARK(sieve(from(tail(z0)))) -> c17(A__SIEVE(a__from(a__tail(mark(z0)))), MARK(from(tail(z0)))) MARK(sieve(from(if(z0, z1, z2)))) -> c17(A__SIEVE(a__from(a__if(mark(z0), z1, z2))), MARK(from(if(z0, z1, z2)))) MARK(sieve(from(filter(z0, z1)))) -> c17(A__SIEVE(a__from(a__filter(mark(z0), mark(z1)))), MARK(from(filter(z0, z1)))) MARK(sieve(from(s(z0)))) -> c17(A__SIEVE(a__from(s(mark(z0)))), MARK(from(s(z0)))) MARK(sieve(from(0))) -> c17(A__SIEVE(a__from(0)), MARK(from(0))) MARK(sieve(from(cons(z0, z1)))) -> c17(A__SIEVE(a__from(cons(mark(z0), z1))), MARK(from(cons(z0, z1)))) MARK(sieve(from(true))) -> c17(A__SIEVE(a__from(true)), MARK(from(true))) MARK(sieve(from(false))) -> c17(A__SIEVE(a__from(false)), MARK(from(false))) MARK(sieve(from(divides(z0, z1)))) -> c17(A__SIEVE(a__from(divides(mark(z0), mark(z1)))), MARK(from(divides(z0, z1)))) MARK(sieve(from(x0))) -> c17(MARK(from(x0))) MARK(sieve(head(primes))) -> c17(A__SIEVE(a__head(a__primes)), MARK(head(primes))) MARK(sieve(head(sieve(z0)))) -> c17(A__SIEVE(a__head(a__sieve(mark(z0)))), MARK(head(sieve(z0)))) MARK(sieve(head(from(z0)))) -> c17(A__SIEVE(a__head(a__from(mark(z0)))), MARK(head(from(z0)))) MARK(sieve(head(head(z0)))) -> c17(A__SIEVE(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(sieve(head(tail(z0)))) -> c17(A__SIEVE(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(sieve(head(if(z0, z1, z2)))) -> c17(A__SIEVE(a__head(a__if(mark(z0), z1, z2))), MARK(head(if(z0, z1, z2)))) MARK(sieve(head(filter(z0, z1)))) -> c17(A__SIEVE(a__head(a__filter(mark(z0), mark(z1)))), MARK(head(filter(z0, z1)))) MARK(sieve(head(s(z0)))) -> c17(A__SIEVE(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(sieve(head(cons(z0, z1)))) -> c17(A__SIEVE(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(sieve(head(divides(z0, z1)))) -> c17(A__SIEVE(a__head(divides(mark(z0), mark(z1)))), MARK(head(divides(z0, z1)))) MARK(sieve(head(x0))) -> c17(MARK(head(x0))) MARK(sieve(head(0))) -> c17(A__SIEVE(a__head(0))) MARK(sieve(head(true))) -> c17(A__SIEVE(a__head(true))) MARK(sieve(head(false))) -> c17(A__SIEVE(a__head(false))) MARK(sieve(tail(primes))) -> c17(A__SIEVE(a__tail(a__primes)), MARK(tail(primes))) MARK(sieve(tail(sieve(z0)))) -> c17(A__SIEVE(a__tail(a__sieve(mark(z0)))), MARK(tail(sieve(z0)))) MARK(sieve(tail(from(z0)))) -> c17(A__SIEVE(a__tail(a__from(mark(z0)))), MARK(tail(from(z0)))) MARK(sieve(tail(head(z0)))) -> c17(A__SIEVE(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(sieve(tail(tail(z0)))) -> c17(A__SIEVE(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(sieve(tail(if(z0, z1, z2)))) -> c17(A__SIEVE(a__tail(a__if(mark(z0), z1, z2))), MARK(tail(if(z0, z1, z2)))) MARK(sieve(tail(filter(z0, z1)))) -> c17(A__SIEVE(a__tail(a__filter(mark(z0), mark(z1)))), MARK(tail(filter(z0, z1)))) MARK(sieve(tail(s(z0)))) -> c17(A__SIEVE(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(sieve(tail(cons(z0, z1)))) -> c17(A__SIEVE(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(sieve(tail(divides(z0, z1)))) -> c17(A__SIEVE(a__tail(divides(mark(z0), mark(z1)))), MARK(tail(divides(z0, z1)))) MARK(sieve(tail(x0))) -> c17(MARK(tail(x0))) MARK(sieve(tail(0))) -> c17(A__SIEVE(a__tail(0))) MARK(sieve(tail(true))) -> c17(A__SIEVE(a__tail(true))) MARK(sieve(tail(false))) -> c17(A__SIEVE(a__tail(false))) MARK(sieve(if(primes, x1, x2))) -> c17(A__SIEVE(a__if(a__primes, x1, x2)), MARK(if(primes, x1, x2))) MARK(sieve(if(sieve(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__sieve(mark(z0)), x1, x2)), MARK(if(sieve(z0), x1, x2))) MARK(sieve(if(from(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__from(mark(z0)), x1, x2)), MARK(if(from(z0), x1, x2))) MARK(sieve(if(head(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__head(mark(z0)), x1, x2)), MARK(if(head(z0), x1, x2))) MARK(sieve(if(tail(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__tail(mark(z0)), x1, x2)), MARK(if(tail(z0), x1, x2))) MARK(sieve(if(if(z0, z1, z2), x1, x2))) -> c17(A__SIEVE(a__if(a__if(mark(z0), z1, z2), x1, x2)), MARK(if(if(z0, z1, z2), x1, x2))) MARK(sieve(if(filter(z0, z1), x1, x2))) -> c17(A__SIEVE(a__if(a__filter(mark(z0), mark(z1)), x1, x2)), MARK(if(filter(z0, z1), x1, x2))) MARK(sieve(if(s(z0), x1, x2))) -> c17(A__SIEVE(a__if(s(mark(z0)), x1, x2)), MARK(if(s(z0), x1, x2))) MARK(sieve(if(cons(z0, z1), x1, x2))) -> c17(A__SIEVE(a__if(cons(mark(z0), z1), x1, x2)), MARK(if(cons(z0, z1), x1, x2))) MARK(sieve(if(true, x1, x2))) -> c17(A__SIEVE(a__if(true, x1, x2)), MARK(if(true, x1, x2))) MARK(sieve(if(false, x1, x2))) -> c17(A__SIEVE(a__if(false, x1, x2)), MARK(if(false, x1, x2))) MARK(sieve(if(divides(z0, z1), x1, x2))) -> c17(A__SIEVE(a__if(divides(mark(z0), mark(z1)), x1, x2)), MARK(if(divides(z0, z1), x1, x2))) MARK(sieve(if(x0, z1, z2))) -> c17(MARK(if(x0, z1, z2))) MARK(sieve(if(0, x1, x2))) -> c17(A__SIEVE(a__if(0, x1, x2))) MARK(sieve(filter(x0, primes))) -> c17(A__SIEVE(a__filter(mark(x0), a__primes)), MARK(filter(x0, primes))) MARK(sieve(filter(x0, sieve(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), a__sieve(mark(z0)))), MARK(filter(x0, sieve(z0)))) MARK(sieve(filter(x0, from(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), a__from(mark(z0)))), MARK(filter(x0, from(z0)))) MARK(sieve(filter(x0, head(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), a__head(mark(z0)))), MARK(filter(x0, head(z0)))) MARK(sieve(filter(x0, tail(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), a__tail(mark(z0)))), MARK(filter(x0, tail(z0)))) MARK(sieve(filter(x0, if(z0, z1, z2)))) -> c17(A__SIEVE(a__filter(mark(x0), a__if(mark(z0), z1, z2))), MARK(filter(x0, if(z0, z1, z2)))) MARK(sieve(filter(x0, filter(z0, z1)))) -> c17(A__SIEVE(a__filter(mark(x0), a__filter(mark(z0), mark(z1)))), MARK(filter(x0, filter(z0, z1)))) MARK(sieve(filter(x0, s(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), s(mark(z0)))), MARK(filter(x0, s(z0)))) MARK(sieve(filter(x0, 0))) -> c17(A__SIEVE(a__filter(mark(x0), 0)), MARK(filter(x0, 0))) MARK(sieve(filter(x0, cons(z0, z1)))) -> c17(A__SIEVE(a__filter(mark(x0), cons(mark(z0), z1))), MARK(filter(x0, cons(z0, z1)))) MARK(sieve(filter(x0, true))) -> c17(A__SIEVE(a__filter(mark(x0), true)), MARK(filter(x0, true))) MARK(sieve(filter(x0, false))) -> c17(A__SIEVE(a__filter(mark(x0), false)), MARK(filter(x0, false))) MARK(sieve(filter(x0, divides(z0, z1)))) -> c17(A__SIEVE(a__filter(mark(x0), divides(mark(z0), mark(z1)))), MARK(filter(x0, divides(z0, z1)))) MARK(sieve(filter(primes, x1))) -> c17(A__SIEVE(a__filter(a__primes, mark(x1))), MARK(filter(primes, x1))) MARK(sieve(filter(sieve(z0), x1))) -> c17(A__SIEVE(a__filter(a__sieve(mark(z0)), mark(x1))), MARK(filter(sieve(z0), x1))) MARK(sieve(filter(from(z0), x1))) -> c17(A__SIEVE(a__filter(a__from(mark(z0)), mark(x1))), MARK(filter(from(z0), x1))) MARK(sieve(filter(head(z0), x1))) -> c17(A__SIEVE(a__filter(a__head(mark(z0)), mark(x1))), MARK(filter(head(z0), x1))) MARK(sieve(filter(tail(z0), x1))) -> c17(A__SIEVE(a__filter(a__tail(mark(z0)), mark(x1))), MARK(filter(tail(z0), x1))) MARK(sieve(filter(if(z0, z1, z2), x1))) -> c17(A__SIEVE(a__filter(a__if(mark(z0), z1, z2), mark(x1))), MARK(filter(if(z0, z1, z2), x1))) MARK(sieve(filter(filter(z0, z1), x1))) -> c17(A__SIEVE(a__filter(a__filter(mark(z0), mark(z1)), mark(x1))), MARK(filter(filter(z0, z1), x1))) MARK(sieve(filter(s(z0), x1))) -> c17(A__SIEVE(a__filter(s(mark(z0)), mark(x1))), MARK(filter(s(z0), x1))) MARK(sieve(filter(0, x1))) -> c17(A__SIEVE(a__filter(0, mark(x1))), MARK(filter(0, x1))) MARK(sieve(filter(cons(z0, z1), x1))) -> c17(A__SIEVE(a__filter(cons(mark(z0), z1), mark(x1))), MARK(filter(cons(z0, z1), x1))) MARK(sieve(filter(true, x1))) -> c17(A__SIEVE(a__filter(true, mark(x1))), MARK(filter(true, x1))) MARK(sieve(filter(false, x1))) -> c17(A__SIEVE(a__filter(false, mark(x1))), MARK(filter(false, x1))) MARK(sieve(filter(divides(z0, z1), x1))) -> c17(A__SIEVE(a__filter(divides(mark(z0), mark(z1)), mark(x1))), MARK(filter(divides(z0, z1), x1))) MARK(sieve(filter(x0, x1))) -> c17(MARK(filter(x0, x1))) MARK(head(primes)) -> c19(A__HEAD(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(head(primes)) -> c19(MARK(primes)) S tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(sieve(z0))) -> c19(A__HEAD(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(head(from(z0))) -> c19(A__HEAD(a__from(mark(z0))), MARK(from(z0))) MARK(head(head(z0))) -> c19(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c19(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(if(z0, z1, z2))) -> c19(A__HEAD(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(head(filter(z0, z1))) -> c19(A__HEAD(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(primes)) -> c20(A__TAIL(a__primes), MARK(primes)) MARK(tail(sieve(z0))) -> c20(A__TAIL(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(tail(from(z0))) -> c20(A__TAIL(a__from(mark(z0))), MARK(from(z0))) MARK(tail(head(z0))) -> c20(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c20(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(if(z0, z1, z2))) -> c20(A__TAIL(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(tail(filter(z0, z1))) -> c20(A__TAIL(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(primes, x1, x2)) -> c21(A__IF(a__primes, x1, x2), MARK(primes)) MARK(if(sieve(z0), x1, x2)) -> c21(A__IF(a__sieve(mark(z0)), x1, x2), MARK(sieve(z0))) MARK(if(from(z0), x1, x2)) -> c21(A__IF(a__from(mark(z0)), x1, x2), MARK(from(z0))) MARK(if(head(z0), x1, x2)) -> c21(A__IF(a__head(mark(z0)), x1, x2), MARK(head(z0))) MARK(if(tail(z0), x1, x2)) -> c21(A__IF(a__tail(mark(z0)), x1, x2), MARK(tail(z0))) MARK(if(if(z0, z1, z2), x1, x2)) -> c21(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(filter(z0, z1), x1, x2)) -> c21(A__IF(a__filter(mark(z0), mark(z1)), x1, x2), MARK(filter(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c22(A__FILTER(mark(x0), a__primes), MARK(x0)) MARK(filter(x0, sieve(z0))) -> c22(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, from(z0))) -> c22(A__FILTER(mark(x0), a__from(mark(z0))), MARK(x0)) MARK(filter(x0, head(z0))) -> c22(A__FILTER(mark(x0), a__head(mark(z0))), MARK(x0)) MARK(filter(x0, tail(z0))) -> c22(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(x0)) MARK(filter(x0, if(z0, z1, z2))) -> c22(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0)) MARK(filter(x0, filter(z0, z1))) -> c22(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(x0)) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(x0, s(z0))) -> c22(MARK(x0)) MARK(filter(x0, 0)) -> c22(MARK(x0)) MARK(filter(x0, true)) -> c22(MARK(x0)) MARK(filter(x0, false)) -> c22(MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), x1)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c23(A__FILTER(mark(x0), a__primes), MARK(primes)) MARK(filter(x0, sieve(z0))) -> c23(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, from(z0))) -> c23(A__FILTER(mark(x0), a__from(mark(z0))), MARK(from(z0))) MARK(filter(x0, head(z0))) -> c23(A__FILTER(mark(x0), a__head(mark(z0))), MARK(head(z0))) MARK(filter(x0, tail(z0))) -> c23(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(tail(z0))) MARK(filter(x0, if(z0, z1, z2))) -> c23(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(filter(x0, filter(z0, z1))) -> c23(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(filter(x0, cons(z0, z1))) -> c23(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(primes, x1)) -> c23(A__FILTER(a__primes, mark(x1)), MARK(x1)) MARK(filter(sieve(z0), x1)) -> c23(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(from(z0), x1)) -> c23(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(head(z0), x1)) -> c23(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(tail(z0), x1)) -> c23(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(if(z0, z1, z2), x1)) -> c23(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1)) MARK(filter(filter(z0, z1), x1)) -> c23(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(x1)) MARK(filter(s(z0), x1)) -> c23(A__FILTER(s(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(x0, s(z0))) -> c23(MARK(s(z0))) MARK(filter(x0, divides(z0, z1))) -> c23(MARK(divides(z0, z1))) MARK(filter(0, x1)) -> c23(MARK(x1)) MARK(filter(cons(z0, z1), x1)) -> c23(MARK(x1)) MARK(filter(true, x1)) -> c23(MARK(x1)) MARK(filter(false, x1)) -> c23(MARK(x1)) MARK(filter(divides(z0, z1), x1)) -> c23(MARK(x1)) A__PRIMES -> c(A__SIEVE(cons(s(mark(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(sieve(primes)) -> c17(MARK(primes)) MARK(sieve(sieve(primes))) -> c17(A__SIEVE(a__sieve(a__primes)), MARK(sieve(primes))) MARK(sieve(sieve(sieve(z0)))) -> c17(A__SIEVE(a__sieve(a__sieve(mark(z0)))), MARK(sieve(sieve(z0)))) MARK(sieve(sieve(from(z0)))) -> c17(A__SIEVE(a__sieve(a__from(mark(z0)))), MARK(sieve(from(z0)))) MARK(sieve(sieve(head(z0)))) -> c17(A__SIEVE(a__sieve(a__head(mark(z0)))), MARK(sieve(head(z0)))) MARK(sieve(sieve(tail(z0)))) -> c17(A__SIEVE(a__sieve(a__tail(mark(z0)))), MARK(sieve(tail(z0)))) MARK(sieve(sieve(if(z0, z1, z2)))) -> c17(A__SIEVE(a__sieve(a__if(mark(z0), z1, z2))), MARK(sieve(if(z0, z1, z2)))) MARK(sieve(sieve(filter(z0, z1)))) -> c17(A__SIEVE(a__sieve(a__filter(mark(z0), mark(z1)))), MARK(sieve(filter(z0, z1)))) MARK(sieve(sieve(s(z0)))) -> c17(A__SIEVE(a__sieve(s(mark(z0)))), MARK(sieve(s(z0)))) MARK(sieve(sieve(cons(z0, z1)))) -> c17(A__SIEVE(a__sieve(cons(mark(z0), z1))), MARK(sieve(cons(z0, z1)))) MARK(sieve(sieve(divides(z0, z1)))) -> c17(A__SIEVE(a__sieve(divides(mark(z0), mark(z1)))), MARK(sieve(divides(z0, z1)))) MARK(sieve(sieve(x0))) -> c17(MARK(sieve(x0))) MARK(sieve(sieve(0))) -> c17(A__SIEVE(a__sieve(0))) MARK(sieve(sieve(true))) -> c17(A__SIEVE(a__sieve(true))) MARK(sieve(sieve(false))) -> c17(A__SIEVE(a__sieve(false))) MARK(sieve(from(x0))) -> c17(A__SIEVE(cons(mark(mark(x0)), from(s(mark(x0))))), MARK(from(x0))) MARK(sieve(from(primes))) -> c17(A__SIEVE(a__from(a__primes)), MARK(from(primes))) MARK(sieve(from(sieve(z0)))) -> c17(A__SIEVE(a__from(a__sieve(mark(z0)))), MARK(from(sieve(z0)))) MARK(sieve(from(from(z0)))) -> c17(A__SIEVE(a__from(a__from(mark(z0)))), MARK(from(from(z0)))) MARK(sieve(from(head(z0)))) -> c17(A__SIEVE(a__from(a__head(mark(z0)))), MARK(from(head(z0)))) MARK(sieve(from(tail(z0)))) -> c17(A__SIEVE(a__from(a__tail(mark(z0)))), MARK(from(tail(z0)))) MARK(sieve(from(if(z0, z1, z2)))) -> c17(A__SIEVE(a__from(a__if(mark(z0), z1, z2))), MARK(from(if(z0, z1, z2)))) MARK(sieve(from(filter(z0, z1)))) -> c17(A__SIEVE(a__from(a__filter(mark(z0), mark(z1)))), MARK(from(filter(z0, z1)))) MARK(sieve(from(s(z0)))) -> c17(A__SIEVE(a__from(s(mark(z0)))), MARK(from(s(z0)))) MARK(sieve(from(0))) -> c17(A__SIEVE(a__from(0)), MARK(from(0))) MARK(sieve(from(cons(z0, z1)))) -> c17(A__SIEVE(a__from(cons(mark(z0), z1))), MARK(from(cons(z0, z1)))) MARK(sieve(from(true))) -> c17(A__SIEVE(a__from(true)), MARK(from(true))) MARK(sieve(from(false))) -> c17(A__SIEVE(a__from(false)), MARK(from(false))) MARK(sieve(from(divides(z0, z1)))) -> c17(A__SIEVE(a__from(divides(mark(z0), mark(z1)))), MARK(from(divides(z0, z1)))) MARK(sieve(from(x0))) -> c17(MARK(from(x0))) MARK(sieve(head(primes))) -> c17(A__SIEVE(a__head(a__primes)), MARK(head(primes))) MARK(sieve(head(sieve(z0)))) -> c17(A__SIEVE(a__head(a__sieve(mark(z0)))), MARK(head(sieve(z0)))) MARK(sieve(head(from(z0)))) -> c17(A__SIEVE(a__head(a__from(mark(z0)))), MARK(head(from(z0)))) MARK(sieve(head(head(z0)))) -> c17(A__SIEVE(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(sieve(head(tail(z0)))) -> c17(A__SIEVE(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(sieve(head(if(z0, z1, z2)))) -> c17(A__SIEVE(a__head(a__if(mark(z0), z1, z2))), MARK(head(if(z0, z1, z2)))) MARK(sieve(head(filter(z0, z1)))) -> c17(A__SIEVE(a__head(a__filter(mark(z0), mark(z1)))), MARK(head(filter(z0, z1)))) MARK(sieve(head(s(z0)))) -> c17(A__SIEVE(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(sieve(head(cons(z0, z1)))) -> c17(A__SIEVE(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(sieve(head(divides(z0, z1)))) -> c17(A__SIEVE(a__head(divides(mark(z0), mark(z1)))), MARK(head(divides(z0, z1)))) MARK(sieve(head(x0))) -> c17(MARK(head(x0))) MARK(sieve(head(0))) -> c17(A__SIEVE(a__head(0))) MARK(sieve(head(true))) -> c17(A__SIEVE(a__head(true))) MARK(sieve(head(false))) -> c17(A__SIEVE(a__head(false))) MARK(sieve(tail(primes))) -> c17(A__SIEVE(a__tail(a__primes)), MARK(tail(primes))) MARK(sieve(tail(sieve(z0)))) -> c17(A__SIEVE(a__tail(a__sieve(mark(z0)))), MARK(tail(sieve(z0)))) MARK(sieve(tail(from(z0)))) -> c17(A__SIEVE(a__tail(a__from(mark(z0)))), MARK(tail(from(z0)))) MARK(sieve(tail(head(z0)))) -> c17(A__SIEVE(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(sieve(tail(tail(z0)))) -> c17(A__SIEVE(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(sieve(tail(if(z0, z1, z2)))) -> c17(A__SIEVE(a__tail(a__if(mark(z0), z1, z2))), MARK(tail(if(z0, z1, z2)))) MARK(sieve(tail(filter(z0, z1)))) -> c17(A__SIEVE(a__tail(a__filter(mark(z0), mark(z1)))), MARK(tail(filter(z0, z1)))) MARK(sieve(tail(s(z0)))) -> c17(A__SIEVE(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(sieve(tail(cons(z0, z1)))) -> c17(A__SIEVE(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(sieve(tail(divides(z0, z1)))) -> c17(A__SIEVE(a__tail(divides(mark(z0), mark(z1)))), MARK(tail(divides(z0, z1)))) MARK(sieve(tail(x0))) -> c17(MARK(tail(x0))) MARK(sieve(tail(0))) -> c17(A__SIEVE(a__tail(0))) MARK(sieve(tail(true))) -> c17(A__SIEVE(a__tail(true))) MARK(sieve(tail(false))) -> c17(A__SIEVE(a__tail(false))) MARK(sieve(if(primes, x1, x2))) -> c17(A__SIEVE(a__if(a__primes, x1, x2)), MARK(if(primes, x1, x2))) MARK(sieve(if(sieve(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__sieve(mark(z0)), x1, x2)), MARK(if(sieve(z0), x1, x2))) MARK(sieve(if(from(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__from(mark(z0)), x1, x2)), MARK(if(from(z0), x1, x2))) MARK(sieve(if(head(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__head(mark(z0)), x1, x2)), MARK(if(head(z0), x1, x2))) MARK(sieve(if(tail(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__tail(mark(z0)), x1, x2)), MARK(if(tail(z0), x1, x2))) MARK(sieve(if(if(z0, z1, z2), x1, x2))) -> c17(A__SIEVE(a__if(a__if(mark(z0), z1, z2), x1, x2)), MARK(if(if(z0, z1, z2), x1, x2))) MARK(sieve(if(filter(z0, z1), x1, x2))) -> c17(A__SIEVE(a__if(a__filter(mark(z0), mark(z1)), x1, x2)), MARK(if(filter(z0, z1), x1, x2))) MARK(sieve(if(s(z0), x1, x2))) -> c17(A__SIEVE(a__if(s(mark(z0)), x1, x2)), MARK(if(s(z0), x1, x2))) MARK(sieve(if(cons(z0, z1), x1, x2))) -> c17(A__SIEVE(a__if(cons(mark(z0), z1), x1, x2)), MARK(if(cons(z0, z1), x1, x2))) MARK(sieve(if(true, x1, x2))) -> c17(A__SIEVE(a__if(true, x1, x2)), MARK(if(true, x1, x2))) MARK(sieve(if(false, x1, x2))) -> c17(A__SIEVE(a__if(false, x1, x2)), MARK(if(false, x1, x2))) MARK(sieve(if(divides(z0, z1), x1, x2))) -> c17(A__SIEVE(a__if(divides(mark(z0), mark(z1)), x1, x2)), MARK(if(divides(z0, z1), x1, x2))) MARK(sieve(if(x0, z1, z2))) -> c17(MARK(if(x0, z1, z2))) MARK(sieve(if(0, x1, x2))) -> c17(A__SIEVE(a__if(0, x1, x2))) MARK(sieve(filter(x0, primes))) -> c17(A__SIEVE(a__filter(mark(x0), a__primes)), MARK(filter(x0, primes))) MARK(sieve(filter(x0, sieve(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), a__sieve(mark(z0)))), MARK(filter(x0, sieve(z0)))) MARK(sieve(filter(x0, from(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), a__from(mark(z0)))), MARK(filter(x0, from(z0)))) MARK(sieve(filter(x0, head(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), a__head(mark(z0)))), MARK(filter(x0, head(z0)))) MARK(sieve(filter(x0, tail(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), a__tail(mark(z0)))), MARK(filter(x0, tail(z0)))) MARK(sieve(filter(x0, if(z0, z1, z2)))) -> c17(A__SIEVE(a__filter(mark(x0), a__if(mark(z0), z1, z2))), MARK(filter(x0, if(z0, z1, z2)))) MARK(sieve(filter(x0, filter(z0, z1)))) -> c17(A__SIEVE(a__filter(mark(x0), a__filter(mark(z0), mark(z1)))), MARK(filter(x0, filter(z0, z1)))) MARK(sieve(filter(x0, s(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), s(mark(z0)))), MARK(filter(x0, s(z0)))) MARK(sieve(filter(x0, 0))) -> c17(A__SIEVE(a__filter(mark(x0), 0)), MARK(filter(x0, 0))) MARK(sieve(filter(x0, cons(z0, z1)))) -> c17(A__SIEVE(a__filter(mark(x0), cons(mark(z0), z1))), MARK(filter(x0, cons(z0, z1)))) MARK(sieve(filter(x0, true))) -> c17(A__SIEVE(a__filter(mark(x0), true)), MARK(filter(x0, true))) MARK(sieve(filter(x0, false))) -> c17(A__SIEVE(a__filter(mark(x0), false)), MARK(filter(x0, false))) MARK(sieve(filter(x0, divides(z0, z1)))) -> c17(A__SIEVE(a__filter(mark(x0), divides(mark(z0), mark(z1)))), MARK(filter(x0, divides(z0, z1)))) MARK(sieve(filter(primes, x1))) -> c17(A__SIEVE(a__filter(a__primes, mark(x1))), MARK(filter(primes, x1))) MARK(sieve(filter(sieve(z0), x1))) -> c17(A__SIEVE(a__filter(a__sieve(mark(z0)), mark(x1))), MARK(filter(sieve(z0), x1))) MARK(sieve(filter(from(z0), x1))) -> c17(A__SIEVE(a__filter(a__from(mark(z0)), mark(x1))), MARK(filter(from(z0), x1))) MARK(sieve(filter(head(z0), x1))) -> c17(A__SIEVE(a__filter(a__head(mark(z0)), mark(x1))), MARK(filter(head(z0), x1))) MARK(sieve(filter(tail(z0), x1))) -> c17(A__SIEVE(a__filter(a__tail(mark(z0)), mark(x1))), MARK(filter(tail(z0), x1))) MARK(sieve(filter(if(z0, z1, z2), x1))) -> c17(A__SIEVE(a__filter(a__if(mark(z0), z1, z2), mark(x1))), MARK(filter(if(z0, z1, z2), x1))) MARK(sieve(filter(filter(z0, z1), x1))) -> c17(A__SIEVE(a__filter(a__filter(mark(z0), mark(z1)), mark(x1))), MARK(filter(filter(z0, z1), x1))) MARK(sieve(filter(s(z0), x1))) -> c17(A__SIEVE(a__filter(s(mark(z0)), mark(x1))), MARK(filter(s(z0), x1))) MARK(sieve(filter(0, x1))) -> c17(A__SIEVE(a__filter(0, mark(x1))), MARK(filter(0, x1))) MARK(sieve(filter(cons(z0, z1), x1))) -> c17(A__SIEVE(a__filter(cons(mark(z0), z1), mark(x1))), MARK(filter(cons(z0, z1), x1))) MARK(sieve(filter(true, x1))) -> c17(A__SIEVE(a__filter(true, mark(x1))), MARK(filter(true, x1))) MARK(sieve(filter(false, x1))) -> c17(A__SIEVE(a__filter(false, mark(x1))), MARK(filter(false, x1))) MARK(sieve(filter(divides(z0, z1), x1))) -> c17(A__SIEVE(a__filter(divides(mark(z0), mark(z1)), mark(x1))), MARK(filter(divides(z0, z1), x1))) MARK(sieve(filter(x0, x1))) -> c17(MARK(filter(x0, x1))) MARK(head(primes)) -> c19(A__HEAD(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(head(primes)) -> c19(MARK(primes)) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c18_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_1, c17_2, c17_1, c19_2, c19_1, c20_2, c20_1, c21_2, c21_1, c22_2, c22_1, c23_2, c23_1, c_2 ---------------------------------------- (131) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(head(sieve(z0))) -> c19(A__HEAD(a__sieve(mark(z0))), MARK(sieve(z0))) by MARK(head(sieve(x0))) -> c19(A__HEAD(sieve(mark(x0))), MARK(sieve(x0))) MARK(head(sieve(primes))) -> c19(A__HEAD(a__sieve(a__primes)), MARK(sieve(primes))) MARK(head(sieve(sieve(z0)))) -> c19(A__HEAD(a__sieve(a__sieve(mark(z0)))), MARK(sieve(sieve(z0)))) MARK(head(sieve(from(z0)))) -> c19(A__HEAD(a__sieve(a__from(mark(z0)))), MARK(sieve(from(z0)))) MARK(head(sieve(head(z0)))) -> c19(A__HEAD(a__sieve(a__head(mark(z0)))), MARK(sieve(head(z0)))) MARK(head(sieve(tail(z0)))) -> c19(A__HEAD(a__sieve(a__tail(mark(z0)))), MARK(sieve(tail(z0)))) MARK(head(sieve(if(z0, z1, z2)))) -> c19(A__HEAD(a__sieve(a__if(mark(z0), z1, z2))), MARK(sieve(if(z0, z1, z2)))) MARK(head(sieve(filter(z0, z1)))) -> c19(A__HEAD(a__sieve(a__filter(mark(z0), mark(z1)))), MARK(sieve(filter(z0, z1)))) MARK(head(sieve(s(z0)))) -> c19(A__HEAD(a__sieve(s(mark(z0)))), MARK(sieve(s(z0)))) MARK(head(sieve(0))) -> c19(A__HEAD(a__sieve(0)), MARK(sieve(0))) MARK(head(sieve(cons(z0, z1)))) -> c19(A__HEAD(a__sieve(cons(mark(z0), z1))), MARK(sieve(cons(z0, z1)))) MARK(head(sieve(true))) -> c19(A__HEAD(a__sieve(true)), MARK(sieve(true))) MARK(head(sieve(false))) -> c19(A__HEAD(a__sieve(false)), MARK(sieve(false))) MARK(head(sieve(divides(z0, z1)))) -> c19(A__HEAD(a__sieve(divides(mark(z0), mark(z1)))), MARK(sieve(divides(z0, z1)))) ---------------------------------------- (132) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(from(z0))) -> c19(A__HEAD(a__from(mark(z0))), MARK(from(z0))) MARK(head(head(z0))) -> c19(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c19(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(if(z0, z1, z2))) -> c19(A__HEAD(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(head(filter(z0, z1))) -> c19(A__HEAD(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(primes)) -> c20(A__TAIL(a__primes), MARK(primes)) MARK(tail(sieve(z0))) -> c20(A__TAIL(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(tail(from(z0))) -> c20(A__TAIL(a__from(mark(z0))), MARK(from(z0))) MARK(tail(head(z0))) -> c20(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c20(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(if(z0, z1, z2))) -> c20(A__TAIL(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(tail(filter(z0, z1))) -> c20(A__TAIL(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(primes, x1, x2)) -> c21(A__IF(a__primes, x1, x2), MARK(primes)) MARK(if(sieve(z0), x1, x2)) -> c21(A__IF(a__sieve(mark(z0)), x1, x2), MARK(sieve(z0))) MARK(if(from(z0), x1, x2)) -> c21(A__IF(a__from(mark(z0)), x1, x2), MARK(from(z0))) MARK(if(head(z0), x1, x2)) -> c21(A__IF(a__head(mark(z0)), x1, x2), MARK(head(z0))) MARK(if(tail(z0), x1, x2)) -> c21(A__IF(a__tail(mark(z0)), x1, x2), MARK(tail(z0))) MARK(if(if(z0, z1, z2), x1, x2)) -> c21(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(filter(z0, z1), x1, x2)) -> c21(A__IF(a__filter(mark(z0), mark(z1)), x1, x2), MARK(filter(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c22(A__FILTER(mark(x0), a__primes), MARK(x0)) MARK(filter(x0, sieve(z0))) -> c22(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, from(z0))) -> c22(A__FILTER(mark(x0), a__from(mark(z0))), MARK(x0)) MARK(filter(x0, head(z0))) -> c22(A__FILTER(mark(x0), a__head(mark(z0))), MARK(x0)) MARK(filter(x0, tail(z0))) -> c22(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(x0)) MARK(filter(x0, if(z0, z1, z2))) -> c22(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0)) MARK(filter(x0, filter(z0, z1))) -> c22(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(x0)) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(x0, s(z0))) -> c22(MARK(x0)) MARK(filter(x0, 0)) -> c22(MARK(x0)) MARK(filter(x0, true)) -> c22(MARK(x0)) MARK(filter(x0, false)) -> c22(MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), x1)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c23(A__FILTER(mark(x0), a__primes), MARK(primes)) MARK(filter(x0, sieve(z0))) -> c23(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, from(z0))) -> c23(A__FILTER(mark(x0), a__from(mark(z0))), MARK(from(z0))) MARK(filter(x0, head(z0))) -> c23(A__FILTER(mark(x0), a__head(mark(z0))), MARK(head(z0))) MARK(filter(x0, tail(z0))) -> c23(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(tail(z0))) MARK(filter(x0, if(z0, z1, z2))) -> c23(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(filter(x0, filter(z0, z1))) -> c23(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(filter(x0, cons(z0, z1))) -> c23(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(primes, x1)) -> c23(A__FILTER(a__primes, mark(x1)), MARK(x1)) MARK(filter(sieve(z0), x1)) -> c23(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(from(z0), x1)) -> c23(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(head(z0), x1)) -> c23(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(tail(z0), x1)) -> c23(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(if(z0, z1, z2), x1)) -> c23(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1)) MARK(filter(filter(z0, z1), x1)) -> c23(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(x1)) MARK(filter(s(z0), x1)) -> c23(A__FILTER(s(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(x0, s(z0))) -> c23(MARK(s(z0))) MARK(filter(x0, divides(z0, z1))) -> c23(MARK(divides(z0, z1))) MARK(filter(0, x1)) -> c23(MARK(x1)) MARK(filter(cons(z0, z1), x1)) -> c23(MARK(x1)) MARK(filter(true, x1)) -> c23(MARK(x1)) MARK(filter(false, x1)) -> c23(MARK(x1)) MARK(filter(divides(z0, z1), x1)) -> c23(MARK(x1)) A__PRIMES -> c(A__SIEVE(cons(s(mark(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(sieve(primes)) -> c17(MARK(primes)) MARK(sieve(sieve(primes))) -> c17(A__SIEVE(a__sieve(a__primes)), MARK(sieve(primes))) MARK(sieve(sieve(sieve(z0)))) -> c17(A__SIEVE(a__sieve(a__sieve(mark(z0)))), MARK(sieve(sieve(z0)))) MARK(sieve(sieve(from(z0)))) -> c17(A__SIEVE(a__sieve(a__from(mark(z0)))), MARK(sieve(from(z0)))) MARK(sieve(sieve(head(z0)))) -> c17(A__SIEVE(a__sieve(a__head(mark(z0)))), MARK(sieve(head(z0)))) MARK(sieve(sieve(tail(z0)))) -> c17(A__SIEVE(a__sieve(a__tail(mark(z0)))), MARK(sieve(tail(z0)))) MARK(sieve(sieve(if(z0, z1, z2)))) -> c17(A__SIEVE(a__sieve(a__if(mark(z0), z1, z2))), MARK(sieve(if(z0, z1, z2)))) MARK(sieve(sieve(filter(z0, z1)))) -> c17(A__SIEVE(a__sieve(a__filter(mark(z0), mark(z1)))), MARK(sieve(filter(z0, z1)))) MARK(sieve(sieve(s(z0)))) -> c17(A__SIEVE(a__sieve(s(mark(z0)))), MARK(sieve(s(z0)))) MARK(sieve(sieve(cons(z0, z1)))) -> c17(A__SIEVE(a__sieve(cons(mark(z0), z1))), MARK(sieve(cons(z0, z1)))) MARK(sieve(sieve(divides(z0, z1)))) -> c17(A__SIEVE(a__sieve(divides(mark(z0), mark(z1)))), MARK(sieve(divides(z0, z1)))) MARK(sieve(sieve(x0))) -> c17(MARK(sieve(x0))) MARK(sieve(sieve(0))) -> c17(A__SIEVE(a__sieve(0))) MARK(sieve(sieve(true))) -> c17(A__SIEVE(a__sieve(true))) MARK(sieve(sieve(false))) -> c17(A__SIEVE(a__sieve(false))) MARK(sieve(from(x0))) -> c17(A__SIEVE(cons(mark(mark(x0)), from(s(mark(x0))))), MARK(from(x0))) MARK(sieve(from(primes))) -> c17(A__SIEVE(a__from(a__primes)), MARK(from(primes))) MARK(sieve(from(sieve(z0)))) -> c17(A__SIEVE(a__from(a__sieve(mark(z0)))), MARK(from(sieve(z0)))) MARK(sieve(from(from(z0)))) -> c17(A__SIEVE(a__from(a__from(mark(z0)))), MARK(from(from(z0)))) MARK(sieve(from(head(z0)))) -> c17(A__SIEVE(a__from(a__head(mark(z0)))), MARK(from(head(z0)))) MARK(sieve(from(tail(z0)))) -> c17(A__SIEVE(a__from(a__tail(mark(z0)))), MARK(from(tail(z0)))) MARK(sieve(from(if(z0, z1, z2)))) -> c17(A__SIEVE(a__from(a__if(mark(z0), z1, z2))), MARK(from(if(z0, z1, z2)))) MARK(sieve(from(filter(z0, z1)))) -> c17(A__SIEVE(a__from(a__filter(mark(z0), mark(z1)))), MARK(from(filter(z0, z1)))) MARK(sieve(from(s(z0)))) -> c17(A__SIEVE(a__from(s(mark(z0)))), MARK(from(s(z0)))) MARK(sieve(from(0))) -> c17(A__SIEVE(a__from(0)), MARK(from(0))) MARK(sieve(from(cons(z0, z1)))) -> c17(A__SIEVE(a__from(cons(mark(z0), z1))), MARK(from(cons(z0, z1)))) MARK(sieve(from(true))) -> c17(A__SIEVE(a__from(true)), MARK(from(true))) MARK(sieve(from(false))) -> c17(A__SIEVE(a__from(false)), MARK(from(false))) MARK(sieve(from(divides(z0, z1)))) -> c17(A__SIEVE(a__from(divides(mark(z0), mark(z1)))), MARK(from(divides(z0, z1)))) MARK(sieve(from(x0))) -> c17(MARK(from(x0))) MARK(sieve(head(primes))) -> c17(A__SIEVE(a__head(a__primes)), MARK(head(primes))) MARK(sieve(head(sieve(z0)))) -> c17(A__SIEVE(a__head(a__sieve(mark(z0)))), MARK(head(sieve(z0)))) MARK(sieve(head(from(z0)))) -> c17(A__SIEVE(a__head(a__from(mark(z0)))), MARK(head(from(z0)))) MARK(sieve(head(head(z0)))) -> c17(A__SIEVE(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(sieve(head(tail(z0)))) -> c17(A__SIEVE(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(sieve(head(if(z0, z1, z2)))) -> c17(A__SIEVE(a__head(a__if(mark(z0), z1, z2))), MARK(head(if(z0, z1, z2)))) MARK(sieve(head(filter(z0, z1)))) -> c17(A__SIEVE(a__head(a__filter(mark(z0), mark(z1)))), MARK(head(filter(z0, z1)))) MARK(sieve(head(s(z0)))) -> c17(A__SIEVE(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(sieve(head(cons(z0, z1)))) -> c17(A__SIEVE(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(sieve(head(divides(z0, z1)))) -> c17(A__SIEVE(a__head(divides(mark(z0), mark(z1)))), MARK(head(divides(z0, z1)))) MARK(sieve(head(x0))) -> c17(MARK(head(x0))) MARK(sieve(head(0))) -> c17(A__SIEVE(a__head(0))) MARK(sieve(head(true))) -> c17(A__SIEVE(a__head(true))) MARK(sieve(head(false))) -> c17(A__SIEVE(a__head(false))) MARK(sieve(tail(primes))) -> c17(A__SIEVE(a__tail(a__primes)), MARK(tail(primes))) MARK(sieve(tail(sieve(z0)))) -> c17(A__SIEVE(a__tail(a__sieve(mark(z0)))), MARK(tail(sieve(z0)))) MARK(sieve(tail(from(z0)))) -> c17(A__SIEVE(a__tail(a__from(mark(z0)))), MARK(tail(from(z0)))) MARK(sieve(tail(head(z0)))) -> c17(A__SIEVE(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(sieve(tail(tail(z0)))) -> c17(A__SIEVE(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(sieve(tail(if(z0, z1, z2)))) -> c17(A__SIEVE(a__tail(a__if(mark(z0), z1, z2))), MARK(tail(if(z0, z1, z2)))) MARK(sieve(tail(filter(z0, z1)))) -> c17(A__SIEVE(a__tail(a__filter(mark(z0), mark(z1)))), MARK(tail(filter(z0, z1)))) MARK(sieve(tail(s(z0)))) -> c17(A__SIEVE(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(sieve(tail(cons(z0, z1)))) -> c17(A__SIEVE(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(sieve(tail(divides(z0, z1)))) -> c17(A__SIEVE(a__tail(divides(mark(z0), mark(z1)))), MARK(tail(divides(z0, z1)))) MARK(sieve(tail(x0))) -> c17(MARK(tail(x0))) MARK(sieve(tail(0))) -> c17(A__SIEVE(a__tail(0))) MARK(sieve(tail(true))) -> c17(A__SIEVE(a__tail(true))) MARK(sieve(tail(false))) -> c17(A__SIEVE(a__tail(false))) MARK(sieve(if(primes, x1, x2))) -> c17(A__SIEVE(a__if(a__primes, x1, x2)), MARK(if(primes, x1, x2))) MARK(sieve(if(sieve(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__sieve(mark(z0)), x1, x2)), MARK(if(sieve(z0), x1, x2))) MARK(sieve(if(from(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__from(mark(z0)), x1, x2)), MARK(if(from(z0), x1, x2))) MARK(sieve(if(head(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__head(mark(z0)), x1, x2)), MARK(if(head(z0), x1, x2))) MARK(sieve(if(tail(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__tail(mark(z0)), x1, x2)), MARK(if(tail(z0), x1, x2))) MARK(sieve(if(if(z0, z1, z2), x1, x2))) -> c17(A__SIEVE(a__if(a__if(mark(z0), z1, z2), x1, x2)), MARK(if(if(z0, z1, z2), x1, x2))) MARK(sieve(if(filter(z0, z1), x1, x2))) -> c17(A__SIEVE(a__if(a__filter(mark(z0), mark(z1)), x1, x2)), MARK(if(filter(z0, z1), x1, x2))) MARK(sieve(if(s(z0), x1, x2))) -> c17(A__SIEVE(a__if(s(mark(z0)), x1, x2)), MARK(if(s(z0), x1, x2))) MARK(sieve(if(cons(z0, z1), x1, x2))) -> c17(A__SIEVE(a__if(cons(mark(z0), z1), x1, x2)), MARK(if(cons(z0, z1), x1, x2))) MARK(sieve(if(true, x1, x2))) -> c17(A__SIEVE(a__if(true, x1, x2)), MARK(if(true, x1, x2))) MARK(sieve(if(false, x1, x2))) -> c17(A__SIEVE(a__if(false, x1, x2)), MARK(if(false, x1, x2))) MARK(sieve(if(divides(z0, z1), x1, x2))) -> c17(A__SIEVE(a__if(divides(mark(z0), mark(z1)), x1, x2)), MARK(if(divides(z0, z1), x1, x2))) MARK(sieve(if(x0, z1, z2))) -> c17(MARK(if(x0, z1, z2))) MARK(sieve(if(0, x1, x2))) -> c17(A__SIEVE(a__if(0, x1, x2))) MARK(sieve(filter(x0, primes))) -> c17(A__SIEVE(a__filter(mark(x0), a__primes)), MARK(filter(x0, primes))) MARK(sieve(filter(x0, sieve(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), a__sieve(mark(z0)))), MARK(filter(x0, sieve(z0)))) MARK(sieve(filter(x0, from(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), a__from(mark(z0)))), MARK(filter(x0, from(z0)))) MARK(sieve(filter(x0, head(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), a__head(mark(z0)))), MARK(filter(x0, head(z0)))) MARK(sieve(filter(x0, tail(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), a__tail(mark(z0)))), MARK(filter(x0, tail(z0)))) MARK(sieve(filter(x0, if(z0, z1, z2)))) -> c17(A__SIEVE(a__filter(mark(x0), a__if(mark(z0), z1, z2))), MARK(filter(x0, if(z0, z1, z2)))) MARK(sieve(filter(x0, filter(z0, z1)))) -> c17(A__SIEVE(a__filter(mark(x0), a__filter(mark(z0), mark(z1)))), MARK(filter(x0, filter(z0, z1)))) MARK(sieve(filter(x0, s(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), s(mark(z0)))), MARK(filter(x0, s(z0)))) MARK(sieve(filter(x0, 0))) -> c17(A__SIEVE(a__filter(mark(x0), 0)), MARK(filter(x0, 0))) MARK(sieve(filter(x0, cons(z0, z1)))) -> c17(A__SIEVE(a__filter(mark(x0), cons(mark(z0), z1))), MARK(filter(x0, cons(z0, z1)))) MARK(sieve(filter(x0, true))) -> c17(A__SIEVE(a__filter(mark(x0), true)), MARK(filter(x0, true))) MARK(sieve(filter(x0, false))) -> c17(A__SIEVE(a__filter(mark(x0), false)), MARK(filter(x0, false))) MARK(sieve(filter(x0, divides(z0, z1)))) -> c17(A__SIEVE(a__filter(mark(x0), divides(mark(z0), mark(z1)))), MARK(filter(x0, divides(z0, z1)))) MARK(sieve(filter(primes, x1))) -> c17(A__SIEVE(a__filter(a__primes, mark(x1))), MARK(filter(primes, x1))) MARK(sieve(filter(sieve(z0), x1))) -> c17(A__SIEVE(a__filter(a__sieve(mark(z0)), mark(x1))), MARK(filter(sieve(z0), x1))) MARK(sieve(filter(from(z0), x1))) -> c17(A__SIEVE(a__filter(a__from(mark(z0)), mark(x1))), MARK(filter(from(z0), x1))) MARK(sieve(filter(head(z0), x1))) -> c17(A__SIEVE(a__filter(a__head(mark(z0)), mark(x1))), MARK(filter(head(z0), x1))) MARK(sieve(filter(tail(z0), x1))) -> c17(A__SIEVE(a__filter(a__tail(mark(z0)), mark(x1))), MARK(filter(tail(z0), x1))) MARK(sieve(filter(if(z0, z1, z2), x1))) -> c17(A__SIEVE(a__filter(a__if(mark(z0), z1, z2), mark(x1))), MARK(filter(if(z0, z1, z2), x1))) MARK(sieve(filter(filter(z0, z1), x1))) -> c17(A__SIEVE(a__filter(a__filter(mark(z0), mark(z1)), mark(x1))), MARK(filter(filter(z0, z1), x1))) MARK(sieve(filter(s(z0), x1))) -> c17(A__SIEVE(a__filter(s(mark(z0)), mark(x1))), MARK(filter(s(z0), x1))) MARK(sieve(filter(0, x1))) -> c17(A__SIEVE(a__filter(0, mark(x1))), MARK(filter(0, x1))) MARK(sieve(filter(cons(z0, z1), x1))) -> c17(A__SIEVE(a__filter(cons(mark(z0), z1), mark(x1))), MARK(filter(cons(z0, z1), x1))) MARK(sieve(filter(true, x1))) -> c17(A__SIEVE(a__filter(true, mark(x1))), MARK(filter(true, x1))) MARK(sieve(filter(false, x1))) -> c17(A__SIEVE(a__filter(false, mark(x1))), MARK(filter(false, x1))) MARK(sieve(filter(divides(z0, z1), x1))) -> c17(A__SIEVE(a__filter(divides(mark(z0), mark(z1)), mark(x1))), MARK(filter(divides(z0, z1), x1))) MARK(sieve(filter(x0, x1))) -> c17(MARK(filter(x0, x1))) MARK(head(primes)) -> c19(A__HEAD(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(head(primes)) -> c19(MARK(primes)) MARK(head(sieve(x0))) -> c19(A__HEAD(sieve(mark(x0))), MARK(sieve(x0))) MARK(head(sieve(primes))) -> c19(A__HEAD(a__sieve(a__primes)), MARK(sieve(primes))) MARK(head(sieve(sieve(z0)))) -> c19(A__HEAD(a__sieve(a__sieve(mark(z0)))), MARK(sieve(sieve(z0)))) MARK(head(sieve(from(z0)))) -> c19(A__HEAD(a__sieve(a__from(mark(z0)))), MARK(sieve(from(z0)))) MARK(head(sieve(head(z0)))) -> c19(A__HEAD(a__sieve(a__head(mark(z0)))), MARK(sieve(head(z0)))) MARK(head(sieve(tail(z0)))) -> c19(A__HEAD(a__sieve(a__tail(mark(z0)))), MARK(sieve(tail(z0)))) MARK(head(sieve(if(z0, z1, z2)))) -> c19(A__HEAD(a__sieve(a__if(mark(z0), z1, z2))), MARK(sieve(if(z0, z1, z2)))) MARK(head(sieve(filter(z0, z1)))) -> c19(A__HEAD(a__sieve(a__filter(mark(z0), mark(z1)))), MARK(sieve(filter(z0, z1)))) MARK(head(sieve(s(z0)))) -> c19(A__HEAD(a__sieve(s(mark(z0)))), MARK(sieve(s(z0)))) MARK(head(sieve(0))) -> c19(A__HEAD(a__sieve(0)), MARK(sieve(0))) MARK(head(sieve(cons(z0, z1)))) -> c19(A__HEAD(a__sieve(cons(mark(z0), z1))), MARK(sieve(cons(z0, z1)))) MARK(head(sieve(true))) -> c19(A__HEAD(a__sieve(true)), MARK(sieve(true))) MARK(head(sieve(false))) -> c19(A__HEAD(a__sieve(false)), MARK(sieve(false))) MARK(head(sieve(divides(z0, z1)))) -> c19(A__HEAD(a__sieve(divides(mark(z0), mark(z1)))), MARK(sieve(divides(z0, z1)))) S tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(from(z0))) -> c19(A__HEAD(a__from(mark(z0))), MARK(from(z0))) MARK(head(head(z0))) -> c19(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c19(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(if(z0, z1, z2))) -> c19(A__HEAD(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(head(filter(z0, z1))) -> c19(A__HEAD(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(primes)) -> c20(A__TAIL(a__primes), MARK(primes)) MARK(tail(sieve(z0))) -> c20(A__TAIL(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(tail(from(z0))) -> c20(A__TAIL(a__from(mark(z0))), MARK(from(z0))) MARK(tail(head(z0))) -> c20(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c20(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(if(z0, z1, z2))) -> c20(A__TAIL(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(tail(filter(z0, z1))) -> c20(A__TAIL(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(primes, x1, x2)) -> c21(A__IF(a__primes, x1, x2), MARK(primes)) MARK(if(sieve(z0), x1, x2)) -> c21(A__IF(a__sieve(mark(z0)), x1, x2), MARK(sieve(z0))) MARK(if(from(z0), x1, x2)) -> c21(A__IF(a__from(mark(z0)), x1, x2), MARK(from(z0))) MARK(if(head(z0), x1, x2)) -> c21(A__IF(a__head(mark(z0)), x1, x2), MARK(head(z0))) MARK(if(tail(z0), x1, x2)) -> c21(A__IF(a__tail(mark(z0)), x1, x2), MARK(tail(z0))) MARK(if(if(z0, z1, z2), x1, x2)) -> c21(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(filter(z0, z1), x1, x2)) -> c21(A__IF(a__filter(mark(z0), mark(z1)), x1, x2), MARK(filter(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c22(A__FILTER(mark(x0), a__primes), MARK(x0)) MARK(filter(x0, sieve(z0))) -> c22(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, from(z0))) -> c22(A__FILTER(mark(x0), a__from(mark(z0))), MARK(x0)) MARK(filter(x0, head(z0))) -> c22(A__FILTER(mark(x0), a__head(mark(z0))), MARK(x0)) MARK(filter(x0, tail(z0))) -> c22(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(x0)) MARK(filter(x0, if(z0, z1, z2))) -> c22(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0)) MARK(filter(x0, filter(z0, z1))) -> c22(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(x0)) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(x0, s(z0))) -> c22(MARK(x0)) MARK(filter(x0, 0)) -> c22(MARK(x0)) MARK(filter(x0, true)) -> c22(MARK(x0)) MARK(filter(x0, false)) -> c22(MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), x1)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c23(A__FILTER(mark(x0), a__primes), MARK(primes)) MARK(filter(x0, sieve(z0))) -> c23(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, from(z0))) -> c23(A__FILTER(mark(x0), a__from(mark(z0))), MARK(from(z0))) MARK(filter(x0, head(z0))) -> c23(A__FILTER(mark(x0), a__head(mark(z0))), MARK(head(z0))) MARK(filter(x0, tail(z0))) -> c23(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(tail(z0))) MARK(filter(x0, if(z0, z1, z2))) -> c23(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(filter(x0, filter(z0, z1))) -> c23(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(filter(x0, cons(z0, z1))) -> c23(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(primes, x1)) -> c23(A__FILTER(a__primes, mark(x1)), MARK(x1)) MARK(filter(sieve(z0), x1)) -> c23(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(from(z0), x1)) -> c23(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(head(z0), x1)) -> c23(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(tail(z0), x1)) -> c23(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(if(z0, z1, z2), x1)) -> c23(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1)) MARK(filter(filter(z0, z1), x1)) -> c23(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(x1)) MARK(filter(s(z0), x1)) -> c23(A__FILTER(s(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(x0, s(z0))) -> c23(MARK(s(z0))) MARK(filter(x0, divides(z0, z1))) -> c23(MARK(divides(z0, z1))) MARK(filter(0, x1)) -> c23(MARK(x1)) MARK(filter(cons(z0, z1), x1)) -> c23(MARK(x1)) MARK(filter(true, x1)) -> c23(MARK(x1)) MARK(filter(false, x1)) -> c23(MARK(x1)) MARK(filter(divides(z0, z1), x1)) -> c23(MARK(x1)) A__PRIMES -> c(A__SIEVE(cons(s(mark(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(sieve(primes)) -> c17(MARK(primes)) MARK(sieve(sieve(primes))) -> c17(A__SIEVE(a__sieve(a__primes)), MARK(sieve(primes))) MARK(sieve(sieve(sieve(z0)))) -> c17(A__SIEVE(a__sieve(a__sieve(mark(z0)))), MARK(sieve(sieve(z0)))) MARK(sieve(sieve(from(z0)))) -> c17(A__SIEVE(a__sieve(a__from(mark(z0)))), MARK(sieve(from(z0)))) MARK(sieve(sieve(head(z0)))) -> c17(A__SIEVE(a__sieve(a__head(mark(z0)))), MARK(sieve(head(z0)))) MARK(sieve(sieve(tail(z0)))) -> c17(A__SIEVE(a__sieve(a__tail(mark(z0)))), MARK(sieve(tail(z0)))) MARK(sieve(sieve(if(z0, z1, z2)))) -> c17(A__SIEVE(a__sieve(a__if(mark(z0), z1, z2))), MARK(sieve(if(z0, z1, z2)))) MARK(sieve(sieve(filter(z0, z1)))) -> c17(A__SIEVE(a__sieve(a__filter(mark(z0), mark(z1)))), MARK(sieve(filter(z0, z1)))) MARK(sieve(sieve(s(z0)))) -> c17(A__SIEVE(a__sieve(s(mark(z0)))), MARK(sieve(s(z0)))) MARK(sieve(sieve(cons(z0, z1)))) -> c17(A__SIEVE(a__sieve(cons(mark(z0), z1))), MARK(sieve(cons(z0, z1)))) MARK(sieve(sieve(divides(z0, z1)))) -> c17(A__SIEVE(a__sieve(divides(mark(z0), mark(z1)))), MARK(sieve(divides(z0, z1)))) MARK(sieve(sieve(x0))) -> c17(MARK(sieve(x0))) MARK(sieve(sieve(0))) -> c17(A__SIEVE(a__sieve(0))) MARK(sieve(sieve(true))) -> c17(A__SIEVE(a__sieve(true))) MARK(sieve(sieve(false))) -> c17(A__SIEVE(a__sieve(false))) MARK(sieve(from(x0))) -> c17(A__SIEVE(cons(mark(mark(x0)), from(s(mark(x0))))), MARK(from(x0))) MARK(sieve(from(primes))) -> c17(A__SIEVE(a__from(a__primes)), MARK(from(primes))) MARK(sieve(from(sieve(z0)))) -> c17(A__SIEVE(a__from(a__sieve(mark(z0)))), MARK(from(sieve(z0)))) MARK(sieve(from(from(z0)))) -> c17(A__SIEVE(a__from(a__from(mark(z0)))), MARK(from(from(z0)))) MARK(sieve(from(head(z0)))) -> c17(A__SIEVE(a__from(a__head(mark(z0)))), MARK(from(head(z0)))) MARK(sieve(from(tail(z0)))) -> c17(A__SIEVE(a__from(a__tail(mark(z0)))), MARK(from(tail(z0)))) MARK(sieve(from(if(z0, z1, z2)))) -> c17(A__SIEVE(a__from(a__if(mark(z0), z1, z2))), MARK(from(if(z0, z1, z2)))) MARK(sieve(from(filter(z0, z1)))) -> c17(A__SIEVE(a__from(a__filter(mark(z0), mark(z1)))), MARK(from(filter(z0, z1)))) MARK(sieve(from(s(z0)))) -> c17(A__SIEVE(a__from(s(mark(z0)))), MARK(from(s(z0)))) MARK(sieve(from(0))) -> c17(A__SIEVE(a__from(0)), MARK(from(0))) MARK(sieve(from(cons(z0, z1)))) -> c17(A__SIEVE(a__from(cons(mark(z0), z1))), MARK(from(cons(z0, z1)))) MARK(sieve(from(true))) -> c17(A__SIEVE(a__from(true)), MARK(from(true))) MARK(sieve(from(false))) -> c17(A__SIEVE(a__from(false)), MARK(from(false))) MARK(sieve(from(divides(z0, z1)))) -> c17(A__SIEVE(a__from(divides(mark(z0), mark(z1)))), MARK(from(divides(z0, z1)))) MARK(sieve(from(x0))) -> c17(MARK(from(x0))) MARK(sieve(head(primes))) -> c17(A__SIEVE(a__head(a__primes)), MARK(head(primes))) MARK(sieve(head(sieve(z0)))) -> c17(A__SIEVE(a__head(a__sieve(mark(z0)))), MARK(head(sieve(z0)))) MARK(sieve(head(from(z0)))) -> c17(A__SIEVE(a__head(a__from(mark(z0)))), MARK(head(from(z0)))) MARK(sieve(head(head(z0)))) -> c17(A__SIEVE(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(sieve(head(tail(z0)))) -> c17(A__SIEVE(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(sieve(head(if(z0, z1, z2)))) -> c17(A__SIEVE(a__head(a__if(mark(z0), z1, z2))), MARK(head(if(z0, z1, z2)))) MARK(sieve(head(filter(z0, z1)))) -> c17(A__SIEVE(a__head(a__filter(mark(z0), mark(z1)))), MARK(head(filter(z0, z1)))) MARK(sieve(head(s(z0)))) -> c17(A__SIEVE(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(sieve(head(cons(z0, z1)))) -> c17(A__SIEVE(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(sieve(head(divides(z0, z1)))) -> c17(A__SIEVE(a__head(divides(mark(z0), mark(z1)))), MARK(head(divides(z0, z1)))) MARK(sieve(head(x0))) -> c17(MARK(head(x0))) MARK(sieve(head(0))) -> c17(A__SIEVE(a__head(0))) MARK(sieve(head(true))) -> c17(A__SIEVE(a__head(true))) MARK(sieve(head(false))) -> c17(A__SIEVE(a__head(false))) MARK(sieve(tail(primes))) -> c17(A__SIEVE(a__tail(a__primes)), MARK(tail(primes))) MARK(sieve(tail(sieve(z0)))) -> c17(A__SIEVE(a__tail(a__sieve(mark(z0)))), MARK(tail(sieve(z0)))) MARK(sieve(tail(from(z0)))) -> c17(A__SIEVE(a__tail(a__from(mark(z0)))), MARK(tail(from(z0)))) MARK(sieve(tail(head(z0)))) -> c17(A__SIEVE(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(sieve(tail(tail(z0)))) -> c17(A__SIEVE(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(sieve(tail(if(z0, z1, z2)))) -> c17(A__SIEVE(a__tail(a__if(mark(z0), z1, z2))), MARK(tail(if(z0, z1, z2)))) MARK(sieve(tail(filter(z0, z1)))) -> c17(A__SIEVE(a__tail(a__filter(mark(z0), mark(z1)))), MARK(tail(filter(z0, z1)))) MARK(sieve(tail(s(z0)))) -> c17(A__SIEVE(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(sieve(tail(cons(z0, z1)))) -> c17(A__SIEVE(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(sieve(tail(divides(z0, z1)))) -> c17(A__SIEVE(a__tail(divides(mark(z0), mark(z1)))), MARK(tail(divides(z0, z1)))) MARK(sieve(tail(x0))) -> c17(MARK(tail(x0))) MARK(sieve(tail(0))) -> c17(A__SIEVE(a__tail(0))) MARK(sieve(tail(true))) -> c17(A__SIEVE(a__tail(true))) MARK(sieve(tail(false))) -> c17(A__SIEVE(a__tail(false))) MARK(sieve(if(primes, x1, x2))) -> c17(A__SIEVE(a__if(a__primes, x1, x2)), MARK(if(primes, x1, x2))) MARK(sieve(if(sieve(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__sieve(mark(z0)), x1, x2)), MARK(if(sieve(z0), x1, x2))) MARK(sieve(if(from(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__from(mark(z0)), x1, x2)), MARK(if(from(z0), x1, x2))) MARK(sieve(if(head(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__head(mark(z0)), x1, x2)), MARK(if(head(z0), x1, x2))) MARK(sieve(if(tail(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__tail(mark(z0)), x1, x2)), MARK(if(tail(z0), x1, x2))) MARK(sieve(if(if(z0, z1, z2), x1, x2))) -> c17(A__SIEVE(a__if(a__if(mark(z0), z1, z2), x1, x2)), MARK(if(if(z0, z1, z2), x1, x2))) MARK(sieve(if(filter(z0, z1), x1, x2))) -> c17(A__SIEVE(a__if(a__filter(mark(z0), mark(z1)), x1, x2)), MARK(if(filter(z0, z1), x1, x2))) MARK(sieve(if(s(z0), x1, x2))) -> c17(A__SIEVE(a__if(s(mark(z0)), x1, x2)), MARK(if(s(z0), x1, x2))) MARK(sieve(if(cons(z0, z1), x1, x2))) -> c17(A__SIEVE(a__if(cons(mark(z0), z1), x1, x2)), MARK(if(cons(z0, z1), x1, x2))) MARK(sieve(if(true, x1, x2))) -> c17(A__SIEVE(a__if(true, x1, x2)), MARK(if(true, x1, x2))) MARK(sieve(if(false, x1, x2))) -> c17(A__SIEVE(a__if(false, x1, x2)), MARK(if(false, x1, x2))) MARK(sieve(if(divides(z0, z1), x1, x2))) -> c17(A__SIEVE(a__if(divides(mark(z0), mark(z1)), x1, x2)), MARK(if(divides(z0, z1), x1, x2))) MARK(sieve(if(x0, z1, z2))) -> c17(MARK(if(x0, z1, z2))) MARK(sieve(if(0, x1, x2))) -> c17(A__SIEVE(a__if(0, x1, x2))) MARK(sieve(filter(x0, primes))) -> c17(A__SIEVE(a__filter(mark(x0), a__primes)), MARK(filter(x0, primes))) MARK(sieve(filter(x0, sieve(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), a__sieve(mark(z0)))), MARK(filter(x0, sieve(z0)))) MARK(sieve(filter(x0, from(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), a__from(mark(z0)))), MARK(filter(x0, from(z0)))) MARK(sieve(filter(x0, head(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), a__head(mark(z0)))), MARK(filter(x0, head(z0)))) MARK(sieve(filter(x0, tail(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), a__tail(mark(z0)))), MARK(filter(x0, tail(z0)))) MARK(sieve(filter(x0, if(z0, z1, z2)))) -> c17(A__SIEVE(a__filter(mark(x0), a__if(mark(z0), z1, z2))), MARK(filter(x0, if(z0, z1, z2)))) MARK(sieve(filter(x0, filter(z0, z1)))) -> c17(A__SIEVE(a__filter(mark(x0), a__filter(mark(z0), mark(z1)))), MARK(filter(x0, filter(z0, z1)))) MARK(sieve(filter(x0, s(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), s(mark(z0)))), MARK(filter(x0, s(z0)))) MARK(sieve(filter(x0, 0))) -> c17(A__SIEVE(a__filter(mark(x0), 0)), MARK(filter(x0, 0))) MARK(sieve(filter(x0, cons(z0, z1)))) -> c17(A__SIEVE(a__filter(mark(x0), cons(mark(z0), z1))), MARK(filter(x0, cons(z0, z1)))) MARK(sieve(filter(x0, true))) -> c17(A__SIEVE(a__filter(mark(x0), true)), MARK(filter(x0, true))) MARK(sieve(filter(x0, false))) -> c17(A__SIEVE(a__filter(mark(x0), false)), MARK(filter(x0, false))) MARK(sieve(filter(x0, divides(z0, z1)))) -> c17(A__SIEVE(a__filter(mark(x0), divides(mark(z0), mark(z1)))), MARK(filter(x0, divides(z0, z1)))) MARK(sieve(filter(primes, x1))) -> c17(A__SIEVE(a__filter(a__primes, mark(x1))), MARK(filter(primes, x1))) MARK(sieve(filter(sieve(z0), x1))) -> c17(A__SIEVE(a__filter(a__sieve(mark(z0)), mark(x1))), MARK(filter(sieve(z0), x1))) MARK(sieve(filter(from(z0), x1))) -> c17(A__SIEVE(a__filter(a__from(mark(z0)), mark(x1))), MARK(filter(from(z0), x1))) MARK(sieve(filter(head(z0), x1))) -> c17(A__SIEVE(a__filter(a__head(mark(z0)), mark(x1))), MARK(filter(head(z0), x1))) MARK(sieve(filter(tail(z0), x1))) -> c17(A__SIEVE(a__filter(a__tail(mark(z0)), mark(x1))), MARK(filter(tail(z0), x1))) MARK(sieve(filter(if(z0, z1, z2), x1))) -> c17(A__SIEVE(a__filter(a__if(mark(z0), z1, z2), mark(x1))), MARK(filter(if(z0, z1, z2), x1))) MARK(sieve(filter(filter(z0, z1), x1))) -> c17(A__SIEVE(a__filter(a__filter(mark(z0), mark(z1)), mark(x1))), MARK(filter(filter(z0, z1), x1))) MARK(sieve(filter(s(z0), x1))) -> c17(A__SIEVE(a__filter(s(mark(z0)), mark(x1))), MARK(filter(s(z0), x1))) MARK(sieve(filter(0, x1))) -> c17(A__SIEVE(a__filter(0, mark(x1))), MARK(filter(0, x1))) MARK(sieve(filter(cons(z0, z1), x1))) -> c17(A__SIEVE(a__filter(cons(mark(z0), z1), mark(x1))), MARK(filter(cons(z0, z1), x1))) MARK(sieve(filter(true, x1))) -> c17(A__SIEVE(a__filter(true, mark(x1))), MARK(filter(true, x1))) MARK(sieve(filter(false, x1))) -> c17(A__SIEVE(a__filter(false, mark(x1))), MARK(filter(false, x1))) MARK(sieve(filter(divides(z0, z1), x1))) -> c17(A__SIEVE(a__filter(divides(mark(z0), mark(z1)), mark(x1))), MARK(filter(divides(z0, z1), x1))) MARK(sieve(filter(x0, x1))) -> c17(MARK(filter(x0, x1))) MARK(head(primes)) -> c19(A__HEAD(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(head(primes)) -> c19(MARK(primes)) MARK(head(sieve(x0))) -> c19(A__HEAD(sieve(mark(x0))), MARK(sieve(x0))) MARK(head(sieve(primes))) -> c19(A__HEAD(a__sieve(a__primes)), MARK(sieve(primes))) MARK(head(sieve(sieve(z0)))) -> c19(A__HEAD(a__sieve(a__sieve(mark(z0)))), MARK(sieve(sieve(z0)))) MARK(head(sieve(from(z0)))) -> c19(A__HEAD(a__sieve(a__from(mark(z0)))), MARK(sieve(from(z0)))) MARK(head(sieve(head(z0)))) -> c19(A__HEAD(a__sieve(a__head(mark(z0)))), MARK(sieve(head(z0)))) MARK(head(sieve(tail(z0)))) -> c19(A__HEAD(a__sieve(a__tail(mark(z0)))), MARK(sieve(tail(z0)))) MARK(head(sieve(if(z0, z1, z2)))) -> c19(A__HEAD(a__sieve(a__if(mark(z0), z1, z2))), MARK(sieve(if(z0, z1, z2)))) MARK(head(sieve(filter(z0, z1)))) -> c19(A__HEAD(a__sieve(a__filter(mark(z0), mark(z1)))), MARK(sieve(filter(z0, z1)))) MARK(head(sieve(s(z0)))) -> c19(A__HEAD(a__sieve(s(mark(z0)))), MARK(sieve(s(z0)))) MARK(head(sieve(0))) -> c19(A__HEAD(a__sieve(0)), MARK(sieve(0))) MARK(head(sieve(cons(z0, z1)))) -> c19(A__HEAD(a__sieve(cons(mark(z0), z1))), MARK(sieve(cons(z0, z1)))) MARK(head(sieve(true))) -> c19(A__HEAD(a__sieve(true)), MARK(sieve(true))) MARK(head(sieve(false))) -> c19(A__HEAD(a__sieve(false)), MARK(sieve(false))) MARK(head(sieve(divides(z0, z1)))) -> c19(A__HEAD(a__sieve(divides(mark(z0), mark(z1)))), MARK(sieve(divides(z0, z1)))) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c18_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_1, c17_2, c17_1, c19_2, c19_1, c20_2, c20_1, c21_2, c21_1, c22_2, c22_1, c23_2, c23_1, c_2 ---------------------------------------- (133) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing tuple parts ---------------------------------------- (134) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(from(z0))) -> c19(A__HEAD(a__from(mark(z0))), MARK(from(z0))) MARK(head(head(z0))) -> c19(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c19(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(if(z0, z1, z2))) -> c19(A__HEAD(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(head(filter(z0, z1))) -> c19(A__HEAD(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(primes)) -> c20(A__TAIL(a__primes), MARK(primes)) MARK(tail(sieve(z0))) -> c20(A__TAIL(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(tail(from(z0))) -> c20(A__TAIL(a__from(mark(z0))), MARK(from(z0))) MARK(tail(head(z0))) -> c20(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c20(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(if(z0, z1, z2))) -> c20(A__TAIL(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(tail(filter(z0, z1))) -> c20(A__TAIL(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(primes, x1, x2)) -> c21(A__IF(a__primes, x1, x2), MARK(primes)) MARK(if(sieve(z0), x1, x2)) -> c21(A__IF(a__sieve(mark(z0)), x1, x2), MARK(sieve(z0))) MARK(if(from(z0), x1, x2)) -> c21(A__IF(a__from(mark(z0)), x1, x2), MARK(from(z0))) MARK(if(head(z0), x1, x2)) -> c21(A__IF(a__head(mark(z0)), x1, x2), MARK(head(z0))) MARK(if(tail(z0), x1, x2)) -> c21(A__IF(a__tail(mark(z0)), x1, x2), MARK(tail(z0))) MARK(if(if(z0, z1, z2), x1, x2)) -> c21(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(filter(z0, z1), x1, x2)) -> c21(A__IF(a__filter(mark(z0), mark(z1)), 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MARK(head(sieve(primes))) -> c19(A__HEAD(a__sieve(a__primes)), MARK(sieve(primes))) MARK(head(sieve(sieve(z0)))) -> c19(A__HEAD(a__sieve(a__sieve(mark(z0)))), MARK(sieve(sieve(z0)))) MARK(head(sieve(from(z0)))) -> c19(A__HEAD(a__sieve(a__from(mark(z0)))), MARK(sieve(from(z0)))) MARK(head(sieve(head(z0)))) -> c19(A__HEAD(a__sieve(a__head(mark(z0)))), MARK(sieve(head(z0)))) MARK(head(sieve(tail(z0)))) -> c19(A__HEAD(a__sieve(a__tail(mark(z0)))), MARK(sieve(tail(z0)))) MARK(head(sieve(if(z0, z1, z2)))) -> c19(A__HEAD(a__sieve(a__if(mark(z0), z1, z2))), MARK(sieve(if(z0, z1, z2)))) MARK(head(sieve(filter(z0, z1)))) -> c19(A__HEAD(a__sieve(a__filter(mark(z0), mark(z1)))), MARK(sieve(filter(z0, z1)))) MARK(head(sieve(s(z0)))) -> c19(A__HEAD(a__sieve(s(mark(z0)))), MARK(sieve(s(z0)))) MARK(head(sieve(cons(z0, z1)))) -> c19(A__HEAD(a__sieve(cons(mark(z0), z1))), MARK(sieve(cons(z0, z1)))) MARK(head(sieve(divides(z0, z1)))) -> c19(A__HEAD(a__sieve(divides(mark(z0), mark(z1)))), MARK(sieve(divides(z0, z1)))) MARK(head(sieve(x0))) -> c19(MARK(sieve(x0))) MARK(head(sieve(0))) -> c19(A__HEAD(a__sieve(0))) MARK(head(sieve(true))) -> c19(A__HEAD(a__sieve(true))) MARK(head(sieve(false))) -> c19(A__HEAD(a__sieve(false))) S tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(from(z0))) -> c19(A__HEAD(a__from(mark(z0))), MARK(from(z0))) MARK(head(head(z0))) -> c19(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c19(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(if(z0, z1, z2))) -> c19(A__HEAD(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(head(filter(z0, z1))) -> c19(A__HEAD(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(primes)) -> c20(A__TAIL(a__primes), MARK(primes)) MARK(tail(sieve(z0))) -> c20(A__TAIL(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(tail(from(z0))) -> c20(A__TAIL(a__from(mark(z0))), MARK(from(z0))) MARK(tail(head(z0))) -> c20(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c20(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(if(z0, z1, z2))) -> c20(A__TAIL(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(tail(filter(z0, z1))) -> c20(A__TAIL(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(primes, x1, x2)) -> c21(A__IF(a__primes, x1, x2), MARK(primes)) MARK(if(sieve(z0), x1, x2)) -> c21(A__IF(a__sieve(mark(z0)), x1, x2), MARK(sieve(z0))) MARK(if(from(z0), x1, x2)) -> c21(A__IF(a__from(mark(z0)), x1, x2), MARK(from(z0))) MARK(if(head(z0), x1, x2)) -> c21(A__IF(a__head(mark(z0)), x1, x2), MARK(head(z0))) MARK(if(tail(z0), x1, x2)) -> c21(A__IF(a__tail(mark(z0)), x1, x2), MARK(tail(z0))) MARK(if(if(z0, z1, z2), x1, x2)) -> c21(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(filter(z0, z1), x1, x2)) -> c21(A__IF(a__filter(mark(z0), mark(z1)), x1, x2), MARK(filter(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c22(A__FILTER(mark(x0), a__primes), MARK(x0)) MARK(filter(x0, sieve(z0))) -> c22(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, from(z0))) -> c22(A__FILTER(mark(x0), a__from(mark(z0))), MARK(x0)) MARK(filter(x0, head(z0))) -> c22(A__FILTER(mark(x0), a__head(mark(z0))), MARK(x0)) MARK(filter(x0, tail(z0))) -> c22(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(x0)) MARK(filter(x0, if(z0, z1, z2))) -> c22(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0)) MARK(filter(x0, filter(z0, z1))) -> c22(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(x0)) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(x0, s(z0))) -> c22(MARK(x0)) MARK(filter(x0, 0)) -> c22(MARK(x0)) MARK(filter(x0, true)) -> c22(MARK(x0)) MARK(filter(x0, false)) -> c22(MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), x1)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c23(A__FILTER(mark(x0), a__primes), MARK(primes)) MARK(filter(x0, sieve(z0))) -> c23(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, from(z0))) -> c23(A__FILTER(mark(x0), a__from(mark(z0))), MARK(from(z0))) MARK(filter(x0, head(z0))) -> c23(A__FILTER(mark(x0), a__head(mark(z0))), MARK(head(z0))) MARK(filter(x0, tail(z0))) -> c23(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(tail(z0))) MARK(filter(x0, if(z0, z1, z2))) -> c23(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(filter(x0, filter(z0, z1))) -> c23(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(filter(x0, cons(z0, z1))) -> c23(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(primes, x1)) -> c23(A__FILTER(a__primes, mark(x1)), MARK(x1)) MARK(filter(sieve(z0), x1)) -> c23(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(from(z0), x1)) -> c23(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(head(z0), x1)) -> c23(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(tail(z0), x1)) -> c23(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(if(z0, z1, z2), x1)) -> c23(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1)) MARK(filter(filter(z0, z1), x1)) -> c23(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(x1)) MARK(filter(s(z0), x1)) -> c23(A__FILTER(s(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(x0, s(z0))) -> c23(MARK(s(z0))) MARK(filter(x0, divides(z0, z1))) -> c23(MARK(divides(z0, z1))) MARK(filter(0, x1)) -> c23(MARK(x1)) MARK(filter(cons(z0, z1), x1)) -> c23(MARK(x1)) MARK(filter(true, x1)) -> c23(MARK(x1)) MARK(filter(false, x1)) -> c23(MARK(x1)) MARK(filter(divides(z0, z1), x1)) -> c23(MARK(x1)) A__PRIMES -> c(A__SIEVE(cons(s(mark(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(sieve(primes)) -> c17(MARK(primes)) MARK(sieve(sieve(primes))) -> c17(A__SIEVE(a__sieve(a__primes)), MARK(sieve(primes))) MARK(sieve(sieve(sieve(z0)))) -> c17(A__SIEVE(a__sieve(a__sieve(mark(z0)))), MARK(sieve(sieve(z0)))) MARK(sieve(sieve(from(z0)))) -> c17(A__SIEVE(a__sieve(a__from(mark(z0)))), MARK(sieve(from(z0)))) MARK(sieve(sieve(head(z0)))) -> c17(A__SIEVE(a__sieve(a__head(mark(z0)))), MARK(sieve(head(z0)))) MARK(sieve(sieve(tail(z0)))) -> c17(A__SIEVE(a__sieve(a__tail(mark(z0)))), MARK(sieve(tail(z0)))) MARK(sieve(sieve(if(z0, z1, z2)))) -> c17(A__SIEVE(a__sieve(a__if(mark(z0), z1, z2))), MARK(sieve(if(z0, z1, z2)))) MARK(sieve(sieve(filter(z0, z1)))) -> c17(A__SIEVE(a__sieve(a__filter(mark(z0), mark(z1)))), MARK(sieve(filter(z0, z1)))) MARK(sieve(sieve(s(z0)))) -> c17(A__SIEVE(a__sieve(s(mark(z0)))), MARK(sieve(s(z0)))) MARK(sieve(sieve(cons(z0, z1)))) -> c17(A__SIEVE(a__sieve(cons(mark(z0), z1))), MARK(sieve(cons(z0, z1)))) MARK(sieve(sieve(divides(z0, z1)))) -> c17(A__SIEVE(a__sieve(divides(mark(z0), mark(z1)))), MARK(sieve(divides(z0, z1)))) MARK(sieve(sieve(x0))) -> c17(MARK(sieve(x0))) MARK(sieve(sieve(0))) -> c17(A__SIEVE(a__sieve(0))) MARK(sieve(sieve(true))) -> c17(A__SIEVE(a__sieve(true))) MARK(sieve(sieve(false))) -> c17(A__SIEVE(a__sieve(false))) MARK(sieve(from(x0))) -> c17(A__SIEVE(cons(mark(mark(x0)), from(s(mark(x0))))), MARK(from(x0))) MARK(sieve(from(primes))) -> c17(A__SIEVE(a__from(a__primes)), MARK(from(primes))) MARK(sieve(from(sieve(z0)))) -> c17(A__SIEVE(a__from(a__sieve(mark(z0)))), MARK(from(sieve(z0)))) MARK(sieve(from(from(z0)))) -> c17(A__SIEVE(a__from(a__from(mark(z0)))), MARK(from(from(z0)))) MARK(sieve(from(head(z0)))) -> c17(A__SIEVE(a__from(a__head(mark(z0)))), MARK(from(head(z0)))) MARK(sieve(from(tail(z0)))) -> c17(A__SIEVE(a__from(a__tail(mark(z0)))), MARK(from(tail(z0)))) MARK(sieve(from(if(z0, z1, z2)))) -> c17(A__SIEVE(a__from(a__if(mark(z0), z1, z2))), MARK(from(if(z0, z1, z2)))) MARK(sieve(from(filter(z0, z1)))) -> c17(A__SIEVE(a__from(a__filter(mark(z0), mark(z1)))), MARK(from(filter(z0, z1)))) MARK(sieve(from(s(z0)))) -> c17(A__SIEVE(a__from(s(mark(z0)))), MARK(from(s(z0)))) MARK(sieve(from(0))) -> c17(A__SIEVE(a__from(0)), MARK(from(0))) MARK(sieve(from(cons(z0, z1)))) -> c17(A__SIEVE(a__from(cons(mark(z0), z1))), MARK(from(cons(z0, z1)))) MARK(sieve(from(true))) -> c17(A__SIEVE(a__from(true)), MARK(from(true))) MARK(sieve(from(false))) -> c17(A__SIEVE(a__from(false)), MARK(from(false))) MARK(sieve(from(divides(z0, z1)))) -> c17(A__SIEVE(a__from(divides(mark(z0), mark(z1)))), MARK(from(divides(z0, z1)))) MARK(sieve(from(x0))) -> c17(MARK(from(x0))) MARK(sieve(head(primes))) -> c17(A__SIEVE(a__head(a__primes)), MARK(head(primes))) MARK(sieve(head(sieve(z0)))) -> c17(A__SIEVE(a__head(a__sieve(mark(z0)))), MARK(head(sieve(z0)))) MARK(sieve(head(from(z0)))) -> c17(A__SIEVE(a__head(a__from(mark(z0)))), MARK(head(from(z0)))) MARK(sieve(head(head(z0)))) -> c17(A__SIEVE(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(sieve(head(tail(z0)))) -> c17(A__SIEVE(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(sieve(head(if(z0, z1, z2)))) -> c17(A__SIEVE(a__head(a__if(mark(z0), z1, z2))), MARK(head(if(z0, z1, z2)))) MARK(sieve(head(filter(z0, z1)))) -> c17(A__SIEVE(a__head(a__filter(mark(z0), mark(z1)))), MARK(head(filter(z0, z1)))) MARK(sieve(head(s(z0)))) -> c17(A__SIEVE(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(sieve(head(cons(z0, z1)))) -> c17(A__SIEVE(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(sieve(head(divides(z0, z1)))) -> c17(A__SIEVE(a__head(divides(mark(z0), mark(z1)))), MARK(head(divides(z0, z1)))) MARK(sieve(head(x0))) -> c17(MARK(head(x0))) MARK(sieve(head(0))) -> c17(A__SIEVE(a__head(0))) MARK(sieve(head(true))) -> c17(A__SIEVE(a__head(true))) MARK(sieve(head(false))) -> c17(A__SIEVE(a__head(false))) MARK(sieve(tail(primes))) -> c17(A__SIEVE(a__tail(a__primes)), MARK(tail(primes))) MARK(sieve(tail(sieve(z0)))) -> c17(A__SIEVE(a__tail(a__sieve(mark(z0)))), MARK(tail(sieve(z0)))) MARK(sieve(tail(from(z0)))) -> c17(A__SIEVE(a__tail(a__from(mark(z0)))), MARK(tail(from(z0)))) MARK(sieve(tail(head(z0)))) -> c17(A__SIEVE(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(sieve(tail(tail(z0)))) -> c17(A__SIEVE(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(sieve(tail(if(z0, z1, z2)))) -> c17(A__SIEVE(a__tail(a__if(mark(z0), z1, z2))), MARK(tail(if(z0, z1, z2)))) MARK(sieve(tail(filter(z0, z1)))) -> c17(A__SIEVE(a__tail(a__filter(mark(z0), mark(z1)))), MARK(tail(filter(z0, z1)))) MARK(sieve(tail(s(z0)))) -> c17(A__SIEVE(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(sieve(tail(cons(z0, z1)))) -> c17(A__SIEVE(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(sieve(tail(divides(z0, z1)))) -> c17(A__SIEVE(a__tail(divides(mark(z0), mark(z1)))), MARK(tail(divides(z0, z1)))) MARK(sieve(tail(x0))) -> c17(MARK(tail(x0))) MARK(sieve(tail(0))) -> c17(A__SIEVE(a__tail(0))) MARK(sieve(tail(true))) -> c17(A__SIEVE(a__tail(true))) MARK(sieve(tail(false))) -> c17(A__SIEVE(a__tail(false))) MARK(sieve(if(primes, x1, x2))) -> c17(A__SIEVE(a__if(a__primes, x1, x2)), MARK(if(primes, x1, x2))) MARK(sieve(if(sieve(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__sieve(mark(z0)), x1, x2)), MARK(if(sieve(z0), x1, x2))) MARK(sieve(if(from(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__from(mark(z0)), x1, x2)), MARK(if(from(z0), x1, x2))) MARK(sieve(if(head(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__head(mark(z0)), x1, x2)), MARK(if(head(z0), x1, x2))) MARK(sieve(if(tail(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__tail(mark(z0)), x1, x2)), MARK(if(tail(z0), x1, x2))) MARK(sieve(if(if(z0, z1, z2), x1, x2))) -> c17(A__SIEVE(a__if(a__if(mark(z0), z1, z2), x1, x2)), MARK(if(if(z0, z1, z2), x1, x2))) MARK(sieve(if(filter(z0, z1), x1, x2))) -> c17(A__SIEVE(a__if(a__filter(mark(z0), mark(z1)), x1, x2)), MARK(if(filter(z0, z1), x1, x2))) MARK(sieve(if(s(z0), x1, x2))) -> c17(A__SIEVE(a__if(s(mark(z0)), x1, x2)), MARK(if(s(z0), x1, x2))) MARK(sieve(if(cons(z0, z1), x1, x2))) -> c17(A__SIEVE(a__if(cons(mark(z0), z1), x1, x2)), MARK(if(cons(z0, z1), x1, x2))) MARK(sieve(if(true, x1, x2))) -> c17(A__SIEVE(a__if(true, x1, x2)), MARK(if(true, x1, x2))) MARK(sieve(if(false, x1, x2))) -> c17(A__SIEVE(a__if(false, x1, x2)), MARK(if(false, x1, x2))) MARK(sieve(if(divides(z0, z1), x1, x2))) -> c17(A__SIEVE(a__if(divides(mark(z0), mark(z1)), x1, x2)), MARK(if(divides(z0, z1), x1, x2))) MARK(sieve(if(x0, z1, z2))) -> c17(MARK(if(x0, z1, z2))) MARK(sieve(if(0, x1, x2))) -> c17(A__SIEVE(a__if(0, x1, x2))) MARK(sieve(filter(x0, primes))) -> c17(A__SIEVE(a__filter(mark(x0), a__primes)), MARK(filter(x0, primes))) MARK(sieve(filter(x0, sieve(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), a__sieve(mark(z0)))), MARK(filter(x0, sieve(z0)))) MARK(sieve(filter(x0, from(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), a__from(mark(z0)))), MARK(filter(x0, from(z0)))) MARK(sieve(filter(x0, head(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), a__head(mark(z0)))), MARK(filter(x0, head(z0)))) MARK(sieve(filter(x0, tail(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), a__tail(mark(z0)))), MARK(filter(x0, tail(z0)))) MARK(sieve(filter(x0, if(z0, z1, z2)))) -> c17(A__SIEVE(a__filter(mark(x0), a__if(mark(z0), z1, z2))), MARK(filter(x0, if(z0, z1, z2)))) MARK(sieve(filter(x0, filter(z0, z1)))) -> c17(A__SIEVE(a__filter(mark(x0), a__filter(mark(z0), mark(z1)))), MARK(filter(x0, filter(z0, z1)))) MARK(sieve(filter(x0, s(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), s(mark(z0)))), MARK(filter(x0, s(z0)))) MARK(sieve(filter(x0, 0))) -> c17(A__SIEVE(a__filter(mark(x0), 0)), MARK(filter(x0, 0))) MARK(sieve(filter(x0, cons(z0, z1)))) -> c17(A__SIEVE(a__filter(mark(x0), cons(mark(z0), z1))), MARK(filter(x0, cons(z0, z1)))) MARK(sieve(filter(x0, true))) -> c17(A__SIEVE(a__filter(mark(x0), true)), MARK(filter(x0, true))) MARK(sieve(filter(x0, false))) -> c17(A__SIEVE(a__filter(mark(x0), false)), MARK(filter(x0, false))) MARK(sieve(filter(x0, divides(z0, z1)))) -> c17(A__SIEVE(a__filter(mark(x0), divides(mark(z0), mark(z1)))), MARK(filter(x0, divides(z0, z1)))) MARK(sieve(filter(primes, x1))) -> c17(A__SIEVE(a__filter(a__primes, mark(x1))), MARK(filter(primes, x1))) MARK(sieve(filter(sieve(z0), x1))) -> c17(A__SIEVE(a__filter(a__sieve(mark(z0)), mark(x1))), MARK(filter(sieve(z0), x1))) MARK(sieve(filter(from(z0), x1))) -> c17(A__SIEVE(a__filter(a__from(mark(z0)), mark(x1))), MARK(filter(from(z0), x1))) MARK(sieve(filter(head(z0), x1))) -> c17(A__SIEVE(a__filter(a__head(mark(z0)), mark(x1))), MARK(filter(head(z0), x1))) MARK(sieve(filter(tail(z0), x1))) -> c17(A__SIEVE(a__filter(a__tail(mark(z0)), mark(x1))), MARK(filter(tail(z0), x1))) MARK(sieve(filter(if(z0, z1, z2), x1))) -> c17(A__SIEVE(a__filter(a__if(mark(z0), z1, z2), mark(x1))), MARK(filter(if(z0, z1, z2), x1))) MARK(sieve(filter(filter(z0, z1), x1))) -> c17(A__SIEVE(a__filter(a__filter(mark(z0), mark(z1)), mark(x1))), MARK(filter(filter(z0, z1), x1))) MARK(sieve(filter(s(z0), x1))) -> c17(A__SIEVE(a__filter(s(mark(z0)), mark(x1))), MARK(filter(s(z0), x1))) MARK(sieve(filter(0, x1))) -> c17(A__SIEVE(a__filter(0, mark(x1))), MARK(filter(0, x1))) MARK(sieve(filter(cons(z0, z1), x1))) -> c17(A__SIEVE(a__filter(cons(mark(z0), z1), mark(x1))), MARK(filter(cons(z0, z1), x1))) MARK(sieve(filter(true, x1))) -> c17(A__SIEVE(a__filter(true, mark(x1))), MARK(filter(true, x1))) MARK(sieve(filter(false, x1))) -> c17(A__SIEVE(a__filter(false, mark(x1))), MARK(filter(false, x1))) MARK(sieve(filter(divides(z0, z1), x1))) -> c17(A__SIEVE(a__filter(divides(mark(z0), mark(z1)), mark(x1))), MARK(filter(divides(z0, z1), x1))) MARK(sieve(filter(x0, x1))) -> c17(MARK(filter(x0, x1))) MARK(head(primes)) -> c19(A__HEAD(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(head(primes)) -> c19(MARK(primes)) MARK(head(sieve(primes))) -> c19(A__HEAD(a__sieve(a__primes)), MARK(sieve(primes))) MARK(head(sieve(sieve(z0)))) -> c19(A__HEAD(a__sieve(a__sieve(mark(z0)))), MARK(sieve(sieve(z0)))) MARK(head(sieve(from(z0)))) -> c19(A__HEAD(a__sieve(a__from(mark(z0)))), MARK(sieve(from(z0)))) MARK(head(sieve(head(z0)))) -> c19(A__HEAD(a__sieve(a__head(mark(z0)))), MARK(sieve(head(z0)))) MARK(head(sieve(tail(z0)))) -> c19(A__HEAD(a__sieve(a__tail(mark(z0)))), MARK(sieve(tail(z0)))) MARK(head(sieve(if(z0, z1, z2)))) -> c19(A__HEAD(a__sieve(a__if(mark(z0), z1, z2))), MARK(sieve(if(z0, z1, z2)))) MARK(head(sieve(filter(z0, z1)))) -> c19(A__HEAD(a__sieve(a__filter(mark(z0), mark(z1)))), MARK(sieve(filter(z0, z1)))) MARK(head(sieve(s(z0)))) -> c19(A__HEAD(a__sieve(s(mark(z0)))), MARK(sieve(s(z0)))) MARK(head(sieve(cons(z0, z1)))) -> c19(A__HEAD(a__sieve(cons(mark(z0), z1))), MARK(sieve(cons(z0, z1)))) MARK(head(sieve(divides(z0, z1)))) -> c19(A__HEAD(a__sieve(divides(mark(z0), mark(z1)))), MARK(sieve(divides(z0, z1)))) MARK(head(sieve(x0))) -> c19(MARK(sieve(x0))) MARK(head(sieve(0))) -> c19(A__HEAD(a__sieve(0))) MARK(head(sieve(true))) -> c19(A__HEAD(a__sieve(true))) MARK(head(sieve(false))) -> c19(A__HEAD(a__sieve(false))) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c18_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_1, c17_2, c17_1, c19_2, c19_1, c20_2, c20_1, c21_2, c21_1, c22_2, c22_1, c23_2, c23_1, c_2 ---------------------------------------- (135) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(head(from(z0))) -> c19(A__HEAD(a__from(mark(z0))), MARK(from(z0))) by MARK(head(from(x0))) -> c19(A__HEAD(cons(mark(mark(x0)), from(s(mark(x0))))), MARK(from(x0))) MARK(head(from(x0))) -> c19(A__HEAD(from(mark(x0))), MARK(from(x0))) MARK(head(from(primes))) -> c19(A__HEAD(a__from(a__primes)), MARK(from(primes))) MARK(head(from(sieve(z0)))) -> c19(A__HEAD(a__from(a__sieve(mark(z0)))), MARK(from(sieve(z0)))) MARK(head(from(from(z0)))) -> c19(A__HEAD(a__from(a__from(mark(z0)))), MARK(from(from(z0)))) MARK(head(from(head(z0)))) -> c19(A__HEAD(a__from(a__head(mark(z0)))), MARK(from(head(z0)))) MARK(head(from(tail(z0)))) -> c19(A__HEAD(a__from(a__tail(mark(z0)))), MARK(from(tail(z0)))) MARK(head(from(if(z0, z1, z2)))) -> c19(A__HEAD(a__from(a__if(mark(z0), z1, z2))), MARK(from(if(z0, z1, z2)))) MARK(head(from(filter(z0, z1)))) -> c19(A__HEAD(a__from(a__filter(mark(z0), mark(z1)))), MARK(from(filter(z0, z1)))) MARK(head(from(s(z0)))) -> c19(A__HEAD(a__from(s(mark(z0)))), MARK(from(s(z0)))) MARK(head(from(0))) -> c19(A__HEAD(a__from(0)), MARK(from(0))) MARK(head(from(cons(z0, z1)))) -> c19(A__HEAD(a__from(cons(mark(z0), z1))), MARK(from(cons(z0, z1)))) MARK(head(from(true))) -> c19(A__HEAD(a__from(true)), MARK(from(true))) MARK(head(from(false))) -> c19(A__HEAD(a__from(false)), MARK(from(false))) MARK(head(from(divides(z0, z1)))) -> c19(A__HEAD(a__from(divides(mark(z0), mark(z1)))), MARK(from(divides(z0, z1)))) MARK(head(from(x0))) -> c19(MARK(from(x0))) ---------------------------------------- (136) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(head(z0))) -> c19(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c19(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(if(z0, z1, z2))) -> c19(A__HEAD(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(head(filter(z0, z1))) -> c19(A__HEAD(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(primes)) -> c20(A__TAIL(a__primes), MARK(primes)) MARK(tail(sieve(z0))) -> c20(A__TAIL(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(tail(from(z0))) -> c20(A__TAIL(a__from(mark(z0))), MARK(from(z0))) MARK(tail(head(z0))) -> c20(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c20(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(if(z0, z1, z2))) -> c20(A__TAIL(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(tail(filter(z0, z1))) -> c20(A__TAIL(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(primes, x1, x2)) -> c21(A__IF(a__primes, x1, x2), MARK(primes)) MARK(if(sieve(z0), x1, x2)) -> c21(A__IF(a__sieve(mark(z0)), x1, x2), MARK(sieve(z0))) MARK(if(from(z0), x1, x2)) -> c21(A__IF(a__from(mark(z0)), x1, x2), MARK(from(z0))) MARK(if(head(z0), x1, x2)) -> c21(A__IF(a__head(mark(z0)), x1, x2), MARK(head(z0))) MARK(if(tail(z0), x1, x2)) -> c21(A__IF(a__tail(mark(z0)), x1, x2), MARK(tail(z0))) MARK(if(if(z0, z1, z2), x1, x2)) -> c21(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(filter(z0, z1), x1, x2)) -> c21(A__IF(a__filter(mark(z0), mark(z1)), x1, x2), MARK(filter(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c22(A__FILTER(mark(x0), a__primes), MARK(x0)) MARK(filter(x0, sieve(z0))) -> c22(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, from(z0))) -> c22(A__FILTER(mark(x0), a__from(mark(z0))), MARK(x0)) MARK(filter(x0, head(z0))) -> c22(A__FILTER(mark(x0), a__head(mark(z0))), MARK(x0)) MARK(filter(x0, tail(z0))) -> c22(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(x0)) MARK(filter(x0, if(z0, z1, z2))) -> c22(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0)) MARK(filter(x0, filter(z0, z1))) -> c22(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(x0)) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(x0, s(z0))) -> c22(MARK(x0)) MARK(filter(x0, 0)) -> c22(MARK(x0)) MARK(filter(x0, true)) -> c22(MARK(x0)) MARK(filter(x0, false)) -> c22(MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), x1)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c23(A__FILTER(mark(x0), a__primes), MARK(primes)) MARK(filter(x0, sieve(z0))) -> c23(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, from(z0))) -> c23(A__FILTER(mark(x0), a__from(mark(z0))), MARK(from(z0))) MARK(filter(x0, head(z0))) -> c23(A__FILTER(mark(x0), a__head(mark(z0))), MARK(head(z0))) MARK(filter(x0, tail(z0))) -> c23(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(tail(z0))) MARK(filter(x0, if(z0, z1, z2))) -> c23(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(filter(x0, filter(z0, z1))) -> c23(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(filter(x0, cons(z0, z1))) -> c23(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(primes, x1)) -> c23(A__FILTER(a__primes, mark(x1)), MARK(x1)) MARK(filter(sieve(z0), x1)) -> c23(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(from(z0), x1)) -> c23(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(head(z0), x1)) -> c23(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(tail(z0), x1)) -> c23(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(if(z0, z1, z2), x1)) -> c23(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1)) MARK(filter(filter(z0, z1), x1)) -> c23(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(x1)) MARK(filter(s(z0), x1)) -> c23(A__FILTER(s(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(x0, s(z0))) -> c23(MARK(s(z0))) MARK(filter(x0, divides(z0, z1))) -> c23(MARK(divides(z0, z1))) MARK(filter(0, x1)) -> c23(MARK(x1)) MARK(filter(cons(z0, z1), x1)) -> c23(MARK(x1)) MARK(filter(true, x1)) -> c23(MARK(x1)) MARK(filter(false, x1)) -> c23(MARK(x1)) MARK(filter(divides(z0, z1), x1)) -> c23(MARK(x1)) A__PRIMES -> c(A__SIEVE(cons(s(mark(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(sieve(primes)) -> c17(MARK(primes)) MARK(sieve(sieve(primes))) -> c17(A__SIEVE(a__sieve(a__primes)), MARK(sieve(primes))) MARK(sieve(sieve(sieve(z0)))) -> c17(A__SIEVE(a__sieve(a__sieve(mark(z0)))), MARK(sieve(sieve(z0)))) MARK(sieve(sieve(from(z0)))) -> c17(A__SIEVE(a__sieve(a__from(mark(z0)))), MARK(sieve(from(z0)))) MARK(sieve(sieve(head(z0)))) -> c17(A__SIEVE(a__sieve(a__head(mark(z0)))), MARK(sieve(head(z0)))) MARK(sieve(sieve(tail(z0)))) -> c17(A__SIEVE(a__sieve(a__tail(mark(z0)))), MARK(sieve(tail(z0)))) MARK(sieve(sieve(if(z0, z1, z2)))) -> c17(A__SIEVE(a__sieve(a__if(mark(z0), z1, z2))), MARK(sieve(if(z0, z1, z2)))) MARK(sieve(sieve(filter(z0, z1)))) -> c17(A__SIEVE(a__sieve(a__filter(mark(z0), mark(z1)))), MARK(sieve(filter(z0, z1)))) MARK(sieve(sieve(s(z0)))) -> c17(A__SIEVE(a__sieve(s(mark(z0)))), MARK(sieve(s(z0)))) MARK(sieve(sieve(cons(z0, z1)))) -> c17(A__SIEVE(a__sieve(cons(mark(z0), 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MARK(sieve(tail(primes))) -> c17(A__SIEVE(a__tail(a__primes)), MARK(tail(primes))) MARK(sieve(tail(sieve(z0)))) -> c17(A__SIEVE(a__tail(a__sieve(mark(z0)))), MARK(tail(sieve(z0)))) MARK(sieve(tail(from(z0)))) -> c17(A__SIEVE(a__tail(a__from(mark(z0)))), MARK(tail(from(z0)))) MARK(sieve(tail(head(z0)))) -> c17(A__SIEVE(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(sieve(tail(tail(z0)))) -> c17(A__SIEVE(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(sieve(tail(if(z0, z1, z2)))) -> c17(A__SIEVE(a__tail(a__if(mark(z0), z1, z2))), MARK(tail(if(z0, z1, z2)))) MARK(sieve(tail(filter(z0, z1)))) -> c17(A__SIEVE(a__tail(a__filter(mark(z0), mark(z1)))), MARK(tail(filter(z0, z1)))) MARK(sieve(tail(s(z0)))) -> c17(A__SIEVE(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(sieve(tail(cons(z0, z1)))) -> c17(A__SIEVE(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(sieve(tail(divides(z0, z1)))) -> c17(A__SIEVE(a__tail(divides(mark(z0), mark(z1)))), MARK(tail(divides(z0, 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x2)), MARK(if(filter(z0, z1), x1, x2))) MARK(sieve(if(s(z0), x1, x2))) -> c17(A__SIEVE(a__if(s(mark(z0)), x1, x2)), MARK(if(s(z0), x1, x2))) MARK(sieve(if(cons(z0, z1), x1, x2))) -> c17(A__SIEVE(a__if(cons(mark(z0), z1), x1, x2)), MARK(if(cons(z0, z1), x1, x2))) MARK(sieve(if(true, x1, x2))) -> c17(A__SIEVE(a__if(true, x1, x2)), MARK(if(true, x1, x2))) MARK(sieve(if(false, x1, x2))) -> c17(A__SIEVE(a__if(false, x1, x2)), MARK(if(false, x1, x2))) MARK(sieve(if(divides(z0, z1), x1, x2))) -> c17(A__SIEVE(a__if(divides(mark(z0), mark(z1)), x1, x2)), MARK(if(divides(z0, z1), x1, x2))) MARK(sieve(if(x0, z1, z2))) -> c17(MARK(if(x0, z1, z2))) MARK(sieve(if(0, x1, x2))) -> c17(A__SIEVE(a__if(0, x1, x2))) MARK(sieve(filter(x0, primes))) -> c17(A__SIEVE(a__filter(mark(x0), a__primes)), MARK(filter(x0, primes))) MARK(sieve(filter(x0, sieve(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), a__sieve(mark(z0)))), MARK(filter(x0, sieve(z0)))) MARK(sieve(filter(x0, from(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), a__from(mark(z0)))), MARK(filter(x0, from(z0)))) MARK(sieve(filter(x0, head(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), a__head(mark(z0)))), MARK(filter(x0, head(z0)))) MARK(sieve(filter(x0, tail(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), a__tail(mark(z0)))), MARK(filter(x0, tail(z0)))) MARK(sieve(filter(x0, if(z0, z1, z2)))) -> c17(A__SIEVE(a__filter(mark(x0), a__if(mark(z0), z1, z2))), MARK(filter(x0, if(z0, z1, z2)))) MARK(sieve(filter(x0, filter(z0, z1)))) -> c17(A__SIEVE(a__filter(mark(x0), a__filter(mark(z0), mark(z1)))), MARK(filter(x0, filter(z0, z1)))) MARK(sieve(filter(x0, s(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), s(mark(z0)))), MARK(filter(x0, s(z0)))) MARK(sieve(filter(x0, 0))) -> c17(A__SIEVE(a__filter(mark(x0), 0)), MARK(filter(x0, 0))) MARK(sieve(filter(x0, cons(z0, z1)))) -> c17(A__SIEVE(a__filter(mark(x0), cons(mark(z0), z1))), MARK(filter(x0, cons(z0, z1)))) MARK(sieve(filter(x0, true))) -> c17(A__SIEVE(a__filter(mark(x0), true)), MARK(filter(x0, true))) MARK(sieve(filter(x0, false))) -> c17(A__SIEVE(a__filter(mark(x0), false)), MARK(filter(x0, false))) MARK(sieve(filter(x0, divides(z0, z1)))) -> c17(A__SIEVE(a__filter(mark(x0), divides(mark(z0), mark(z1)))), MARK(filter(x0, divides(z0, z1)))) MARK(sieve(filter(primes, x1))) -> c17(A__SIEVE(a__filter(a__primes, mark(x1))), MARK(filter(primes, x1))) MARK(sieve(filter(sieve(z0), x1))) -> c17(A__SIEVE(a__filter(a__sieve(mark(z0)), mark(x1))), MARK(filter(sieve(z0), x1))) MARK(sieve(filter(from(z0), x1))) -> c17(A__SIEVE(a__filter(a__from(mark(z0)), mark(x1))), MARK(filter(from(z0), x1))) MARK(sieve(filter(head(z0), x1))) -> c17(A__SIEVE(a__filter(a__head(mark(z0)), mark(x1))), MARK(filter(head(z0), x1))) MARK(sieve(filter(tail(z0), x1))) -> c17(A__SIEVE(a__filter(a__tail(mark(z0)), mark(x1))), MARK(filter(tail(z0), x1))) MARK(sieve(filter(if(z0, z1, z2), x1))) -> c17(A__SIEVE(a__filter(a__if(mark(z0), z1, z2), mark(x1))), MARK(filter(if(z0, z1, z2), x1))) MARK(sieve(filter(filter(z0, z1), x1))) -> c17(A__SIEVE(a__filter(a__filter(mark(z0), mark(z1)), mark(x1))), MARK(filter(filter(z0, z1), x1))) MARK(sieve(filter(s(z0), x1))) -> c17(A__SIEVE(a__filter(s(mark(z0)), mark(x1))), MARK(filter(s(z0), x1))) MARK(sieve(filter(0, x1))) -> c17(A__SIEVE(a__filter(0, mark(x1))), MARK(filter(0, x1))) MARK(sieve(filter(cons(z0, z1), x1))) -> c17(A__SIEVE(a__filter(cons(mark(z0), z1), mark(x1))), MARK(filter(cons(z0, z1), x1))) MARK(sieve(filter(true, x1))) -> c17(A__SIEVE(a__filter(true, mark(x1))), MARK(filter(true, x1))) MARK(sieve(filter(false, x1))) -> c17(A__SIEVE(a__filter(false, mark(x1))), MARK(filter(false, x1))) MARK(sieve(filter(divides(z0, z1), x1))) -> c17(A__SIEVE(a__filter(divides(mark(z0), mark(z1)), mark(x1))), MARK(filter(divides(z0, z1), x1))) MARK(sieve(filter(x0, x1))) -> c17(MARK(filter(x0, x1))) MARK(head(primes)) -> c19(A__HEAD(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(head(primes)) -> c19(MARK(primes)) MARK(head(sieve(primes))) -> c19(A__HEAD(a__sieve(a__primes)), MARK(sieve(primes))) MARK(head(sieve(sieve(z0)))) -> c19(A__HEAD(a__sieve(a__sieve(mark(z0)))), MARK(sieve(sieve(z0)))) MARK(head(sieve(from(z0)))) -> c19(A__HEAD(a__sieve(a__from(mark(z0)))), MARK(sieve(from(z0)))) MARK(head(sieve(head(z0)))) -> c19(A__HEAD(a__sieve(a__head(mark(z0)))), MARK(sieve(head(z0)))) MARK(head(sieve(tail(z0)))) -> c19(A__HEAD(a__sieve(a__tail(mark(z0)))), MARK(sieve(tail(z0)))) MARK(head(sieve(if(z0, z1, z2)))) -> c19(A__HEAD(a__sieve(a__if(mark(z0), z1, z2))), MARK(sieve(if(z0, z1, z2)))) MARK(head(sieve(filter(z0, z1)))) -> c19(A__HEAD(a__sieve(a__filter(mark(z0), mark(z1)))), MARK(sieve(filter(z0, z1)))) MARK(head(sieve(s(z0)))) -> c19(A__HEAD(a__sieve(s(mark(z0)))), MARK(sieve(s(z0)))) MARK(head(sieve(cons(z0, z1)))) -> c19(A__HEAD(a__sieve(cons(mark(z0), z1))), MARK(sieve(cons(z0, z1)))) MARK(head(sieve(divides(z0, z1)))) -> c19(A__HEAD(a__sieve(divides(mark(z0), mark(z1)))), MARK(sieve(divides(z0, z1)))) MARK(head(sieve(x0))) -> c19(MARK(sieve(x0))) MARK(head(sieve(0))) -> c19(A__HEAD(a__sieve(0))) MARK(head(sieve(true))) -> c19(A__HEAD(a__sieve(true))) MARK(head(sieve(false))) -> c19(A__HEAD(a__sieve(false))) MARK(head(from(x0))) -> c19(A__HEAD(cons(mark(mark(x0)), from(s(mark(x0))))), MARK(from(x0))) MARK(head(from(x0))) -> c19(A__HEAD(from(mark(x0))), MARK(from(x0))) MARK(head(from(primes))) -> c19(A__HEAD(a__from(a__primes)), MARK(from(primes))) MARK(head(from(sieve(z0)))) -> c19(A__HEAD(a__from(a__sieve(mark(z0)))), MARK(from(sieve(z0)))) MARK(head(from(from(z0)))) -> c19(A__HEAD(a__from(a__from(mark(z0)))), MARK(from(from(z0)))) MARK(head(from(head(z0)))) -> c19(A__HEAD(a__from(a__head(mark(z0)))), MARK(from(head(z0)))) MARK(head(from(tail(z0)))) -> c19(A__HEAD(a__from(a__tail(mark(z0)))), MARK(from(tail(z0)))) MARK(head(from(if(z0, z1, z2)))) -> c19(A__HEAD(a__from(a__if(mark(z0), z1, z2))), MARK(from(if(z0, z1, z2)))) MARK(head(from(filter(z0, z1)))) -> c19(A__HEAD(a__from(a__filter(mark(z0), mark(z1)))), MARK(from(filter(z0, z1)))) MARK(head(from(s(z0)))) -> c19(A__HEAD(a__from(s(mark(z0)))), MARK(from(s(z0)))) MARK(head(from(0))) -> c19(A__HEAD(a__from(0)), MARK(from(0))) MARK(head(from(cons(z0, z1)))) -> c19(A__HEAD(a__from(cons(mark(z0), z1))), MARK(from(cons(z0, z1)))) MARK(head(from(true))) -> c19(A__HEAD(a__from(true)), MARK(from(true))) MARK(head(from(false))) -> c19(A__HEAD(a__from(false)), MARK(from(false))) MARK(head(from(divides(z0, z1)))) -> c19(A__HEAD(a__from(divides(mark(z0), mark(z1)))), MARK(from(divides(z0, z1)))) MARK(head(from(x0))) -> c19(MARK(from(x0))) S tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, 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x1))) -> c17(MARK(filter(x0, x1))) MARK(head(primes)) -> c19(A__HEAD(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(head(primes)) -> c19(MARK(primes)) MARK(head(sieve(primes))) -> c19(A__HEAD(a__sieve(a__primes)), MARK(sieve(primes))) MARK(head(sieve(sieve(z0)))) -> c19(A__HEAD(a__sieve(a__sieve(mark(z0)))), MARK(sieve(sieve(z0)))) MARK(head(sieve(from(z0)))) -> c19(A__HEAD(a__sieve(a__from(mark(z0)))), MARK(sieve(from(z0)))) MARK(head(sieve(head(z0)))) -> c19(A__HEAD(a__sieve(a__head(mark(z0)))), MARK(sieve(head(z0)))) MARK(head(sieve(tail(z0)))) -> c19(A__HEAD(a__sieve(a__tail(mark(z0)))), MARK(sieve(tail(z0)))) MARK(head(sieve(if(z0, z1, z2)))) -> c19(A__HEAD(a__sieve(a__if(mark(z0), z1, z2))), MARK(sieve(if(z0, z1, z2)))) MARK(head(sieve(filter(z0, z1)))) -> c19(A__HEAD(a__sieve(a__filter(mark(z0), mark(z1)))), MARK(sieve(filter(z0, z1)))) MARK(head(sieve(s(z0)))) -> c19(A__HEAD(a__sieve(s(mark(z0)))), MARK(sieve(s(z0)))) MARK(head(sieve(cons(z0, z1)))) -> c19(A__HEAD(a__sieve(cons(mark(z0), z1))), MARK(sieve(cons(z0, z1)))) MARK(head(sieve(divides(z0, z1)))) -> c19(A__HEAD(a__sieve(divides(mark(z0), mark(z1)))), MARK(sieve(divides(z0, z1)))) MARK(head(sieve(x0))) -> c19(MARK(sieve(x0))) MARK(head(sieve(0))) -> c19(A__HEAD(a__sieve(0))) MARK(head(sieve(true))) -> c19(A__HEAD(a__sieve(true))) MARK(head(sieve(false))) -> c19(A__HEAD(a__sieve(false))) MARK(head(from(x0))) -> c19(A__HEAD(cons(mark(mark(x0)), from(s(mark(x0))))), MARK(from(x0))) MARK(head(from(x0))) -> c19(A__HEAD(from(mark(x0))), MARK(from(x0))) MARK(head(from(primes))) -> c19(A__HEAD(a__from(a__primes)), MARK(from(primes))) MARK(head(from(sieve(z0)))) -> c19(A__HEAD(a__from(a__sieve(mark(z0)))), MARK(from(sieve(z0)))) MARK(head(from(from(z0)))) -> c19(A__HEAD(a__from(a__from(mark(z0)))), MARK(from(from(z0)))) MARK(head(from(head(z0)))) -> c19(A__HEAD(a__from(a__head(mark(z0)))), MARK(from(head(z0)))) MARK(head(from(tail(z0)))) -> c19(A__HEAD(a__from(a__tail(mark(z0)))), MARK(from(tail(z0)))) MARK(head(from(if(z0, z1, z2)))) -> c19(A__HEAD(a__from(a__if(mark(z0), z1, z2))), MARK(from(if(z0, z1, z2)))) MARK(head(from(filter(z0, z1)))) -> c19(A__HEAD(a__from(a__filter(mark(z0), mark(z1)))), MARK(from(filter(z0, z1)))) MARK(head(from(s(z0)))) -> c19(A__HEAD(a__from(s(mark(z0)))), MARK(from(s(z0)))) MARK(head(from(0))) -> c19(A__HEAD(a__from(0)), MARK(from(0))) MARK(head(from(cons(z0, z1)))) -> c19(A__HEAD(a__from(cons(mark(z0), z1))), MARK(from(cons(z0, z1)))) MARK(head(from(true))) -> c19(A__HEAD(a__from(true)), MARK(from(true))) MARK(head(from(false))) -> c19(A__HEAD(a__from(false)), MARK(from(false))) MARK(head(from(divides(z0, z1)))) -> c19(A__HEAD(a__from(divides(mark(z0), mark(z1)))), MARK(from(divides(z0, z1)))) MARK(head(from(x0))) -> c19(MARK(from(x0))) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c18_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_1, c17_2, c17_1, c19_2, c19_1, c20_2, c20_1, c21_2, c21_1, c22_2, c22_1, c23_2, c23_1, c_2 ---------------------------------------- (137) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (138) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(head(z0))) -> c19(A__HEAD(a__head(mark(z0))), MARK(head(z0))) MARK(head(tail(z0))) -> c19(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(if(z0, z1, z2))) -> c19(A__HEAD(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(head(filter(z0, z1))) -> c19(A__HEAD(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(primes)) -> c20(A__TAIL(a__primes), MARK(primes)) MARK(tail(sieve(z0))) -> c20(A__TAIL(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(tail(from(z0))) -> c20(A__TAIL(a__from(mark(z0))), MARK(from(z0))) MARK(tail(head(z0))) -> c20(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c20(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(if(z0, z1, z2))) -> c20(A__TAIL(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(tail(filter(z0, z1))) -> c20(A__TAIL(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(primes, x1, x2)) -> c21(A__IF(a__primes, x1, x2), MARK(primes)) MARK(if(sieve(z0), x1, x2)) -> c21(A__IF(a__sieve(mark(z0)), x1, x2), MARK(sieve(z0))) MARK(if(from(z0), x1, x2)) -> c21(A__IF(a__from(mark(z0)), x1, x2), MARK(from(z0))) MARK(if(head(z0), x1, x2)) -> c21(A__IF(a__head(mark(z0)), x1, x2), MARK(head(z0))) MARK(if(tail(z0), x1, x2)) -> c21(A__IF(a__tail(mark(z0)), x1, x2), MARK(tail(z0))) MARK(if(if(z0, z1, z2), x1, x2)) -> c21(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(filter(z0, z1), x1, x2)) -> c21(A__IF(a__filter(mark(z0), mark(z1)), x1, x2), MARK(filter(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c22(A__FILTER(mark(x0), a__primes), MARK(x0)) MARK(filter(x0, sieve(z0))) -> c22(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, from(z0))) -> c22(A__FILTER(mark(x0), a__from(mark(z0))), MARK(x0)) MARK(filter(x0, head(z0))) -> c22(A__FILTER(mark(x0), a__head(mark(z0))), MARK(x0)) MARK(filter(x0, tail(z0))) -> c22(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(x0)) MARK(filter(x0, if(z0, z1, z2))) -> c22(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0)) MARK(filter(x0, filter(z0, z1))) -> c22(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(x0)) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(x0, s(z0))) -> c22(MARK(x0)) MARK(filter(x0, 0)) -> c22(MARK(x0)) MARK(filter(x0, true)) -> c22(MARK(x0)) MARK(filter(x0, false)) -> c22(MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), x1)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c23(A__FILTER(mark(x0), a__primes), MARK(primes)) MARK(filter(x0, sieve(z0))) -> c23(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, from(z0))) -> c23(A__FILTER(mark(x0), a__from(mark(z0))), MARK(from(z0))) MARK(filter(x0, head(z0))) -> c23(A__FILTER(mark(x0), a__head(mark(z0))), MARK(head(z0))) MARK(filter(x0, tail(z0))) -> c23(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(tail(z0))) MARK(filter(x0, if(z0, z1, z2))) -> c23(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(filter(x0, filter(z0, z1))) -> c23(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(filter(x0, cons(z0, z1))) -> c23(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(primes, x1)) -> c23(A__FILTER(a__primes, mark(x1)), MARK(x1)) MARK(filter(sieve(z0), x1)) -> c23(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(from(z0), x1)) -> c23(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(head(z0), x1)) -> c23(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(tail(z0), x1)) -> c23(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(if(z0, z1, z2), x1)) -> c23(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1)) MARK(filter(filter(z0, z1), x1)) -> c23(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(x1)) MARK(filter(s(z0), x1)) -> c23(A__FILTER(s(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(x0, s(z0))) -> c23(MARK(s(z0))) MARK(filter(x0, divides(z0, z1))) -> c23(MARK(divides(z0, z1))) MARK(filter(0, x1)) -> c23(MARK(x1)) MARK(filter(cons(z0, z1), x1)) -> c23(MARK(x1)) MARK(filter(true, x1)) -> c23(MARK(x1)) MARK(filter(false, x1)) -> c23(MARK(x1)) MARK(filter(divides(z0, z1), x1)) -> c23(MARK(x1)) A__PRIMES -> c(A__SIEVE(cons(s(mark(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(sieve(primes)) -> c17(MARK(primes)) MARK(sieve(sieve(primes))) -> c17(A__SIEVE(a__sieve(a__primes)), MARK(sieve(primes))) MARK(sieve(sieve(sieve(z0)))) -> c17(A__SIEVE(a__sieve(a__sieve(mark(z0)))), MARK(sieve(sieve(z0)))) MARK(sieve(sieve(from(z0)))) -> c17(A__SIEVE(a__sieve(a__from(mark(z0)))), MARK(sieve(from(z0)))) MARK(sieve(sieve(head(z0)))) -> c17(A__SIEVE(a__sieve(a__head(mark(z0)))), MARK(sieve(head(z0)))) MARK(sieve(sieve(tail(z0)))) -> c17(A__SIEVE(a__sieve(a__tail(mark(z0)))), MARK(sieve(tail(z0)))) MARK(sieve(sieve(if(z0, z1, z2)))) -> c17(A__SIEVE(a__sieve(a__if(mark(z0), z1, z2))), MARK(sieve(if(z0, z1, z2)))) MARK(sieve(sieve(filter(z0, z1)))) -> c17(A__SIEVE(a__sieve(a__filter(mark(z0), mark(z1)))), MARK(sieve(filter(z0, z1)))) MARK(sieve(sieve(s(z0)))) -> c17(A__SIEVE(a__sieve(s(mark(z0)))), MARK(sieve(s(z0)))) MARK(sieve(sieve(cons(z0, z1)))) -> c17(A__SIEVE(a__sieve(cons(mark(z0), z1))), MARK(sieve(cons(z0, z1)))) MARK(sieve(sieve(divides(z0, z1)))) -> c17(A__SIEVE(a__sieve(divides(mark(z0), mark(z1)))), MARK(sieve(divides(z0, z1)))) MARK(sieve(sieve(x0))) -> c17(MARK(sieve(x0))) MARK(sieve(sieve(0))) -> c17(A__SIEVE(a__sieve(0))) MARK(sieve(sieve(true))) -> c17(A__SIEVE(a__sieve(true))) MARK(sieve(sieve(false))) -> c17(A__SIEVE(a__sieve(false))) MARK(sieve(from(x0))) -> c17(A__SIEVE(cons(mark(mark(x0)), from(s(mark(x0))))), MARK(from(x0))) MARK(sieve(from(primes))) -> c17(A__SIEVE(a__from(a__primes)), MARK(from(primes))) MARK(sieve(from(sieve(z0)))) -> c17(A__SIEVE(a__from(a__sieve(mark(z0)))), MARK(from(sieve(z0)))) MARK(sieve(from(from(z0)))) -> c17(A__SIEVE(a__from(a__from(mark(z0)))), MARK(from(from(z0)))) MARK(sieve(from(head(z0)))) -> c17(A__SIEVE(a__from(a__head(mark(z0)))), MARK(from(head(z0)))) MARK(sieve(from(tail(z0)))) -> c17(A__SIEVE(a__from(a__tail(mark(z0)))), MARK(from(tail(z0)))) MARK(sieve(from(if(z0, z1, z2)))) -> c17(A__SIEVE(a__from(a__if(mark(z0), z1, z2))), MARK(from(if(z0, z1, z2)))) MARK(sieve(from(filter(z0, z1)))) -> c17(A__SIEVE(a__from(a__filter(mark(z0), mark(z1)))), MARK(from(filter(z0, z1)))) MARK(sieve(from(s(z0)))) -> c17(A__SIEVE(a__from(s(mark(z0)))), MARK(from(s(z0)))) MARK(sieve(from(0))) -> c17(A__SIEVE(a__from(0)), MARK(from(0))) MARK(sieve(from(cons(z0, z1)))) -> c17(A__SIEVE(a__from(cons(mark(z0), z1))), MARK(from(cons(z0, z1)))) MARK(sieve(from(true))) -> c17(A__SIEVE(a__from(true)), MARK(from(true))) MARK(sieve(from(false))) -> c17(A__SIEVE(a__from(false)), MARK(from(false))) MARK(sieve(from(divides(z0, z1)))) -> c17(A__SIEVE(a__from(divides(mark(z0), mark(z1)))), MARK(from(divides(z0, z1)))) MARK(sieve(from(x0))) -> c17(MARK(from(x0))) MARK(sieve(head(primes))) -> c17(A__SIEVE(a__head(a__primes)), MARK(head(primes))) MARK(sieve(head(sieve(z0)))) -> 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MARK(head(from(if(z0, z1, z2)))) -> c19(A__HEAD(a__from(a__if(mark(z0), z1, z2))), MARK(from(if(z0, z1, z2)))) MARK(head(from(filter(z0, z1)))) -> c19(A__HEAD(a__from(a__filter(mark(z0), mark(z1)))), MARK(from(filter(z0, z1)))) MARK(head(from(s(z0)))) -> c19(A__HEAD(a__from(s(mark(z0)))), MARK(from(s(z0)))) MARK(head(from(0))) -> c19(A__HEAD(a__from(0)), MARK(from(0))) MARK(head(from(cons(z0, z1)))) -> c19(A__HEAD(a__from(cons(mark(z0), z1))), MARK(from(cons(z0, z1)))) MARK(head(from(true))) -> c19(A__HEAD(a__from(true)), MARK(from(true))) MARK(head(from(false))) -> c19(A__HEAD(a__from(false)), MARK(from(false))) MARK(head(from(divides(z0, z1)))) -> c19(A__HEAD(a__from(divides(mark(z0), mark(z1)))), MARK(from(divides(z0, z1)))) MARK(head(from(x0))) -> c19(MARK(from(x0))) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c18_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_1, c17_2, c17_1, c19_2, c19_1, c20_2, c20_1, c21_2, c21_1, c22_2, c22_1, c23_2, c23_1, c_2 ---------------------------------------- (139) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(head(head(z0))) -> c19(A__HEAD(a__head(mark(z0))), MARK(head(z0))) by MARK(head(head(x0))) -> c19(A__HEAD(head(mark(x0))), MARK(head(x0))) MARK(head(head(primes))) -> c19(A__HEAD(a__head(a__primes)), MARK(head(primes))) MARK(head(head(sieve(z0)))) -> c19(A__HEAD(a__head(a__sieve(mark(z0)))), MARK(head(sieve(z0)))) MARK(head(head(from(z0)))) -> c19(A__HEAD(a__head(a__from(mark(z0)))), MARK(head(from(z0)))) MARK(head(head(head(z0)))) -> c19(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c19(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(if(z0, z1, z2)))) -> c19(A__HEAD(a__head(a__if(mark(z0), z1, z2))), MARK(head(if(z0, z1, z2)))) MARK(head(head(filter(z0, z1)))) -> c19(A__HEAD(a__head(a__filter(mark(z0), mark(z1)))), MARK(head(filter(z0, z1)))) MARK(head(head(s(z0)))) -> c19(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(0))) -> c19(A__HEAD(a__head(0)), MARK(head(0))) MARK(head(head(cons(z0, z1)))) -> c19(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(true))) -> c19(A__HEAD(a__head(true)), MARK(head(true))) MARK(head(head(false))) -> c19(A__HEAD(a__head(false)), MARK(head(false))) MARK(head(head(divides(z0, z1)))) -> c19(A__HEAD(a__head(divides(mark(z0), mark(z1)))), MARK(head(divides(z0, z1)))) ---------------------------------------- (140) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(tail(z0))) -> c19(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(if(z0, z1, z2))) -> c19(A__HEAD(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(head(filter(z0, z1))) -> c19(A__HEAD(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(primes)) -> c20(A__TAIL(a__primes), MARK(primes)) MARK(tail(sieve(z0))) -> c20(A__TAIL(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(tail(from(z0))) -> c20(A__TAIL(a__from(mark(z0))), MARK(from(z0))) MARK(tail(head(z0))) -> c20(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c20(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(if(z0, z1, z2))) -> c20(A__TAIL(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(tail(filter(z0, z1))) -> c20(A__TAIL(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(primes, x1, x2)) -> c21(A__IF(a__primes, x1, x2), MARK(primes)) MARK(if(sieve(z0), x1, x2)) -> c21(A__IF(a__sieve(mark(z0)), x1, x2), MARK(sieve(z0))) MARK(if(from(z0), x1, x2)) -> c21(A__IF(a__from(mark(z0)), x1, x2), MARK(from(z0))) MARK(if(head(z0), x1, x2)) -> c21(A__IF(a__head(mark(z0)), x1, x2), MARK(head(z0))) MARK(if(tail(z0), x1, x2)) -> c21(A__IF(a__tail(mark(z0)), x1, x2), MARK(tail(z0))) MARK(if(if(z0, z1, z2), x1, x2)) -> c21(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(filter(z0, z1), x1, x2)) -> c21(A__IF(a__filter(mark(z0), mark(z1)), x1, x2), MARK(filter(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c22(A__FILTER(mark(x0), a__primes), MARK(x0)) MARK(filter(x0, sieve(z0))) -> c22(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, from(z0))) -> c22(A__FILTER(mark(x0), a__from(mark(z0))), MARK(x0)) MARK(filter(x0, head(z0))) -> c22(A__FILTER(mark(x0), a__head(mark(z0))), MARK(x0)) MARK(filter(x0, tail(z0))) -> c22(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(x0)) MARK(filter(x0, if(z0, z1, z2))) -> c22(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0)) MARK(filter(x0, filter(z0, z1))) -> c22(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(x0)) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(x0, s(z0))) -> c22(MARK(x0)) MARK(filter(x0, 0)) -> c22(MARK(x0)) MARK(filter(x0, true)) -> c22(MARK(x0)) MARK(filter(x0, false)) -> c22(MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), x1)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c23(A__FILTER(mark(x0), a__primes), MARK(primes)) MARK(filter(x0, sieve(z0))) -> c23(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, from(z0))) -> c23(A__FILTER(mark(x0), a__from(mark(z0))), MARK(from(z0))) MARK(filter(x0, head(z0))) -> c23(A__FILTER(mark(x0), a__head(mark(z0))), MARK(head(z0))) MARK(filter(x0, tail(z0))) -> c23(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(tail(z0))) MARK(filter(x0, if(z0, z1, z2))) -> c23(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(filter(x0, filter(z0, z1))) -> c23(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(filter(x0, cons(z0, z1))) -> c23(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(primes, x1)) -> c23(A__FILTER(a__primes, mark(x1)), MARK(x1)) MARK(filter(sieve(z0), x1)) -> c23(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(from(z0), x1)) -> c23(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(head(z0), x1)) -> c23(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(tail(z0), x1)) -> c23(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(if(z0, z1, z2), x1)) -> c23(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1)) MARK(filter(filter(z0, z1), x1)) -> c23(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(x1)) MARK(filter(s(z0), x1)) -> c23(A__FILTER(s(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(x0, s(z0))) -> c23(MARK(s(z0))) MARK(filter(x0, divides(z0, z1))) -> c23(MARK(divides(z0, z1))) MARK(filter(0, x1)) -> c23(MARK(x1)) MARK(filter(cons(z0, z1), x1)) -> c23(MARK(x1)) MARK(filter(true, x1)) -> c23(MARK(x1)) MARK(filter(false, x1)) -> c23(MARK(x1)) MARK(filter(divides(z0, z1), x1)) -> c23(MARK(x1)) A__PRIMES -> c(A__SIEVE(cons(s(mark(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(sieve(primes)) -> c17(MARK(primes)) MARK(sieve(sieve(primes))) -> c17(A__SIEVE(a__sieve(a__primes)), MARK(sieve(primes))) MARK(sieve(sieve(sieve(z0)))) -> c17(A__SIEVE(a__sieve(a__sieve(mark(z0)))), MARK(sieve(sieve(z0)))) MARK(sieve(sieve(from(z0)))) -> c17(A__SIEVE(a__sieve(a__from(mark(z0)))), MARK(sieve(from(z0)))) MARK(sieve(sieve(head(z0)))) -> c17(A__SIEVE(a__sieve(a__head(mark(z0)))), MARK(sieve(head(z0)))) MARK(sieve(sieve(tail(z0)))) -> c17(A__SIEVE(a__sieve(a__tail(mark(z0)))), MARK(sieve(tail(z0)))) MARK(sieve(sieve(if(z0, z1, z2)))) -> c17(A__SIEVE(a__sieve(a__if(mark(z0), z1, z2))), MARK(sieve(if(z0, z1, z2)))) MARK(sieve(sieve(filter(z0, z1)))) -> c17(A__SIEVE(a__sieve(a__filter(mark(z0), mark(z1)))), MARK(sieve(filter(z0, z1)))) MARK(sieve(sieve(s(z0)))) -> c17(A__SIEVE(a__sieve(s(mark(z0)))), MARK(sieve(s(z0)))) MARK(sieve(sieve(cons(z0, z1)))) -> c17(A__SIEVE(a__sieve(cons(mark(z0), z1))), MARK(sieve(cons(z0, z1)))) MARK(sieve(sieve(divides(z0, z1)))) -> c17(A__SIEVE(a__sieve(divides(mark(z0), mark(z1)))), MARK(sieve(divides(z0, z1)))) MARK(sieve(sieve(x0))) -> c17(MARK(sieve(x0))) MARK(sieve(sieve(0))) -> c17(A__SIEVE(a__sieve(0))) MARK(sieve(sieve(true))) -> c17(A__SIEVE(a__sieve(true))) MARK(sieve(sieve(false))) -> c17(A__SIEVE(a__sieve(false))) MARK(sieve(from(x0))) -> c17(A__SIEVE(cons(mark(mark(x0)), from(s(mark(x0))))), MARK(from(x0))) MARK(sieve(from(primes))) -> c17(A__SIEVE(a__from(a__primes)), MARK(from(primes))) MARK(sieve(from(sieve(z0)))) -> c17(A__SIEVE(a__from(a__sieve(mark(z0)))), MARK(from(sieve(z0)))) MARK(sieve(from(from(z0)))) -> c17(A__SIEVE(a__from(a__from(mark(z0)))), MARK(from(from(z0)))) MARK(sieve(from(head(z0)))) -> c17(A__SIEVE(a__from(a__head(mark(z0)))), MARK(from(head(z0)))) MARK(sieve(from(tail(z0)))) -> c17(A__SIEVE(a__from(a__tail(mark(z0)))), MARK(from(tail(z0)))) MARK(sieve(from(if(z0, z1, z2)))) -> c17(A__SIEVE(a__from(a__if(mark(z0), z1, z2))), MARK(from(if(z0, z1, z2)))) MARK(sieve(from(filter(z0, z1)))) -> c17(A__SIEVE(a__from(a__filter(mark(z0), mark(z1)))), MARK(from(filter(z0, z1)))) MARK(sieve(from(s(z0)))) -> c17(A__SIEVE(a__from(s(mark(z0)))), MARK(from(s(z0)))) MARK(sieve(from(0))) -> c17(A__SIEVE(a__from(0)), MARK(from(0))) MARK(sieve(from(cons(z0, z1)))) -> c17(A__SIEVE(a__from(cons(mark(z0), z1))), MARK(from(cons(z0, z1)))) MARK(sieve(from(true))) -> c17(A__SIEVE(a__from(true)), MARK(from(true))) MARK(sieve(from(false))) -> c17(A__SIEVE(a__from(false)), MARK(from(false))) MARK(sieve(from(divides(z0, z1)))) -> c17(A__SIEVE(a__from(divides(mark(z0), mark(z1)))), MARK(from(divides(z0, 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c19(A__HEAD(a__from(a__primes)), MARK(from(primes))) MARK(head(from(sieve(z0)))) -> c19(A__HEAD(a__from(a__sieve(mark(z0)))), MARK(from(sieve(z0)))) MARK(head(from(from(z0)))) -> c19(A__HEAD(a__from(a__from(mark(z0)))), MARK(from(from(z0)))) MARK(head(from(head(z0)))) -> c19(A__HEAD(a__from(a__head(mark(z0)))), MARK(from(head(z0)))) MARK(head(from(tail(z0)))) -> c19(A__HEAD(a__from(a__tail(mark(z0)))), MARK(from(tail(z0)))) MARK(head(from(if(z0, z1, z2)))) -> c19(A__HEAD(a__from(a__if(mark(z0), z1, z2))), MARK(from(if(z0, z1, z2)))) MARK(head(from(filter(z0, z1)))) -> c19(A__HEAD(a__from(a__filter(mark(z0), mark(z1)))), MARK(from(filter(z0, z1)))) MARK(head(from(s(z0)))) -> c19(A__HEAD(a__from(s(mark(z0)))), MARK(from(s(z0)))) MARK(head(from(0))) -> c19(A__HEAD(a__from(0)), MARK(from(0))) MARK(head(from(cons(z0, z1)))) -> c19(A__HEAD(a__from(cons(mark(z0), z1))), MARK(from(cons(z0, z1)))) MARK(head(from(true))) -> c19(A__HEAD(a__from(true)), MARK(from(true))) MARK(head(from(false))) -> c19(A__HEAD(a__from(false)), MARK(from(false))) MARK(head(from(divides(z0, z1)))) -> c19(A__HEAD(a__from(divides(mark(z0), mark(z1)))), MARK(from(divides(z0, z1)))) MARK(head(from(x0))) -> c19(MARK(from(x0))) MARK(head(head(x0))) -> c19(A__HEAD(head(mark(x0))), MARK(head(x0))) MARK(head(head(primes))) -> c19(A__HEAD(a__head(a__primes)), MARK(head(primes))) MARK(head(head(sieve(z0)))) -> c19(A__HEAD(a__head(a__sieve(mark(z0)))), MARK(head(sieve(z0)))) MARK(head(head(from(z0)))) -> c19(A__HEAD(a__head(a__from(mark(z0)))), MARK(head(from(z0)))) MARK(head(head(head(z0)))) -> c19(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c19(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(if(z0, z1, z2)))) -> c19(A__HEAD(a__head(a__if(mark(z0), z1, z2))), MARK(head(if(z0, z1, z2)))) MARK(head(head(filter(z0, z1)))) -> c19(A__HEAD(a__head(a__filter(mark(z0), mark(z1)))), MARK(head(filter(z0, z1)))) MARK(head(head(s(z0)))) -> c19(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(0))) -> c19(A__HEAD(a__head(0)), MARK(head(0))) MARK(head(head(cons(z0, z1)))) -> c19(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(true))) -> c19(A__HEAD(a__head(true)), MARK(head(true))) MARK(head(head(false))) -> c19(A__HEAD(a__head(false)), MARK(head(false))) MARK(head(head(divides(z0, z1)))) -> c19(A__HEAD(a__head(divides(mark(z0), mark(z1)))), MARK(head(divides(z0, z1)))) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c18_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_1, c17_2, c17_1, c19_2, c19_1, c20_2, c20_1, c21_2, c21_1, c22_2, c22_1, c23_2, c23_1, c_2 ---------------------------------------- (141) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing tuple parts ---------------------------------------- (142) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(tail(z0))) -> c19(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) MARK(head(if(z0, z1, z2))) -> c19(A__HEAD(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(head(filter(z0, z1))) -> c19(A__HEAD(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(primes)) -> c20(A__TAIL(a__primes), MARK(primes)) MARK(tail(sieve(z0))) -> c20(A__TAIL(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(tail(from(z0))) -> c20(A__TAIL(a__from(mark(z0))), MARK(from(z0))) MARK(tail(head(z0))) -> c20(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c20(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(if(z0, z1, z2))) -> c20(A__TAIL(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(tail(filter(z0, z1))) -> c20(A__TAIL(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(primes, x1, x2)) -> c21(A__IF(a__primes, x1, x2), MARK(primes)) MARK(if(sieve(z0), x1, x2)) -> c21(A__IF(a__sieve(mark(z0)), x1, x2), MARK(sieve(z0))) MARK(if(from(z0), x1, x2)) -> c21(A__IF(a__from(mark(z0)), x1, x2), MARK(from(z0))) MARK(if(head(z0), x1, x2)) -> c21(A__IF(a__head(mark(z0)), x1, x2), MARK(head(z0))) MARK(if(tail(z0), x1, x2)) -> c21(A__IF(a__tail(mark(z0)), x1, x2), MARK(tail(z0))) MARK(if(if(z0, z1, z2), x1, x2)) -> c21(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(filter(z0, z1), x1, x2)) -> c21(A__IF(a__filter(mark(z0), mark(z1)), x1, x2), MARK(filter(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c22(A__FILTER(mark(x0), a__primes), MARK(x0)) MARK(filter(x0, sieve(z0))) -> c22(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, from(z0))) -> c22(A__FILTER(mark(x0), a__from(mark(z0))), MARK(x0)) MARK(filter(x0, head(z0))) -> c22(A__FILTER(mark(x0), a__head(mark(z0))), MARK(x0)) MARK(filter(x0, tail(z0))) -> c22(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(x0)) MARK(filter(x0, if(z0, z1, z2))) -> c22(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0)) MARK(filter(x0, filter(z0, z1))) -> c22(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(x0)) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(x0, s(z0))) -> c22(MARK(x0)) MARK(filter(x0, 0)) -> c22(MARK(x0)) MARK(filter(x0, true)) -> c22(MARK(x0)) MARK(filter(x0, false)) -> c22(MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), x1)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c23(A__FILTER(mark(x0), a__primes), MARK(primes)) MARK(filter(x0, sieve(z0))) -> c23(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, from(z0))) -> c23(A__FILTER(mark(x0), a__from(mark(z0))), MARK(from(z0))) MARK(filter(x0, head(z0))) -> c23(A__FILTER(mark(x0), a__head(mark(z0))), MARK(head(z0))) MARK(filter(x0, tail(z0))) -> c23(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(tail(z0))) MARK(filter(x0, if(z0, z1, z2))) -> c23(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(filter(x0, filter(z0, z1))) -> c23(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(filter(x0, cons(z0, z1))) -> c23(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(primes, x1)) -> c23(A__FILTER(a__primes, mark(x1)), MARK(x1)) MARK(filter(sieve(z0), x1)) -> c23(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(from(z0), x1)) -> c23(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(head(z0), x1)) -> c23(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(tail(z0), x1)) -> c23(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(if(z0, z1, z2), x1)) -> c23(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1)) MARK(filter(filter(z0, z1), x1)) -> c23(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(x1)) MARK(filter(s(z0), x1)) -> c23(A__FILTER(s(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(x0, s(z0))) -> c23(MARK(s(z0))) MARK(filter(x0, divides(z0, z1))) -> c23(MARK(divides(z0, z1))) MARK(filter(0, x1)) -> c23(MARK(x1)) MARK(filter(cons(z0, z1), x1)) -> c23(MARK(x1)) MARK(filter(true, x1)) -> c23(MARK(x1)) MARK(filter(false, x1)) -> c23(MARK(x1)) MARK(filter(divides(z0, z1), x1)) -> c23(MARK(x1)) A__PRIMES -> c(A__SIEVE(cons(s(mark(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(sieve(primes)) -> c17(MARK(primes)) MARK(sieve(sieve(primes))) -> c17(A__SIEVE(a__sieve(a__primes)), MARK(sieve(primes))) MARK(sieve(sieve(sieve(z0)))) -> c17(A__SIEVE(a__sieve(a__sieve(mark(z0)))), MARK(sieve(sieve(z0)))) MARK(sieve(sieve(from(z0)))) -> c17(A__SIEVE(a__sieve(a__from(mark(z0)))), MARK(sieve(from(z0)))) MARK(sieve(sieve(head(z0)))) -> c17(A__SIEVE(a__sieve(a__head(mark(z0)))), MARK(sieve(head(z0)))) MARK(sieve(sieve(tail(z0)))) -> c17(A__SIEVE(a__sieve(a__tail(mark(z0)))), MARK(sieve(tail(z0)))) MARK(sieve(sieve(if(z0, z1, z2)))) -> c17(A__SIEVE(a__sieve(a__if(mark(z0), z1, z2))), MARK(sieve(if(z0, z1, z2)))) MARK(sieve(sieve(filter(z0, z1)))) -> c17(A__SIEVE(a__sieve(a__filter(mark(z0), mark(z1)))), MARK(sieve(filter(z0, z1)))) MARK(sieve(sieve(s(z0)))) -> c17(A__SIEVE(a__sieve(s(mark(z0)))), MARK(sieve(s(z0)))) MARK(sieve(sieve(cons(z0, z1)))) -> c17(A__SIEVE(a__sieve(cons(mark(z0), z1))), MARK(sieve(cons(z0, z1)))) MARK(sieve(sieve(divides(z0, z1)))) -> c17(A__SIEVE(a__sieve(divides(mark(z0), mark(z1)))), MARK(sieve(divides(z0, z1)))) MARK(sieve(sieve(x0))) -> c17(MARK(sieve(x0))) MARK(sieve(sieve(0))) -> c17(A__SIEVE(a__sieve(0))) MARK(sieve(sieve(true))) -> 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MARK(head(sieve(x0))) -> c19(MARK(sieve(x0))) MARK(head(sieve(0))) -> c19(A__HEAD(a__sieve(0))) MARK(head(sieve(true))) -> c19(A__HEAD(a__sieve(true))) MARK(head(sieve(false))) -> c19(A__HEAD(a__sieve(false))) MARK(head(from(x0))) -> c19(A__HEAD(cons(mark(mark(x0)), from(s(mark(x0))))), MARK(from(x0))) MARK(head(from(primes))) -> c19(A__HEAD(a__from(a__primes)), MARK(from(primes))) MARK(head(from(sieve(z0)))) -> c19(A__HEAD(a__from(a__sieve(mark(z0)))), MARK(from(sieve(z0)))) MARK(head(from(from(z0)))) -> c19(A__HEAD(a__from(a__from(mark(z0)))), MARK(from(from(z0)))) MARK(head(from(head(z0)))) -> c19(A__HEAD(a__from(a__head(mark(z0)))), MARK(from(head(z0)))) MARK(head(from(tail(z0)))) -> c19(A__HEAD(a__from(a__tail(mark(z0)))), MARK(from(tail(z0)))) MARK(head(from(if(z0, z1, z2)))) -> c19(A__HEAD(a__from(a__if(mark(z0), z1, z2))), MARK(from(if(z0, z1, z2)))) MARK(head(from(filter(z0, z1)))) -> c19(A__HEAD(a__from(a__filter(mark(z0), mark(z1)))), MARK(from(filter(z0, z1)))) MARK(head(from(s(z0)))) -> c19(A__HEAD(a__from(s(mark(z0)))), MARK(from(s(z0)))) MARK(head(from(0))) -> c19(A__HEAD(a__from(0)), MARK(from(0))) MARK(head(from(cons(z0, z1)))) -> c19(A__HEAD(a__from(cons(mark(z0), z1))), MARK(from(cons(z0, z1)))) MARK(head(from(true))) -> c19(A__HEAD(a__from(true)), MARK(from(true))) MARK(head(from(false))) -> c19(A__HEAD(a__from(false)), MARK(from(false))) MARK(head(from(divides(z0, z1)))) -> c19(A__HEAD(a__from(divides(mark(z0), mark(z1)))), MARK(from(divides(z0, z1)))) MARK(head(from(x0))) -> c19(MARK(from(x0))) MARK(head(head(primes))) -> c19(A__HEAD(a__head(a__primes)), MARK(head(primes))) MARK(head(head(sieve(z0)))) -> c19(A__HEAD(a__head(a__sieve(mark(z0)))), MARK(head(sieve(z0)))) MARK(head(head(from(z0)))) -> c19(A__HEAD(a__head(a__from(mark(z0)))), MARK(head(from(z0)))) MARK(head(head(head(z0)))) -> c19(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c19(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(if(z0, z1, z2)))) -> c19(A__HEAD(a__head(a__if(mark(z0), z1, z2))), MARK(head(if(z0, z1, z2)))) MARK(head(head(filter(z0, z1)))) -> c19(A__HEAD(a__head(a__filter(mark(z0), mark(z1)))), MARK(head(filter(z0, z1)))) MARK(head(head(s(z0)))) -> c19(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(cons(z0, z1)))) -> c19(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(divides(z0, z1)))) -> c19(A__HEAD(a__head(divides(mark(z0), mark(z1)))), MARK(head(divides(z0, z1)))) MARK(head(head(x0))) -> c19(MARK(head(x0))) MARK(head(head(0))) -> c19(A__HEAD(a__head(0))) MARK(head(head(true))) -> c19(A__HEAD(a__head(true))) MARK(head(head(false))) -> c19(A__HEAD(a__head(false))) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c18_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_1, c17_2, c17_1, c19_2, c19_1, c20_2, c20_1, c21_2, c21_1, c22_2, c22_1, c23_2, c23_1, c_2 ---------------------------------------- (143) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(head(tail(z0))) -> c19(A__HEAD(a__tail(mark(z0))), MARK(tail(z0))) by MARK(head(tail(x0))) -> c19(A__HEAD(tail(mark(x0))), MARK(tail(x0))) MARK(head(tail(primes))) -> c19(A__HEAD(a__tail(a__primes)), MARK(tail(primes))) MARK(head(tail(sieve(z0)))) -> c19(A__HEAD(a__tail(a__sieve(mark(z0)))), MARK(tail(sieve(z0)))) MARK(head(tail(from(z0)))) -> c19(A__HEAD(a__tail(a__from(mark(z0)))), MARK(tail(from(z0)))) MARK(head(tail(head(z0)))) -> c19(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c19(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(if(z0, z1, z2)))) -> c19(A__HEAD(a__tail(a__if(mark(z0), z1, z2))), MARK(tail(if(z0, z1, z2)))) MARK(head(tail(filter(z0, z1)))) -> c19(A__HEAD(a__tail(a__filter(mark(z0), mark(z1)))), MARK(tail(filter(z0, z1)))) MARK(head(tail(s(z0)))) -> c19(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(0))) -> c19(A__HEAD(a__tail(0)), MARK(tail(0))) MARK(head(tail(cons(z0, z1)))) -> c19(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(true))) -> c19(A__HEAD(a__tail(true)), MARK(tail(true))) MARK(head(tail(false))) -> c19(A__HEAD(a__tail(false)), MARK(tail(false))) MARK(head(tail(divides(z0, z1)))) -> c19(A__HEAD(a__tail(divides(mark(z0), mark(z1)))), MARK(tail(divides(z0, z1)))) ---------------------------------------- (144) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(if(z0, z1, z2))) -> c19(A__HEAD(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(head(filter(z0, z1))) -> c19(A__HEAD(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(primes)) -> c20(A__TAIL(a__primes), MARK(primes)) MARK(tail(sieve(z0))) -> c20(A__TAIL(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(tail(from(z0))) -> c20(A__TAIL(a__from(mark(z0))), MARK(from(z0))) MARK(tail(head(z0))) -> c20(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c20(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(if(z0, z1, z2))) -> c20(A__TAIL(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(tail(filter(z0, z1))) -> c20(A__TAIL(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(primes, x1, x2)) -> c21(A__IF(a__primes, x1, x2), MARK(primes)) MARK(if(sieve(z0), x1, x2)) -> c21(A__IF(a__sieve(mark(z0)), x1, x2), MARK(sieve(z0))) MARK(if(from(z0), x1, x2)) -> c21(A__IF(a__from(mark(z0)), x1, x2), MARK(from(z0))) MARK(if(head(z0), x1, x2)) -> c21(A__IF(a__head(mark(z0)), x1, x2), MARK(head(z0))) MARK(if(tail(z0), x1, x2)) -> c21(A__IF(a__tail(mark(z0)), x1, x2), MARK(tail(z0))) MARK(if(if(z0, z1, z2), x1, x2)) -> c21(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(filter(z0, z1), x1, x2)) -> c21(A__IF(a__filter(mark(z0), mark(z1)), x1, x2), MARK(filter(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c22(A__FILTER(mark(x0), a__primes), MARK(x0)) MARK(filter(x0, sieve(z0))) -> c22(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, from(z0))) -> c22(A__FILTER(mark(x0), a__from(mark(z0))), MARK(x0)) MARK(filter(x0, head(z0))) -> c22(A__FILTER(mark(x0), a__head(mark(z0))), MARK(x0)) MARK(filter(x0, tail(z0))) -> c22(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(x0)) MARK(filter(x0, if(z0, z1, z2))) -> c22(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0)) MARK(filter(x0, filter(z0, z1))) -> c22(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(x0)) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(x0, s(z0))) -> c22(MARK(x0)) MARK(filter(x0, 0)) -> c22(MARK(x0)) MARK(filter(x0, true)) -> c22(MARK(x0)) MARK(filter(x0, false)) -> c22(MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), x1)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c23(A__FILTER(mark(x0), a__primes), MARK(primes)) MARK(filter(x0, sieve(z0))) -> c23(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, from(z0))) -> c23(A__FILTER(mark(x0), a__from(mark(z0))), MARK(from(z0))) MARK(filter(x0, head(z0))) -> c23(A__FILTER(mark(x0), a__head(mark(z0))), MARK(head(z0))) MARK(filter(x0, tail(z0))) -> c23(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(tail(z0))) MARK(filter(x0, if(z0, z1, z2))) -> c23(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(filter(x0, filter(z0, z1))) -> c23(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(filter(x0, cons(z0, z1))) -> c23(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(primes, x1)) -> c23(A__FILTER(a__primes, mark(x1)), MARK(x1)) MARK(filter(sieve(z0), x1)) -> c23(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(from(z0), x1)) -> c23(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(head(z0), x1)) -> c23(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(tail(z0), x1)) -> c23(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(if(z0, z1, z2), x1)) -> c23(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1)) MARK(filter(filter(z0, z1), x1)) -> c23(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(x1)) MARK(filter(s(z0), x1)) -> c23(A__FILTER(s(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(x0, s(z0))) -> c23(MARK(s(z0))) MARK(filter(x0, divides(z0, z1))) -> c23(MARK(divides(z0, z1))) MARK(filter(0, x1)) -> c23(MARK(x1)) MARK(filter(cons(z0, z1), x1)) -> c23(MARK(x1)) MARK(filter(true, x1)) -> c23(MARK(x1)) MARK(filter(false, x1)) -> c23(MARK(x1)) MARK(filter(divides(z0, z1), x1)) -> c23(MARK(x1)) A__PRIMES -> c(A__SIEVE(cons(s(mark(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(sieve(primes)) -> c17(MARK(primes)) MARK(sieve(sieve(primes))) -> c17(A__SIEVE(a__sieve(a__primes)), MARK(sieve(primes))) MARK(sieve(sieve(sieve(z0)))) -> c17(A__SIEVE(a__sieve(a__sieve(mark(z0)))), MARK(sieve(sieve(z0)))) MARK(sieve(sieve(from(z0)))) -> c17(A__SIEVE(a__sieve(a__from(mark(z0)))), MARK(sieve(from(z0)))) MARK(sieve(sieve(head(z0)))) -> c17(A__SIEVE(a__sieve(a__head(mark(z0)))), MARK(sieve(head(z0)))) MARK(sieve(sieve(tail(z0)))) -> c17(A__SIEVE(a__sieve(a__tail(mark(z0)))), MARK(sieve(tail(z0)))) MARK(sieve(sieve(if(z0, z1, z2)))) -> c17(A__SIEVE(a__sieve(a__if(mark(z0), z1, z2))), MARK(sieve(if(z0, z1, z2)))) MARK(sieve(sieve(filter(z0, z1)))) -> c17(A__SIEVE(a__sieve(a__filter(mark(z0), mark(z1)))), MARK(sieve(filter(z0, z1)))) MARK(sieve(sieve(s(z0)))) -> c17(A__SIEVE(a__sieve(s(mark(z0)))), MARK(sieve(s(z0)))) MARK(sieve(sieve(cons(z0, z1)))) -> c17(A__SIEVE(a__sieve(cons(mark(z0), z1))), MARK(sieve(cons(z0, z1)))) MARK(sieve(sieve(divides(z0, z1)))) -> c17(A__SIEVE(a__sieve(divides(mark(z0), mark(z1)))), MARK(sieve(divides(z0, z1)))) MARK(sieve(sieve(x0))) -> c17(MARK(sieve(x0))) MARK(sieve(sieve(0))) -> c17(A__SIEVE(a__sieve(0))) MARK(sieve(sieve(true))) -> c17(A__SIEVE(a__sieve(true))) MARK(sieve(sieve(false))) -> c17(A__SIEVE(a__sieve(false))) MARK(sieve(from(x0))) -> c17(A__SIEVE(cons(mark(mark(x0)), from(s(mark(x0))))), MARK(from(x0))) MARK(sieve(from(primes))) -> c17(A__SIEVE(a__from(a__primes)), MARK(from(primes))) MARK(sieve(from(sieve(z0)))) -> c17(A__SIEVE(a__from(a__sieve(mark(z0)))), MARK(from(sieve(z0)))) MARK(sieve(from(from(z0)))) -> c17(A__SIEVE(a__from(a__from(mark(z0)))), MARK(from(from(z0)))) MARK(sieve(from(head(z0)))) -> c17(A__SIEVE(a__from(a__head(mark(z0)))), MARK(from(head(z0)))) MARK(sieve(from(tail(z0)))) -> c17(A__SIEVE(a__from(a__tail(mark(z0)))), MARK(from(tail(z0)))) MARK(sieve(from(if(z0, z1, z2)))) -> c17(A__SIEVE(a__from(a__if(mark(z0), z1, z2))), MARK(from(if(z0, z1, z2)))) MARK(sieve(from(filter(z0, z1)))) -> c17(A__SIEVE(a__from(a__filter(mark(z0), mark(z1)))), MARK(from(filter(z0, z1)))) MARK(sieve(from(s(z0)))) -> c17(A__SIEVE(a__from(s(mark(z0)))), MARK(from(s(z0)))) MARK(sieve(from(0))) -> c17(A__SIEVE(a__from(0)), MARK(from(0))) MARK(sieve(from(cons(z0, z1)))) -> c17(A__SIEVE(a__from(cons(mark(z0), z1))), MARK(from(cons(z0, z1)))) MARK(sieve(from(true))) -> c17(A__SIEVE(a__from(true)), MARK(from(true))) MARK(sieve(from(false))) -> c17(A__SIEVE(a__from(false)), MARK(from(false))) MARK(sieve(from(divides(z0, z1)))) -> c17(A__SIEVE(a__from(divides(mark(z0), mark(z1)))), MARK(from(divides(z0, z1)))) MARK(sieve(from(x0))) -> c17(MARK(from(x0))) MARK(sieve(head(primes))) -> 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c17(A__SIEVE(a__filter(a__sieve(mark(z0)), mark(x1))), MARK(filter(sieve(z0), x1))) MARK(sieve(filter(from(z0), x1))) -> c17(A__SIEVE(a__filter(a__from(mark(z0)), mark(x1))), MARK(filter(from(z0), x1))) MARK(sieve(filter(head(z0), x1))) -> c17(A__SIEVE(a__filter(a__head(mark(z0)), mark(x1))), MARK(filter(head(z0), x1))) MARK(sieve(filter(tail(z0), x1))) -> c17(A__SIEVE(a__filter(a__tail(mark(z0)), mark(x1))), MARK(filter(tail(z0), x1))) MARK(sieve(filter(if(z0, z1, z2), x1))) -> c17(A__SIEVE(a__filter(a__if(mark(z0), z1, z2), mark(x1))), MARK(filter(if(z0, z1, z2), x1))) MARK(sieve(filter(filter(z0, z1), x1))) -> c17(A__SIEVE(a__filter(a__filter(mark(z0), mark(z1)), mark(x1))), MARK(filter(filter(z0, z1), x1))) MARK(sieve(filter(s(z0), x1))) -> c17(A__SIEVE(a__filter(s(mark(z0)), mark(x1))), MARK(filter(s(z0), x1))) MARK(sieve(filter(0, x1))) -> c17(A__SIEVE(a__filter(0, mark(x1))), MARK(filter(0, x1))) MARK(sieve(filter(cons(z0, z1), x1))) -> c17(A__SIEVE(a__filter(cons(mark(z0), z1), mark(x1))), MARK(filter(cons(z0, z1), x1))) MARK(sieve(filter(true, x1))) -> c17(A__SIEVE(a__filter(true, mark(x1))), MARK(filter(true, x1))) MARK(sieve(filter(false, x1))) -> c17(A__SIEVE(a__filter(false, mark(x1))), MARK(filter(false, x1))) MARK(sieve(filter(divides(z0, z1), x1))) -> c17(A__SIEVE(a__filter(divides(mark(z0), mark(z1)), mark(x1))), MARK(filter(divides(z0, z1), x1))) MARK(sieve(filter(x0, x1))) -> c17(MARK(filter(x0, x1))) MARK(head(primes)) -> c19(A__HEAD(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(head(primes)) -> c19(MARK(primes)) MARK(head(sieve(primes))) -> c19(A__HEAD(a__sieve(a__primes)), MARK(sieve(primes))) MARK(head(sieve(sieve(z0)))) -> c19(A__HEAD(a__sieve(a__sieve(mark(z0)))), MARK(sieve(sieve(z0)))) MARK(head(sieve(from(z0)))) -> c19(A__HEAD(a__sieve(a__from(mark(z0)))), MARK(sieve(from(z0)))) MARK(head(sieve(head(z0)))) -> c19(A__HEAD(a__sieve(a__head(mark(z0)))), MARK(sieve(head(z0)))) MARK(head(sieve(tail(z0)))) -> c19(A__HEAD(a__sieve(a__tail(mark(z0)))), MARK(sieve(tail(z0)))) MARK(head(sieve(if(z0, z1, z2)))) -> c19(A__HEAD(a__sieve(a__if(mark(z0), z1, z2))), MARK(sieve(if(z0, z1, z2)))) MARK(head(sieve(filter(z0, z1)))) -> c19(A__HEAD(a__sieve(a__filter(mark(z0), mark(z1)))), MARK(sieve(filter(z0, z1)))) MARK(head(sieve(s(z0)))) -> c19(A__HEAD(a__sieve(s(mark(z0)))), MARK(sieve(s(z0)))) MARK(head(sieve(cons(z0, z1)))) -> c19(A__HEAD(a__sieve(cons(mark(z0), z1))), MARK(sieve(cons(z0, z1)))) MARK(head(sieve(divides(z0, z1)))) -> c19(A__HEAD(a__sieve(divides(mark(z0), mark(z1)))), MARK(sieve(divides(z0, z1)))) MARK(head(sieve(x0))) -> c19(MARK(sieve(x0))) MARK(head(sieve(0))) -> c19(A__HEAD(a__sieve(0))) MARK(head(sieve(true))) -> c19(A__HEAD(a__sieve(true))) MARK(head(sieve(false))) -> c19(A__HEAD(a__sieve(false))) MARK(head(from(x0))) -> c19(A__HEAD(cons(mark(mark(x0)), from(s(mark(x0))))), MARK(from(x0))) MARK(head(from(primes))) -> c19(A__HEAD(a__from(a__primes)), MARK(from(primes))) MARK(head(from(sieve(z0)))) -> c19(A__HEAD(a__from(a__sieve(mark(z0)))), MARK(from(sieve(z0)))) MARK(head(from(from(z0)))) -> c19(A__HEAD(a__from(a__from(mark(z0)))), MARK(from(from(z0)))) MARK(head(from(head(z0)))) -> c19(A__HEAD(a__from(a__head(mark(z0)))), MARK(from(head(z0)))) MARK(head(from(tail(z0)))) -> c19(A__HEAD(a__from(a__tail(mark(z0)))), MARK(from(tail(z0)))) MARK(head(from(if(z0, z1, z2)))) -> c19(A__HEAD(a__from(a__if(mark(z0), z1, z2))), MARK(from(if(z0, z1, z2)))) MARK(head(from(filter(z0, z1)))) -> c19(A__HEAD(a__from(a__filter(mark(z0), mark(z1)))), MARK(from(filter(z0, z1)))) MARK(head(from(s(z0)))) -> c19(A__HEAD(a__from(s(mark(z0)))), MARK(from(s(z0)))) MARK(head(from(0))) -> c19(A__HEAD(a__from(0)), MARK(from(0))) MARK(head(from(cons(z0, z1)))) -> c19(A__HEAD(a__from(cons(mark(z0), z1))), MARK(from(cons(z0, z1)))) MARK(head(from(true))) -> c19(A__HEAD(a__from(true)), MARK(from(true))) MARK(head(from(false))) -> c19(A__HEAD(a__from(false)), MARK(from(false))) MARK(head(from(divides(z0, z1)))) -> c19(A__HEAD(a__from(divides(mark(z0), mark(z1)))), MARK(from(divides(z0, z1)))) MARK(head(from(x0))) -> c19(MARK(from(x0))) MARK(head(head(primes))) -> c19(A__HEAD(a__head(a__primes)), MARK(head(primes))) MARK(head(head(sieve(z0)))) -> c19(A__HEAD(a__head(a__sieve(mark(z0)))), MARK(head(sieve(z0)))) MARK(head(head(from(z0)))) -> c19(A__HEAD(a__head(a__from(mark(z0)))), MARK(head(from(z0)))) MARK(head(head(head(z0)))) -> c19(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c19(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(if(z0, z1, z2)))) -> c19(A__HEAD(a__head(a__if(mark(z0), z1, z2))), MARK(head(if(z0, z1, z2)))) MARK(head(head(filter(z0, z1)))) -> c19(A__HEAD(a__head(a__filter(mark(z0), mark(z1)))), MARK(head(filter(z0, z1)))) MARK(head(head(s(z0)))) -> c19(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(cons(z0, z1)))) -> c19(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(divides(z0, z1)))) -> c19(A__HEAD(a__head(divides(mark(z0), mark(z1)))), MARK(head(divides(z0, z1)))) MARK(head(head(x0))) -> c19(MARK(head(x0))) MARK(head(head(0))) -> c19(A__HEAD(a__head(0))) MARK(head(head(true))) -> c19(A__HEAD(a__head(true))) MARK(head(head(false))) -> c19(A__HEAD(a__head(false))) MARK(head(tail(x0))) -> c19(A__HEAD(tail(mark(x0))), MARK(tail(x0))) MARK(head(tail(primes))) -> c19(A__HEAD(a__tail(a__primes)), MARK(tail(primes))) MARK(head(tail(sieve(z0)))) -> c19(A__HEAD(a__tail(a__sieve(mark(z0)))), MARK(tail(sieve(z0)))) MARK(head(tail(from(z0)))) -> c19(A__HEAD(a__tail(a__from(mark(z0)))), MARK(tail(from(z0)))) MARK(head(tail(head(z0)))) -> c19(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c19(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(if(z0, z1, z2)))) -> c19(A__HEAD(a__tail(a__if(mark(z0), z1, z2))), MARK(tail(if(z0, z1, z2)))) MARK(head(tail(filter(z0, z1)))) -> c19(A__HEAD(a__tail(a__filter(mark(z0), mark(z1)))), MARK(tail(filter(z0, z1)))) MARK(head(tail(s(z0)))) -> c19(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(0))) -> c19(A__HEAD(a__tail(0)), MARK(tail(0))) MARK(head(tail(cons(z0, z1)))) -> c19(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(true))) -> c19(A__HEAD(a__tail(true)), MARK(tail(true))) MARK(head(tail(false))) -> c19(A__HEAD(a__tail(false)), MARK(tail(false))) MARK(head(tail(divides(z0, z1)))) -> c19(A__HEAD(a__tail(divides(mark(z0), mark(z1)))), MARK(tail(divides(z0, z1)))) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c18_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_1, c17_2, c17_1, c19_2, c19_1, c20_2, c20_1, c21_2, c21_1, c22_2, c22_1, c23_2, c23_1, c_2 ---------------------------------------- (145) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing tuple parts ---------------------------------------- (146) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(if(z0, z1, z2))) -> c19(A__HEAD(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(head(filter(z0, z1))) -> c19(A__HEAD(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(primes)) -> c20(A__TAIL(a__primes), MARK(primes)) MARK(tail(sieve(z0))) -> c20(A__TAIL(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(tail(from(z0))) -> c20(A__TAIL(a__from(mark(z0))), MARK(from(z0))) MARK(tail(head(z0))) -> c20(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c20(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(if(z0, z1, z2))) -> c20(A__TAIL(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(tail(filter(z0, z1))) -> c20(A__TAIL(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(primes, x1, x2)) -> c21(A__IF(a__primes, x1, x2), MARK(primes)) MARK(if(sieve(z0), x1, x2)) -> c21(A__IF(a__sieve(mark(z0)), x1, x2), MARK(sieve(z0))) MARK(if(from(z0), x1, x2)) -> c21(A__IF(a__from(mark(z0)), x1, x2), MARK(from(z0))) MARK(if(head(z0), x1, x2)) -> c21(A__IF(a__head(mark(z0)), x1, x2), MARK(head(z0))) MARK(if(tail(z0), x1, x2)) -> c21(A__IF(a__tail(mark(z0)), x1, x2), MARK(tail(z0))) MARK(if(if(z0, z1, z2), x1, x2)) -> c21(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(filter(z0, z1), x1, x2)) -> c21(A__IF(a__filter(mark(z0), mark(z1)), x1, x2), MARK(filter(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c22(A__FILTER(mark(x0), a__primes), MARK(x0)) MARK(filter(x0, sieve(z0))) -> c22(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, from(z0))) -> c22(A__FILTER(mark(x0), a__from(mark(z0))), MARK(x0)) MARK(filter(x0, head(z0))) -> c22(A__FILTER(mark(x0), a__head(mark(z0))), MARK(x0)) MARK(filter(x0, tail(z0))) -> c22(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(x0)) MARK(filter(x0, if(z0, z1, z2))) -> c22(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0)) MARK(filter(x0, filter(z0, z1))) -> c22(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(x0)) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(x0, s(z0))) -> c22(MARK(x0)) MARK(filter(x0, 0)) -> c22(MARK(x0)) MARK(filter(x0, true)) -> c22(MARK(x0)) MARK(filter(x0, false)) -> c22(MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), x1)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c23(A__FILTER(mark(x0), a__primes), MARK(primes)) MARK(filter(x0, sieve(z0))) -> c23(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, from(z0))) -> c23(A__FILTER(mark(x0), a__from(mark(z0))), MARK(from(z0))) MARK(filter(x0, head(z0))) -> c23(A__FILTER(mark(x0), a__head(mark(z0))), MARK(head(z0))) MARK(filter(x0, tail(z0))) -> c23(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(tail(z0))) MARK(filter(x0, if(z0, z1, z2))) -> c23(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(filter(x0, filter(z0, z1))) -> c23(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(filter(x0, cons(z0, z1))) -> c23(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(primes, x1)) -> c23(A__FILTER(a__primes, mark(x1)), MARK(x1)) MARK(filter(sieve(z0), x1)) -> c23(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(from(z0), x1)) -> c23(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(head(z0), x1)) -> c23(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(tail(z0), x1)) -> c23(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(if(z0, z1, z2), x1)) -> c23(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1)) MARK(filter(filter(z0, z1), x1)) -> c23(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(x1)) MARK(filter(s(z0), x1)) -> c23(A__FILTER(s(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(x0, s(z0))) -> c23(MARK(s(z0))) MARK(filter(x0, divides(z0, z1))) -> c23(MARK(divides(z0, z1))) MARK(filter(0, x1)) -> c23(MARK(x1)) MARK(filter(cons(z0, z1), x1)) -> c23(MARK(x1)) MARK(filter(true, x1)) -> c23(MARK(x1)) MARK(filter(false, x1)) -> c23(MARK(x1)) MARK(filter(divides(z0, z1), x1)) -> c23(MARK(x1)) A__PRIMES -> c(A__SIEVE(cons(s(mark(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(sieve(primes)) -> c17(MARK(primes)) MARK(sieve(sieve(primes))) -> c17(A__SIEVE(a__sieve(a__primes)), MARK(sieve(primes))) MARK(sieve(sieve(sieve(z0)))) 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c17(A__SIEVE(a__from(a__head(mark(z0)))), MARK(from(head(z0)))) MARK(sieve(from(tail(z0)))) -> c17(A__SIEVE(a__from(a__tail(mark(z0)))), MARK(from(tail(z0)))) MARK(sieve(from(if(z0, z1, z2)))) -> c17(A__SIEVE(a__from(a__if(mark(z0), z1, z2))), MARK(from(if(z0, z1, z2)))) MARK(sieve(from(filter(z0, z1)))) -> c17(A__SIEVE(a__from(a__filter(mark(z0), mark(z1)))), MARK(from(filter(z0, z1)))) MARK(sieve(from(s(z0)))) -> c17(A__SIEVE(a__from(s(mark(z0)))), MARK(from(s(z0)))) MARK(sieve(from(0))) -> c17(A__SIEVE(a__from(0)), MARK(from(0))) MARK(sieve(from(cons(z0, z1)))) -> c17(A__SIEVE(a__from(cons(mark(z0), z1))), MARK(from(cons(z0, z1)))) MARK(sieve(from(true))) -> c17(A__SIEVE(a__from(true)), MARK(from(true))) MARK(sieve(from(false))) -> c17(A__SIEVE(a__from(false)), MARK(from(false))) MARK(sieve(from(divides(z0, z1)))) -> c17(A__SIEVE(a__from(divides(mark(z0), mark(z1)))), MARK(from(divides(z0, z1)))) MARK(sieve(from(x0))) -> c17(MARK(from(x0))) MARK(sieve(head(primes))) -> 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MARK(sieve(filter(x0, sieve(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), a__sieve(mark(z0)))), MARK(filter(x0, sieve(z0)))) MARK(sieve(filter(x0, from(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), a__from(mark(z0)))), MARK(filter(x0, from(z0)))) MARK(sieve(filter(x0, head(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), a__head(mark(z0)))), MARK(filter(x0, head(z0)))) MARK(sieve(filter(x0, tail(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), a__tail(mark(z0)))), MARK(filter(x0, tail(z0)))) MARK(sieve(filter(x0, if(z0, z1, z2)))) -> c17(A__SIEVE(a__filter(mark(x0), a__if(mark(z0), z1, z2))), MARK(filter(x0, if(z0, z1, z2)))) MARK(sieve(filter(x0, filter(z0, z1)))) -> c17(A__SIEVE(a__filter(mark(x0), a__filter(mark(z0), mark(z1)))), MARK(filter(x0, filter(z0, z1)))) MARK(sieve(filter(x0, s(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), s(mark(z0)))), MARK(filter(x0, s(z0)))) MARK(sieve(filter(x0, 0))) -> c17(A__SIEVE(a__filter(mark(x0), 0)), MARK(filter(x0, 0))) MARK(sieve(filter(x0, cons(z0, z1)))) -> c17(A__SIEVE(a__filter(mark(x0), cons(mark(z0), z1))), MARK(filter(x0, cons(z0, z1)))) MARK(sieve(filter(x0, true))) -> c17(A__SIEVE(a__filter(mark(x0), true)), MARK(filter(x0, true))) MARK(sieve(filter(x0, false))) -> c17(A__SIEVE(a__filter(mark(x0), false)), MARK(filter(x0, false))) MARK(sieve(filter(x0, divides(z0, z1)))) -> c17(A__SIEVE(a__filter(mark(x0), divides(mark(z0), mark(z1)))), MARK(filter(x0, divides(z0, z1)))) MARK(sieve(filter(primes, x1))) -> c17(A__SIEVE(a__filter(a__primes, mark(x1))), MARK(filter(primes, x1))) MARK(sieve(filter(sieve(z0), x1))) -> c17(A__SIEVE(a__filter(a__sieve(mark(z0)), mark(x1))), MARK(filter(sieve(z0), x1))) MARK(sieve(filter(from(z0), x1))) -> c17(A__SIEVE(a__filter(a__from(mark(z0)), mark(x1))), MARK(filter(from(z0), x1))) MARK(sieve(filter(head(z0), x1))) -> c17(A__SIEVE(a__filter(a__head(mark(z0)), mark(x1))), MARK(filter(head(z0), x1))) MARK(sieve(filter(tail(z0), x1))) -> c17(A__SIEVE(a__filter(a__tail(mark(z0)), mark(x1))), MARK(filter(tail(z0), x1))) MARK(sieve(filter(if(z0, z1, z2), x1))) -> c17(A__SIEVE(a__filter(a__if(mark(z0), z1, z2), mark(x1))), MARK(filter(if(z0, z1, z2), x1))) MARK(sieve(filter(filter(z0, z1), x1))) -> c17(A__SIEVE(a__filter(a__filter(mark(z0), mark(z1)), mark(x1))), MARK(filter(filter(z0, z1), x1))) MARK(sieve(filter(s(z0), x1))) -> c17(A__SIEVE(a__filter(s(mark(z0)), mark(x1))), MARK(filter(s(z0), x1))) MARK(sieve(filter(0, x1))) -> c17(A__SIEVE(a__filter(0, mark(x1))), MARK(filter(0, x1))) MARK(sieve(filter(cons(z0, z1), x1))) -> c17(A__SIEVE(a__filter(cons(mark(z0), z1), mark(x1))), MARK(filter(cons(z0, z1), x1))) MARK(sieve(filter(true, x1))) -> c17(A__SIEVE(a__filter(true, mark(x1))), MARK(filter(true, x1))) MARK(sieve(filter(false, x1))) -> c17(A__SIEVE(a__filter(false, mark(x1))), MARK(filter(false, x1))) MARK(sieve(filter(divides(z0, z1), x1))) -> c17(A__SIEVE(a__filter(divides(mark(z0), mark(z1)), mark(x1))), MARK(filter(divides(z0, z1), x1))) MARK(sieve(filter(x0, x1))) -> c17(MARK(filter(x0, x1))) MARK(head(primes)) -> c19(A__HEAD(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(head(primes)) -> c19(MARK(primes)) MARK(head(sieve(primes))) -> c19(A__HEAD(a__sieve(a__primes)), MARK(sieve(primes))) MARK(head(sieve(sieve(z0)))) -> c19(A__HEAD(a__sieve(a__sieve(mark(z0)))), MARK(sieve(sieve(z0)))) MARK(head(sieve(from(z0)))) -> c19(A__HEAD(a__sieve(a__from(mark(z0)))), MARK(sieve(from(z0)))) MARK(head(sieve(head(z0)))) -> c19(A__HEAD(a__sieve(a__head(mark(z0)))), MARK(sieve(head(z0)))) MARK(head(sieve(tail(z0)))) -> c19(A__HEAD(a__sieve(a__tail(mark(z0)))), MARK(sieve(tail(z0)))) MARK(head(sieve(if(z0, z1, z2)))) -> c19(A__HEAD(a__sieve(a__if(mark(z0), z1, z2))), MARK(sieve(if(z0, z1, z2)))) MARK(head(sieve(filter(z0, z1)))) -> c19(A__HEAD(a__sieve(a__filter(mark(z0), mark(z1)))), MARK(sieve(filter(z0, z1)))) MARK(head(sieve(s(z0)))) -> c19(A__HEAD(a__sieve(s(mark(z0)))), MARK(sieve(s(z0)))) MARK(head(sieve(cons(z0, z1)))) -> c19(A__HEAD(a__sieve(cons(mark(z0), z1))), MARK(sieve(cons(z0, z1)))) MARK(head(sieve(divides(z0, z1)))) -> c19(A__HEAD(a__sieve(divides(mark(z0), mark(z1)))), MARK(sieve(divides(z0, z1)))) MARK(head(sieve(x0))) -> c19(MARK(sieve(x0))) MARK(head(sieve(0))) -> c19(A__HEAD(a__sieve(0))) MARK(head(sieve(true))) -> c19(A__HEAD(a__sieve(true))) MARK(head(sieve(false))) -> c19(A__HEAD(a__sieve(false))) MARK(head(from(x0))) -> c19(A__HEAD(cons(mark(mark(x0)), from(s(mark(x0))))), MARK(from(x0))) MARK(head(from(primes))) -> c19(A__HEAD(a__from(a__primes)), MARK(from(primes))) MARK(head(from(sieve(z0)))) -> c19(A__HEAD(a__from(a__sieve(mark(z0)))), MARK(from(sieve(z0)))) MARK(head(from(from(z0)))) -> c19(A__HEAD(a__from(a__from(mark(z0)))), MARK(from(from(z0)))) MARK(head(from(head(z0)))) -> c19(A__HEAD(a__from(a__head(mark(z0)))), MARK(from(head(z0)))) MARK(head(from(tail(z0)))) -> c19(A__HEAD(a__from(a__tail(mark(z0)))), MARK(from(tail(z0)))) MARK(head(from(if(z0, z1, z2)))) -> c19(A__HEAD(a__from(a__if(mark(z0), z1, z2))), MARK(from(if(z0, z1, z2)))) MARK(head(from(filter(z0, z1)))) -> c19(A__HEAD(a__from(a__filter(mark(z0), mark(z1)))), MARK(from(filter(z0, z1)))) MARK(head(from(s(z0)))) -> c19(A__HEAD(a__from(s(mark(z0)))), MARK(from(s(z0)))) MARK(head(from(0))) -> c19(A__HEAD(a__from(0)), MARK(from(0))) MARK(head(from(cons(z0, z1)))) -> c19(A__HEAD(a__from(cons(mark(z0), z1))), MARK(from(cons(z0, z1)))) MARK(head(from(true))) -> c19(A__HEAD(a__from(true)), MARK(from(true))) MARK(head(from(false))) -> c19(A__HEAD(a__from(false)), MARK(from(false))) MARK(head(from(divides(z0, z1)))) -> c19(A__HEAD(a__from(divides(mark(z0), mark(z1)))), MARK(from(divides(z0, z1)))) MARK(head(from(x0))) -> c19(MARK(from(x0))) MARK(head(head(primes))) -> c19(A__HEAD(a__head(a__primes)), MARK(head(primes))) MARK(head(head(sieve(z0)))) -> c19(A__HEAD(a__head(a__sieve(mark(z0)))), MARK(head(sieve(z0)))) MARK(head(head(from(z0)))) -> c19(A__HEAD(a__head(a__from(mark(z0)))), MARK(head(from(z0)))) MARK(head(head(head(z0)))) -> c19(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c19(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(if(z0, z1, z2)))) -> c19(A__HEAD(a__head(a__if(mark(z0), z1, z2))), MARK(head(if(z0, z1, z2)))) MARK(head(head(filter(z0, z1)))) -> c19(A__HEAD(a__head(a__filter(mark(z0), mark(z1)))), MARK(head(filter(z0, z1)))) MARK(head(head(s(z0)))) -> c19(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(cons(z0, z1)))) -> c19(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(divides(z0, z1)))) -> c19(A__HEAD(a__head(divides(mark(z0), mark(z1)))), MARK(head(divides(z0, z1)))) MARK(head(head(x0))) -> c19(MARK(head(x0))) MARK(head(head(0))) -> c19(A__HEAD(a__head(0))) MARK(head(head(true))) -> c19(A__HEAD(a__head(true))) MARK(head(head(false))) -> c19(A__HEAD(a__head(false))) MARK(head(tail(primes))) -> c19(A__HEAD(a__tail(a__primes)), MARK(tail(primes))) MARK(head(tail(sieve(z0)))) -> c19(A__HEAD(a__tail(a__sieve(mark(z0)))), MARK(tail(sieve(z0)))) MARK(head(tail(from(z0)))) -> c19(A__HEAD(a__tail(a__from(mark(z0)))), MARK(tail(from(z0)))) MARK(head(tail(head(z0)))) -> c19(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c19(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(if(z0, z1, z2)))) -> c19(A__HEAD(a__tail(a__if(mark(z0), z1, z2))), MARK(tail(if(z0, z1, z2)))) MARK(head(tail(filter(z0, z1)))) -> c19(A__HEAD(a__tail(a__filter(mark(z0), mark(z1)))), MARK(tail(filter(z0, z1)))) MARK(head(tail(s(z0)))) -> c19(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(cons(z0, z1)))) -> c19(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(divides(z0, z1)))) -> c19(A__HEAD(a__tail(divides(mark(z0), mark(z1)))), MARK(tail(divides(z0, z1)))) MARK(head(tail(x0))) -> c19(MARK(tail(x0))) MARK(head(tail(0))) -> c19(A__HEAD(a__tail(0))) MARK(head(tail(true))) -> c19(A__HEAD(a__tail(true))) MARK(head(tail(false))) -> c19(A__HEAD(a__tail(false))) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c18_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_1, c17_2, c17_1, c19_2, c19_1, c20_2, c20_1, c21_2, c21_1, c22_2, c22_1, c23_2, c23_1, c_2 ---------------------------------------- (147) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(head(if(z0, z1, z2))) -> c19(A__HEAD(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) by MARK(head(if(x0, z1, z2))) -> c19(A__HEAD(if(mark(x0), z1, z2)), MARK(if(x0, z1, z2))) MARK(head(if(primes, x1, x2))) -> c19(A__HEAD(a__if(a__primes, x1, x2)), MARK(if(primes, x1, x2))) MARK(head(if(sieve(z0), x1, x2))) -> c19(A__HEAD(a__if(a__sieve(mark(z0)), x1, x2)), MARK(if(sieve(z0), x1, x2))) MARK(head(if(from(z0), x1, x2))) -> c19(A__HEAD(a__if(a__from(mark(z0)), x1, x2)), MARK(if(from(z0), x1, x2))) MARK(head(if(head(z0), x1, x2))) -> c19(A__HEAD(a__if(a__head(mark(z0)), x1, x2)), MARK(if(head(z0), x1, x2))) MARK(head(if(tail(z0), x1, x2))) -> c19(A__HEAD(a__if(a__tail(mark(z0)), x1, x2)), MARK(if(tail(z0), x1, x2))) MARK(head(if(if(z0, z1, z2), x1, x2))) -> c19(A__HEAD(a__if(a__if(mark(z0), z1, z2), x1, x2)), MARK(if(if(z0, z1, z2), x1, x2))) MARK(head(if(filter(z0, z1), x1, x2))) -> c19(A__HEAD(a__if(a__filter(mark(z0), mark(z1)), x1, x2)), MARK(if(filter(z0, z1), x1, x2))) MARK(head(if(s(z0), x1, x2))) -> c19(A__HEAD(a__if(s(mark(z0)), x1, x2)), MARK(if(s(z0), x1, x2))) MARK(head(if(0, x1, x2))) -> c19(A__HEAD(a__if(0, x1, x2)), MARK(if(0, x1, x2))) MARK(head(if(cons(z0, z1), x1, x2))) -> c19(A__HEAD(a__if(cons(mark(z0), z1), x1, x2)), MARK(if(cons(z0, z1), x1, x2))) MARK(head(if(true, x1, x2))) -> c19(A__HEAD(a__if(true, x1, x2)), MARK(if(true, x1, x2))) MARK(head(if(false, x1, x2))) -> c19(A__HEAD(a__if(false, x1, x2)), MARK(if(false, x1, x2))) MARK(head(if(divides(z0, z1), x1, x2))) -> c19(A__HEAD(a__if(divides(mark(z0), mark(z1)), x1, x2)), MARK(if(divides(z0, z1), x1, x2))) ---------------------------------------- (148) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(filter(z0, z1))) -> c19(A__HEAD(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(primes)) -> c20(A__TAIL(a__primes), MARK(primes)) MARK(tail(sieve(z0))) -> c20(A__TAIL(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(tail(from(z0))) -> c20(A__TAIL(a__from(mark(z0))), MARK(from(z0))) MARK(tail(head(z0))) -> c20(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c20(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(if(z0, z1, z2))) -> c20(A__TAIL(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(tail(filter(z0, z1))) -> c20(A__TAIL(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(primes, x1, x2)) -> c21(A__IF(a__primes, x1, x2), MARK(primes)) MARK(if(sieve(z0), x1, x2)) -> c21(A__IF(a__sieve(mark(z0)), x1, x2), MARK(sieve(z0))) MARK(if(from(z0), x1, x2)) -> c21(A__IF(a__from(mark(z0)), x1, x2), MARK(from(z0))) MARK(if(head(z0), x1, x2)) -> c21(A__IF(a__head(mark(z0)), x1, x2), MARK(head(z0))) MARK(if(tail(z0), x1, x2)) -> c21(A__IF(a__tail(mark(z0)), x1, x2), MARK(tail(z0))) MARK(if(if(z0, z1, z2), x1, x2)) -> c21(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(filter(z0, z1), x1, x2)) -> c21(A__IF(a__filter(mark(z0), mark(z1)), x1, x2), MARK(filter(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, 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MARK(sieve(tail(tail(z0)))) -> c17(A__SIEVE(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(sieve(tail(if(z0, z1, z2)))) -> c17(A__SIEVE(a__tail(a__if(mark(z0), z1, z2))), MARK(tail(if(z0, z1, z2)))) MARK(sieve(tail(filter(z0, z1)))) -> c17(A__SIEVE(a__tail(a__filter(mark(z0), mark(z1)))), MARK(tail(filter(z0, z1)))) MARK(sieve(tail(s(z0)))) -> c17(A__SIEVE(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(sieve(tail(cons(z0, z1)))) -> c17(A__SIEVE(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(sieve(tail(divides(z0, z1)))) -> c17(A__SIEVE(a__tail(divides(mark(z0), mark(z1)))), MARK(tail(divides(z0, z1)))) MARK(sieve(tail(x0))) -> c17(MARK(tail(x0))) MARK(sieve(tail(0))) -> c17(A__SIEVE(a__tail(0))) MARK(sieve(tail(true))) -> c17(A__SIEVE(a__tail(true))) MARK(sieve(tail(false))) -> c17(A__SIEVE(a__tail(false))) MARK(sieve(if(primes, x1, x2))) -> c17(A__SIEVE(a__if(a__primes, x1, x2)), MARK(if(primes, x1, x2))) MARK(sieve(if(sieve(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__sieve(mark(z0)), x1, x2)), MARK(if(sieve(z0), x1, x2))) MARK(sieve(if(from(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__from(mark(z0)), x1, x2)), MARK(if(from(z0), x1, x2))) MARK(sieve(if(head(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__head(mark(z0)), x1, x2)), MARK(if(head(z0), x1, x2))) MARK(sieve(if(tail(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__tail(mark(z0)), x1, x2)), MARK(if(tail(z0), x1, x2))) MARK(sieve(if(if(z0, z1, z2), x1, x2))) -> c17(A__SIEVE(a__if(a__if(mark(z0), z1, z2), x1, x2)), MARK(if(if(z0, z1, z2), x1, x2))) MARK(sieve(if(filter(z0, z1), x1, x2))) -> c17(A__SIEVE(a__if(a__filter(mark(z0), mark(z1)), x1, x2)), MARK(if(filter(z0, z1), x1, x2))) MARK(sieve(if(s(z0), x1, x2))) -> c17(A__SIEVE(a__if(s(mark(z0)), x1, x2)), MARK(if(s(z0), x1, x2))) MARK(sieve(if(cons(z0, z1), x1, x2))) -> c17(A__SIEVE(a__if(cons(mark(z0), z1), x1, x2)), MARK(if(cons(z0, z1), x1, x2))) MARK(sieve(if(true, x1, x2))) -> c17(A__SIEVE(a__if(true, x1, x2)), MARK(if(true, x1, x2))) MARK(sieve(if(false, x1, x2))) -> c17(A__SIEVE(a__if(false, x1, x2)), MARK(if(false, x1, x2))) MARK(sieve(if(divides(z0, z1), x1, x2))) -> c17(A__SIEVE(a__if(divides(mark(z0), mark(z1)), x1, x2)), MARK(if(divides(z0, z1), x1, x2))) MARK(sieve(if(x0, z1, z2))) -> c17(MARK(if(x0, z1, z2))) MARK(sieve(if(0, x1, x2))) -> c17(A__SIEVE(a__if(0, x1, x2))) MARK(sieve(filter(x0, primes))) -> c17(A__SIEVE(a__filter(mark(x0), a__primes)), MARK(filter(x0, primes))) MARK(sieve(filter(x0, sieve(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), a__sieve(mark(z0)))), MARK(filter(x0, sieve(z0)))) MARK(sieve(filter(x0, from(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), a__from(mark(z0)))), MARK(filter(x0, from(z0)))) MARK(sieve(filter(x0, head(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), a__head(mark(z0)))), MARK(filter(x0, head(z0)))) MARK(sieve(filter(x0, tail(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), a__tail(mark(z0)))), MARK(filter(x0, tail(z0)))) MARK(sieve(filter(x0, if(z0, z1, z2)))) -> c17(A__SIEVE(a__filter(mark(x0), a__if(mark(z0), z1, z2))), MARK(filter(x0, if(z0, z1, z2)))) MARK(sieve(filter(x0, filter(z0, z1)))) -> c17(A__SIEVE(a__filter(mark(x0), a__filter(mark(z0), mark(z1)))), MARK(filter(x0, filter(z0, z1)))) MARK(sieve(filter(x0, s(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), s(mark(z0)))), MARK(filter(x0, s(z0)))) MARK(sieve(filter(x0, 0))) -> c17(A__SIEVE(a__filter(mark(x0), 0)), MARK(filter(x0, 0))) MARK(sieve(filter(x0, cons(z0, z1)))) -> c17(A__SIEVE(a__filter(mark(x0), cons(mark(z0), z1))), MARK(filter(x0, cons(z0, z1)))) MARK(sieve(filter(x0, true))) -> c17(A__SIEVE(a__filter(mark(x0), true)), MARK(filter(x0, true))) MARK(sieve(filter(x0, false))) -> c17(A__SIEVE(a__filter(mark(x0), false)), MARK(filter(x0, false))) MARK(sieve(filter(x0, divides(z0, z1)))) -> c17(A__SIEVE(a__filter(mark(x0), divides(mark(z0), mark(z1)))), MARK(filter(x0, divides(z0, z1)))) MARK(sieve(filter(primes, x1))) -> c17(A__SIEVE(a__filter(a__primes, mark(x1))), MARK(filter(primes, x1))) MARK(sieve(filter(sieve(z0), x1))) -> c17(A__SIEVE(a__filter(a__sieve(mark(z0)), mark(x1))), MARK(filter(sieve(z0), x1))) MARK(sieve(filter(from(z0), x1))) -> c17(A__SIEVE(a__filter(a__from(mark(z0)), mark(x1))), MARK(filter(from(z0), x1))) MARK(sieve(filter(head(z0), x1))) -> c17(A__SIEVE(a__filter(a__head(mark(z0)), mark(x1))), MARK(filter(head(z0), x1))) MARK(sieve(filter(tail(z0), x1))) -> c17(A__SIEVE(a__filter(a__tail(mark(z0)), mark(x1))), MARK(filter(tail(z0), x1))) MARK(sieve(filter(if(z0, z1, z2), x1))) -> c17(A__SIEVE(a__filter(a__if(mark(z0), z1, z2), mark(x1))), MARK(filter(if(z0, z1, z2), x1))) MARK(sieve(filter(filter(z0, z1), x1))) -> c17(A__SIEVE(a__filter(a__filter(mark(z0), mark(z1)), mark(x1))), MARK(filter(filter(z0, z1), x1))) MARK(sieve(filter(s(z0), x1))) -> c17(A__SIEVE(a__filter(s(mark(z0)), mark(x1))), MARK(filter(s(z0), x1))) MARK(sieve(filter(0, x1))) -> c17(A__SIEVE(a__filter(0, mark(x1))), MARK(filter(0, x1))) MARK(sieve(filter(cons(z0, z1), x1))) -> c17(A__SIEVE(a__filter(cons(mark(z0), z1), mark(x1))), MARK(filter(cons(z0, z1), x1))) MARK(sieve(filter(true, x1))) -> c17(A__SIEVE(a__filter(true, mark(x1))), MARK(filter(true, x1))) MARK(sieve(filter(false, x1))) -> c17(A__SIEVE(a__filter(false, mark(x1))), MARK(filter(false, x1))) MARK(sieve(filter(divides(z0, z1), x1))) -> c17(A__SIEVE(a__filter(divides(mark(z0), mark(z1)), mark(x1))), MARK(filter(divides(z0, z1), x1))) MARK(sieve(filter(x0, x1))) -> c17(MARK(filter(x0, x1))) MARK(head(primes)) -> c19(A__HEAD(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(head(primes)) -> c19(MARK(primes)) MARK(head(sieve(primes))) -> c19(A__HEAD(a__sieve(a__primes)), MARK(sieve(primes))) MARK(head(sieve(sieve(z0)))) -> c19(A__HEAD(a__sieve(a__sieve(mark(z0)))), MARK(sieve(sieve(z0)))) MARK(head(sieve(from(z0)))) -> c19(A__HEAD(a__sieve(a__from(mark(z0)))), MARK(sieve(from(z0)))) MARK(head(sieve(head(z0)))) -> c19(A__HEAD(a__sieve(a__head(mark(z0)))), MARK(sieve(head(z0)))) MARK(head(sieve(tail(z0)))) -> c19(A__HEAD(a__sieve(a__tail(mark(z0)))), MARK(sieve(tail(z0)))) MARK(head(sieve(if(z0, z1, z2)))) -> c19(A__HEAD(a__sieve(a__if(mark(z0), z1, z2))), MARK(sieve(if(z0, z1, z2)))) MARK(head(sieve(filter(z0, z1)))) -> c19(A__HEAD(a__sieve(a__filter(mark(z0), mark(z1)))), MARK(sieve(filter(z0, z1)))) MARK(head(sieve(s(z0)))) -> c19(A__HEAD(a__sieve(s(mark(z0)))), MARK(sieve(s(z0)))) MARK(head(sieve(cons(z0, z1)))) -> c19(A__HEAD(a__sieve(cons(mark(z0), z1))), MARK(sieve(cons(z0, z1)))) MARK(head(sieve(divides(z0, z1)))) -> c19(A__HEAD(a__sieve(divides(mark(z0), mark(z1)))), MARK(sieve(divides(z0, z1)))) MARK(head(sieve(x0))) -> c19(MARK(sieve(x0))) MARK(head(sieve(0))) -> c19(A__HEAD(a__sieve(0))) MARK(head(sieve(true))) -> c19(A__HEAD(a__sieve(true))) MARK(head(sieve(false))) -> c19(A__HEAD(a__sieve(false))) MARK(head(from(x0))) -> c19(A__HEAD(cons(mark(mark(x0)), from(s(mark(x0))))), MARK(from(x0))) MARK(head(from(primes))) -> c19(A__HEAD(a__from(a__primes)), MARK(from(primes))) MARK(head(from(sieve(z0)))) -> c19(A__HEAD(a__from(a__sieve(mark(z0)))), MARK(from(sieve(z0)))) MARK(head(from(from(z0)))) -> c19(A__HEAD(a__from(a__from(mark(z0)))), MARK(from(from(z0)))) MARK(head(from(head(z0)))) -> c19(A__HEAD(a__from(a__head(mark(z0)))), MARK(from(head(z0)))) MARK(head(from(tail(z0)))) -> c19(A__HEAD(a__from(a__tail(mark(z0)))), MARK(from(tail(z0)))) MARK(head(from(if(z0, z1, z2)))) -> c19(A__HEAD(a__from(a__if(mark(z0), z1, z2))), MARK(from(if(z0, z1, z2)))) MARK(head(from(filter(z0, z1)))) -> c19(A__HEAD(a__from(a__filter(mark(z0), mark(z1)))), MARK(from(filter(z0, z1)))) MARK(head(from(s(z0)))) -> c19(A__HEAD(a__from(s(mark(z0)))), MARK(from(s(z0)))) MARK(head(from(0))) -> c19(A__HEAD(a__from(0)), MARK(from(0))) MARK(head(from(cons(z0, z1)))) -> c19(A__HEAD(a__from(cons(mark(z0), z1))), MARK(from(cons(z0, z1)))) MARK(head(from(true))) -> c19(A__HEAD(a__from(true)), MARK(from(true))) MARK(head(from(false))) -> c19(A__HEAD(a__from(false)), MARK(from(false))) MARK(head(from(divides(z0, z1)))) -> c19(A__HEAD(a__from(divides(mark(z0), mark(z1)))), MARK(from(divides(z0, z1)))) MARK(head(from(x0))) -> c19(MARK(from(x0))) MARK(head(head(primes))) -> c19(A__HEAD(a__head(a__primes)), MARK(head(primes))) MARK(head(head(sieve(z0)))) -> c19(A__HEAD(a__head(a__sieve(mark(z0)))), MARK(head(sieve(z0)))) MARK(head(head(from(z0)))) -> c19(A__HEAD(a__head(a__from(mark(z0)))), MARK(head(from(z0)))) MARK(head(head(head(z0)))) -> c19(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c19(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(if(z0, z1, z2)))) -> c19(A__HEAD(a__head(a__if(mark(z0), z1, z2))), MARK(head(if(z0, z1, z2)))) MARK(head(head(filter(z0, z1)))) -> c19(A__HEAD(a__head(a__filter(mark(z0), mark(z1)))), MARK(head(filter(z0, z1)))) MARK(head(head(s(z0)))) -> c19(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(cons(z0, z1)))) -> c19(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(divides(z0, z1)))) -> c19(A__HEAD(a__head(divides(mark(z0), mark(z1)))), MARK(head(divides(z0, z1)))) MARK(head(head(x0))) -> c19(MARK(head(x0))) MARK(head(head(0))) -> c19(A__HEAD(a__head(0))) MARK(head(head(true))) -> c19(A__HEAD(a__head(true))) MARK(head(head(false))) -> c19(A__HEAD(a__head(false))) MARK(head(tail(primes))) -> c19(A__HEAD(a__tail(a__primes)), MARK(tail(primes))) MARK(head(tail(sieve(z0)))) -> c19(A__HEAD(a__tail(a__sieve(mark(z0)))), MARK(tail(sieve(z0)))) MARK(head(tail(from(z0)))) -> c19(A__HEAD(a__tail(a__from(mark(z0)))), MARK(tail(from(z0)))) MARK(head(tail(head(z0)))) -> c19(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c19(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(if(z0, z1, z2)))) -> c19(A__HEAD(a__tail(a__if(mark(z0), z1, z2))), MARK(tail(if(z0, z1, z2)))) MARK(head(tail(filter(z0, z1)))) -> c19(A__HEAD(a__tail(a__filter(mark(z0), mark(z1)))), MARK(tail(filter(z0, z1)))) MARK(head(tail(s(z0)))) -> c19(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(cons(z0, z1)))) -> c19(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(divides(z0, z1)))) -> c19(A__HEAD(a__tail(divides(mark(z0), mark(z1)))), MARK(tail(divides(z0, z1)))) MARK(head(tail(x0))) -> c19(MARK(tail(x0))) MARK(head(tail(0))) -> c19(A__HEAD(a__tail(0))) MARK(head(tail(true))) -> c19(A__HEAD(a__tail(true))) MARK(head(tail(false))) -> c19(A__HEAD(a__tail(false))) MARK(head(if(x0, z1, z2))) -> c19(A__HEAD(if(mark(x0), z1, z2)), MARK(if(x0, z1, z2))) MARK(head(if(primes, x1, x2))) -> c19(A__HEAD(a__if(a__primes, x1, x2)), MARK(if(primes, x1, x2))) MARK(head(if(sieve(z0), x1, x2))) -> c19(A__HEAD(a__if(a__sieve(mark(z0)), x1, x2)), MARK(if(sieve(z0), x1, x2))) MARK(head(if(from(z0), x1, x2))) -> c19(A__HEAD(a__if(a__from(mark(z0)), x1, x2)), MARK(if(from(z0), x1, x2))) MARK(head(if(head(z0), x1, x2))) -> c19(A__HEAD(a__if(a__head(mark(z0)), x1, x2)), MARK(if(head(z0), x1, x2))) MARK(head(if(tail(z0), x1, x2))) -> c19(A__HEAD(a__if(a__tail(mark(z0)), x1, x2)), MARK(if(tail(z0), x1, x2))) MARK(head(if(if(z0, z1, z2), x1, x2))) -> c19(A__HEAD(a__if(a__if(mark(z0), z1, z2), x1, x2)), MARK(if(if(z0, z1, z2), x1, x2))) MARK(head(if(filter(z0, z1), x1, x2))) -> c19(A__HEAD(a__if(a__filter(mark(z0), mark(z1)), x1, x2)), MARK(if(filter(z0, z1), x1, x2))) MARK(head(if(s(z0), x1, x2))) -> c19(A__HEAD(a__if(s(mark(z0)), x1, x2)), MARK(if(s(z0), x1, x2))) MARK(head(if(0, x1, x2))) -> c19(A__HEAD(a__if(0, x1, x2)), MARK(if(0, x1, x2))) MARK(head(if(cons(z0, z1), x1, x2))) -> c19(A__HEAD(a__if(cons(mark(z0), z1), x1, x2)), MARK(if(cons(z0, z1), x1, x2))) MARK(head(if(true, x1, x2))) -> c19(A__HEAD(a__if(true, x1, x2)), MARK(if(true, x1, x2))) MARK(head(if(false, x1, x2))) -> c19(A__HEAD(a__if(false, x1, x2)), MARK(if(false, x1, x2))) MARK(head(if(divides(z0, z1), x1, x2))) -> c19(A__HEAD(a__if(divides(mark(z0), mark(z1)), x1, x2)), MARK(if(divides(z0, z1), x1, x2))) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c18_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_1, c17_2, c17_1, c19_2, c19_1, c20_2, c20_1, c21_2, c21_1, c22_2, c22_1, c23_2, c23_1, c_2 ---------------------------------------- (149) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (150) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(filter(z0, z1))) -> c19(A__HEAD(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(primes)) -> c20(A__TAIL(a__primes), MARK(primes)) MARK(tail(sieve(z0))) -> c20(A__TAIL(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(tail(from(z0))) -> c20(A__TAIL(a__from(mark(z0))), MARK(from(z0))) MARK(tail(head(z0))) -> c20(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c20(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(if(z0, z1, z2))) -> c20(A__TAIL(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(tail(filter(z0, z1))) -> c20(A__TAIL(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(primes, x1, x2)) -> c21(A__IF(a__primes, x1, x2), MARK(primes)) MARK(if(sieve(z0), x1, x2)) -> c21(A__IF(a__sieve(mark(z0)), x1, x2), MARK(sieve(z0))) MARK(if(from(z0), x1, x2)) -> c21(A__IF(a__from(mark(z0)), x1, x2), MARK(from(z0))) MARK(if(head(z0), x1, x2)) -> c21(A__IF(a__head(mark(z0)), x1, x2), MARK(head(z0))) MARK(if(tail(z0), x1, x2)) -> c21(A__IF(a__tail(mark(z0)), x1, x2), MARK(tail(z0))) MARK(if(if(z0, z1, z2), x1, x2)) -> c21(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(filter(z0, z1), x1, x2)) -> c21(A__IF(a__filter(mark(z0), mark(z1)), x1, x2), MARK(filter(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c22(A__FILTER(mark(x0), a__primes), MARK(x0)) MARK(filter(x0, sieve(z0))) -> c22(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, from(z0))) -> c22(A__FILTER(mark(x0), a__from(mark(z0))), MARK(x0)) MARK(filter(x0, head(z0))) -> c22(A__FILTER(mark(x0), a__head(mark(z0))), MARK(x0)) MARK(filter(x0, tail(z0))) -> c22(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(x0)) MARK(filter(x0, if(z0, z1, z2))) -> c22(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0)) MARK(filter(x0, filter(z0, z1))) -> c22(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(x0)) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(x0, s(z0))) -> c22(MARK(x0)) MARK(filter(x0, 0)) -> c22(MARK(x0)) MARK(filter(x0, true)) -> c22(MARK(x0)) MARK(filter(x0, false)) -> c22(MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), x1)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c23(A__FILTER(mark(x0), a__primes), MARK(primes)) MARK(filter(x0, sieve(z0))) -> c23(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, from(z0))) -> c23(A__FILTER(mark(x0), a__from(mark(z0))), MARK(from(z0))) MARK(filter(x0, head(z0))) -> c23(A__FILTER(mark(x0), a__head(mark(z0))), MARK(head(z0))) MARK(filter(x0, tail(z0))) -> c23(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(tail(z0))) MARK(filter(x0, if(z0, z1, z2))) -> c23(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(filter(x0, filter(z0, z1))) -> c23(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(filter(x0, cons(z0, z1))) -> c23(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(primes, x1)) -> c23(A__FILTER(a__primes, mark(x1)), MARK(x1)) MARK(filter(sieve(z0), x1)) -> c23(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(from(z0), x1)) -> c23(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(head(z0), x1)) -> c23(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(tail(z0), x1)) -> c23(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(if(z0, z1, z2), x1)) -> c23(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1)) MARK(filter(filter(z0, z1), x1)) -> c23(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(x1)) MARK(filter(s(z0), x1)) -> c23(A__FILTER(s(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(x0, s(z0))) -> c23(MARK(s(z0))) MARK(filter(x0, divides(z0, z1))) -> c23(MARK(divides(z0, z1))) MARK(filter(0, x1)) -> c23(MARK(x1)) MARK(filter(cons(z0, z1), x1)) -> c23(MARK(x1)) MARK(filter(true, x1)) -> c23(MARK(x1)) MARK(filter(false, x1)) -> c23(MARK(x1)) MARK(filter(divides(z0, z1), x1)) -> c23(MARK(x1)) A__PRIMES -> c(A__SIEVE(cons(s(mark(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(sieve(primes)) -> c17(MARK(primes)) MARK(sieve(sieve(primes))) -> c17(A__SIEVE(a__sieve(a__primes)), MARK(sieve(primes))) MARK(sieve(sieve(sieve(z0)))) -> c17(A__SIEVE(a__sieve(a__sieve(mark(z0)))), MARK(sieve(sieve(z0)))) MARK(sieve(sieve(from(z0)))) -> c17(A__SIEVE(a__sieve(a__from(mark(z0)))), MARK(sieve(from(z0)))) MARK(sieve(sieve(head(z0)))) -> c17(A__SIEVE(a__sieve(a__head(mark(z0)))), MARK(sieve(head(z0)))) MARK(sieve(sieve(tail(z0)))) -> c17(A__SIEVE(a__sieve(a__tail(mark(z0)))), MARK(sieve(tail(z0)))) MARK(sieve(sieve(if(z0, z1, z2)))) -> c17(A__SIEVE(a__sieve(a__if(mark(z0), z1, z2))), MARK(sieve(if(z0, z1, z2)))) MARK(sieve(sieve(filter(z0, z1)))) -> c17(A__SIEVE(a__sieve(a__filter(mark(z0), mark(z1)))), MARK(sieve(filter(z0, z1)))) MARK(sieve(sieve(s(z0)))) -> c17(A__SIEVE(a__sieve(s(mark(z0)))), MARK(sieve(s(z0)))) MARK(sieve(sieve(cons(z0, z1)))) -> c17(A__SIEVE(a__sieve(cons(mark(z0), z1))), MARK(sieve(cons(z0, z1)))) MARK(sieve(sieve(divides(z0, z1)))) -> c17(A__SIEVE(a__sieve(divides(mark(z0), mark(z1)))), MARK(sieve(divides(z0, z1)))) MARK(sieve(sieve(x0))) -> c17(MARK(sieve(x0))) MARK(sieve(sieve(0))) -> c17(A__SIEVE(a__sieve(0))) MARK(sieve(sieve(true))) -> c17(A__SIEVE(a__sieve(true))) MARK(sieve(sieve(false))) -> c17(A__SIEVE(a__sieve(false))) MARK(sieve(from(x0))) -> c17(A__SIEVE(cons(mark(mark(x0)), from(s(mark(x0))))), MARK(from(x0))) MARK(sieve(from(primes))) -> c17(A__SIEVE(a__from(a__primes)), MARK(from(primes))) MARK(sieve(from(sieve(z0)))) -> c17(A__SIEVE(a__from(a__sieve(mark(z0)))), MARK(from(sieve(z0)))) MARK(sieve(from(from(z0)))) -> c17(A__SIEVE(a__from(a__from(mark(z0)))), MARK(from(from(z0)))) MARK(sieve(from(head(z0)))) -> c17(A__SIEVE(a__from(a__head(mark(z0)))), MARK(from(head(z0)))) MARK(sieve(from(tail(z0)))) -> c17(A__SIEVE(a__from(a__tail(mark(z0)))), MARK(from(tail(z0)))) MARK(sieve(from(if(z0, z1, z2)))) -> c17(A__SIEVE(a__from(a__if(mark(z0), z1, z2))), MARK(from(if(z0, z1, z2)))) MARK(sieve(from(filter(z0, z1)))) -> c17(A__SIEVE(a__from(a__filter(mark(z0), mark(z1)))), MARK(from(filter(z0, z1)))) MARK(sieve(from(s(z0)))) -> c17(A__SIEVE(a__from(s(mark(z0)))), MARK(from(s(z0)))) MARK(sieve(from(0))) -> c17(A__SIEVE(a__from(0)), MARK(from(0))) MARK(sieve(from(cons(z0, z1)))) -> c17(A__SIEVE(a__from(cons(mark(z0), z1))), MARK(from(cons(z0, z1)))) MARK(sieve(from(true))) -> c17(A__SIEVE(a__from(true)), MARK(from(true))) MARK(sieve(from(false))) -> c17(A__SIEVE(a__from(false)), MARK(from(false))) MARK(sieve(from(divides(z0, z1)))) -> c17(A__SIEVE(a__from(divides(mark(z0), mark(z1)))), MARK(from(divides(z0, z1)))) MARK(sieve(from(x0))) -> c17(MARK(from(x0))) MARK(sieve(head(primes))) -> c17(A__SIEVE(a__head(a__primes)), MARK(head(primes))) MARK(sieve(head(sieve(z0)))) -> c17(A__SIEVE(a__head(a__sieve(mark(z0)))), MARK(head(sieve(z0)))) MARK(sieve(head(from(z0)))) -> c17(A__SIEVE(a__head(a__from(mark(z0)))), MARK(head(from(z0)))) MARK(sieve(head(head(z0)))) -> c17(A__SIEVE(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(sieve(head(tail(z0)))) -> c17(A__SIEVE(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(sieve(head(if(z0, z1, z2)))) -> c17(A__SIEVE(a__head(a__if(mark(z0), z1, z2))), MARK(head(if(z0, z1, z2)))) MARK(sieve(head(filter(z0, z1)))) -> c17(A__SIEVE(a__head(a__filter(mark(z0), mark(z1)))), MARK(head(filter(z0, z1)))) MARK(sieve(head(s(z0)))) -> c17(A__SIEVE(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(sieve(head(cons(z0, z1)))) -> c17(A__SIEVE(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(sieve(head(divides(z0, z1)))) -> c17(A__SIEVE(a__head(divides(mark(z0), mark(z1)))), MARK(head(divides(z0, z1)))) MARK(sieve(head(x0))) -> c17(MARK(head(x0))) MARK(sieve(head(0))) -> c17(A__SIEVE(a__head(0))) MARK(sieve(head(true))) -> c17(A__SIEVE(a__head(true))) MARK(sieve(head(false))) -> c17(A__SIEVE(a__head(false))) MARK(sieve(tail(primes))) -> c17(A__SIEVE(a__tail(a__primes)), MARK(tail(primes))) MARK(sieve(tail(sieve(z0)))) -> c17(A__SIEVE(a__tail(a__sieve(mark(z0)))), MARK(tail(sieve(z0)))) MARK(sieve(tail(from(z0)))) -> c17(A__SIEVE(a__tail(a__from(mark(z0)))), MARK(tail(from(z0)))) MARK(sieve(tail(head(z0)))) -> c17(A__SIEVE(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(sieve(tail(tail(z0)))) -> c17(A__SIEVE(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(sieve(tail(if(z0, z1, z2)))) -> c17(A__SIEVE(a__tail(a__if(mark(z0), z1, z2))), MARK(tail(if(z0, z1, z2)))) MARK(sieve(tail(filter(z0, z1)))) -> c17(A__SIEVE(a__tail(a__filter(mark(z0), mark(z1)))), MARK(tail(filter(z0, z1)))) MARK(sieve(tail(s(z0)))) -> c17(A__SIEVE(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(sieve(tail(cons(z0, z1)))) -> c17(A__SIEVE(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(sieve(tail(divides(z0, z1)))) -> c17(A__SIEVE(a__tail(divides(mark(z0), mark(z1)))), MARK(tail(divides(z0, z1)))) MARK(sieve(tail(x0))) -> c17(MARK(tail(x0))) MARK(sieve(tail(0))) -> c17(A__SIEVE(a__tail(0))) MARK(sieve(tail(true))) -> c17(A__SIEVE(a__tail(true))) MARK(sieve(tail(false))) -> c17(A__SIEVE(a__tail(false))) MARK(sieve(if(primes, x1, x2))) -> c17(A__SIEVE(a__if(a__primes, x1, x2)), MARK(if(primes, x1, x2))) MARK(sieve(if(sieve(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__sieve(mark(z0)), x1, x2)), MARK(if(sieve(z0), x1, x2))) MARK(sieve(if(from(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__from(mark(z0)), x1, x2)), MARK(if(from(z0), x1, x2))) MARK(sieve(if(head(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__head(mark(z0)), x1, x2)), MARK(if(head(z0), x1, x2))) MARK(sieve(if(tail(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__tail(mark(z0)), x1, x2)), MARK(if(tail(z0), x1, x2))) MARK(sieve(if(if(z0, z1, z2), x1, x2))) -> c17(A__SIEVE(a__if(a__if(mark(z0), z1, z2), x1, x2)), MARK(if(if(z0, z1, 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MARK(head(from(s(z0)))) -> c19(A__HEAD(a__from(s(mark(z0)))), MARK(from(s(z0)))) MARK(head(from(0))) -> c19(A__HEAD(a__from(0)), MARK(from(0))) MARK(head(from(cons(z0, z1)))) -> c19(A__HEAD(a__from(cons(mark(z0), z1))), MARK(from(cons(z0, z1)))) MARK(head(from(true))) -> c19(A__HEAD(a__from(true)), MARK(from(true))) MARK(head(from(false))) -> c19(A__HEAD(a__from(false)), MARK(from(false))) MARK(head(from(divides(z0, z1)))) -> c19(A__HEAD(a__from(divides(mark(z0), mark(z1)))), MARK(from(divides(z0, z1)))) MARK(head(from(x0))) -> c19(MARK(from(x0))) MARK(head(head(primes))) -> c19(A__HEAD(a__head(a__primes)), MARK(head(primes))) MARK(head(head(sieve(z0)))) -> c19(A__HEAD(a__head(a__sieve(mark(z0)))), MARK(head(sieve(z0)))) MARK(head(head(from(z0)))) -> c19(A__HEAD(a__head(a__from(mark(z0)))), MARK(head(from(z0)))) MARK(head(head(head(z0)))) -> c19(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c19(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(if(z0, z1, z2)))) -> c19(A__HEAD(a__head(a__if(mark(z0), z1, z2))), MARK(head(if(z0, z1, z2)))) MARK(head(head(filter(z0, z1)))) -> c19(A__HEAD(a__head(a__filter(mark(z0), mark(z1)))), MARK(head(filter(z0, z1)))) MARK(head(head(s(z0)))) -> c19(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(cons(z0, z1)))) -> c19(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(divides(z0, z1)))) -> c19(A__HEAD(a__head(divides(mark(z0), mark(z1)))), MARK(head(divides(z0, z1)))) MARK(head(head(x0))) -> c19(MARK(head(x0))) MARK(head(head(0))) -> c19(A__HEAD(a__head(0))) MARK(head(head(true))) -> c19(A__HEAD(a__head(true))) MARK(head(head(false))) -> c19(A__HEAD(a__head(false))) MARK(head(tail(primes))) -> c19(A__HEAD(a__tail(a__primes)), MARK(tail(primes))) MARK(head(tail(sieve(z0)))) -> c19(A__HEAD(a__tail(a__sieve(mark(z0)))), MARK(tail(sieve(z0)))) MARK(head(tail(from(z0)))) -> c19(A__HEAD(a__tail(a__from(mark(z0)))), MARK(tail(from(z0)))) MARK(head(tail(head(z0)))) -> c19(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c19(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(if(z0, z1, z2)))) -> c19(A__HEAD(a__tail(a__if(mark(z0), z1, z2))), MARK(tail(if(z0, z1, z2)))) MARK(head(tail(filter(z0, z1)))) -> c19(A__HEAD(a__tail(a__filter(mark(z0), mark(z1)))), MARK(tail(filter(z0, z1)))) MARK(head(tail(s(z0)))) -> c19(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(cons(z0, z1)))) -> c19(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(divides(z0, z1)))) -> c19(A__HEAD(a__tail(divides(mark(z0), mark(z1)))), MARK(tail(divides(z0, z1)))) MARK(head(tail(x0))) -> c19(MARK(tail(x0))) MARK(head(tail(0))) -> c19(A__HEAD(a__tail(0))) MARK(head(tail(true))) -> c19(A__HEAD(a__tail(true))) MARK(head(tail(false))) -> c19(A__HEAD(a__tail(false))) MARK(head(if(primes, x1, x2))) -> c19(A__HEAD(a__if(a__primes, x1, x2)), MARK(if(primes, x1, x2))) MARK(head(if(sieve(z0), x1, x2))) -> c19(A__HEAD(a__if(a__sieve(mark(z0)), x1, x2)), MARK(if(sieve(z0), x1, x2))) MARK(head(if(from(z0), x1, x2))) -> c19(A__HEAD(a__if(a__from(mark(z0)), x1, x2)), MARK(if(from(z0), x1, x2))) MARK(head(if(head(z0), x1, x2))) -> c19(A__HEAD(a__if(a__head(mark(z0)), x1, x2)), MARK(if(head(z0), x1, x2))) MARK(head(if(tail(z0), x1, x2))) -> c19(A__HEAD(a__if(a__tail(mark(z0)), x1, x2)), MARK(if(tail(z0), x1, x2))) MARK(head(if(if(z0, z1, z2), x1, x2))) -> c19(A__HEAD(a__if(a__if(mark(z0), z1, z2), x1, x2)), MARK(if(if(z0, z1, z2), x1, x2))) MARK(head(if(filter(z0, z1), x1, x2))) -> c19(A__HEAD(a__if(a__filter(mark(z0), mark(z1)), x1, x2)), MARK(if(filter(z0, z1), x1, x2))) MARK(head(if(s(z0), x1, x2))) -> c19(A__HEAD(a__if(s(mark(z0)), x1, x2)), MARK(if(s(z0), x1, x2))) MARK(head(if(cons(z0, z1), x1, x2))) -> c19(A__HEAD(a__if(cons(mark(z0), z1), x1, x2)), MARK(if(cons(z0, z1), x1, x2))) MARK(head(if(true, x1, x2))) -> c19(A__HEAD(a__if(true, x1, x2)), MARK(if(true, x1, x2))) MARK(head(if(false, x1, x2))) -> c19(A__HEAD(a__if(false, x1, x2)), MARK(if(false, x1, x2))) MARK(head(if(divides(z0, z1), x1, x2))) -> c19(A__HEAD(a__if(divides(mark(z0), mark(z1)), x1, x2)), MARK(if(divides(z0, z1), x1, x2))) MARK(head(if(x0, z1, z2))) -> c19(MARK(if(x0, z1, z2))) MARK(head(if(0, x1, x2))) -> c19(A__HEAD(a__if(0, x1, x2))) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c18_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_1, c17_2, c17_1, c19_2, c19_1, c20_2, c20_1, c21_2, c21_1, c22_2, c22_1, c23_2, c23_1, c_2 ---------------------------------------- (151) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(head(filter(z0, z1))) -> c19(A__HEAD(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) by MARK(head(filter(x0, x1))) -> c19(A__HEAD(filter(mark(x0), mark(x1))), MARK(filter(x0, x1))) MARK(head(filter(x0, primes))) -> c19(A__HEAD(a__filter(mark(x0), a__primes)), MARK(filter(x0, primes))) MARK(head(filter(x0, sieve(z0)))) -> c19(A__HEAD(a__filter(mark(x0), a__sieve(mark(z0)))), MARK(filter(x0, sieve(z0)))) MARK(head(filter(x0, from(z0)))) -> c19(A__HEAD(a__filter(mark(x0), a__from(mark(z0)))), MARK(filter(x0, from(z0)))) MARK(head(filter(x0, head(z0)))) -> c19(A__HEAD(a__filter(mark(x0), a__head(mark(z0)))), MARK(filter(x0, head(z0)))) MARK(head(filter(x0, tail(z0)))) -> c19(A__HEAD(a__filter(mark(x0), a__tail(mark(z0)))), MARK(filter(x0, tail(z0)))) MARK(head(filter(x0, if(z0, z1, z2)))) -> c19(A__HEAD(a__filter(mark(x0), a__if(mark(z0), z1, z2))), MARK(filter(x0, if(z0, z1, z2)))) MARK(head(filter(x0, filter(z0, z1)))) -> c19(A__HEAD(a__filter(mark(x0), a__filter(mark(z0), mark(z1)))), MARK(filter(x0, filter(z0, z1)))) MARK(head(filter(x0, s(z0)))) -> c19(A__HEAD(a__filter(mark(x0), s(mark(z0)))), MARK(filter(x0, s(z0)))) MARK(head(filter(x0, 0))) -> c19(A__HEAD(a__filter(mark(x0), 0)), MARK(filter(x0, 0))) MARK(head(filter(x0, cons(z0, z1)))) -> c19(A__HEAD(a__filter(mark(x0), cons(mark(z0), z1))), MARK(filter(x0, cons(z0, z1)))) MARK(head(filter(x0, true))) -> c19(A__HEAD(a__filter(mark(x0), true)), MARK(filter(x0, true))) MARK(head(filter(x0, false))) -> c19(A__HEAD(a__filter(mark(x0), false)), MARK(filter(x0, false))) MARK(head(filter(x0, divides(z0, z1)))) -> c19(A__HEAD(a__filter(mark(x0), divides(mark(z0), mark(z1)))), MARK(filter(x0, divides(z0, z1)))) MARK(head(filter(primes, x1))) -> c19(A__HEAD(a__filter(a__primes, mark(x1))), MARK(filter(primes, x1))) MARK(head(filter(sieve(z0), x1))) -> c19(A__HEAD(a__filter(a__sieve(mark(z0)), mark(x1))), MARK(filter(sieve(z0), x1))) MARK(head(filter(from(z0), x1))) -> c19(A__HEAD(a__filter(a__from(mark(z0)), mark(x1))), MARK(filter(from(z0), x1))) MARK(head(filter(head(z0), x1))) -> c19(A__HEAD(a__filter(a__head(mark(z0)), mark(x1))), MARK(filter(head(z0), x1))) MARK(head(filter(tail(z0), x1))) -> c19(A__HEAD(a__filter(a__tail(mark(z0)), mark(x1))), MARK(filter(tail(z0), x1))) MARK(head(filter(if(z0, z1, z2), x1))) -> c19(A__HEAD(a__filter(a__if(mark(z0), z1, z2), mark(x1))), MARK(filter(if(z0, z1, z2), x1))) MARK(head(filter(filter(z0, z1), x1))) -> c19(A__HEAD(a__filter(a__filter(mark(z0), mark(z1)), mark(x1))), MARK(filter(filter(z0, z1), x1))) MARK(head(filter(s(z0), x1))) -> c19(A__HEAD(a__filter(s(mark(z0)), mark(x1))), MARK(filter(s(z0), x1))) MARK(head(filter(0, x1))) -> c19(A__HEAD(a__filter(0, mark(x1))), MARK(filter(0, x1))) MARK(head(filter(cons(z0, z1), x1))) -> c19(A__HEAD(a__filter(cons(mark(z0), z1), mark(x1))), MARK(filter(cons(z0, z1), x1))) MARK(head(filter(true, x1))) -> c19(A__HEAD(a__filter(true, mark(x1))), MARK(filter(true, x1))) MARK(head(filter(false, x1))) -> c19(A__HEAD(a__filter(false, mark(x1))), MARK(filter(false, x1))) MARK(head(filter(divides(z0, z1), x1))) -> c19(A__HEAD(a__filter(divides(mark(z0), mark(z1)), mark(x1))), MARK(filter(divides(z0, z1), x1))) ---------------------------------------- (152) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(primes)) -> c20(A__TAIL(a__primes), MARK(primes)) MARK(tail(sieve(z0))) -> c20(A__TAIL(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(tail(from(z0))) -> c20(A__TAIL(a__from(mark(z0))), MARK(from(z0))) MARK(tail(head(z0))) -> c20(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c20(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(if(z0, z1, z2))) -> c20(A__TAIL(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(tail(filter(z0, z1))) -> c20(A__TAIL(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(primes, x1, x2)) -> c21(A__IF(a__primes, x1, x2), MARK(primes)) MARK(if(sieve(z0), x1, x2)) -> c21(A__IF(a__sieve(mark(z0)), x1, x2), MARK(sieve(z0))) MARK(if(from(z0), x1, x2)) -> c21(A__IF(a__from(mark(z0)), x1, x2), MARK(from(z0))) MARK(if(head(z0), x1, x2)) -> c21(A__IF(a__head(mark(z0)), x1, x2), MARK(head(z0))) MARK(if(tail(z0), x1, x2)) -> c21(A__IF(a__tail(mark(z0)), x1, x2), MARK(tail(z0))) MARK(if(if(z0, z1, z2), x1, x2)) -> c21(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(filter(z0, z1), x1, x2)) -> c21(A__IF(a__filter(mark(z0), mark(z1)), x1, x2), MARK(filter(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c22(A__FILTER(mark(x0), a__primes), MARK(x0)) MARK(filter(x0, sieve(z0))) -> c22(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, from(z0))) -> c22(A__FILTER(mark(x0), a__from(mark(z0))), MARK(x0)) MARK(filter(x0, head(z0))) -> c22(A__FILTER(mark(x0), a__head(mark(z0))), MARK(x0)) MARK(filter(x0, tail(z0))) -> c22(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(x0)) MARK(filter(x0, if(z0, z1, z2))) -> c22(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0)) MARK(filter(x0, filter(z0, z1))) -> c22(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(x0)) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(x0, s(z0))) -> c22(MARK(x0)) MARK(filter(x0, 0)) -> c22(MARK(x0)) MARK(filter(x0, true)) -> c22(MARK(x0)) MARK(filter(x0, false)) -> c22(MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), x1)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c23(A__FILTER(mark(x0), a__primes), MARK(primes)) MARK(filter(x0, sieve(z0))) -> c23(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, from(z0))) -> c23(A__FILTER(mark(x0), a__from(mark(z0))), MARK(from(z0))) MARK(filter(x0, 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MARK(head(filter(if(z0, z1, z2), x1))) -> c19(A__HEAD(a__filter(a__if(mark(z0), z1, z2), mark(x1))), MARK(filter(if(z0, z1, z2), x1))) MARK(head(filter(filter(z0, z1), x1))) -> c19(A__HEAD(a__filter(a__filter(mark(z0), mark(z1)), mark(x1))), MARK(filter(filter(z0, z1), x1))) MARK(head(filter(s(z0), x1))) -> c19(A__HEAD(a__filter(s(mark(z0)), mark(x1))), MARK(filter(s(z0), x1))) MARK(head(filter(0, x1))) -> c19(A__HEAD(a__filter(0, mark(x1))), MARK(filter(0, x1))) MARK(head(filter(cons(z0, z1), x1))) -> c19(A__HEAD(a__filter(cons(mark(z0), z1), mark(x1))), MARK(filter(cons(z0, z1), x1))) MARK(head(filter(true, x1))) -> c19(A__HEAD(a__filter(true, mark(x1))), MARK(filter(true, x1))) MARK(head(filter(false, x1))) -> c19(A__HEAD(a__filter(false, mark(x1))), MARK(filter(false, x1))) MARK(head(filter(divides(z0, z1), x1))) -> c19(A__HEAD(a__filter(divides(mark(z0), mark(z1)), mark(x1))), MARK(filter(divides(z0, z1), x1))) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c18_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_1, c17_2, c17_1, c19_2, c19_1, c20_2, c20_1, c21_2, c21_1, c22_2, c22_1, c23_2, c23_1, c_2 ---------------------------------------- (153) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (154) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(primes)) -> c20(A__TAIL(a__primes), MARK(primes)) MARK(tail(sieve(z0))) -> c20(A__TAIL(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(tail(from(z0))) -> c20(A__TAIL(a__from(mark(z0))), MARK(from(z0))) MARK(tail(head(z0))) -> c20(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c20(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(if(z0, z1, z2))) -> c20(A__TAIL(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(tail(filter(z0, z1))) -> c20(A__TAIL(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(primes, x1, x2)) -> c21(A__IF(a__primes, x1, x2), MARK(primes)) MARK(if(sieve(z0), x1, x2)) -> c21(A__IF(a__sieve(mark(z0)), x1, x2), MARK(sieve(z0))) MARK(if(from(z0), x1, x2)) -> c21(A__IF(a__from(mark(z0)), x1, x2), MARK(from(z0))) MARK(if(head(z0), x1, x2)) -> c21(A__IF(a__head(mark(z0)), x1, x2), MARK(head(z0))) MARK(if(tail(z0), x1, x2)) -> c21(A__IF(a__tail(mark(z0)), x1, x2), MARK(tail(z0))) MARK(if(if(z0, z1, z2), x1, x2)) -> c21(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(filter(z0, z1), x1, x2)) -> c21(A__IF(a__filter(mark(z0), mark(z1)), x1, x2), MARK(filter(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c22(A__FILTER(mark(x0), a__primes), MARK(x0)) MARK(filter(x0, sieve(z0))) -> c22(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, from(z0))) -> c22(A__FILTER(mark(x0), a__from(mark(z0))), MARK(x0)) MARK(filter(x0, head(z0))) -> c22(A__FILTER(mark(x0), a__head(mark(z0))), MARK(x0)) MARK(filter(x0, tail(z0))) -> c22(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(x0)) MARK(filter(x0, if(z0, z1, z2))) -> c22(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0)) MARK(filter(x0, filter(z0, z1))) -> c22(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(x0)) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(x0, s(z0))) -> c22(MARK(x0)) MARK(filter(x0, 0)) -> c22(MARK(x0)) MARK(filter(x0, true)) -> c22(MARK(x0)) MARK(filter(x0, false)) -> c22(MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), x1)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c23(A__FILTER(mark(x0), a__primes), MARK(primes)) MARK(filter(x0, sieve(z0))) -> c23(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, from(z0))) -> c23(A__FILTER(mark(x0), a__from(mark(z0))), MARK(from(z0))) MARK(filter(x0, head(z0))) -> c23(A__FILTER(mark(x0), a__head(mark(z0))), MARK(head(z0))) MARK(filter(x0, tail(z0))) -> c23(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(tail(z0))) MARK(filter(x0, if(z0, z1, z2))) -> c23(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(filter(x0, filter(z0, z1))) -> c23(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(filter(x0, cons(z0, z1))) -> c23(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(primes, x1)) -> c23(A__FILTER(a__primes, mark(x1)), MARK(x1)) MARK(filter(sieve(z0), x1)) -> c23(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(from(z0), x1)) -> c23(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(head(z0), x1)) -> c23(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(tail(z0), x1)) -> c23(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(if(z0, z1, z2), x1)) -> c23(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1)) MARK(filter(filter(z0, z1), x1)) -> c23(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(x1)) MARK(filter(s(z0), x1)) -> c23(A__FILTER(s(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(x0, s(z0))) -> c23(MARK(s(z0))) MARK(filter(x0, divides(z0, z1))) -> c23(MARK(divides(z0, z1))) MARK(filter(0, x1)) -> c23(MARK(x1)) MARK(filter(cons(z0, z1), x1)) -> c23(MARK(x1)) MARK(filter(true, x1)) -> c23(MARK(x1)) MARK(filter(false, x1)) -> c23(MARK(x1)) MARK(filter(divides(z0, z1), x1)) -> c23(MARK(x1)) A__PRIMES -> c(A__SIEVE(cons(s(mark(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(sieve(primes)) -> c17(MARK(primes)) MARK(sieve(sieve(primes))) -> c17(A__SIEVE(a__sieve(a__primes)), MARK(sieve(primes))) MARK(sieve(sieve(sieve(z0)))) -> c17(A__SIEVE(a__sieve(a__sieve(mark(z0)))), MARK(sieve(sieve(z0)))) MARK(sieve(sieve(from(z0)))) -> c17(A__SIEVE(a__sieve(a__from(mark(z0)))), MARK(sieve(from(z0)))) MARK(sieve(sieve(head(z0)))) -> c17(A__SIEVE(a__sieve(a__head(mark(z0)))), MARK(sieve(head(z0)))) MARK(sieve(sieve(tail(z0)))) -> c17(A__SIEVE(a__sieve(a__tail(mark(z0)))), MARK(sieve(tail(z0)))) MARK(sieve(sieve(if(z0, z1, z2)))) -> c17(A__SIEVE(a__sieve(a__if(mark(z0), z1, z2))), MARK(sieve(if(z0, z1, z2)))) MARK(sieve(sieve(filter(z0, z1)))) -> c17(A__SIEVE(a__sieve(a__filter(mark(z0), mark(z1)))), MARK(sieve(filter(z0, z1)))) MARK(sieve(sieve(s(z0)))) -> c17(A__SIEVE(a__sieve(s(mark(z0)))), MARK(sieve(s(z0)))) MARK(sieve(sieve(cons(z0, z1)))) -> c17(A__SIEVE(a__sieve(cons(mark(z0), z1))), MARK(sieve(cons(z0, z1)))) MARK(sieve(sieve(divides(z0, z1)))) -> c17(A__SIEVE(a__sieve(divides(mark(z0), mark(z1)))), MARK(sieve(divides(z0, z1)))) MARK(sieve(sieve(x0))) -> c17(MARK(sieve(x0))) MARK(sieve(sieve(0))) -> c17(A__SIEVE(a__sieve(0))) MARK(sieve(sieve(true))) -> c17(A__SIEVE(a__sieve(true))) MARK(sieve(sieve(false))) -> c17(A__SIEVE(a__sieve(false))) MARK(sieve(from(x0))) -> c17(A__SIEVE(cons(mark(mark(x0)), from(s(mark(x0))))), MARK(from(x0))) MARK(sieve(from(primes))) -> c17(A__SIEVE(a__from(a__primes)), MARK(from(primes))) MARK(sieve(from(sieve(z0)))) -> c17(A__SIEVE(a__from(a__sieve(mark(z0)))), MARK(from(sieve(z0)))) MARK(sieve(from(from(z0)))) -> c17(A__SIEVE(a__from(a__from(mark(z0)))), MARK(from(from(z0)))) MARK(sieve(from(head(z0)))) -> c17(A__SIEVE(a__from(a__head(mark(z0)))), MARK(from(head(z0)))) MARK(sieve(from(tail(z0)))) -> c17(A__SIEVE(a__from(a__tail(mark(z0)))), MARK(from(tail(z0)))) MARK(sieve(from(if(z0, z1, z2)))) -> c17(A__SIEVE(a__from(a__if(mark(z0), z1, z2))), MARK(from(if(z0, z1, z2)))) MARK(sieve(from(filter(z0, z1)))) -> c17(A__SIEVE(a__from(a__filter(mark(z0), mark(z1)))), MARK(from(filter(z0, z1)))) MARK(sieve(from(s(z0)))) -> c17(A__SIEVE(a__from(s(mark(z0)))), MARK(from(s(z0)))) MARK(sieve(from(0))) -> c17(A__SIEVE(a__from(0)), MARK(from(0))) MARK(sieve(from(cons(z0, z1)))) -> c17(A__SIEVE(a__from(cons(mark(z0), z1))), MARK(from(cons(z0, z1)))) MARK(sieve(from(true))) -> c17(A__SIEVE(a__from(true)), MARK(from(true))) MARK(sieve(from(false))) -> c17(A__SIEVE(a__from(false)), MARK(from(false))) MARK(sieve(from(divides(z0, z1)))) -> c17(A__SIEVE(a__from(divides(mark(z0), mark(z1)))), MARK(from(divides(z0, z1)))) MARK(sieve(from(x0))) -> c17(MARK(from(x0))) MARK(sieve(head(primes))) -> c17(A__SIEVE(a__head(a__primes)), MARK(head(primes))) MARK(sieve(head(sieve(z0)))) -> c17(A__SIEVE(a__head(a__sieve(mark(z0)))), MARK(head(sieve(z0)))) MARK(sieve(head(from(z0)))) -> c17(A__SIEVE(a__head(a__from(mark(z0)))), MARK(head(from(z0)))) MARK(sieve(head(head(z0)))) -> 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A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(primes)) -> c20(A__TAIL(a__primes), MARK(primes)) MARK(tail(sieve(z0))) -> c20(A__TAIL(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(tail(from(z0))) -> c20(A__TAIL(a__from(mark(z0))), MARK(from(z0))) MARK(tail(head(z0))) -> c20(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c20(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(if(z0, z1, z2))) -> c20(A__TAIL(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(tail(filter(z0, z1))) -> c20(A__TAIL(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(tail(cons(z0, 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MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c22(A__FILTER(mark(x0), a__primes), MARK(x0)) MARK(filter(x0, sieve(z0))) -> c22(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, from(z0))) -> c22(A__FILTER(mark(x0), a__from(mark(z0))), MARK(x0)) MARK(filter(x0, head(z0))) -> c22(A__FILTER(mark(x0), a__head(mark(z0))), MARK(x0)) MARK(filter(x0, tail(z0))) -> c22(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(x0)) MARK(filter(x0, if(z0, z1, z2))) -> c22(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0)) MARK(filter(x0, filter(z0, z1))) -> c22(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(x0)) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(x0, s(z0))) -> c22(MARK(x0)) MARK(filter(x0, 0)) -> c22(MARK(x0)) MARK(filter(x0, true)) -> c22(MARK(x0)) MARK(filter(x0, false)) -> c22(MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), x1)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c23(A__FILTER(mark(x0), a__primes), MARK(primes)) MARK(filter(x0, sieve(z0))) -> c23(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, from(z0))) -> c23(A__FILTER(mark(x0), a__from(mark(z0))), MARK(from(z0))) MARK(filter(x0, head(z0))) -> c23(A__FILTER(mark(x0), a__head(mark(z0))), MARK(head(z0))) MARK(filter(x0, tail(z0))) -> c23(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(tail(z0))) MARK(filter(x0, if(z0, z1, z2))) -> c23(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(filter(x0, filter(z0, z1))) -> c23(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(filter(x0, cons(z0, z1))) -> c23(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(primes, x1)) -> c23(A__FILTER(a__primes, mark(x1)), MARK(x1)) MARK(filter(sieve(z0), x1)) -> c23(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(from(z0), x1)) -> c23(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(head(z0), x1)) -> c23(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(tail(z0), x1)) -> c23(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(if(z0, z1, z2), x1)) -> c23(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1)) MARK(filter(filter(z0, z1), x1)) -> c23(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(x1)) MARK(filter(s(z0), x1)) -> c23(A__FILTER(s(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(x0, s(z0))) -> c23(MARK(s(z0))) MARK(filter(x0, divides(z0, z1))) -> c23(MARK(divides(z0, z1))) MARK(filter(0, x1)) -> c23(MARK(x1)) MARK(filter(cons(z0, z1), x1)) -> c23(MARK(x1)) MARK(filter(true, x1)) -> c23(MARK(x1)) MARK(filter(false, x1)) -> c23(MARK(x1)) MARK(filter(divides(z0, z1), x1)) -> c23(MARK(x1)) A__PRIMES -> c(A__SIEVE(cons(s(mark(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(sieve(primes)) -> c17(MARK(primes)) MARK(sieve(sieve(primes))) -> c17(A__SIEVE(a__sieve(a__primes)), MARK(sieve(primes))) MARK(sieve(sieve(sieve(z0)))) -> c17(A__SIEVE(a__sieve(a__sieve(mark(z0)))), MARK(sieve(sieve(z0)))) MARK(sieve(sieve(from(z0)))) -> c17(A__SIEVE(a__sieve(a__from(mark(z0)))), MARK(sieve(from(z0)))) MARK(sieve(sieve(head(z0)))) -> c17(A__SIEVE(a__sieve(a__head(mark(z0)))), MARK(sieve(head(z0)))) MARK(sieve(sieve(tail(z0)))) -> c17(A__SIEVE(a__sieve(a__tail(mark(z0)))), MARK(sieve(tail(z0)))) MARK(sieve(sieve(if(z0, z1, z2)))) -> c17(A__SIEVE(a__sieve(a__if(mark(z0), z1, z2))), MARK(sieve(if(z0, z1, z2)))) MARK(sieve(sieve(filter(z0, z1)))) -> c17(A__SIEVE(a__sieve(a__filter(mark(z0), mark(z1)))), MARK(sieve(filter(z0, z1)))) MARK(sieve(sieve(s(z0)))) -> c17(A__SIEVE(a__sieve(s(mark(z0)))), MARK(sieve(s(z0)))) MARK(sieve(sieve(cons(z0, z1)))) -> c17(A__SIEVE(a__sieve(cons(mark(z0), z1))), MARK(sieve(cons(z0, z1)))) MARK(sieve(sieve(divides(z0, z1)))) -> c17(A__SIEVE(a__sieve(divides(mark(z0), mark(z1)))), MARK(sieve(divides(z0, z1)))) MARK(sieve(sieve(x0))) -> c17(MARK(sieve(x0))) MARK(sieve(sieve(0))) -> c17(A__SIEVE(a__sieve(0))) MARK(sieve(sieve(true))) -> c17(A__SIEVE(a__sieve(true))) MARK(sieve(sieve(false))) -> c17(A__SIEVE(a__sieve(false))) MARK(sieve(from(x0))) -> c17(A__SIEVE(cons(mark(mark(x0)), from(s(mark(x0))))), MARK(from(x0))) MARK(sieve(from(primes))) -> c17(A__SIEVE(a__from(a__primes)), MARK(from(primes))) MARK(sieve(from(sieve(z0)))) -> c17(A__SIEVE(a__from(a__sieve(mark(z0)))), MARK(from(sieve(z0)))) MARK(sieve(from(from(z0)))) -> c17(A__SIEVE(a__from(a__from(mark(z0)))), MARK(from(from(z0)))) MARK(sieve(from(head(z0)))) -> c17(A__SIEVE(a__from(a__head(mark(z0)))), MARK(from(head(z0)))) MARK(sieve(from(tail(z0)))) -> c17(A__SIEVE(a__from(a__tail(mark(z0)))), MARK(from(tail(z0)))) MARK(sieve(from(if(z0, z1, z2)))) -> c17(A__SIEVE(a__from(a__if(mark(z0), z1, z2))), MARK(from(if(z0, z1, z2)))) MARK(sieve(from(filter(z0, z1)))) -> c17(A__SIEVE(a__from(a__filter(mark(z0), mark(z1)))), MARK(from(filter(z0, z1)))) MARK(sieve(from(s(z0)))) -> c17(A__SIEVE(a__from(s(mark(z0)))), MARK(from(s(z0)))) MARK(sieve(from(0))) -> c17(A__SIEVE(a__from(0)), MARK(from(0))) MARK(sieve(from(cons(z0, z1)))) -> c17(A__SIEVE(a__from(cons(mark(z0), z1))), MARK(from(cons(z0, z1)))) MARK(sieve(from(true))) -> c17(A__SIEVE(a__from(true)), MARK(from(true))) MARK(sieve(from(false))) -> c17(A__SIEVE(a__from(false)), MARK(from(false))) MARK(sieve(from(divides(z0, z1)))) -> c17(A__SIEVE(a__from(divides(mark(z0), mark(z1)))), MARK(from(divides(z0, z1)))) MARK(sieve(from(x0))) -> c17(MARK(from(x0))) MARK(sieve(head(primes))) -> c17(A__SIEVE(a__head(a__primes)), MARK(head(primes))) MARK(sieve(head(sieve(z0)))) -> c17(A__SIEVE(a__head(a__sieve(mark(z0)))), MARK(head(sieve(z0)))) MARK(sieve(head(from(z0)))) -> c17(A__SIEVE(a__head(a__from(mark(z0)))), MARK(head(from(z0)))) MARK(sieve(head(head(z0)))) -> c17(A__SIEVE(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(sieve(head(tail(z0)))) -> c17(A__SIEVE(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(sieve(head(if(z0, z1, z2)))) -> c17(A__SIEVE(a__head(a__if(mark(z0), 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c17(A__SIEVE(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(sieve(tail(tail(z0)))) -> c17(A__SIEVE(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(sieve(tail(if(z0, z1, z2)))) -> c17(A__SIEVE(a__tail(a__if(mark(z0), z1, z2))), MARK(tail(if(z0, z1, z2)))) MARK(sieve(tail(filter(z0, z1)))) -> c17(A__SIEVE(a__tail(a__filter(mark(z0), mark(z1)))), MARK(tail(filter(z0, z1)))) MARK(sieve(tail(s(z0)))) -> c17(A__SIEVE(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(sieve(tail(cons(z0, z1)))) -> c17(A__SIEVE(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(sieve(tail(divides(z0, z1)))) -> c17(A__SIEVE(a__tail(divides(mark(z0), mark(z1)))), MARK(tail(divides(z0, z1)))) MARK(sieve(tail(x0))) -> c17(MARK(tail(x0))) MARK(sieve(tail(0))) -> c17(A__SIEVE(a__tail(0))) MARK(sieve(tail(true))) -> c17(A__SIEVE(a__tail(true))) MARK(sieve(tail(false))) -> c17(A__SIEVE(a__tail(false))) MARK(sieve(if(primes, x1, x2))) -> c17(A__SIEVE(a__if(a__primes, x1, x2)), MARK(if(primes, x1, 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c17(A__SIEVE(a__if(true, x1, x2)), MARK(if(true, x1, x2))) MARK(sieve(if(false, x1, x2))) -> c17(A__SIEVE(a__if(false, x1, x2)), MARK(if(false, x1, x2))) MARK(sieve(if(divides(z0, z1), x1, x2))) -> c17(A__SIEVE(a__if(divides(mark(z0), mark(z1)), x1, x2)), MARK(if(divides(z0, z1), x1, x2))) MARK(sieve(if(x0, z1, z2))) -> c17(MARK(if(x0, z1, z2))) MARK(sieve(if(0, x1, x2))) -> c17(A__SIEVE(a__if(0, x1, x2))) MARK(sieve(filter(x0, primes))) -> c17(A__SIEVE(a__filter(mark(x0), a__primes)), MARK(filter(x0, primes))) MARK(sieve(filter(x0, sieve(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), a__sieve(mark(z0)))), MARK(filter(x0, sieve(z0)))) MARK(sieve(filter(x0, from(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), a__from(mark(z0)))), MARK(filter(x0, from(z0)))) MARK(sieve(filter(x0, head(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), a__head(mark(z0)))), MARK(filter(x0, head(z0)))) MARK(sieve(filter(x0, tail(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), a__tail(mark(z0)))), MARK(filter(x0, tail(z0)))) MARK(sieve(filter(x0, if(z0, z1, z2)))) -> c17(A__SIEVE(a__filter(mark(x0), a__if(mark(z0), z1, z2))), MARK(filter(x0, if(z0, z1, z2)))) MARK(sieve(filter(x0, filter(z0, z1)))) -> c17(A__SIEVE(a__filter(mark(x0), a__filter(mark(z0), mark(z1)))), MARK(filter(x0, filter(z0, z1)))) MARK(sieve(filter(x0, s(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), s(mark(z0)))), MARK(filter(x0, s(z0)))) MARK(sieve(filter(x0, 0))) -> c17(A__SIEVE(a__filter(mark(x0), 0)), MARK(filter(x0, 0))) MARK(sieve(filter(x0, cons(z0, z1)))) -> c17(A__SIEVE(a__filter(mark(x0), cons(mark(z0), z1))), MARK(filter(x0, cons(z0, z1)))) MARK(sieve(filter(x0, true))) -> c17(A__SIEVE(a__filter(mark(x0), true)), MARK(filter(x0, true))) MARK(sieve(filter(x0, false))) -> c17(A__SIEVE(a__filter(mark(x0), false)), MARK(filter(x0, false))) MARK(sieve(filter(x0, divides(z0, z1)))) -> c17(A__SIEVE(a__filter(mark(x0), divides(mark(z0), mark(z1)))), MARK(filter(x0, divides(z0, z1)))) MARK(sieve(filter(primes, x1))) -> c17(A__SIEVE(a__filter(a__primes, mark(x1))), MARK(filter(primes, x1))) MARK(sieve(filter(sieve(z0), x1))) -> c17(A__SIEVE(a__filter(a__sieve(mark(z0)), mark(x1))), MARK(filter(sieve(z0), x1))) MARK(sieve(filter(from(z0), x1))) -> c17(A__SIEVE(a__filter(a__from(mark(z0)), mark(x1))), MARK(filter(from(z0), x1))) MARK(sieve(filter(head(z0), x1))) -> c17(A__SIEVE(a__filter(a__head(mark(z0)), mark(x1))), MARK(filter(head(z0), x1))) MARK(sieve(filter(tail(z0), x1))) -> c17(A__SIEVE(a__filter(a__tail(mark(z0)), mark(x1))), MARK(filter(tail(z0), x1))) MARK(sieve(filter(if(z0, z1, z2), x1))) -> c17(A__SIEVE(a__filter(a__if(mark(z0), z1, z2), mark(x1))), MARK(filter(if(z0, z1, z2), x1))) MARK(sieve(filter(filter(z0, z1), x1))) -> c17(A__SIEVE(a__filter(a__filter(mark(z0), mark(z1)), mark(x1))), MARK(filter(filter(z0, z1), x1))) MARK(sieve(filter(s(z0), x1))) -> c17(A__SIEVE(a__filter(s(mark(z0)), mark(x1))), MARK(filter(s(z0), x1))) MARK(sieve(filter(0, x1))) -> c17(A__SIEVE(a__filter(0, mark(x1))), MARK(filter(0, x1))) MARK(sieve(filter(cons(z0, z1), x1))) -> c17(A__SIEVE(a__filter(cons(mark(z0), z1), mark(x1))), MARK(filter(cons(z0, z1), x1))) MARK(sieve(filter(true, x1))) -> c17(A__SIEVE(a__filter(true, mark(x1))), MARK(filter(true, x1))) MARK(sieve(filter(false, x1))) -> c17(A__SIEVE(a__filter(false, mark(x1))), MARK(filter(false, x1))) MARK(sieve(filter(divides(z0, z1), x1))) -> c17(A__SIEVE(a__filter(divides(mark(z0), mark(z1)), mark(x1))), MARK(filter(divides(z0, z1), x1))) MARK(sieve(filter(x0, x1))) -> c17(MARK(filter(x0, x1))) MARK(head(primes)) -> c19(A__HEAD(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(head(primes)) -> c19(MARK(primes)) MARK(head(sieve(primes))) -> c19(A__HEAD(a__sieve(a__primes)), MARK(sieve(primes))) MARK(head(sieve(sieve(z0)))) -> c19(A__HEAD(a__sieve(a__sieve(mark(z0)))), MARK(sieve(sieve(z0)))) MARK(head(sieve(from(z0)))) -> c19(A__HEAD(a__sieve(a__from(mark(z0)))), MARK(sieve(from(z0)))) MARK(head(sieve(head(z0)))) -> c19(A__HEAD(a__sieve(a__head(mark(z0)))), MARK(sieve(head(z0)))) MARK(head(sieve(tail(z0)))) -> c19(A__HEAD(a__sieve(a__tail(mark(z0)))), MARK(sieve(tail(z0)))) MARK(head(sieve(if(z0, z1, z2)))) -> c19(A__HEAD(a__sieve(a__if(mark(z0), z1, z2))), MARK(sieve(if(z0, z1, z2)))) MARK(head(sieve(filter(z0, z1)))) -> c19(A__HEAD(a__sieve(a__filter(mark(z0), mark(z1)))), MARK(sieve(filter(z0, z1)))) MARK(head(sieve(s(z0)))) -> c19(A__HEAD(a__sieve(s(mark(z0)))), MARK(sieve(s(z0)))) MARK(head(sieve(cons(z0, z1)))) -> c19(A__HEAD(a__sieve(cons(mark(z0), z1))), MARK(sieve(cons(z0, z1)))) MARK(head(sieve(divides(z0, z1)))) -> c19(A__HEAD(a__sieve(divides(mark(z0), mark(z1)))), MARK(sieve(divides(z0, z1)))) MARK(head(sieve(x0))) -> c19(MARK(sieve(x0))) MARK(head(sieve(0))) -> c19(A__HEAD(a__sieve(0))) MARK(head(sieve(true))) -> c19(A__HEAD(a__sieve(true))) MARK(head(sieve(false))) -> c19(A__HEAD(a__sieve(false))) MARK(head(from(x0))) -> c19(A__HEAD(cons(mark(mark(x0)), from(s(mark(x0))))), MARK(from(x0))) MARK(head(from(primes))) -> c19(A__HEAD(a__from(a__primes)), MARK(from(primes))) MARK(head(from(sieve(z0)))) -> c19(A__HEAD(a__from(a__sieve(mark(z0)))), MARK(from(sieve(z0)))) MARK(head(from(from(z0)))) -> c19(A__HEAD(a__from(a__from(mark(z0)))), MARK(from(from(z0)))) MARK(head(from(head(z0)))) -> c19(A__HEAD(a__from(a__head(mark(z0)))), MARK(from(head(z0)))) MARK(head(from(tail(z0)))) -> c19(A__HEAD(a__from(a__tail(mark(z0)))), MARK(from(tail(z0)))) MARK(head(from(if(z0, z1, z2)))) -> c19(A__HEAD(a__from(a__if(mark(z0), z1, z2))), MARK(from(if(z0, z1, z2)))) MARK(head(from(filter(z0, z1)))) -> c19(A__HEAD(a__from(a__filter(mark(z0), mark(z1)))), MARK(from(filter(z0, z1)))) MARK(head(from(s(z0)))) -> c19(A__HEAD(a__from(s(mark(z0)))), MARK(from(s(z0)))) MARK(head(from(0))) -> c19(A__HEAD(a__from(0)), MARK(from(0))) MARK(head(from(cons(z0, z1)))) -> c19(A__HEAD(a__from(cons(mark(z0), z1))), MARK(from(cons(z0, z1)))) MARK(head(from(true))) -> c19(A__HEAD(a__from(true)), MARK(from(true))) MARK(head(from(false))) -> c19(A__HEAD(a__from(false)), MARK(from(false))) MARK(head(from(divides(z0, z1)))) -> c19(A__HEAD(a__from(divides(mark(z0), mark(z1)))), MARK(from(divides(z0, z1)))) MARK(head(from(x0))) -> c19(MARK(from(x0))) MARK(head(head(primes))) -> c19(A__HEAD(a__head(a__primes)), MARK(head(primes))) MARK(head(head(sieve(z0)))) -> c19(A__HEAD(a__head(a__sieve(mark(z0)))), MARK(head(sieve(z0)))) MARK(head(head(from(z0)))) -> c19(A__HEAD(a__head(a__from(mark(z0)))), MARK(head(from(z0)))) MARK(head(head(head(z0)))) -> c19(A__HEAD(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(head(head(tail(z0)))) -> c19(A__HEAD(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(head(head(if(z0, z1, z2)))) -> c19(A__HEAD(a__head(a__if(mark(z0), z1, z2))), MARK(head(if(z0, z1, z2)))) MARK(head(head(filter(z0, z1)))) -> c19(A__HEAD(a__head(a__filter(mark(z0), mark(z1)))), MARK(head(filter(z0, z1)))) MARK(head(head(s(z0)))) -> c19(A__HEAD(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(head(head(cons(z0, z1)))) -> c19(A__HEAD(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(head(head(divides(z0, z1)))) -> c19(A__HEAD(a__head(divides(mark(z0), mark(z1)))), MARK(head(divides(z0, z1)))) MARK(head(head(x0))) -> c19(MARK(head(x0))) MARK(head(head(0))) -> c19(A__HEAD(a__head(0))) MARK(head(head(true))) -> c19(A__HEAD(a__head(true))) MARK(head(head(false))) -> c19(A__HEAD(a__head(false))) MARK(head(tail(primes))) -> c19(A__HEAD(a__tail(a__primes)), MARK(tail(primes))) MARK(head(tail(sieve(z0)))) -> c19(A__HEAD(a__tail(a__sieve(mark(z0)))), MARK(tail(sieve(z0)))) MARK(head(tail(from(z0)))) -> c19(A__HEAD(a__tail(a__from(mark(z0)))), MARK(tail(from(z0)))) MARK(head(tail(head(z0)))) -> c19(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c19(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(if(z0, z1, z2)))) -> c19(A__HEAD(a__tail(a__if(mark(z0), z1, z2))), MARK(tail(if(z0, z1, z2)))) MARK(head(tail(filter(z0, z1)))) -> c19(A__HEAD(a__tail(a__filter(mark(z0), mark(z1)))), MARK(tail(filter(z0, z1)))) MARK(head(tail(s(z0)))) -> c19(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(cons(z0, z1)))) -> c19(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(divides(z0, z1)))) -> c19(A__HEAD(a__tail(divides(mark(z0), mark(z1)))), MARK(tail(divides(z0, z1)))) MARK(head(tail(x0))) -> c19(MARK(tail(x0))) MARK(head(tail(0))) -> c19(A__HEAD(a__tail(0))) MARK(head(tail(true))) -> c19(A__HEAD(a__tail(true))) MARK(head(tail(false))) -> c19(A__HEAD(a__tail(false))) MARK(head(if(primes, x1, x2))) -> c19(A__HEAD(a__if(a__primes, x1, x2)), MARK(if(primes, x1, x2))) MARK(head(if(sieve(z0), x1, x2))) -> c19(A__HEAD(a__if(a__sieve(mark(z0)), x1, x2)), MARK(if(sieve(z0), x1, x2))) MARK(head(if(from(z0), x1, x2))) -> c19(A__HEAD(a__if(a__from(mark(z0)), x1, x2)), MARK(if(from(z0), x1, x2))) MARK(head(if(head(z0), x1, x2))) -> c19(A__HEAD(a__if(a__head(mark(z0)), x1, x2)), MARK(if(head(z0), x1, x2))) MARK(head(if(tail(z0), x1, x2))) -> c19(A__HEAD(a__if(a__tail(mark(z0)), x1, x2)), MARK(if(tail(z0), x1, x2))) MARK(head(if(if(z0, z1, z2), x1, x2))) -> c19(A__HEAD(a__if(a__if(mark(z0), z1, z2), x1, x2)), MARK(if(if(z0, z1, z2), x1, x2))) MARK(head(if(filter(z0, z1), x1, x2))) -> c19(A__HEAD(a__if(a__filter(mark(z0), mark(z1)), x1, x2)), MARK(if(filter(z0, z1), x1, x2))) MARK(head(if(s(z0), x1, x2))) -> c19(A__HEAD(a__if(s(mark(z0)), x1, x2)), MARK(if(s(z0), x1, x2))) MARK(head(if(cons(z0, z1), x1, x2))) -> c19(A__HEAD(a__if(cons(mark(z0), z1), x1, x2)), MARK(if(cons(z0, z1), x1, x2))) MARK(head(if(true, x1, x2))) -> c19(A__HEAD(a__if(true, x1, x2)), MARK(if(true, x1, x2))) MARK(head(if(false, x1, x2))) -> c19(A__HEAD(a__if(false, x1, x2)), MARK(if(false, x1, x2))) MARK(head(if(divides(z0, z1), x1, x2))) -> c19(A__HEAD(a__if(divides(mark(z0), mark(z1)), x1, x2)), MARK(if(divides(z0, z1), x1, x2))) MARK(head(if(x0, z1, z2))) -> c19(MARK(if(x0, z1, z2))) MARK(head(if(0, x1, x2))) -> c19(A__HEAD(a__if(0, x1, x2))) MARK(head(filter(x0, primes))) -> c19(A__HEAD(a__filter(mark(x0), a__primes)), MARK(filter(x0, primes))) MARK(head(filter(x0, sieve(z0)))) -> c19(A__HEAD(a__filter(mark(x0), a__sieve(mark(z0)))), MARK(filter(x0, sieve(z0)))) MARK(head(filter(x0, from(z0)))) -> c19(A__HEAD(a__filter(mark(x0), a__from(mark(z0)))), MARK(filter(x0, from(z0)))) MARK(head(filter(x0, head(z0)))) -> c19(A__HEAD(a__filter(mark(x0), a__head(mark(z0)))), MARK(filter(x0, head(z0)))) MARK(head(filter(x0, tail(z0)))) -> c19(A__HEAD(a__filter(mark(x0), a__tail(mark(z0)))), MARK(filter(x0, tail(z0)))) MARK(head(filter(x0, if(z0, z1, z2)))) -> c19(A__HEAD(a__filter(mark(x0), a__if(mark(z0), z1, z2))), MARK(filter(x0, if(z0, z1, z2)))) MARK(head(filter(x0, filter(z0, z1)))) -> c19(A__HEAD(a__filter(mark(x0), a__filter(mark(z0), mark(z1)))), MARK(filter(x0, filter(z0, z1)))) MARK(head(filter(x0, s(z0)))) -> c19(A__HEAD(a__filter(mark(x0), s(mark(z0)))), MARK(filter(x0, s(z0)))) MARK(head(filter(x0, 0))) -> c19(A__HEAD(a__filter(mark(x0), 0)), MARK(filter(x0, 0))) MARK(head(filter(x0, cons(z0, z1)))) -> c19(A__HEAD(a__filter(mark(x0), cons(mark(z0), z1))), MARK(filter(x0, cons(z0, z1)))) MARK(head(filter(x0, true))) -> c19(A__HEAD(a__filter(mark(x0), true)), MARK(filter(x0, true))) MARK(head(filter(x0, false))) -> c19(A__HEAD(a__filter(mark(x0), false)), MARK(filter(x0, false))) MARK(head(filter(x0, divides(z0, z1)))) -> c19(A__HEAD(a__filter(mark(x0), divides(mark(z0), mark(z1)))), MARK(filter(x0, divides(z0, z1)))) MARK(head(filter(primes, x1))) -> c19(A__HEAD(a__filter(a__primes, mark(x1))), MARK(filter(primes, x1))) MARK(head(filter(sieve(z0), x1))) -> c19(A__HEAD(a__filter(a__sieve(mark(z0)), mark(x1))), MARK(filter(sieve(z0), x1))) MARK(head(filter(from(z0), x1))) -> c19(A__HEAD(a__filter(a__from(mark(z0)), mark(x1))), MARK(filter(from(z0), x1))) MARK(head(filter(head(z0), x1))) -> c19(A__HEAD(a__filter(a__head(mark(z0)), mark(x1))), MARK(filter(head(z0), x1))) MARK(head(filter(tail(z0), x1))) -> c19(A__HEAD(a__filter(a__tail(mark(z0)), mark(x1))), MARK(filter(tail(z0), x1))) MARK(head(filter(if(z0, z1, z2), x1))) -> c19(A__HEAD(a__filter(a__if(mark(z0), z1, z2), mark(x1))), MARK(filter(if(z0, z1, z2), x1))) MARK(head(filter(filter(z0, z1), x1))) -> c19(A__HEAD(a__filter(a__filter(mark(z0), mark(z1)), mark(x1))), MARK(filter(filter(z0, z1), x1))) MARK(head(filter(s(z0), x1))) -> c19(A__HEAD(a__filter(s(mark(z0)), mark(x1))), MARK(filter(s(z0), x1))) MARK(head(filter(0, x1))) -> c19(A__HEAD(a__filter(0, mark(x1))), MARK(filter(0, x1))) MARK(head(filter(cons(z0, z1), x1))) -> c19(A__HEAD(a__filter(cons(mark(z0), z1), mark(x1))), MARK(filter(cons(z0, z1), x1))) MARK(head(filter(true, x1))) -> c19(A__HEAD(a__filter(true, mark(x1))), MARK(filter(true, x1))) MARK(head(filter(false, x1))) -> c19(A__HEAD(a__filter(false, mark(x1))), MARK(filter(false, x1))) MARK(head(filter(divides(z0, z1), x1))) -> c19(A__HEAD(a__filter(divides(mark(z0), mark(z1)), mark(x1))), MARK(filter(divides(z0, z1), x1))) MARK(head(filter(x0, x1))) -> c19(MARK(filter(x0, x1))) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c18_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_1, c17_2, c17_1, c19_2, c19_1, c20_2, c20_1, c21_2, c21_1, c22_2, c22_1, c23_2, c23_1, c_2 ---------------------------------------- (155) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(tail(primes)) -> c20(A__TAIL(a__primes), MARK(primes)) by MARK(tail(primes)) -> c20(A__TAIL(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(tail(primes)) -> c20(A__TAIL(primes), MARK(primes)) ---------------------------------------- (156) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(sieve(z0))) -> c20(A__TAIL(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(tail(from(z0))) -> c20(A__TAIL(a__from(mark(z0))), MARK(from(z0))) MARK(tail(head(z0))) -> c20(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c20(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(if(z0, z1, z2))) -> c20(A__TAIL(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(tail(filter(z0, z1))) -> c20(A__TAIL(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(primes, x1, x2)) -> c21(A__IF(a__primes, x1, x2), MARK(primes)) MARK(if(sieve(z0), x1, x2)) -> c21(A__IF(a__sieve(mark(z0)), x1, x2), MARK(sieve(z0))) MARK(if(from(z0), x1, x2)) -> c21(A__IF(a__from(mark(z0)), x1, x2), MARK(from(z0))) MARK(if(head(z0), x1, x2)) -> c21(A__IF(a__head(mark(z0)), x1, x2), MARK(head(z0))) MARK(if(tail(z0), x1, x2)) -> c21(A__IF(a__tail(mark(z0)), x1, x2), MARK(tail(z0))) MARK(if(if(z0, z1, z2), x1, x2)) -> c21(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(filter(z0, z1), x1, x2)) -> c21(A__IF(a__filter(mark(z0), mark(z1)), x1, x2), MARK(filter(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c22(A__FILTER(mark(x0), a__primes), MARK(x0)) MARK(filter(x0, sieve(z0))) -> c22(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, from(z0))) -> c22(A__FILTER(mark(x0), a__from(mark(z0))), MARK(x0)) MARK(filter(x0, head(z0))) -> c22(A__FILTER(mark(x0), a__head(mark(z0))), MARK(x0)) MARK(filter(x0, tail(z0))) -> c22(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(x0)) MARK(filter(x0, if(z0, z1, z2))) -> c22(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0)) MARK(filter(x0, filter(z0, z1))) -> c22(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(x0)) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(x0, s(z0))) -> c22(MARK(x0)) MARK(filter(x0, 0)) -> c22(MARK(x0)) MARK(filter(x0, true)) -> c22(MARK(x0)) MARK(filter(x0, false)) -> c22(MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), x1)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c23(A__FILTER(mark(x0), a__primes), MARK(primes)) MARK(filter(x0, sieve(z0))) -> c23(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, from(z0))) -> c23(A__FILTER(mark(x0), a__from(mark(z0))), MARK(from(z0))) MARK(filter(x0, head(z0))) -> c23(A__FILTER(mark(x0), a__head(mark(z0))), MARK(head(z0))) MARK(filter(x0, tail(z0))) -> c23(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(tail(z0))) MARK(filter(x0, if(z0, z1, z2))) -> c23(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(filter(x0, filter(z0, z1))) -> c23(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(filter(x0, cons(z0, z1))) -> c23(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(primes, x1)) -> c23(A__FILTER(a__primes, mark(x1)), MARK(x1)) MARK(filter(sieve(z0), x1)) -> c23(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(from(z0), x1)) -> c23(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(head(z0), x1)) -> c23(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(tail(z0), x1)) -> c23(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(if(z0, z1, z2), x1)) -> c23(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1)) MARK(filter(filter(z0, z1), x1)) -> c23(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(x1)) MARK(filter(s(z0), x1)) -> c23(A__FILTER(s(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(x0, s(z0))) -> c23(MARK(s(z0))) MARK(filter(x0, divides(z0, z1))) -> c23(MARK(divides(z0, z1))) MARK(filter(0, x1)) -> c23(MARK(x1)) MARK(filter(cons(z0, z1), x1)) -> c23(MARK(x1)) MARK(filter(true, x1)) -> c23(MARK(x1)) MARK(filter(false, x1)) -> c23(MARK(x1)) MARK(filter(divides(z0, z1), x1)) -> c23(MARK(x1)) A__PRIMES -> c(A__SIEVE(cons(s(mark(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(sieve(primes)) -> c17(MARK(primes)) MARK(sieve(sieve(primes))) -> c17(A__SIEVE(a__sieve(a__primes)), MARK(sieve(primes))) MARK(sieve(sieve(sieve(z0)))) -> c17(A__SIEVE(a__sieve(a__sieve(mark(z0)))), MARK(sieve(sieve(z0)))) MARK(sieve(sieve(from(z0)))) -> c17(A__SIEVE(a__sieve(a__from(mark(z0)))), MARK(sieve(from(z0)))) MARK(sieve(sieve(head(z0)))) -> c17(A__SIEVE(a__sieve(a__head(mark(z0)))), MARK(sieve(head(z0)))) MARK(sieve(sieve(tail(z0)))) -> c17(A__SIEVE(a__sieve(a__tail(mark(z0)))), MARK(sieve(tail(z0)))) MARK(sieve(sieve(if(z0, z1, z2)))) -> c17(A__SIEVE(a__sieve(a__if(mark(z0), z1, z2))), MARK(sieve(if(z0, z1, z2)))) MARK(sieve(sieve(filter(z0, z1)))) -> c17(A__SIEVE(a__sieve(a__filter(mark(z0), mark(z1)))), MARK(sieve(filter(z0, z1)))) MARK(sieve(sieve(s(z0)))) -> c17(A__SIEVE(a__sieve(s(mark(z0)))), MARK(sieve(s(z0)))) MARK(sieve(sieve(cons(z0, z1)))) -> c17(A__SIEVE(a__sieve(cons(mark(z0), z1))), MARK(sieve(cons(z0, z1)))) MARK(sieve(sieve(divides(z0, z1)))) -> c17(A__SIEVE(a__sieve(divides(mark(z0), mark(z1)))), MARK(sieve(divides(z0, z1)))) MARK(sieve(sieve(x0))) -> c17(MARK(sieve(x0))) MARK(sieve(sieve(0))) -> c17(A__SIEVE(a__sieve(0))) MARK(sieve(sieve(true))) -> c17(A__SIEVE(a__sieve(true))) MARK(sieve(sieve(false))) -> c17(A__SIEVE(a__sieve(false))) MARK(sieve(from(x0))) -> c17(A__SIEVE(cons(mark(mark(x0)), from(s(mark(x0))))), MARK(from(x0))) MARK(sieve(from(primes))) -> c17(A__SIEVE(a__from(a__primes)), MARK(from(primes))) MARK(sieve(from(sieve(z0)))) -> c17(A__SIEVE(a__from(a__sieve(mark(z0)))), MARK(from(sieve(z0)))) MARK(sieve(from(from(z0)))) -> c17(A__SIEVE(a__from(a__from(mark(z0)))), MARK(from(from(z0)))) MARK(sieve(from(head(z0)))) -> c17(A__SIEVE(a__from(a__head(mark(z0)))), MARK(from(head(z0)))) MARK(sieve(from(tail(z0)))) -> c17(A__SIEVE(a__from(a__tail(mark(z0)))), MARK(from(tail(z0)))) MARK(sieve(from(if(z0, z1, z2)))) -> c17(A__SIEVE(a__from(a__if(mark(z0), z1, z2))), MARK(from(if(z0, z1, z2)))) MARK(sieve(from(filter(z0, z1)))) -> c17(A__SIEVE(a__from(a__filter(mark(z0), mark(z1)))), MARK(from(filter(z0, z1)))) MARK(sieve(from(s(z0)))) -> c17(A__SIEVE(a__from(s(mark(z0)))), MARK(from(s(z0)))) MARK(sieve(from(0))) -> c17(A__SIEVE(a__from(0)), MARK(from(0))) MARK(sieve(from(cons(z0, z1)))) -> c17(A__SIEVE(a__from(cons(mark(z0), z1))), MARK(from(cons(z0, z1)))) MARK(sieve(from(true))) -> c17(A__SIEVE(a__from(true)), MARK(from(true))) MARK(sieve(from(false))) -> c17(A__SIEVE(a__from(false)), MARK(from(false))) MARK(sieve(from(divides(z0, z1)))) -> c17(A__SIEVE(a__from(divides(mark(z0), mark(z1)))), MARK(from(divides(z0, z1)))) MARK(sieve(from(x0))) -> c17(MARK(from(x0))) MARK(sieve(head(primes))) -> c17(A__SIEVE(a__head(a__primes)), MARK(head(primes))) MARK(sieve(head(sieve(z0)))) -> c17(A__SIEVE(a__head(a__sieve(mark(z0)))), MARK(head(sieve(z0)))) MARK(sieve(head(from(z0)))) -> c17(A__SIEVE(a__head(a__from(mark(z0)))), MARK(head(from(z0)))) MARK(sieve(head(head(z0)))) -> c17(A__SIEVE(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(sieve(head(tail(z0)))) -> c17(A__SIEVE(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(sieve(head(if(z0, z1, z2)))) -> c17(A__SIEVE(a__head(a__if(mark(z0), z1, z2))), MARK(head(if(z0, z1, z2)))) MARK(sieve(head(filter(z0, z1)))) -> c17(A__SIEVE(a__head(a__filter(mark(z0), mark(z1)))), MARK(head(filter(z0, z1)))) MARK(sieve(head(s(z0)))) -> c17(A__SIEVE(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(sieve(head(cons(z0, z1)))) -> c17(A__SIEVE(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(sieve(head(divides(z0, z1)))) -> c17(A__SIEVE(a__head(divides(mark(z0), mark(z1)))), MARK(head(divides(z0, z1)))) MARK(sieve(head(x0))) -> c17(MARK(head(x0))) MARK(sieve(head(0))) -> c17(A__SIEVE(a__head(0))) MARK(sieve(head(true))) -> c17(A__SIEVE(a__head(true))) MARK(sieve(head(false))) -> c17(A__SIEVE(a__head(false))) MARK(sieve(tail(primes))) -> c17(A__SIEVE(a__tail(a__primes)), MARK(tail(primes))) MARK(sieve(tail(sieve(z0)))) -> c17(A__SIEVE(a__tail(a__sieve(mark(z0)))), MARK(tail(sieve(z0)))) MARK(sieve(tail(from(z0)))) -> c17(A__SIEVE(a__tail(a__from(mark(z0)))), MARK(tail(from(z0)))) MARK(sieve(tail(head(z0)))) -> c17(A__SIEVE(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(sieve(tail(tail(z0)))) -> c17(A__SIEVE(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(sieve(tail(if(z0, z1, z2)))) -> c17(A__SIEVE(a__tail(a__if(mark(z0), z1, z2))), MARK(tail(if(z0, z1, z2)))) MARK(sieve(tail(filter(z0, z1)))) -> c17(A__SIEVE(a__tail(a__filter(mark(z0), mark(z1)))), MARK(tail(filter(z0, z1)))) MARK(sieve(tail(s(z0)))) -> c17(A__SIEVE(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(sieve(tail(cons(z0, z1)))) -> c17(A__SIEVE(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(sieve(tail(divides(z0, z1)))) -> c17(A__SIEVE(a__tail(divides(mark(z0), mark(z1)))), MARK(tail(divides(z0, z1)))) MARK(sieve(tail(x0))) -> c17(MARK(tail(x0))) MARK(sieve(tail(0))) -> c17(A__SIEVE(a__tail(0))) MARK(sieve(tail(true))) -> c17(A__SIEVE(a__tail(true))) MARK(sieve(tail(false))) -> c17(A__SIEVE(a__tail(false))) MARK(sieve(if(primes, x1, x2))) -> c17(A__SIEVE(a__if(a__primes, x1, x2)), MARK(if(primes, x1, x2))) MARK(sieve(if(sieve(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__sieve(mark(z0)), x1, x2)), MARK(if(sieve(z0), x1, x2))) MARK(sieve(if(from(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__from(mark(z0)), x1, x2)), MARK(if(from(z0), x1, x2))) MARK(sieve(if(head(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__head(mark(z0)), x1, x2)), MARK(if(head(z0), x1, x2))) MARK(sieve(if(tail(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__tail(mark(z0)), x1, x2)), MARK(if(tail(z0), x1, x2))) MARK(sieve(if(if(z0, z1, z2), x1, x2))) -> c17(A__SIEVE(a__if(a__if(mark(z0), z1, z2), x1, x2)), MARK(if(if(z0, z1, z2), x1, x2))) MARK(sieve(if(filter(z0, z1), x1, x2))) -> c17(A__SIEVE(a__if(a__filter(mark(z0), mark(z1)), x1, x2)), MARK(if(filter(z0, z1), x1, x2))) MARK(sieve(if(s(z0), x1, x2))) -> c17(A__SIEVE(a__if(s(mark(z0)), x1, x2)), MARK(if(s(z0), x1, x2))) MARK(sieve(if(cons(z0, z1), x1, x2))) -> c17(A__SIEVE(a__if(cons(mark(z0), z1), x1, x2)), MARK(if(cons(z0, z1), x1, x2))) MARK(sieve(if(true, x1, x2))) -> c17(A__SIEVE(a__if(true, x1, x2)), MARK(if(true, x1, x2))) MARK(sieve(if(false, x1, x2))) -> c17(A__SIEVE(a__if(false, x1, x2)), MARK(if(false, x1, x2))) MARK(sieve(if(divides(z0, z1), x1, x2))) -> c17(A__SIEVE(a__if(divides(mark(z0), mark(z1)), x1, x2)), MARK(if(divides(z0, z1), x1, x2))) MARK(sieve(if(x0, z1, z2))) -> c17(MARK(if(x0, z1, z2))) MARK(sieve(if(0, x1, x2))) -> c17(A__SIEVE(a__if(0, x1, x2))) MARK(sieve(filter(x0, primes))) -> c17(A__SIEVE(a__filter(mark(x0), a__primes)), MARK(filter(x0, primes))) MARK(sieve(filter(x0, sieve(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), a__sieve(mark(z0)))), MARK(filter(x0, sieve(z0)))) MARK(sieve(filter(x0, from(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), a__from(mark(z0)))), MARK(filter(x0, from(z0)))) MARK(sieve(filter(x0, head(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), a__head(mark(z0)))), MARK(filter(x0, head(z0)))) MARK(sieve(filter(x0, tail(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), a__tail(mark(z0)))), MARK(filter(x0, tail(z0)))) MARK(sieve(filter(x0, if(z0, z1, z2)))) -> c17(A__SIEVE(a__filter(mark(x0), a__if(mark(z0), z1, z2))), MARK(filter(x0, if(z0, z1, z2)))) MARK(sieve(filter(x0, filter(z0, z1)))) -> c17(A__SIEVE(a__filter(mark(x0), a__filter(mark(z0), mark(z1)))), MARK(filter(x0, filter(z0, z1)))) MARK(sieve(filter(x0, s(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), s(mark(z0)))), MARK(filter(x0, s(z0)))) MARK(sieve(filter(x0, 0))) -> c17(A__SIEVE(a__filter(mark(x0), 0)), MARK(filter(x0, 0))) MARK(sieve(filter(x0, cons(z0, z1)))) -> c17(A__SIEVE(a__filter(mark(x0), cons(mark(z0), z1))), MARK(filter(x0, cons(z0, z1)))) MARK(sieve(filter(x0, true))) -> c17(A__SIEVE(a__filter(mark(x0), true)), MARK(filter(x0, true))) MARK(sieve(filter(x0, false))) -> c17(A__SIEVE(a__filter(mark(x0), false)), MARK(filter(x0, false))) MARK(sieve(filter(x0, divides(z0, z1)))) -> c17(A__SIEVE(a__filter(mark(x0), divides(mark(z0), mark(z1)))), MARK(filter(x0, divides(z0, z1)))) MARK(sieve(filter(primes, x1))) -> c17(A__SIEVE(a__filter(a__primes, mark(x1))), MARK(filter(primes, x1))) MARK(sieve(filter(sieve(z0), x1))) -> c17(A__SIEVE(a__filter(a__sieve(mark(z0)), mark(x1))), MARK(filter(sieve(z0), x1))) MARK(sieve(filter(from(z0), x1))) -> c17(A__SIEVE(a__filter(a__from(mark(z0)), mark(x1))), MARK(filter(from(z0), x1))) MARK(sieve(filter(head(z0), x1))) -> c17(A__SIEVE(a__filter(a__head(mark(z0)), mark(x1))), MARK(filter(head(z0), x1))) MARK(sieve(filter(tail(z0), x1))) -> c17(A__SIEVE(a__filter(a__tail(mark(z0)), mark(x1))), MARK(filter(tail(z0), x1))) MARK(sieve(filter(if(z0, z1, z2), x1))) -> c17(A__SIEVE(a__filter(a__if(mark(z0), z1, z2), mark(x1))), MARK(filter(if(z0, z1, z2), x1))) MARK(sieve(filter(filter(z0, z1), x1))) -> c17(A__SIEVE(a__filter(a__filter(mark(z0), mark(z1)), mark(x1))), MARK(filter(filter(z0, z1), x1))) MARK(sieve(filter(s(z0), x1))) -> c17(A__SIEVE(a__filter(s(mark(z0)), mark(x1))), MARK(filter(s(z0), x1))) MARK(sieve(filter(0, x1))) -> c17(A__SIEVE(a__filter(0, mark(x1))), MARK(filter(0, x1))) MARK(sieve(filter(cons(z0, z1), x1))) -> c17(A__SIEVE(a__filter(cons(mark(z0), z1), mark(x1))), MARK(filter(cons(z0, z1), x1))) MARK(sieve(filter(true, x1))) -> c17(A__SIEVE(a__filter(true, mark(x1))), MARK(filter(true, x1))) MARK(sieve(filter(false, x1))) -> c17(A__SIEVE(a__filter(false, mark(x1))), MARK(filter(false, x1))) MARK(sieve(filter(divides(z0, z1), x1))) -> 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c19(A__HEAD(a__tail(a__sieve(mark(z0)))), MARK(tail(sieve(z0)))) MARK(head(tail(from(z0)))) -> c19(A__HEAD(a__tail(a__from(mark(z0)))), MARK(tail(from(z0)))) MARK(head(tail(head(z0)))) -> c19(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c19(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(if(z0, z1, z2)))) -> c19(A__HEAD(a__tail(a__if(mark(z0), z1, z2))), MARK(tail(if(z0, z1, z2)))) MARK(head(tail(filter(z0, z1)))) -> c19(A__HEAD(a__tail(a__filter(mark(z0), mark(z1)))), MARK(tail(filter(z0, z1)))) MARK(head(tail(s(z0)))) -> c19(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(cons(z0, z1)))) -> c19(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(divides(z0, z1)))) -> c19(A__HEAD(a__tail(divides(mark(z0), mark(z1)))), MARK(tail(divides(z0, z1)))) MARK(head(tail(x0))) -> c19(MARK(tail(x0))) MARK(head(tail(0))) -> c19(A__HEAD(a__tail(0))) MARK(head(tail(true))) -> c19(A__HEAD(a__tail(true))) MARK(head(tail(false))) -> c19(A__HEAD(a__tail(false))) MARK(head(if(primes, x1, x2))) -> c19(A__HEAD(a__if(a__primes, x1, x2)), MARK(if(primes, x1, x2))) MARK(head(if(sieve(z0), x1, x2))) -> c19(A__HEAD(a__if(a__sieve(mark(z0)), x1, x2)), MARK(if(sieve(z0), x1, x2))) MARK(head(if(from(z0), x1, x2))) -> c19(A__HEAD(a__if(a__from(mark(z0)), x1, x2)), MARK(if(from(z0), x1, x2))) MARK(head(if(head(z0), x1, x2))) -> c19(A__HEAD(a__if(a__head(mark(z0)), x1, x2)), MARK(if(head(z0), x1, x2))) MARK(head(if(tail(z0), x1, x2))) -> c19(A__HEAD(a__if(a__tail(mark(z0)), x1, x2)), MARK(if(tail(z0), x1, x2))) MARK(head(if(if(z0, z1, z2), x1, x2))) -> c19(A__HEAD(a__if(a__if(mark(z0), z1, z2), x1, x2)), MARK(if(if(z0, z1, z2), x1, x2))) MARK(head(if(filter(z0, z1), x1, x2))) -> c19(A__HEAD(a__if(a__filter(mark(z0), mark(z1)), x1, x2)), MARK(if(filter(z0, z1), x1, x2))) MARK(head(if(s(z0), x1, x2))) -> c19(A__HEAD(a__if(s(mark(z0)), x1, x2)), MARK(if(s(z0), x1, x2))) MARK(head(if(cons(z0, z1), x1, x2))) -> c19(A__HEAD(a__if(cons(mark(z0), z1), x1, x2)), MARK(if(cons(z0, z1), x1, x2))) MARK(head(if(true, x1, x2))) -> c19(A__HEAD(a__if(true, x1, x2)), MARK(if(true, x1, x2))) MARK(head(if(false, x1, x2))) -> c19(A__HEAD(a__if(false, x1, x2)), MARK(if(false, x1, x2))) MARK(head(if(divides(z0, z1), x1, x2))) -> c19(A__HEAD(a__if(divides(mark(z0), mark(z1)), x1, x2)), MARK(if(divides(z0, z1), x1, x2))) MARK(head(if(x0, z1, z2))) -> c19(MARK(if(x0, z1, z2))) MARK(head(if(0, x1, x2))) -> c19(A__HEAD(a__if(0, x1, x2))) MARK(head(filter(x0, primes))) -> c19(A__HEAD(a__filter(mark(x0), a__primes)), MARK(filter(x0, primes))) MARK(head(filter(x0, sieve(z0)))) -> c19(A__HEAD(a__filter(mark(x0), a__sieve(mark(z0)))), MARK(filter(x0, sieve(z0)))) MARK(head(filter(x0, from(z0)))) -> c19(A__HEAD(a__filter(mark(x0), a__from(mark(z0)))), MARK(filter(x0, from(z0)))) MARK(head(filter(x0, head(z0)))) -> c19(A__HEAD(a__filter(mark(x0), a__head(mark(z0)))), MARK(filter(x0, head(z0)))) MARK(head(filter(x0, tail(z0)))) -> c19(A__HEAD(a__filter(mark(x0), a__tail(mark(z0)))), MARK(filter(x0, tail(z0)))) MARK(head(filter(x0, if(z0, z1, z2)))) -> c19(A__HEAD(a__filter(mark(x0), a__if(mark(z0), z1, z2))), MARK(filter(x0, if(z0, z1, z2)))) MARK(head(filter(x0, filter(z0, z1)))) -> c19(A__HEAD(a__filter(mark(x0), a__filter(mark(z0), mark(z1)))), MARK(filter(x0, filter(z0, z1)))) MARK(head(filter(x0, s(z0)))) -> c19(A__HEAD(a__filter(mark(x0), s(mark(z0)))), MARK(filter(x0, s(z0)))) MARK(head(filter(x0, 0))) -> c19(A__HEAD(a__filter(mark(x0), 0)), MARK(filter(x0, 0))) MARK(head(filter(x0, cons(z0, z1)))) -> c19(A__HEAD(a__filter(mark(x0), cons(mark(z0), z1))), MARK(filter(x0, cons(z0, z1)))) MARK(head(filter(x0, true))) -> c19(A__HEAD(a__filter(mark(x0), true)), MARK(filter(x0, true))) MARK(head(filter(x0, false))) -> c19(A__HEAD(a__filter(mark(x0), false)), MARK(filter(x0, false))) MARK(head(filter(x0, divides(z0, z1)))) -> c19(A__HEAD(a__filter(mark(x0), divides(mark(z0), mark(z1)))), MARK(filter(x0, divides(z0, z1)))) MARK(head(filter(primes, x1))) -> c19(A__HEAD(a__filter(a__primes, mark(x1))), MARK(filter(primes, x1))) MARK(head(filter(sieve(z0), x1))) -> c19(A__HEAD(a__filter(a__sieve(mark(z0)), mark(x1))), MARK(filter(sieve(z0), x1))) MARK(head(filter(from(z0), x1))) -> c19(A__HEAD(a__filter(a__from(mark(z0)), mark(x1))), MARK(filter(from(z0), x1))) MARK(head(filter(head(z0), x1))) -> c19(A__HEAD(a__filter(a__head(mark(z0)), mark(x1))), MARK(filter(head(z0), x1))) MARK(head(filter(tail(z0), x1))) -> c19(A__HEAD(a__filter(a__tail(mark(z0)), mark(x1))), MARK(filter(tail(z0), x1))) MARK(head(filter(if(z0, z1, z2), x1))) -> c19(A__HEAD(a__filter(a__if(mark(z0), z1, z2), mark(x1))), MARK(filter(if(z0, z1, z2), x1))) MARK(head(filter(filter(z0, z1), x1))) -> c19(A__HEAD(a__filter(a__filter(mark(z0), mark(z1)), mark(x1))), MARK(filter(filter(z0, z1), x1))) MARK(head(filter(s(z0), x1))) -> c19(A__HEAD(a__filter(s(mark(z0)), mark(x1))), MARK(filter(s(z0), x1))) MARK(head(filter(0, x1))) -> c19(A__HEAD(a__filter(0, mark(x1))), MARK(filter(0, x1))) MARK(head(filter(cons(z0, z1), x1))) -> c19(A__HEAD(a__filter(cons(mark(z0), z1), mark(x1))), MARK(filter(cons(z0, z1), x1))) MARK(head(filter(true, x1))) -> c19(A__HEAD(a__filter(true, mark(x1))), MARK(filter(true, x1))) MARK(head(filter(false, x1))) -> c19(A__HEAD(a__filter(false, mark(x1))), MARK(filter(false, x1))) MARK(head(filter(divides(z0, z1), x1))) -> c19(A__HEAD(a__filter(divides(mark(z0), mark(z1)), mark(x1))), MARK(filter(divides(z0, z1), x1))) MARK(head(filter(x0, x1))) -> c19(MARK(filter(x0, x1))) MARK(tail(primes)) -> c20(A__TAIL(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(tail(primes)) -> c20(A__TAIL(primes), MARK(primes)) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c18_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_1, c17_2, c17_1, c19_2, c19_1, c20_2, c20_1, c21_2, c21_1, c22_2, c22_1, c23_2, c23_1, c_2 ---------------------------------------- (157) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (158) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(sieve(z0))) -> c20(A__TAIL(a__sieve(mark(z0))), MARK(sieve(z0))) MARK(tail(from(z0))) -> c20(A__TAIL(a__from(mark(z0))), MARK(from(z0))) MARK(tail(head(z0))) -> c20(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c20(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(if(z0, z1, z2))) -> c20(A__TAIL(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(tail(filter(z0, z1))) -> c20(A__TAIL(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(primes, x1, x2)) -> c21(A__IF(a__primes, x1, x2), MARK(primes)) MARK(if(sieve(z0), x1, x2)) -> c21(A__IF(a__sieve(mark(z0)), x1, x2), MARK(sieve(z0))) MARK(if(from(z0), x1, x2)) -> c21(A__IF(a__from(mark(z0)), x1, x2), MARK(from(z0))) MARK(if(head(z0), x1, x2)) -> c21(A__IF(a__head(mark(z0)), x1, x2), MARK(head(z0))) MARK(if(tail(z0), x1, x2)) -> c21(A__IF(a__tail(mark(z0)), x1, x2), MARK(tail(z0))) MARK(if(if(z0, z1, z2), x1, x2)) -> c21(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(filter(z0, z1), x1, x2)) -> c21(A__IF(a__filter(mark(z0), mark(z1)), x1, x2), MARK(filter(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c22(A__FILTER(mark(x0), a__primes), MARK(x0)) MARK(filter(x0, sieve(z0))) -> c22(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, from(z0))) -> c22(A__FILTER(mark(x0), a__from(mark(z0))), MARK(x0)) MARK(filter(x0, head(z0))) -> c22(A__FILTER(mark(x0), a__head(mark(z0))), MARK(x0)) MARK(filter(x0, tail(z0))) -> c22(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(x0)) MARK(filter(x0, if(z0, z1, z2))) -> c22(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0)) MARK(filter(x0, filter(z0, z1))) -> c22(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(x0)) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(x0, s(z0))) -> c22(MARK(x0)) MARK(filter(x0, 0)) -> c22(MARK(x0)) MARK(filter(x0, true)) -> c22(MARK(x0)) MARK(filter(x0, false)) -> c22(MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), x1)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c23(A__FILTER(mark(x0), a__primes), MARK(primes)) MARK(filter(x0, sieve(z0))) -> c23(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, from(z0))) -> 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MARK(head(filter(x0, sieve(z0)))) -> c19(A__HEAD(a__filter(mark(x0), a__sieve(mark(z0)))), MARK(filter(x0, sieve(z0)))) MARK(head(filter(x0, from(z0)))) -> c19(A__HEAD(a__filter(mark(x0), a__from(mark(z0)))), MARK(filter(x0, from(z0)))) MARK(head(filter(x0, head(z0)))) -> c19(A__HEAD(a__filter(mark(x0), a__head(mark(z0)))), MARK(filter(x0, head(z0)))) MARK(head(filter(x0, tail(z0)))) -> c19(A__HEAD(a__filter(mark(x0), a__tail(mark(z0)))), MARK(filter(x0, tail(z0)))) MARK(head(filter(x0, if(z0, z1, z2)))) -> c19(A__HEAD(a__filter(mark(x0), a__if(mark(z0), z1, z2))), MARK(filter(x0, if(z0, z1, z2)))) MARK(head(filter(x0, filter(z0, z1)))) -> c19(A__HEAD(a__filter(mark(x0), a__filter(mark(z0), mark(z1)))), MARK(filter(x0, filter(z0, z1)))) MARK(head(filter(x0, s(z0)))) -> c19(A__HEAD(a__filter(mark(x0), s(mark(z0)))), MARK(filter(x0, s(z0)))) MARK(head(filter(x0, 0))) -> c19(A__HEAD(a__filter(mark(x0), 0)), MARK(filter(x0, 0))) MARK(head(filter(x0, cons(z0, z1)))) -> c19(A__HEAD(a__filter(mark(x0), cons(mark(z0), z1))), MARK(filter(x0, cons(z0, z1)))) MARK(head(filter(x0, true))) -> c19(A__HEAD(a__filter(mark(x0), true)), MARK(filter(x0, true))) MARK(head(filter(x0, false))) -> c19(A__HEAD(a__filter(mark(x0), false)), MARK(filter(x0, false))) MARK(head(filter(x0, divides(z0, z1)))) -> c19(A__HEAD(a__filter(mark(x0), divides(mark(z0), mark(z1)))), MARK(filter(x0, divides(z0, z1)))) MARK(head(filter(primes, x1))) -> c19(A__HEAD(a__filter(a__primes, mark(x1))), MARK(filter(primes, x1))) MARK(head(filter(sieve(z0), x1))) -> c19(A__HEAD(a__filter(a__sieve(mark(z0)), mark(x1))), MARK(filter(sieve(z0), x1))) MARK(head(filter(from(z0), x1))) -> c19(A__HEAD(a__filter(a__from(mark(z0)), mark(x1))), MARK(filter(from(z0), x1))) MARK(head(filter(head(z0), x1))) -> c19(A__HEAD(a__filter(a__head(mark(z0)), mark(x1))), MARK(filter(head(z0), x1))) MARK(head(filter(tail(z0), x1))) -> c19(A__HEAD(a__filter(a__tail(mark(z0)), mark(x1))), MARK(filter(tail(z0), x1))) MARK(head(filter(if(z0, z1, z2), x1))) -> c19(A__HEAD(a__filter(a__if(mark(z0), z1, z2), mark(x1))), MARK(filter(if(z0, z1, z2), x1))) MARK(head(filter(filter(z0, z1), x1))) -> c19(A__HEAD(a__filter(a__filter(mark(z0), mark(z1)), mark(x1))), MARK(filter(filter(z0, z1), x1))) MARK(head(filter(s(z0), x1))) -> c19(A__HEAD(a__filter(s(mark(z0)), mark(x1))), MARK(filter(s(z0), x1))) MARK(head(filter(0, x1))) -> c19(A__HEAD(a__filter(0, mark(x1))), MARK(filter(0, x1))) MARK(head(filter(cons(z0, z1), x1))) -> c19(A__HEAD(a__filter(cons(mark(z0), z1), mark(x1))), MARK(filter(cons(z0, z1), x1))) MARK(head(filter(true, x1))) -> c19(A__HEAD(a__filter(true, mark(x1))), MARK(filter(true, x1))) MARK(head(filter(false, x1))) -> c19(A__HEAD(a__filter(false, mark(x1))), MARK(filter(false, x1))) MARK(head(filter(divides(z0, z1), x1))) -> c19(A__HEAD(a__filter(divides(mark(z0), mark(z1)), mark(x1))), MARK(filter(divides(z0, z1), x1))) MARK(head(filter(x0, x1))) -> c19(MARK(filter(x0, x1))) MARK(tail(primes)) -> c20(A__TAIL(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(tail(primes)) -> c20(MARK(primes)) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c18_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_1, c17_2, c17_1, c19_2, c19_1, c20_2, c20_1, c21_2, c21_1, c22_2, c22_1, c23_2, c23_1, c_2 ---------------------------------------- (159) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(tail(sieve(z0))) -> c20(A__TAIL(a__sieve(mark(z0))), MARK(sieve(z0))) by MARK(tail(sieve(x0))) -> c20(A__TAIL(sieve(mark(x0))), MARK(sieve(x0))) MARK(tail(sieve(primes))) -> c20(A__TAIL(a__sieve(a__primes)), MARK(sieve(primes))) MARK(tail(sieve(sieve(z0)))) -> c20(A__TAIL(a__sieve(a__sieve(mark(z0)))), MARK(sieve(sieve(z0)))) MARK(tail(sieve(from(z0)))) -> c20(A__TAIL(a__sieve(a__from(mark(z0)))), MARK(sieve(from(z0)))) MARK(tail(sieve(head(z0)))) -> c20(A__TAIL(a__sieve(a__head(mark(z0)))), MARK(sieve(head(z0)))) MARK(tail(sieve(tail(z0)))) -> c20(A__TAIL(a__sieve(a__tail(mark(z0)))), MARK(sieve(tail(z0)))) MARK(tail(sieve(if(z0, z1, z2)))) -> c20(A__TAIL(a__sieve(a__if(mark(z0), z1, z2))), MARK(sieve(if(z0, z1, z2)))) MARK(tail(sieve(filter(z0, z1)))) -> c20(A__TAIL(a__sieve(a__filter(mark(z0), mark(z1)))), MARK(sieve(filter(z0, z1)))) MARK(tail(sieve(s(z0)))) -> c20(A__TAIL(a__sieve(s(mark(z0)))), MARK(sieve(s(z0)))) MARK(tail(sieve(0))) -> c20(A__TAIL(a__sieve(0)), MARK(sieve(0))) MARK(tail(sieve(cons(z0, z1)))) -> c20(A__TAIL(a__sieve(cons(mark(z0), z1))), MARK(sieve(cons(z0, z1)))) MARK(tail(sieve(true))) -> c20(A__TAIL(a__sieve(true)), MARK(sieve(true))) MARK(tail(sieve(false))) -> c20(A__TAIL(a__sieve(false)), MARK(sieve(false))) MARK(tail(sieve(divides(z0, z1)))) -> c20(A__TAIL(a__sieve(divides(mark(z0), mark(z1)))), MARK(sieve(divides(z0, z1)))) ---------------------------------------- (160) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(from(z0))) -> c20(A__TAIL(a__from(mark(z0))), MARK(from(z0))) MARK(tail(head(z0))) -> c20(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c20(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(if(z0, z1, z2))) -> c20(A__TAIL(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(tail(filter(z0, z1))) -> c20(A__TAIL(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(primes, x1, x2)) -> c21(A__IF(a__primes, x1, x2), MARK(primes)) MARK(if(sieve(z0), x1, x2)) -> c21(A__IF(a__sieve(mark(z0)), x1, x2), MARK(sieve(z0))) MARK(if(from(z0), x1, x2)) -> c21(A__IF(a__from(mark(z0)), x1, x2), MARK(from(z0))) MARK(if(head(z0), x1, x2)) -> c21(A__IF(a__head(mark(z0)), x1, x2), MARK(head(z0))) MARK(if(tail(z0), x1, x2)) -> c21(A__IF(a__tail(mark(z0)), x1, x2), MARK(tail(z0))) MARK(if(if(z0, z1, z2), x1, x2)) -> c21(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(filter(z0, z1), x1, x2)) -> c21(A__IF(a__filter(mark(z0), mark(z1)), x1, x2), MARK(filter(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c22(A__FILTER(mark(x0), a__primes), MARK(x0)) MARK(filter(x0, sieve(z0))) -> c22(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, from(z0))) -> c22(A__FILTER(mark(x0), a__from(mark(z0))), MARK(x0)) MARK(filter(x0, head(z0))) -> c22(A__FILTER(mark(x0), a__head(mark(z0))), MARK(x0)) MARK(filter(x0, tail(z0))) -> c22(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(x0)) MARK(filter(x0, if(z0, z1, z2))) -> c22(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0)) MARK(filter(x0, filter(z0, z1))) -> c22(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(x0)) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(x0, s(z0))) -> c22(MARK(x0)) MARK(filter(x0, 0)) -> c22(MARK(x0)) MARK(filter(x0, true)) -> c22(MARK(x0)) MARK(filter(x0, false)) -> c22(MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), x1)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c23(A__FILTER(mark(x0), a__primes), MARK(primes)) MARK(filter(x0, sieve(z0))) -> c23(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, from(z0))) -> c23(A__FILTER(mark(x0), a__from(mark(z0))), MARK(from(z0))) MARK(filter(x0, head(z0))) -> c23(A__FILTER(mark(x0), a__head(mark(z0))), MARK(head(z0))) MARK(filter(x0, tail(z0))) -> c23(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(tail(z0))) MARK(filter(x0, if(z0, z1, z2))) -> c23(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(filter(x0, filter(z0, z1))) -> c23(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(filter(x0, cons(z0, z1))) -> c23(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(primes, x1)) -> c23(A__FILTER(a__primes, mark(x1)), MARK(x1)) MARK(filter(sieve(z0), x1)) -> c23(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(from(z0), x1)) -> c23(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(head(z0), x1)) -> c23(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(tail(z0), x1)) -> c23(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(if(z0, z1, z2), x1)) -> c23(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1)) MARK(filter(filter(z0, z1), x1)) -> c23(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(x1)) MARK(filter(s(z0), x1)) -> c23(A__FILTER(s(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(x0, s(z0))) -> c23(MARK(s(z0))) MARK(filter(x0, divides(z0, z1))) -> c23(MARK(divides(z0, 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a__sieve(mark(z0)))), MARK(filter(x0, sieve(z0)))) MARK(head(filter(x0, from(z0)))) -> c19(A__HEAD(a__filter(mark(x0), a__from(mark(z0)))), MARK(filter(x0, from(z0)))) MARK(head(filter(x0, head(z0)))) -> c19(A__HEAD(a__filter(mark(x0), a__head(mark(z0)))), MARK(filter(x0, head(z0)))) MARK(head(filter(x0, tail(z0)))) -> c19(A__HEAD(a__filter(mark(x0), a__tail(mark(z0)))), MARK(filter(x0, tail(z0)))) MARK(head(filter(x0, if(z0, z1, z2)))) -> c19(A__HEAD(a__filter(mark(x0), a__if(mark(z0), z1, z2))), MARK(filter(x0, if(z0, z1, z2)))) MARK(head(filter(x0, filter(z0, z1)))) -> c19(A__HEAD(a__filter(mark(x0), a__filter(mark(z0), mark(z1)))), MARK(filter(x0, filter(z0, z1)))) MARK(head(filter(x0, s(z0)))) -> c19(A__HEAD(a__filter(mark(x0), s(mark(z0)))), MARK(filter(x0, s(z0)))) MARK(head(filter(x0, 0))) -> c19(A__HEAD(a__filter(mark(x0), 0)), MARK(filter(x0, 0))) MARK(head(filter(x0, cons(z0, z1)))) -> c19(A__HEAD(a__filter(mark(x0), cons(mark(z0), z1))), MARK(filter(x0, cons(z0, z1)))) MARK(head(filter(x0, true))) -> c19(A__HEAD(a__filter(mark(x0), true)), MARK(filter(x0, true))) MARK(head(filter(x0, false))) -> c19(A__HEAD(a__filter(mark(x0), false)), MARK(filter(x0, false))) MARK(head(filter(x0, divides(z0, z1)))) -> c19(A__HEAD(a__filter(mark(x0), divides(mark(z0), mark(z1)))), MARK(filter(x0, divides(z0, z1)))) MARK(head(filter(primes, x1))) -> c19(A__HEAD(a__filter(a__primes, mark(x1))), MARK(filter(primes, x1))) MARK(head(filter(sieve(z0), x1))) -> c19(A__HEAD(a__filter(a__sieve(mark(z0)), mark(x1))), MARK(filter(sieve(z0), x1))) MARK(head(filter(from(z0), x1))) -> c19(A__HEAD(a__filter(a__from(mark(z0)), mark(x1))), MARK(filter(from(z0), x1))) MARK(head(filter(head(z0), x1))) -> c19(A__HEAD(a__filter(a__head(mark(z0)), mark(x1))), MARK(filter(head(z0), x1))) MARK(head(filter(tail(z0), x1))) -> c19(A__HEAD(a__filter(a__tail(mark(z0)), mark(x1))), MARK(filter(tail(z0), x1))) MARK(head(filter(if(z0, z1, z2), x1))) -> c19(A__HEAD(a__filter(a__if(mark(z0), z1, z2), mark(x1))), MARK(filter(if(z0, z1, z2), x1))) MARK(head(filter(filter(z0, z1), x1))) -> c19(A__HEAD(a__filter(a__filter(mark(z0), mark(z1)), mark(x1))), MARK(filter(filter(z0, z1), x1))) MARK(head(filter(s(z0), x1))) -> c19(A__HEAD(a__filter(s(mark(z0)), mark(x1))), MARK(filter(s(z0), x1))) MARK(head(filter(0, x1))) -> c19(A__HEAD(a__filter(0, mark(x1))), MARK(filter(0, x1))) MARK(head(filter(cons(z0, z1), x1))) -> c19(A__HEAD(a__filter(cons(mark(z0), z1), mark(x1))), MARK(filter(cons(z0, z1), x1))) MARK(head(filter(true, x1))) -> c19(A__HEAD(a__filter(true, mark(x1))), MARK(filter(true, x1))) MARK(head(filter(false, x1))) -> c19(A__HEAD(a__filter(false, mark(x1))), MARK(filter(false, x1))) MARK(head(filter(divides(z0, z1), x1))) -> c19(A__HEAD(a__filter(divides(mark(z0), mark(z1)), mark(x1))), MARK(filter(divides(z0, z1), x1))) MARK(head(filter(x0, x1))) -> c19(MARK(filter(x0, x1))) MARK(tail(primes)) -> c20(A__TAIL(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(tail(primes)) -> c20(MARK(primes)) MARK(tail(sieve(x0))) -> c20(A__TAIL(sieve(mark(x0))), MARK(sieve(x0))) MARK(tail(sieve(primes))) -> c20(A__TAIL(a__sieve(a__primes)), MARK(sieve(primes))) MARK(tail(sieve(sieve(z0)))) -> c20(A__TAIL(a__sieve(a__sieve(mark(z0)))), MARK(sieve(sieve(z0)))) MARK(tail(sieve(from(z0)))) -> c20(A__TAIL(a__sieve(a__from(mark(z0)))), MARK(sieve(from(z0)))) MARK(tail(sieve(head(z0)))) -> c20(A__TAIL(a__sieve(a__head(mark(z0)))), MARK(sieve(head(z0)))) MARK(tail(sieve(tail(z0)))) -> c20(A__TAIL(a__sieve(a__tail(mark(z0)))), MARK(sieve(tail(z0)))) MARK(tail(sieve(if(z0, z1, z2)))) -> c20(A__TAIL(a__sieve(a__if(mark(z0), z1, z2))), MARK(sieve(if(z0, z1, z2)))) MARK(tail(sieve(filter(z0, z1)))) -> c20(A__TAIL(a__sieve(a__filter(mark(z0), mark(z1)))), MARK(sieve(filter(z0, z1)))) MARK(tail(sieve(s(z0)))) -> c20(A__TAIL(a__sieve(s(mark(z0)))), MARK(sieve(s(z0)))) MARK(tail(sieve(0))) -> c20(A__TAIL(a__sieve(0)), MARK(sieve(0))) MARK(tail(sieve(cons(z0, z1)))) -> c20(A__TAIL(a__sieve(cons(mark(z0), z1))), MARK(sieve(cons(z0, z1)))) MARK(tail(sieve(true))) -> c20(A__TAIL(a__sieve(true)), MARK(sieve(true))) MARK(tail(sieve(false))) -> c20(A__TAIL(a__sieve(false)), MARK(sieve(false))) MARK(tail(sieve(divides(z0, z1)))) -> c20(A__TAIL(a__sieve(divides(mark(z0), mark(z1)))), MARK(sieve(divides(z0, z1)))) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c18_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_1, c17_2, c17_1, c19_2, c19_1, c20_2, c20_1, c21_2, c21_1, c22_2, c22_1, c23_2, c23_1, c_2 ---------------------------------------- (161) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing tuple parts ---------------------------------------- (162) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(from(z0))) -> c20(A__TAIL(a__from(mark(z0))), MARK(from(z0))) MARK(tail(head(z0))) -> c20(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c20(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(if(z0, z1, z2))) -> c20(A__TAIL(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(tail(filter(z0, z1))) -> c20(A__TAIL(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(primes, x1, x2)) -> c21(A__IF(a__primes, x1, x2), MARK(primes)) MARK(if(sieve(z0), x1, x2)) -> c21(A__IF(a__sieve(mark(z0)), x1, x2), MARK(sieve(z0))) MARK(if(from(z0), x1, x2)) -> c21(A__IF(a__from(mark(z0)), x1, x2), MARK(from(z0))) MARK(if(head(z0), x1, x2)) -> c21(A__IF(a__head(mark(z0)), x1, x2), MARK(head(z0))) MARK(if(tail(z0), x1, x2)) -> c21(A__IF(a__tail(mark(z0)), x1, x2), MARK(tail(z0))) MARK(if(if(z0, z1, z2), x1, x2)) -> c21(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(filter(z0, z1), x1, x2)) -> c21(A__IF(a__filter(mark(z0), mark(z1)), x1, x2), MARK(filter(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c22(A__FILTER(mark(x0), a__primes), MARK(x0)) MARK(filter(x0, sieve(z0))) -> c22(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, from(z0))) -> c22(A__FILTER(mark(x0), a__from(mark(z0))), MARK(x0)) MARK(filter(x0, head(z0))) -> c22(A__FILTER(mark(x0), a__head(mark(z0))), MARK(x0)) MARK(filter(x0, tail(z0))) -> c22(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(x0)) MARK(filter(x0, if(z0, z1, z2))) -> c22(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0)) MARK(filter(x0, filter(z0, z1))) -> c22(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(x0)) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(x0, s(z0))) -> c22(MARK(x0)) MARK(filter(x0, 0)) -> c22(MARK(x0)) MARK(filter(x0, true)) -> c22(MARK(x0)) MARK(filter(x0, false)) -> c22(MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), x1)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c23(A__FILTER(mark(x0), a__primes), MARK(primes)) MARK(filter(x0, sieve(z0))) -> c23(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, from(z0))) -> c23(A__FILTER(mark(x0), a__from(mark(z0))), MARK(from(z0))) MARK(filter(x0, head(z0))) -> c23(A__FILTER(mark(x0), a__head(mark(z0))), MARK(head(z0))) MARK(filter(x0, tail(z0))) -> c23(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(tail(z0))) MARK(filter(x0, if(z0, z1, z2))) -> c23(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(filter(x0, filter(z0, z1))) -> c23(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(filter(x0, cons(z0, z1))) -> c23(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(primes, x1)) -> c23(A__FILTER(a__primes, mark(x1)), MARK(x1)) MARK(filter(sieve(z0), x1)) -> c23(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(from(z0), x1)) -> c23(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(head(z0), x1)) -> c23(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(tail(z0), x1)) -> c23(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(if(z0, z1, z2), x1)) -> c23(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1)) MARK(filter(filter(z0, z1), x1)) -> c23(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(x1)) MARK(filter(s(z0), x1)) -> c23(A__FILTER(s(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(x0, s(z0))) -> c23(MARK(s(z0))) MARK(filter(x0, divides(z0, z1))) -> c23(MARK(divides(z0, z1))) MARK(filter(0, x1)) -> c23(MARK(x1)) MARK(filter(cons(z0, z1), x1)) -> c23(MARK(x1)) MARK(filter(true, x1)) -> c23(MARK(x1)) MARK(filter(false, x1)) -> c23(MARK(x1)) 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MARK(filter(primes, x1)) -> c23(A__FILTER(a__primes, mark(x1)), MARK(x1)) MARK(filter(sieve(z0), x1)) -> c23(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(from(z0), x1)) -> c23(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(head(z0), x1)) -> c23(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(tail(z0), x1)) -> c23(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(if(z0, z1, z2), x1)) -> c23(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1)) MARK(filter(filter(z0, z1), x1)) -> c23(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(x1)) MARK(filter(s(z0), x1)) -> c23(A__FILTER(s(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(x0, s(z0))) -> c23(MARK(s(z0))) MARK(filter(x0, divides(z0, z1))) -> c23(MARK(divides(z0, z1))) MARK(filter(0, x1)) -> c23(MARK(x1)) MARK(filter(cons(z0, z1), x1)) -> c23(MARK(x1)) MARK(filter(true, x1)) -> c23(MARK(x1)) MARK(filter(false, x1)) -> c23(MARK(x1)) MARK(filter(divides(z0, z1), x1)) -> 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divides(mark(z0), mark(z1)))), MARK(filter(x0, divides(z0, z1)))) MARK(head(filter(primes, x1))) -> c19(A__HEAD(a__filter(a__primes, mark(x1))), MARK(filter(primes, x1))) MARK(head(filter(sieve(z0), x1))) -> c19(A__HEAD(a__filter(a__sieve(mark(z0)), mark(x1))), MARK(filter(sieve(z0), x1))) MARK(head(filter(from(z0), x1))) -> c19(A__HEAD(a__filter(a__from(mark(z0)), mark(x1))), MARK(filter(from(z0), x1))) MARK(head(filter(head(z0), x1))) -> c19(A__HEAD(a__filter(a__head(mark(z0)), mark(x1))), MARK(filter(head(z0), x1))) MARK(head(filter(tail(z0), x1))) -> c19(A__HEAD(a__filter(a__tail(mark(z0)), mark(x1))), MARK(filter(tail(z0), x1))) MARK(head(filter(if(z0, z1, z2), x1))) -> c19(A__HEAD(a__filter(a__if(mark(z0), z1, z2), mark(x1))), MARK(filter(if(z0, z1, z2), x1))) MARK(head(filter(filter(z0, z1), x1))) -> c19(A__HEAD(a__filter(a__filter(mark(z0), mark(z1)), mark(x1))), MARK(filter(filter(z0, z1), x1))) MARK(head(filter(s(z0), x1))) -> c19(A__HEAD(a__filter(s(mark(z0)), mark(x1))), MARK(filter(s(z0), x1))) MARK(head(filter(0, x1))) -> c19(A__HEAD(a__filter(0, mark(x1))), MARK(filter(0, x1))) MARK(head(filter(cons(z0, z1), x1))) -> c19(A__HEAD(a__filter(cons(mark(z0), z1), mark(x1))), MARK(filter(cons(z0, z1), x1))) MARK(head(filter(true, x1))) -> c19(A__HEAD(a__filter(true, mark(x1))), MARK(filter(true, x1))) MARK(head(filter(false, x1))) -> c19(A__HEAD(a__filter(false, mark(x1))), MARK(filter(false, x1))) MARK(head(filter(divides(z0, z1), x1))) -> c19(A__HEAD(a__filter(divides(mark(z0), mark(z1)), mark(x1))), MARK(filter(divides(z0, z1), x1))) MARK(head(filter(x0, x1))) -> c19(MARK(filter(x0, x1))) MARK(tail(primes)) -> c20(A__TAIL(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(tail(primes)) -> c20(MARK(primes)) MARK(tail(sieve(primes))) -> c20(A__TAIL(a__sieve(a__primes)), MARK(sieve(primes))) MARK(tail(sieve(sieve(z0)))) -> c20(A__TAIL(a__sieve(a__sieve(mark(z0)))), MARK(sieve(sieve(z0)))) MARK(tail(sieve(from(z0)))) -> c20(A__TAIL(a__sieve(a__from(mark(z0)))), MARK(sieve(from(z0)))) MARK(tail(sieve(head(z0)))) -> c20(A__TAIL(a__sieve(a__head(mark(z0)))), MARK(sieve(head(z0)))) MARK(tail(sieve(tail(z0)))) -> c20(A__TAIL(a__sieve(a__tail(mark(z0)))), MARK(sieve(tail(z0)))) MARK(tail(sieve(if(z0, z1, z2)))) -> c20(A__TAIL(a__sieve(a__if(mark(z0), z1, z2))), MARK(sieve(if(z0, z1, z2)))) MARK(tail(sieve(filter(z0, z1)))) -> c20(A__TAIL(a__sieve(a__filter(mark(z0), mark(z1)))), MARK(sieve(filter(z0, z1)))) MARK(tail(sieve(s(z0)))) -> c20(A__TAIL(a__sieve(s(mark(z0)))), MARK(sieve(s(z0)))) MARK(tail(sieve(cons(z0, z1)))) -> c20(A__TAIL(a__sieve(cons(mark(z0), z1))), MARK(sieve(cons(z0, z1)))) MARK(tail(sieve(divides(z0, z1)))) -> c20(A__TAIL(a__sieve(divides(mark(z0), mark(z1)))), MARK(sieve(divides(z0, z1)))) MARK(tail(sieve(x0))) -> c20(MARK(sieve(x0))) MARK(tail(sieve(0))) -> c20(A__TAIL(a__sieve(0))) MARK(tail(sieve(true))) -> c20(A__TAIL(a__sieve(true))) MARK(tail(sieve(false))) -> c20(A__TAIL(a__sieve(false))) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c18_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_1, c17_2, c17_1, c19_2, c19_1, c20_2, c20_1, c21_2, c21_1, c22_2, c22_1, c23_2, c23_1, c_2 ---------------------------------------- (163) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(tail(from(z0))) -> c20(A__TAIL(a__from(mark(z0))), MARK(from(z0))) by MARK(tail(from(x0))) -> c20(A__TAIL(cons(mark(mark(x0)), from(s(mark(x0))))), MARK(from(x0))) MARK(tail(from(x0))) -> c20(A__TAIL(from(mark(x0))), MARK(from(x0))) MARK(tail(from(primes))) -> c20(A__TAIL(a__from(a__primes)), MARK(from(primes))) MARK(tail(from(sieve(z0)))) -> c20(A__TAIL(a__from(a__sieve(mark(z0)))), MARK(from(sieve(z0)))) MARK(tail(from(from(z0)))) -> c20(A__TAIL(a__from(a__from(mark(z0)))), MARK(from(from(z0)))) MARK(tail(from(head(z0)))) -> c20(A__TAIL(a__from(a__head(mark(z0)))), MARK(from(head(z0)))) MARK(tail(from(tail(z0)))) -> c20(A__TAIL(a__from(a__tail(mark(z0)))), MARK(from(tail(z0)))) MARK(tail(from(if(z0, z1, z2)))) -> c20(A__TAIL(a__from(a__if(mark(z0), z1, z2))), MARK(from(if(z0, z1, z2)))) MARK(tail(from(filter(z0, z1)))) -> c20(A__TAIL(a__from(a__filter(mark(z0), mark(z1)))), MARK(from(filter(z0, z1)))) MARK(tail(from(s(z0)))) -> c20(A__TAIL(a__from(s(mark(z0)))), MARK(from(s(z0)))) MARK(tail(from(0))) -> c20(A__TAIL(a__from(0)), MARK(from(0))) MARK(tail(from(cons(z0, z1)))) -> c20(A__TAIL(a__from(cons(mark(z0), z1))), MARK(from(cons(z0, z1)))) MARK(tail(from(true))) -> c20(A__TAIL(a__from(true)), MARK(from(true))) MARK(tail(from(false))) -> c20(A__TAIL(a__from(false)), MARK(from(false))) MARK(tail(from(divides(z0, z1)))) -> c20(A__TAIL(a__from(divides(mark(z0), mark(z1)))), MARK(from(divides(z0, z1)))) MARK(tail(from(x0))) -> c20(MARK(from(x0))) ---------------------------------------- (164) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(head(z0))) -> c20(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c20(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(if(z0, z1, z2))) -> c20(A__TAIL(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(tail(filter(z0, z1))) -> c20(A__TAIL(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(primes, x1, x2)) -> c21(A__IF(a__primes, x1, x2), MARK(primes)) MARK(if(sieve(z0), x1, x2)) -> c21(A__IF(a__sieve(mark(z0)), x1, x2), MARK(sieve(z0))) MARK(if(from(z0), x1, x2)) -> c21(A__IF(a__from(mark(z0)), x1, x2), MARK(from(z0))) MARK(if(head(z0), x1, x2)) -> c21(A__IF(a__head(mark(z0)), x1, x2), MARK(head(z0))) MARK(if(tail(z0), x1, x2)) -> c21(A__IF(a__tail(mark(z0)), x1, x2), MARK(tail(z0))) MARK(if(if(z0, z1, z2), x1, x2)) -> c21(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(filter(z0, z1), x1, x2)) -> c21(A__IF(a__filter(mark(z0), mark(z1)), x1, x2), MARK(filter(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c22(A__FILTER(mark(x0), a__primes), MARK(x0)) MARK(filter(x0, sieve(z0))) -> c22(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, from(z0))) -> c22(A__FILTER(mark(x0), a__from(mark(z0))), MARK(x0)) MARK(filter(x0, head(z0))) -> c22(A__FILTER(mark(x0), a__head(mark(z0))), MARK(x0)) MARK(filter(x0, tail(z0))) -> c22(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(x0)) MARK(filter(x0, if(z0, z1, z2))) -> c22(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0)) MARK(filter(x0, filter(z0, z1))) -> c22(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(x0)) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(x0, s(z0))) -> c22(MARK(x0)) MARK(filter(x0, 0)) -> c22(MARK(x0)) MARK(filter(x0, true)) -> c22(MARK(x0)) MARK(filter(x0, false)) -> c22(MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), x1)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c23(A__FILTER(mark(x0), a__primes), MARK(primes)) MARK(filter(x0, sieve(z0))) -> c23(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, from(z0))) -> c23(A__FILTER(mark(x0), a__from(mark(z0))), MARK(from(z0))) MARK(filter(x0, head(z0))) -> c23(A__FILTER(mark(x0), a__head(mark(z0))), MARK(head(z0))) MARK(filter(x0, tail(z0))) -> c23(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(tail(z0))) MARK(filter(x0, if(z0, z1, z2))) -> c23(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(filter(x0, filter(z0, z1))) -> c23(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(filter(x0, cons(z0, z1))) -> c23(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(primes, x1)) -> c23(A__FILTER(a__primes, mark(x1)), MARK(x1)) MARK(filter(sieve(z0), x1)) -> c23(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(from(z0), x1)) -> c23(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(head(z0), x1)) -> c23(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(tail(z0), x1)) -> c23(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(if(z0, z1, z2), x1)) -> c23(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1)) MARK(filter(filter(z0, z1), x1)) -> c23(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(x1)) MARK(filter(s(z0), x1)) -> c23(A__FILTER(s(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(x0, s(z0))) -> c23(MARK(s(z0))) MARK(filter(x0, divides(z0, z1))) -> c23(MARK(divides(z0, z1))) MARK(filter(0, x1)) -> c23(MARK(x1)) MARK(filter(cons(z0, z1), x1)) -> c23(MARK(x1)) MARK(filter(true, x1)) -> c23(MARK(x1)) MARK(filter(false, x1)) -> c23(MARK(x1)) MARK(filter(divides(z0, z1), x1)) -> c23(MARK(x1)) A__PRIMES -> c(A__SIEVE(cons(s(mark(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(sieve(primes)) -> c17(MARK(primes)) MARK(sieve(sieve(primes))) -> c17(A__SIEVE(a__sieve(a__primes)), MARK(sieve(primes))) MARK(sieve(sieve(sieve(z0)))) -> c17(A__SIEVE(a__sieve(a__sieve(mark(z0)))), MARK(sieve(sieve(z0)))) MARK(sieve(sieve(from(z0)))) -> c17(A__SIEVE(a__sieve(a__from(mark(z0)))), MARK(sieve(from(z0)))) MARK(sieve(sieve(head(z0)))) -> c17(A__SIEVE(a__sieve(a__head(mark(z0)))), MARK(sieve(head(z0)))) MARK(sieve(sieve(tail(z0)))) -> c17(A__SIEVE(a__sieve(a__tail(mark(z0)))), MARK(sieve(tail(z0)))) MARK(sieve(sieve(if(z0, z1, z2)))) -> c17(A__SIEVE(a__sieve(a__if(mark(z0), z1, z2))), MARK(sieve(if(z0, z1, z2)))) 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mark(x1))), MARK(filter(head(z0), x1))) MARK(head(filter(tail(z0), x1))) -> c19(A__HEAD(a__filter(a__tail(mark(z0)), mark(x1))), MARK(filter(tail(z0), x1))) MARK(head(filter(if(z0, z1, z2), x1))) -> c19(A__HEAD(a__filter(a__if(mark(z0), z1, z2), mark(x1))), MARK(filter(if(z0, z1, z2), x1))) MARK(head(filter(filter(z0, z1), x1))) -> c19(A__HEAD(a__filter(a__filter(mark(z0), mark(z1)), mark(x1))), MARK(filter(filter(z0, z1), x1))) MARK(head(filter(s(z0), x1))) -> c19(A__HEAD(a__filter(s(mark(z0)), mark(x1))), MARK(filter(s(z0), x1))) MARK(head(filter(0, x1))) -> c19(A__HEAD(a__filter(0, mark(x1))), MARK(filter(0, x1))) MARK(head(filter(cons(z0, z1), x1))) -> c19(A__HEAD(a__filter(cons(mark(z0), z1), mark(x1))), MARK(filter(cons(z0, z1), x1))) MARK(head(filter(true, x1))) -> c19(A__HEAD(a__filter(true, mark(x1))), MARK(filter(true, x1))) MARK(head(filter(false, x1))) -> c19(A__HEAD(a__filter(false, mark(x1))), MARK(filter(false, x1))) MARK(head(filter(divides(z0, z1), x1))) -> c19(A__HEAD(a__filter(divides(mark(z0), mark(z1)), mark(x1))), MARK(filter(divides(z0, z1), x1))) MARK(head(filter(x0, x1))) -> c19(MARK(filter(x0, x1))) MARK(tail(primes)) -> c20(A__TAIL(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(tail(primes)) -> c20(MARK(primes)) MARK(tail(sieve(primes))) -> c20(A__TAIL(a__sieve(a__primes)), MARK(sieve(primes))) MARK(tail(sieve(sieve(z0)))) -> c20(A__TAIL(a__sieve(a__sieve(mark(z0)))), MARK(sieve(sieve(z0)))) MARK(tail(sieve(from(z0)))) -> c20(A__TAIL(a__sieve(a__from(mark(z0)))), MARK(sieve(from(z0)))) MARK(tail(sieve(head(z0)))) -> c20(A__TAIL(a__sieve(a__head(mark(z0)))), MARK(sieve(head(z0)))) MARK(tail(sieve(tail(z0)))) -> c20(A__TAIL(a__sieve(a__tail(mark(z0)))), MARK(sieve(tail(z0)))) MARK(tail(sieve(if(z0, z1, z2)))) -> c20(A__TAIL(a__sieve(a__if(mark(z0), z1, z2))), MARK(sieve(if(z0, z1, z2)))) MARK(tail(sieve(filter(z0, z1)))) -> c20(A__TAIL(a__sieve(a__filter(mark(z0), mark(z1)))), MARK(sieve(filter(z0, z1)))) MARK(tail(sieve(s(z0)))) -> c20(A__TAIL(a__sieve(s(mark(z0)))), MARK(sieve(s(z0)))) MARK(tail(sieve(cons(z0, z1)))) -> c20(A__TAIL(a__sieve(cons(mark(z0), z1))), MARK(sieve(cons(z0, z1)))) MARK(tail(sieve(divides(z0, z1)))) -> c20(A__TAIL(a__sieve(divides(mark(z0), mark(z1)))), MARK(sieve(divides(z0, z1)))) MARK(tail(sieve(x0))) -> c20(MARK(sieve(x0))) MARK(tail(sieve(0))) -> c20(A__TAIL(a__sieve(0))) MARK(tail(sieve(true))) -> c20(A__TAIL(a__sieve(true))) MARK(tail(sieve(false))) -> c20(A__TAIL(a__sieve(false))) MARK(tail(from(x0))) -> c20(A__TAIL(cons(mark(mark(x0)), from(s(mark(x0))))), MARK(from(x0))) MARK(tail(from(x0))) -> c20(A__TAIL(from(mark(x0))), MARK(from(x0))) MARK(tail(from(primes))) -> c20(A__TAIL(a__from(a__primes)), MARK(from(primes))) MARK(tail(from(sieve(z0)))) -> c20(A__TAIL(a__from(a__sieve(mark(z0)))), MARK(from(sieve(z0)))) MARK(tail(from(from(z0)))) -> c20(A__TAIL(a__from(a__from(mark(z0)))), MARK(from(from(z0)))) MARK(tail(from(head(z0)))) -> c20(A__TAIL(a__from(a__head(mark(z0)))), MARK(from(head(z0)))) MARK(tail(from(tail(z0)))) -> c20(A__TAIL(a__from(a__tail(mark(z0)))), MARK(from(tail(z0)))) MARK(tail(from(if(z0, z1, z2)))) -> c20(A__TAIL(a__from(a__if(mark(z0), z1, z2))), MARK(from(if(z0, z1, z2)))) MARK(tail(from(filter(z0, z1)))) -> c20(A__TAIL(a__from(a__filter(mark(z0), mark(z1)))), MARK(from(filter(z0, z1)))) MARK(tail(from(s(z0)))) -> c20(A__TAIL(a__from(s(mark(z0)))), MARK(from(s(z0)))) MARK(tail(from(0))) -> c20(A__TAIL(a__from(0)), MARK(from(0))) MARK(tail(from(cons(z0, z1)))) -> c20(A__TAIL(a__from(cons(mark(z0), z1))), MARK(from(cons(z0, z1)))) MARK(tail(from(true))) -> c20(A__TAIL(a__from(true)), MARK(from(true))) MARK(tail(from(false))) -> c20(A__TAIL(a__from(false)), MARK(from(false))) MARK(tail(from(divides(z0, z1)))) -> c20(A__TAIL(a__from(divides(mark(z0), mark(z1)))), MARK(from(divides(z0, z1)))) MARK(tail(from(x0))) -> c20(MARK(from(x0))) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c18_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_1, c17_2, c17_1, c19_2, c19_1, c20_2, c20_1, c21_2, c21_1, c22_2, c22_1, c23_2, c23_1, c_2 ---------------------------------------- (165) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (166) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(head(z0))) -> c20(A__TAIL(a__head(mark(z0))), MARK(head(z0))) MARK(tail(tail(z0))) -> c20(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(if(z0, z1, z2))) -> c20(A__TAIL(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(tail(filter(z0, z1))) -> c20(A__TAIL(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(primes, x1, x2)) -> c21(A__IF(a__primes, x1, x2), MARK(primes)) MARK(if(sieve(z0), x1, x2)) -> c21(A__IF(a__sieve(mark(z0)), x1, x2), MARK(sieve(z0))) MARK(if(from(z0), x1, x2)) -> c21(A__IF(a__from(mark(z0)), x1, x2), MARK(from(z0))) MARK(if(head(z0), x1, x2)) -> c21(A__IF(a__head(mark(z0)), x1, x2), MARK(head(z0))) MARK(if(tail(z0), x1, x2)) -> c21(A__IF(a__tail(mark(z0)), x1, x2), MARK(tail(z0))) MARK(if(if(z0, z1, z2), x1, x2)) -> c21(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(filter(z0, z1), x1, x2)) -> c21(A__IF(a__filter(mark(z0), mark(z1)), x1, x2), MARK(filter(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c22(A__FILTER(mark(x0), a__primes), MARK(x0)) MARK(filter(x0, sieve(z0))) -> c22(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, from(z0))) -> c22(A__FILTER(mark(x0), a__from(mark(z0))), MARK(x0)) MARK(filter(x0, head(z0))) -> c22(A__FILTER(mark(x0), a__head(mark(z0))), MARK(x0)) MARK(filter(x0, tail(z0))) -> c22(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(x0)) MARK(filter(x0, if(z0, z1, z2))) -> c22(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0)) MARK(filter(x0, filter(z0, z1))) -> c22(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(x0)) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(x0, s(z0))) -> c22(MARK(x0)) MARK(filter(x0, 0)) -> c22(MARK(x0)) MARK(filter(x0, true)) -> c22(MARK(x0)) MARK(filter(x0, false)) -> c22(MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), x1)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c23(A__FILTER(mark(x0), a__primes), MARK(primes)) MARK(filter(x0, sieve(z0))) -> c23(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, from(z0))) -> c23(A__FILTER(mark(x0), a__from(mark(z0))), MARK(from(z0))) MARK(filter(x0, head(z0))) -> c23(A__FILTER(mark(x0), a__head(mark(z0))), MARK(head(z0))) MARK(filter(x0, tail(z0))) -> c23(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(tail(z0))) MARK(filter(x0, if(z0, z1, z2))) -> c23(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(filter(x0, filter(z0, z1))) -> c23(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(filter(x0, cons(z0, z1))) -> c23(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(primes, x1)) -> c23(A__FILTER(a__primes, mark(x1)), MARK(x1)) MARK(filter(sieve(z0), x1)) -> c23(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(from(z0), x1)) -> c23(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(head(z0), x1)) -> c23(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(tail(z0), x1)) -> c23(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(if(z0, z1, z2), x1)) -> c23(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1)) MARK(filter(filter(z0, z1), x1)) -> c23(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(x1)) MARK(filter(s(z0), x1)) -> c23(A__FILTER(s(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(x0, s(z0))) -> c23(MARK(s(z0))) MARK(filter(x0, divides(z0, z1))) -> c23(MARK(divides(z0, z1))) MARK(filter(0, x1)) -> c23(MARK(x1)) MARK(filter(cons(z0, z1), x1)) -> c23(MARK(x1)) MARK(filter(true, x1)) -> c23(MARK(x1)) MARK(filter(false, x1)) -> c23(MARK(x1)) MARK(filter(divides(z0, z1), x1)) -> c23(MARK(x1)) A__PRIMES -> c(A__SIEVE(cons(s(mark(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(sieve(primes)) -> c17(MARK(primes)) MARK(sieve(sieve(primes))) -> c17(A__SIEVE(a__sieve(a__primes)), MARK(sieve(primes))) MARK(sieve(sieve(sieve(z0)))) -> c17(A__SIEVE(a__sieve(a__sieve(mark(z0)))), MARK(sieve(sieve(z0)))) MARK(sieve(sieve(from(z0)))) -> c17(A__SIEVE(a__sieve(a__from(mark(z0)))), 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MARK(head(tail(tail(z0)))) -> c19(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(if(z0, z1, z2)))) -> c19(A__HEAD(a__tail(a__if(mark(z0), z1, z2))), MARK(tail(if(z0, z1, z2)))) MARK(head(tail(filter(z0, z1)))) -> c19(A__HEAD(a__tail(a__filter(mark(z0), mark(z1)))), MARK(tail(filter(z0, z1)))) MARK(head(tail(s(z0)))) -> c19(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(cons(z0, z1)))) -> c19(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(divides(z0, z1)))) -> c19(A__HEAD(a__tail(divides(mark(z0), mark(z1)))), MARK(tail(divides(z0, z1)))) MARK(head(tail(x0))) -> c19(MARK(tail(x0))) MARK(head(tail(0))) -> c19(A__HEAD(a__tail(0))) MARK(head(tail(true))) -> c19(A__HEAD(a__tail(true))) MARK(head(tail(false))) -> c19(A__HEAD(a__tail(false))) MARK(head(if(primes, x1, x2))) -> c19(A__HEAD(a__if(a__primes, x1, x2)), MARK(if(primes, x1, x2))) MARK(head(if(sieve(z0), x1, x2))) -> c19(A__HEAD(a__if(a__sieve(mark(z0)), x1, x2)), MARK(if(sieve(z0), x1, x2))) MARK(head(if(from(z0), x1, x2))) -> c19(A__HEAD(a__if(a__from(mark(z0)), x1, x2)), MARK(if(from(z0), x1, x2))) MARK(head(if(head(z0), x1, x2))) -> c19(A__HEAD(a__if(a__head(mark(z0)), x1, x2)), MARK(if(head(z0), x1, x2))) MARK(head(if(tail(z0), x1, x2))) -> c19(A__HEAD(a__if(a__tail(mark(z0)), x1, x2)), MARK(if(tail(z0), x1, x2))) MARK(head(if(if(z0, z1, z2), x1, x2))) -> c19(A__HEAD(a__if(a__if(mark(z0), z1, z2), x1, x2)), MARK(if(if(z0, z1, z2), x1, x2))) MARK(head(if(filter(z0, z1), x1, x2))) -> c19(A__HEAD(a__if(a__filter(mark(z0), mark(z1)), x1, x2)), MARK(if(filter(z0, z1), x1, x2))) MARK(head(if(s(z0), x1, x2))) -> c19(A__HEAD(a__if(s(mark(z0)), x1, x2)), MARK(if(s(z0), x1, x2))) MARK(head(if(cons(z0, z1), x1, x2))) -> c19(A__HEAD(a__if(cons(mark(z0), z1), x1, x2)), MARK(if(cons(z0, z1), x1, x2))) MARK(head(if(true, x1, x2))) -> c19(A__HEAD(a__if(true, x1, x2)), MARK(if(true, x1, x2))) MARK(head(if(false, x1, x2))) -> c19(A__HEAD(a__if(false, x1, x2)), MARK(if(false, x1, x2))) MARK(head(if(divides(z0, z1), x1, x2))) -> c19(A__HEAD(a__if(divides(mark(z0), mark(z1)), x1, x2)), MARK(if(divides(z0, z1), x1, x2))) MARK(head(if(x0, z1, z2))) -> c19(MARK(if(x0, z1, z2))) MARK(head(if(0, x1, x2))) -> c19(A__HEAD(a__if(0, x1, x2))) MARK(head(filter(x0, primes))) -> c19(A__HEAD(a__filter(mark(x0), a__primes)), MARK(filter(x0, primes))) MARK(head(filter(x0, sieve(z0)))) -> c19(A__HEAD(a__filter(mark(x0), a__sieve(mark(z0)))), MARK(filter(x0, sieve(z0)))) MARK(head(filter(x0, from(z0)))) -> c19(A__HEAD(a__filter(mark(x0), a__from(mark(z0)))), MARK(filter(x0, from(z0)))) MARK(head(filter(x0, head(z0)))) -> c19(A__HEAD(a__filter(mark(x0), a__head(mark(z0)))), MARK(filter(x0, head(z0)))) MARK(head(filter(x0, tail(z0)))) -> c19(A__HEAD(a__filter(mark(x0), a__tail(mark(z0)))), MARK(filter(x0, tail(z0)))) MARK(head(filter(x0, if(z0, z1, z2)))) -> c19(A__HEAD(a__filter(mark(x0), a__if(mark(z0), z1, z2))), MARK(filter(x0, if(z0, z1, z2)))) MARK(head(filter(x0, filter(z0, z1)))) -> c19(A__HEAD(a__filter(mark(x0), a__filter(mark(z0), mark(z1)))), MARK(filter(x0, filter(z0, z1)))) MARK(head(filter(x0, s(z0)))) -> c19(A__HEAD(a__filter(mark(x0), s(mark(z0)))), MARK(filter(x0, s(z0)))) MARK(head(filter(x0, 0))) -> c19(A__HEAD(a__filter(mark(x0), 0)), MARK(filter(x0, 0))) MARK(head(filter(x0, cons(z0, z1)))) -> c19(A__HEAD(a__filter(mark(x0), cons(mark(z0), z1))), MARK(filter(x0, cons(z0, z1)))) MARK(head(filter(x0, true))) -> c19(A__HEAD(a__filter(mark(x0), true)), MARK(filter(x0, true))) MARK(head(filter(x0, false))) -> c19(A__HEAD(a__filter(mark(x0), false)), MARK(filter(x0, false))) MARK(head(filter(x0, divides(z0, z1)))) -> c19(A__HEAD(a__filter(mark(x0), divides(mark(z0), mark(z1)))), MARK(filter(x0, divides(z0, z1)))) MARK(head(filter(primes, x1))) -> c19(A__HEAD(a__filter(a__primes, mark(x1))), MARK(filter(primes, x1))) MARK(head(filter(sieve(z0), x1))) -> c19(A__HEAD(a__filter(a__sieve(mark(z0)), mark(x1))), MARK(filter(sieve(z0), x1))) MARK(head(filter(from(z0), x1))) -> c19(A__HEAD(a__filter(a__from(mark(z0)), mark(x1))), MARK(filter(from(z0), x1))) MARK(head(filter(head(z0), x1))) -> c19(A__HEAD(a__filter(a__head(mark(z0)), mark(x1))), MARK(filter(head(z0), x1))) MARK(head(filter(tail(z0), x1))) -> c19(A__HEAD(a__filter(a__tail(mark(z0)), mark(x1))), MARK(filter(tail(z0), x1))) MARK(head(filter(if(z0, z1, z2), x1))) -> c19(A__HEAD(a__filter(a__if(mark(z0), z1, z2), mark(x1))), MARK(filter(if(z0, z1, z2), x1))) MARK(head(filter(filter(z0, z1), x1))) -> c19(A__HEAD(a__filter(a__filter(mark(z0), mark(z1)), mark(x1))), MARK(filter(filter(z0, z1), x1))) MARK(head(filter(s(z0), x1))) -> c19(A__HEAD(a__filter(s(mark(z0)), mark(x1))), MARK(filter(s(z0), x1))) MARK(head(filter(0, x1))) -> c19(A__HEAD(a__filter(0, mark(x1))), MARK(filter(0, x1))) MARK(head(filter(cons(z0, z1), x1))) -> c19(A__HEAD(a__filter(cons(mark(z0), z1), mark(x1))), MARK(filter(cons(z0, z1), x1))) MARK(head(filter(true, x1))) -> c19(A__HEAD(a__filter(true, mark(x1))), MARK(filter(true, x1))) MARK(head(filter(false, x1))) -> c19(A__HEAD(a__filter(false, mark(x1))), MARK(filter(false, x1))) MARK(head(filter(divides(z0, z1), x1))) -> c19(A__HEAD(a__filter(divides(mark(z0), mark(z1)), mark(x1))), MARK(filter(divides(z0, z1), x1))) MARK(head(filter(x0, x1))) -> c19(MARK(filter(x0, x1))) MARK(tail(primes)) -> c20(A__TAIL(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(tail(primes)) -> c20(MARK(primes)) MARK(tail(sieve(primes))) -> c20(A__TAIL(a__sieve(a__primes)), MARK(sieve(primes))) MARK(tail(sieve(sieve(z0)))) -> c20(A__TAIL(a__sieve(a__sieve(mark(z0)))), MARK(sieve(sieve(z0)))) MARK(tail(sieve(from(z0)))) -> c20(A__TAIL(a__sieve(a__from(mark(z0)))), MARK(sieve(from(z0)))) MARK(tail(sieve(head(z0)))) -> c20(A__TAIL(a__sieve(a__head(mark(z0)))), MARK(sieve(head(z0)))) MARK(tail(sieve(tail(z0)))) -> c20(A__TAIL(a__sieve(a__tail(mark(z0)))), MARK(sieve(tail(z0)))) MARK(tail(sieve(if(z0, z1, z2)))) -> c20(A__TAIL(a__sieve(a__if(mark(z0), z1, z2))), MARK(sieve(if(z0, z1, z2)))) MARK(tail(sieve(filter(z0, z1)))) -> c20(A__TAIL(a__sieve(a__filter(mark(z0), mark(z1)))), MARK(sieve(filter(z0, z1)))) MARK(tail(sieve(s(z0)))) -> c20(A__TAIL(a__sieve(s(mark(z0)))), MARK(sieve(s(z0)))) MARK(tail(sieve(cons(z0, z1)))) -> c20(A__TAIL(a__sieve(cons(mark(z0), z1))), MARK(sieve(cons(z0, z1)))) MARK(tail(sieve(divides(z0, z1)))) -> c20(A__TAIL(a__sieve(divides(mark(z0), mark(z1)))), MARK(sieve(divides(z0, z1)))) MARK(tail(sieve(x0))) -> c20(MARK(sieve(x0))) MARK(tail(sieve(0))) -> c20(A__TAIL(a__sieve(0))) MARK(tail(sieve(true))) -> c20(A__TAIL(a__sieve(true))) MARK(tail(sieve(false))) -> c20(A__TAIL(a__sieve(false))) MARK(tail(from(x0))) -> c20(A__TAIL(cons(mark(mark(x0)), from(s(mark(x0))))), MARK(from(x0))) MARK(tail(from(primes))) -> c20(A__TAIL(a__from(a__primes)), MARK(from(primes))) MARK(tail(from(sieve(z0)))) -> c20(A__TAIL(a__from(a__sieve(mark(z0)))), MARK(from(sieve(z0)))) MARK(tail(from(from(z0)))) -> c20(A__TAIL(a__from(a__from(mark(z0)))), MARK(from(from(z0)))) MARK(tail(from(head(z0)))) -> c20(A__TAIL(a__from(a__head(mark(z0)))), MARK(from(head(z0)))) MARK(tail(from(tail(z0)))) -> c20(A__TAIL(a__from(a__tail(mark(z0)))), MARK(from(tail(z0)))) MARK(tail(from(if(z0, z1, z2)))) -> c20(A__TAIL(a__from(a__if(mark(z0), z1, z2))), MARK(from(if(z0, z1, z2)))) MARK(tail(from(filter(z0, z1)))) -> c20(A__TAIL(a__from(a__filter(mark(z0), mark(z1)))), MARK(from(filter(z0, z1)))) MARK(tail(from(s(z0)))) -> c20(A__TAIL(a__from(s(mark(z0)))), MARK(from(s(z0)))) MARK(tail(from(0))) -> c20(A__TAIL(a__from(0)), MARK(from(0))) MARK(tail(from(cons(z0, z1)))) -> c20(A__TAIL(a__from(cons(mark(z0), z1))), MARK(from(cons(z0, z1)))) MARK(tail(from(true))) -> c20(A__TAIL(a__from(true)), MARK(from(true))) MARK(tail(from(false))) -> c20(A__TAIL(a__from(false)), MARK(from(false))) MARK(tail(from(divides(z0, z1)))) -> c20(A__TAIL(a__from(divides(mark(z0), mark(z1)))), MARK(from(divides(z0, z1)))) MARK(tail(from(x0))) -> c20(MARK(from(x0))) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c18_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_1, c17_2, c17_1, c19_2, c19_1, c20_2, c20_1, c21_2, c21_1, c22_2, c22_1, c23_2, c23_1, c_2 ---------------------------------------- (167) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(tail(head(z0))) -> c20(A__TAIL(a__head(mark(z0))), MARK(head(z0))) by MARK(tail(head(x0))) -> c20(A__TAIL(head(mark(x0))), MARK(head(x0))) MARK(tail(head(primes))) -> c20(A__TAIL(a__head(a__primes)), MARK(head(primes))) MARK(tail(head(sieve(z0)))) -> c20(A__TAIL(a__head(a__sieve(mark(z0)))), MARK(head(sieve(z0)))) MARK(tail(head(from(z0)))) -> c20(A__TAIL(a__head(a__from(mark(z0)))), MARK(head(from(z0)))) MARK(tail(head(head(z0)))) -> c20(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c20(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(if(z0, z1, z2)))) -> c20(A__TAIL(a__head(a__if(mark(z0), z1, z2))), MARK(head(if(z0, z1, z2)))) MARK(tail(head(filter(z0, z1)))) -> c20(A__TAIL(a__head(a__filter(mark(z0), mark(z1)))), MARK(head(filter(z0, z1)))) MARK(tail(head(s(z0)))) -> c20(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(0))) -> c20(A__TAIL(a__head(0)), MARK(head(0))) MARK(tail(head(cons(z0, z1)))) -> c20(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(true))) -> c20(A__TAIL(a__head(true)), MARK(head(true))) MARK(tail(head(false))) -> c20(A__TAIL(a__head(false)), MARK(head(false))) MARK(tail(head(divides(z0, z1)))) -> c20(A__TAIL(a__head(divides(mark(z0), mark(z1)))), MARK(head(divides(z0, z1)))) ---------------------------------------- (168) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(tail(z0))) -> c20(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(if(z0, z1, z2))) -> c20(A__TAIL(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(tail(filter(z0, z1))) -> c20(A__TAIL(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(primes, x1, x2)) -> c21(A__IF(a__primes, x1, x2), MARK(primes)) MARK(if(sieve(z0), x1, x2)) -> c21(A__IF(a__sieve(mark(z0)), x1, x2), MARK(sieve(z0))) MARK(if(from(z0), x1, x2)) -> c21(A__IF(a__from(mark(z0)), x1, x2), MARK(from(z0))) MARK(if(head(z0), x1, x2)) -> c21(A__IF(a__head(mark(z0)), x1, x2), MARK(head(z0))) MARK(if(tail(z0), x1, x2)) -> c21(A__IF(a__tail(mark(z0)), x1, x2), MARK(tail(z0))) MARK(if(if(z0, z1, z2), x1, x2)) -> c21(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(filter(z0, z1), x1, x2)) -> c21(A__IF(a__filter(mark(z0), mark(z1)), x1, x2), MARK(filter(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c22(A__FILTER(mark(x0), a__primes), MARK(x0)) MARK(filter(x0, sieve(z0))) -> c22(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, from(z0))) -> c22(A__FILTER(mark(x0), a__from(mark(z0))), MARK(x0)) MARK(filter(x0, head(z0))) -> c22(A__FILTER(mark(x0), a__head(mark(z0))), MARK(x0)) MARK(filter(x0, tail(z0))) -> c22(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(x0)) MARK(filter(x0, if(z0, z1, z2))) -> c22(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0)) MARK(filter(x0, filter(z0, z1))) -> 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c20(A__TAIL(a__head(a__sieve(mark(z0)))), MARK(head(sieve(z0)))) MARK(tail(head(from(z0)))) -> c20(A__TAIL(a__head(a__from(mark(z0)))), MARK(head(from(z0)))) MARK(tail(head(head(z0)))) -> c20(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c20(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(if(z0, z1, z2)))) -> c20(A__TAIL(a__head(a__if(mark(z0), z1, z2))), MARK(head(if(z0, z1, z2)))) MARK(tail(head(filter(z0, z1)))) -> c20(A__TAIL(a__head(a__filter(mark(z0), mark(z1)))), MARK(head(filter(z0, z1)))) MARK(tail(head(s(z0)))) -> c20(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(0))) -> c20(A__TAIL(a__head(0)), MARK(head(0))) MARK(tail(head(cons(z0, z1)))) -> c20(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(true))) -> c20(A__TAIL(a__head(true)), MARK(head(true))) MARK(tail(head(false))) -> c20(A__TAIL(a__head(false)), MARK(head(false))) MARK(tail(head(divides(z0, z1)))) -> c20(A__TAIL(a__head(divides(mark(z0), mark(z1)))), MARK(head(divides(z0, z1)))) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c18_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_1, c17_2, c17_1, c19_2, c19_1, c20_2, c20_1, c21_2, c21_1, c22_2, c22_1, c23_2, c23_1, c_2 ---------------------------------------- (169) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing tuple parts ---------------------------------------- (170) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(tail(z0))) -> c20(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) MARK(tail(if(z0, z1, z2))) -> c20(A__TAIL(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(tail(filter(z0, z1))) -> c20(A__TAIL(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(primes, x1, x2)) -> c21(A__IF(a__primes, x1, x2), MARK(primes)) MARK(if(sieve(z0), x1, x2)) -> c21(A__IF(a__sieve(mark(z0)), x1, x2), MARK(sieve(z0))) MARK(if(from(z0), x1, x2)) -> c21(A__IF(a__from(mark(z0)), x1, x2), MARK(from(z0))) MARK(if(head(z0), x1, x2)) -> c21(A__IF(a__head(mark(z0)), x1, x2), MARK(head(z0))) MARK(if(tail(z0), x1, x2)) -> c21(A__IF(a__tail(mark(z0)), x1, x2), MARK(tail(z0))) MARK(if(if(z0, z1, z2), x1, x2)) -> c21(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(filter(z0, z1), x1, x2)) -> c21(A__IF(a__filter(mark(z0), mark(z1)), x1, x2), MARK(filter(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> 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MARK(head(tail(primes))) -> c19(A__HEAD(a__tail(a__primes)), MARK(tail(primes))) MARK(head(tail(sieve(z0)))) -> c19(A__HEAD(a__tail(a__sieve(mark(z0)))), MARK(tail(sieve(z0)))) MARK(head(tail(from(z0)))) -> c19(A__HEAD(a__tail(a__from(mark(z0)))), MARK(tail(from(z0)))) MARK(head(tail(head(z0)))) -> c19(A__HEAD(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(head(tail(tail(z0)))) -> c19(A__HEAD(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(head(tail(if(z0, z1, z2)))) -> c19(A__HEAD(a__tail(a__if(mark(z0), z1, z2))), MARK(tail(if(z0, z1, z2)))) MARK(head(tail(filter(z0, z1)))) -> c19(A__HEAD(a__tail(a__filter(mark(z0), mark(z1)))), MARK(tail(filter(z0, z1)))) MARK(head(tail(s(z0)))) -> c19(A__HEAD(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(head(tail(cons(z0, z1)))) -> c19(A__HEAD(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(head(tail(divides(z0, z1)))) -> c19(A__HEAD(a__tail(divides(mark(z0), mark(z1)))), MARK(tail(divides(z0, z1)))) MARK(head(tail(x0))) -> c19(MARK(tail(x0))) MARK(head(tail(0))) -> c19(A__HEAD(a__tail(0))) MARK(head(tail(true))) -> c19(A__HEAD(a__tail(true))) MARK(head(tail(false))) -> c19(A__HEAD(a__tail(false))) MARK(head(if(primes, x1, x2))) -> c19(A__HEAD(a__if(a__primes, x1, x2)), MARK(if(primes, x1, x2))) MARK(head(if(sieve(z0), x1, x2))) -> c19(A__HEAD(a__if(a__sieve(mark(z0)), x1, x2)), MARK(if(sieve(z0), x1, x2))) MARK(head(if(from(z0), x1, x2))) -> c19(A__HEAD(a__if(a__from(mark(z0)), x1, x2)), MARK(if(from(z0), x1, x2))) MARK(head(if(head(z0), x1, x2))) -> c19(A__HEAD(a__if(a__head(mark(z0)), x1, x2)), MARK(if(head(z0), x1, x2))) MARK(head(if(tail(z0), x1, x2))) -> c19(A__HEAD(a__if(a__tail(mark(z0)), x1, x2)), MARK(if(tail(z0), x1, x2))) MARK(head(if(if(z0, z1, z2), x1, x2))) -> c19(A__HEAD(a__if(a__if(mark(z0), z1, z2), x1, x2)), MARK(if(if(z0, z1, z2), x1, x2))) MARK(head(if(filter(z0, z1), x1, x2))) -> c19(A__HEAD(a__if(a__filter(mark(z0), mark(z1)), x1, x2)), MARK(if(filter(z0, z1), x1, x2))) MARK(head(if(s(z0), x1, x2))) -> c19(A__HEAD(a__if(s(mark(z0)), x1, x2)), MARK(if(s(z0), x1, x2))) MARK(head(if(cons(z0, z1), x1, x2))) -> c19(A__HEAD(a__if(cons(mark(z0), z1), x1, x2)), MARK(if(cons(z0, z1), x1, x2))) MARK(head(if(true, x1, x2))) -> c19(A__HEAD(a__if(true, x1, x2)), MARK(if(true, x1, x2))) MARK(head(if(false, x1, x2))) -> c19(A__HEAD(a__if(false, x1, x2)), MARK(if(false, x1, x2))) MARK(head(if(divides(z0, z1), x1, x2))) -> c19(A__HEAD(a__if(divides(mark(z0), mark(z1)), x1, x2)), MARK(if(divides(z0, z1), x1, x2))) MARK(head(if(x0, z1, z2))) -> c19(MARK(if(x0, z1, z2))) MARK(head(if(0, x1, x2))) -> c19(A__HEAD(a__if(0, x1, x2))) MARK(head(filter(x0, primes))) -> c19(A__HEAD(a__filter(mark(x0), a__primes)), MARK(filter(x0, primes))) MARK(head(filter(x0, sieve(z0)))) -> c19(A__HEAD(a__filter(mark(x0), a__sieve(mark(z0)))), MARK(filter(x0, sieve(z0)))) MARK(head(filter(x0, from(z0)))) -> c19(A__HEAD(a__filter(mark(x0), a__from(mark(z0)))), MARK(filter(x0, from(z0)))) MARK(head(filter(x0, head(z0)))) -> c19(A__HEAD(a__filter(mark(x0), a__head(mark(z0)))), MARK(filter(x0, head(z0)))) MARK(head(filter(x0, tail(z0)))) -> c19(A__HEAD(a__filter(mark(x0), a__tail(mark(z0)))), MARK(filter(x0, tail(z0)))) MARK(head(filter(x0, if(z0, z1, z2)))) -> c19(A__HEAD(a__filter(mark(x0), a__if(mark(z0), z1, z2))), MARK(filter(x0, if(z0, z1, z2)))) MARK(head(filter(x0, filter(z0, z1)))) -> c19(A__HEAD(a__filter(mark(x0), a__filter(mark(z0), mark(z1)))), MARK(filter(x0, filter(z0, z1)))) MARK(head(filter(x0, s(z0)))) -> c19(A__HEAD(a__filter(mark(x0), s(mark(z0)))), MARK(filter(x0, s(z0)))) MARK(head(filter(x0, 0))) -> c19(A__HEAD(a__filter(mark(x0), 0)), MARK(filter(x0, 0))) MARK(head(filter(x0, cons(z0, z1)))) -> c19(A__HEAD(a__filter(mark(x0), cons(mark(z0), z1))), MARK(filter(x0, cons(z0, z1)))) MARK(head(filter(x0, true))) -> c19(A__HEAD(a__filter(mark(x0), true)), MARK(filter(x0, true))) MARK(head(filter(x0, false))) -> c19(A__HEAD(a__filter(mark(x0), false)), MARK(filter(x0, false))) MARK(head(filter(x0, divides(z0, z1)))) -> c19(A__HEAD(a__filter(mark(x0), divides(mark(z0), mark(z1)))), MARK(filter(x0, divides(z0, z1)))) MARK(head(filter(primes, x1))) -> c19(A__HEAD(a__filter(a__primes, mark(x1))), MARK(filter(primes, x1))) MARK(head(filter(sieve(z0), x1))) -> c19(A__HEAD(a__filter(a__sieve(mark(z0)), mark(x1))), MARK(filter(sieve(z0), x1))) MARK(head(filter(from(z0), x1))) -> c19(A__HEAD(a__filter(a__from(mark(z0)), mark(x1))), MARK(filter(from(z0), x1))) MARK(head(filter(head(z0), x1))) -> c19(A__HEAD(a__filter(a__head(mark(z0)), mark(x1))), MARK(filter(head(z0), x1))) MARK(head(filter(tail(z0), x1))) -> c19(A__HEAD(a__filter(a__tail(mark(z0)), mark(x1))), MARK(filter(tail(z0), x1))) MARK(head(filter(if(z0, z1, z2), x1))) -> c19(A__HEAD(a__filter(a__if(mark(z0), z1, z2), mark(x1))), MARK(filter(if(z0, z1, z2), x1))) MARK(head(filter(filter(z0, z1), x1))) -> c19(A__HEAD(a__filter(a__filter(mark(z0), mark(z1)), mark(x1))), MARK(filter(filter(z0, z1), x1))) MARK(head(filter(s(z0), x1))) -> c19(A__HEAD(a__filter(s(mark(z0)), mark(x1))), MARK(filter(s(z0), x1))) MARK(head(filter(0, x1))) -> c19(A__HEAD(a__filter(0, mark(x1))), MARK(filter(0, x1))) MARK(head(filter(cons(z0, z1), x1))) -> c19(A__HEAD(a__filter(cons(mark(z0), z1), mark(x1))), MARK(filter(cons(z0, z1), x1))) MARK(head(filter(true, x1))) -> c19(A__HEAD(a__filter(true, mark(x1))), MARK(filter(true, x1))) MARK(head(filter(false, x1))) -> c19(A__HEAD(a__filter(false, mark(x1))), MARK(filter(false, x1))) MARK(head(filter(divides(z0, z1), x1))) -> c19(A__HEAD(a__filter(divides(mark(z0), mark(z1)), mark(x1))), MARK(filter(divides(z0, z1), x1))) MARK(head(filter(x0, x1))) -> c19(MARK(filter(x0, x1))) MARK(tail(primes)) -> c20(A__TAIL(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(tail(primes)) -> c20(MARK(primes)) MARK(tail(sieve(primes))) -> c20(A__TAIL(a__sieve(a__primes)), MARK(sieve(primes))) MARK(tail(sieve(sieve(z0)))) -> c20(A__TAIL(a__sieve(a__sieve(mark(z0)))), MARK(sieve(sieve(z0)))) MARK(tail(sieve(from(z0)))) -> c20(A__TAIL(a__sieve(a__from(mark(z0)))), MARK(sieve(from(z0)))) MARK(tail(sieve(head(z0)))) -> c20(A__TAIL(a__sieve(a__head(mark(z0)))), MARK(sieve(head(z0)))) MARK(tail(sieve(tail(z0)))) -> c20(A__TAIL(a__sieve(a__tail(mark(z0)))), MARK(sieve(tail(z0)))) MARK(tail(sieve(if(z0, z1, z2)))) -> c20(A__TAIL(a__sieve(a__if(mark(z0), z1, z2))), MARK(sieve(if(z0, z1, z2)))) MARK(tail(sieve(filter(z0, z1)))) -> c20(A__TAIL(a__sieve(a__filter(mark(z0), mark(z1)))), MARK(sieve(filter(z0, z1)))) MARK(tail(sieve(s(z0)))) -> c20(A__TAIL(a__sieve(s(mark(z0)))), MARK(sieve(s(z0)))) MARK(tail(sieve(cons(z0, z1)))) -> c20(A__TAIL(a__sieve(cons(mark(z0), z1))), MARK(sieve(cons(z0, z1)))) MARK(tail(sieve(divides(z0, z1)))) -> c20(A__TAIL(a__sieve(divides(mark(z0), mark(z1)))), MARK(sieve(divides(z0, z1)))) MARK(tail(sieve(x0))) -> c20(MARK(sieve(x0))) MARK(tail(sieve(0))) -> c20(A__TAIL(a__sieve(0))) MARK(tail(sieve(true))) -> c20(A__TAIL(a__sieve(true))) MARK(tail(sieve(false))) -> c20(A__TAIL(a__sieve(false))) MARK(tail(from(x0))) -> c20(A__TAIL(cons(mark(mark(x0)), from(s(mark(x0))))), MARK(from(x0))) MARK(tail(from(primes))) -> c20(A__TAIL(a__from(a__primes)), MARK(from(primes))) MARK(tail(from(sieve(z0)))) -> c20(A__TAIL(a__from(a__sieve(mark(z0)))), MARK(from(sieve(z0)))) MARK(tail(from(from(z0)))) -> c20(A__TAIL(a__from(a__from(mark(z0)))), MARK(from(from(z0)))) MARK(tail(from(head(z0)))) -> c20(A__TAIL(a__from(a__head(mark(z0)))), MARK(from(head(z0)))) MARK(tail(from(tail(z0)))) -> c20(A__TAIL(a__from(a__tail(mark(z0)))), MARK(from(tail(z0)))) MARK(tail(from(if(z0, z1, z2)))) -> c20(A__TAIL(a__from(a__if(mark(z0), z1, z2))), MARK(from(if(z0, z1, z2)))) MARK(tail(from(filter(z0, z1)))) -> c20(A__TAIL(a__from(a__filter(mark(z0), mark(z1)))), MARK(from(filter(z0, z1)))) MARK(tail(from(s(z0)))) -> c20(A__TAIL(a__from(s(mark(z0)))), MARK(from(s(z0)))) MARK(tail(from(0))) -> c20(A__TAIL(a__from(0)), MARK(from(0))) MARK(tail(from(cons(z0, z1)))) -> c20(A__TAIL(a__from(cons(mark(z0), z1))), MARK(from(cons(z0, z1)))) MARK(tail(from(true))) -> c20(A__TAIL(a__from(true)), MARK(from(true))) MARK(tail(from(false))) -> c20(A__TAIL(a__from(false)), MARK(from(false))) MARK(tail(from(divides(z0, z1)))) -> c20(A__TAIL(a__from(divides(mark(z0), mark(z1)))), MARK(from(divides(z0, z1)))) MARK(tail(from(x0))) -> c20(MARK(from(x0))) MARK(tail(head(primes))) -> c20(A__TAIL(a__head(a__primes)), MARK(head(primes))) MARK(tail(head(sieve(z0)))) -> c20(A__TAIL(a__head(a__sieve(mark(z0)))), MARK(head(sieve(z0)))) MARK(tail(head(from(z0)))) -> c20(A__TAIL(a__head(a__from(mark(z0)))), MARK(head(from(z0)))) MARK(tail(head(head(z0)))) -> c20(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c20(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(if(z0, z1, z2)))) -> c20(A__TAIL(a__head(a__if(mark(z0), z1, z2))), MARK(head(if(z0, z1, z2)))) MARK(tail(head(filter(z0, z1)))) -> c20(A__TAIL(a__head(a__filter(mark(z0), mark(z1)))), MARK(head(filter(z0, z1)))) MARK(tail(head(s(z0)))) -> c20(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(cons(z0, z1)))) -> c20(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(divides(z0, z1)))) -> c20(A__TAIL(a__head(divides(mark(z0), mark(z1)))), MARK(head(divides(z0, z1)))) MARK(tail(head(x0))) -> c20(MARK(head(x0))) MARK(tail(head(0))) -> c20(A__TAIL(a__head(0))) MARK(tail(head(true))) -> c20(A__TAIL(a__head(true))) MARK(tail(head(false))) -> c20(A__TAIL(a__head(false))) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c18_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_1, c17_2, c17_1, c19_2, c19_1, c20_2, c20_1, c21_2, c21_1, c22_2, c22_1, c23_2, c23_1, c_2 ---------------------------------------- (171) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(tail(tail(z0))) -> c20(A__TAIL(a__tail(mark(z0))), MARK(tail(z0))) by MARK(tail(tail(x0))) -> c20(A__TAIL(tail(mark(x0))), MARK(tail(x0))) MARK(tail(tail(primes))) -> c20(A__TAIL(a__tail(a__primes)), MARK(tail(primes))) MARK(tail(tail(sieve(z0)))) -> c20(A__TAIL(a__tail(a__sieve(mark(z0)))), MARK(tail(sieve(z0)))) MARK(tail(tail(from(z0)))) -> c20(A__TAIL(a__tail(a__from(mark(z0)))), MARK(tail(from(z0)))) MARK(tail(tail(head(z0)))) -> c20(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c20(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(if(z0, z1, z2)))) -> c20(A__TAIL(a__tail(a__if(mark(z0), z1, z2))), MARK(tail(if(z0, z1, z2)))) MARK(tail(tail(filter(z0, z1)))) -> c20(A__TAIL(a__tail(a__filter(mark(z0), mark(z1)))), MARK(tail(filter(z0, z1)))) MARK(tail(tail(s(z0)))) -> c20(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(0))) -> c20(A__TAIL(a__tail(0)), MARK(tail(0))) MARK(tail(tail(cons(z0, z1)))) -> c20(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(true))) -> c20(A__TAIL(a__tail(true)), MARK(tail(true))) MARK(tail(tail(false))) -> c20(A__TAIL(a__tail(false)), MARK(tail(false))) MARK(tail(tail(divides(z0, z1)))) -> c20(A__TAIL(a__tail(divides(mark(z0), mark(z1)))), MARK(tail(divides(z0, z1)))) ---------------------------------------- (172) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> 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c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(if(z0, z1, z2))) -> c20(A__TAIL(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(tail(filter(z0, z1))) -> c20(A__TAIL(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(primes, x1, x2)) -> c21(A__IF(a__primes, x1, x2), MARK(primes)) MARK(if(sieve(z0), x1, x2)) -> c21(A__IF(a__sieve(mark(z0)), x1, x2), MARK(sieve(z0))) MARK(if(from(z0), x1, x2)) -> c21(A__IF(a__from(mark(z0)), x1, x2), MARK(from(z0))) MARK(if(head(z0), x1, x2)) -> c21(A__IF(a__head(mark(z0)), x1, x2), MARK(head(z0))) MARK(if(tail(z0), x1, x2)) -> c21(A__IF(a__tail(mark(z0)), x1, x2), MARK(tail(z0))) MARK(if(if(z0, z1, z2), x1, x2)) -> c21(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(filter(z0, z1), x1, x2)) -> c21(A__IF(a__filter(mark(z0), mark(z1)), x1, x2), MARK(filter(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c22(A__FILTER(mark(x0), a__primes), MARK(x0)) MARK(filter(x0, sieve(z0))) -> c22(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, from(z0))) -> c22(A__FILTER(mark(x0), a__from(mark(z0))), MARK(x0)) MARK(filter(x0, head(z0))) -> c22(A__FILTER(mark(x0), a__head(mark(z0))), MARK(x0)) MARK(filter(x0, tail(z0))) -> c22(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(x0)) MARK(filter(x0, if(z0, z1, z2))) -> c22(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0)) MARK(filter(x0, filter(z0, z1))) -> c22(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(x0)) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(x0, s(z0))) -> c22(MARK(x0)) MARK(filter(x0, 0)) -> c22(MARK(x0)) MARK(filter(x0, true)) -> c22(MARK(x0)) MARK(filter(x0, false)) -> c22(MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), x1)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c23(A__FILTER(mark(x0), a__primes), MARK(primes)) MARK(filter(x0, sieve(z0))) -> c23(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, from(z0))) -> c23(A__FILTER(mark(x0), a__from(mark(z0))), MARK(from(z0))) MARK(filter(x0, head(z0))) -> c23(A__FILTER(mark(x0), a__head(mark(z0))), MARK(head(z0))) MARK(filter(x0, tail(z0))) -> c23(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(tail(z0))) MARK(filter(x0, if(z0, z1, z2))) -> c23(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(filter(x0, filter(z0, z1))) -> c23(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(filter(x0, cons(z0, z1))) -> c23(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(primes, x1)) -> c23(A__FILTER(a__primes, mark(x1)), MARK(x1)) MARK(filter(sieve(z0), x1)) -> c23(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(from(z0), x1)) -> c23(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(head(z0), x1)) -> c23(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(tail(z0), x1)) -> c23(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(if(z0, z1, z2), x1)) -> c23(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1)) MARK(filter(filter(z0, z1), x1)) -> c23(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(x1)) MARK(filter(s(z0), x1)) -> c23(A__FILTER(s(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(x0, s(z0))) -> c23(MARK(s(z0))) MARK(filter(x0, divides(z0, z1))) -> c23(MARK(divides(z0, z1))) MARK(filter(0, x1)) -> c23(MARK(x1)) MARK(filter(cons(z0, z1), x1)) -> c23(MARK(x1)) MARK(filter(true, x1)) -> c23(MARK(x1)) MARK(filter(false, x1)) -> c23(MARK(x1)) MARK(filter(divides(z0, z1), x1)) -> c23(MARK(x1)) A__PRIMES -> c(A__SIEVE(cons(s(mark(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(sieve(primes)) -> c17(MARK(primes)) MARK(sieve(sieve(primes))) -> c17(A__SIEVE(a__sieve(a__primes)), MARK(sieve(primes))) MARK(sieve(sieve(sieve(z0)))) -> c17(A__SIEVE(a__sieve(a__sieve(mark(z0)))), MARK(sieve(sieve(z0)))) MARK(sieve(sieve(from(z0)))) -> c17(A__SIEVE(a__sieve(a__from(mark(z0)))), MARK(sieve(from(z0)))) MARK(sieve(sieve(head(z0)))) -> c17(A__SIEVE(a__sieve(a__head(mark(z0)))), MARK(sieve(head(z0)))) MARK(sieve(sieve(tail(z0)))) -> c17(A__SIEVE(a__sieve(a__tail(mark(z0)))), MARK(sieve(tail(z0)))) MARK(sieve(sieve(if(z0, z1, z2)))) -> c17(A__SIEVE(a__sieve(a__if(mark(z0), z1, z2))), MARK(sieve(if(z0, z1, z2)))) MARK(sieve(sieve(filter(z0, z1)))) -> c17(A__SIEVE(a__sieve(a__filter(mark(z0), mark(z1)))), 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c17(A__SIEVE(a__from(a__head(mark(z0)))), MARK(from(head(z0)))) MARK(sieve(from(tail(z0)))) -> c17(A__SIEVE(a__from(a__tail(mark(z0)))), MARK(from(tail(z0)))) MARK(sieve(from(if(z0, z1, z2)))) -> c17(A__SIEVE(a__from(a__if(mark(z0), z1, z2))), MARK(from(if(z0, z1, z2)))) MARK(sieve(from(filter(z0, z1)))) -> c17(A__SIEVE(a__from(a__filter(mark(z0), mark(z1)))), MARK(from(filter(z0, z1)))) MARK(sieve(from(s(z0)))) -> c17(A__SIEVE(a__from(s(mark(z0)))), MARK(from(s(z0)))) MARK(sieve(from(0))) -> c17(A__SIEVE(a__from(0)), MARK(from(0))) MARK(sieve(from(cons(z0, z1)))) -> c17(A__SIEVE(a__from(cons(mark(z0), z1))), MARK(from(cons(z0, z1)))) MARK(sieve(from(true))) -> c17(A__SIEVE(a__from(true)), MARK(from(true))) MARK(sieve(from(false))) -> c17(A__SIEVE(a__from(false)), MARK(from(false))) MARK(sieve(from(divides(z0, z1)))) -> c17(A__SIEVE(a__from(divides(mark(z0), mark(z1)))), MARK(from(divides(z0, z1)))) MARK(sieve(from(x0))) -> c17(MARK(from(x0))) MARK(sieve(head(primes))) -> c17(A__SIEVE(a__head(a__primes)), MARK(head(primes))) MARK(sieve(head(sieve(z0)))) -> c17(A__SIEVE(a__head(a__sieve(mark(z0)))), MARK(head(sieve(z0)))) MARK(sieve(head(from(z0)))) -> c17(A__SIEVE(a__head(a__from(mark(z0)))), MARK(head(from(z0)))) MARK(sieve(head(head(z0)))) -> c17(A__SIEVE(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(sieve(head(tail(z0)))) -> c17(A__SIEVE(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(sieve(head(if(z0, z1, z2)))) -> c17(A__SIEVE(a__head(a__if(mark(z0), z1, z2))), MARK(head(if(z0, z1, z2)))) MARK(sieve(head(filter(z0, z1)))) -> c17(A__SIEVE(a__head(a__filter(mark(z0), mark(z1)))), MARK(head(filter(z0, z1)))) MARK(sieve(head(s(z0)))) -> c17(A__SIEVE(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(sieve(head(cons(z0, z1)))) -> c17(A__SIEVE(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(sieve(head(divides(z0, z1)))) -> c17(A__SIEVE(a__head(divides(mark(z0), mark(z1)))), MARK(head(divides(z0, z1)))) MARK(sieve(head(x0))) -> c17(MARK(head(x0))) MARK(sieve(head(0))) -> c17(A__SIEVE(a__head(0))) MARK(sieve(head(true))) -> c17(A__SIEVE(a__head(true))) MARK(sieve(head(false))) -> c17(A__SIEVE(a__head(false))) MARK(sieve(tail(primes))) -> c17(A__SIEVE(a__tail(a__primes)), MARK(tail(primes))) MARK(sieve(tail(sieve(z0)))) -> c17(A__SIEVE(a__tail(a__sieve(mark(z0)))), MARK(tail(sieve(z0)))) MARK(sieve(tail(from(z0)))) -> c17(A__SIEVE(a__tail(a__from(mark(z0)))), MARK(tail(from(z0)))) MARK(sieve(tail(head(z0)))) -> c17(A__SIEVE(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(sieve(tail(tail(z0)))) -> c17(A__SIEVE(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(sieve(tail(if(z0, z1, z2)))) -> c17(A__SIEVE(a__tail(a__if(mark(z0), z1, z2))), MARK(tail(if(z0, z1, z2)))) MARK(sieve(tail(filter(z0, z1)))) -> c17(A__SIEVE(a__tail(a__filter(mark(z0), mark(z1)))), MARK(tail(filter(z0, z1)))) MARK(sieve(tail(s(z0)))) -> c17(A__SIEVE(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(sieve(tail(cons(z0, z1)))) -> c17(A__SIEVE(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(sieve(tail(divides(z0, z1)))) -> c17(A__SIEVE(a__tail(divides(mark(z0), mark(z1)))), MARK(tail(divides(z0, z1)))) MARK(sieve(tail(x0))) -> c17(MARK(tail(x0))) MARK(sieve(tail(0))) -> c17(A__SIEVE(a__tail(0))) MARK(sieve(tail(true))) -> c17(A__SIEVE(a__tail(true))) MARK(sieve(tail(false))) -> c17(A__SIEVE(a__tail(false))) MARK(sieve(if(primes, x1, x2))) -> c17(A__SIEVE(a__if(a__primes, x1, x2)), MARK(if(primes, x1, x2))) MARK(sieve(if(sieve(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__sieve(mark(z0)), x1, x2)), MARK(if(sieve(z0), x1, x2))) MARK(sieve(if(from(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__from(mark(z0)), x1, x2)), MARK(if(from(z0), x1, x2))) MARK(sieve(if(head(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__head(mark(z0)), x1, x2)), MARK(if(head(z0), x1, x2))) MARK(sieve(if(tail(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__tail(mark(z0)), x1, x2)), MARK(if(tail(z0), x1, x2))) MARK(sieve(if(if(z0, z1, z2), x1, x2))) -> c17(A__SIEVE(a__if(a__if(mark(z0), z1, z2), x1, x2)), MARK(if(if(z0, z1, z2), x1, x2))) MARK(sieve(if(filter(z0, z1), x1, x2))) -> c17(A__SIEVE(a__if(a__filter(mark(z0), mark(z1)), x1, x2)), MARK(if(filter(z0, z1), x1, x2))) MARK(sieve(if(s(z0), x1, x2))) -> c17(A__SIEVE(a__if(s(mark(z0)), x1, x2)), MARK(if(s(z0), x1, x2))) MARK(sieve(if(cons(z0, z1), x1, x2))) -> c17(A__SIEVE(a__if(cons(mark(z0), z1), x1, x2)), MARK(if(cons(z0, z1), x1, x2))) MARK(sieve(if(true, x1, x2))) -> c17(A__SIEVE(a__if(true, x1, x2)), MARK(if(true, x1, x2))) MARK(sieve(if(false, x1, x2))) -> c17(A__SIEVE(a__if(false, x1, x2)), MARK(if(false, x1, x2))) MARK(sieve(if(divides(z0, z1), x1, x2))) -> c17(A__SIEVE(a__if(divides(mark(z0), mark(z1)), x1, x2)), MARK(if(divides(z0, z1), x1, x2))) MARK(sieve(if(x0, z1, z2))) -> c17(MARK(if(x0, z1, z2))) MARK(sieve(if(0, x1, x2))) -> c17(A__SIEVE(a__if(0, x1, x2))) MARK(sieve(filter(x0, primes))) -> c17(A__SIEVE(a__filter(mark(x0), a__primes)), MARK(filter(x0, primes))) 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c20(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c20(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(if(z0, z1, z2)))) -> c20(A__TAIL(a__tail(a__if(mark(z0), z1, z2))), MARK(tail(if(z0, z1, z2)))) MARK(tail(tail(filter(z0, z1)))) -> c20(A__TAIL(a__tail(a__filter(mark(z0), mark(z1)))), MARK(tail(filter(z0, z1)))) MARK(tail(tail(s(z0)))) -> c20(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(0))) -> c20(A__TAIL(a__tail(0)), MARK(tail(0))) MARK(tail(tail(cons(z0, z1)))) -> c20(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(true))) -> c20(A__TAIL(a__tail(true)), MARK(tail(true))) MARK(tail(tail(false))) -> c20(A__TAIL(a__tail(false)), MARK(tail(false))) MARK(tail(tail(divides(z0, z1)))) -> c20(A__TAIL(a__tail(divides(mark(z0), mark(z1)))), MARK(tail(divides(z0, z1)))) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c18_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_1, c17_2, c17_1, c19_2, c19_1, c20_2, c20_1, c21_2, c21_1, c22_2, c22_1, c23_2, c23_1, c_2 ---------------------------------------- (173) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing tuple parts ---------------------------------------- (174) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(if(z0, z1, z2))) -> c20(A__TAIL(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(tail(filter(z0, z1))) -> c20(A__TAIL(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(primes, x1, x2)) -> c21(A__IF(a__primes, x1, x2), MARK(primes)) MARK(if(sieve(z0), x1, x2)) -> c21(A__IF(a__sieve(mark(z0)), x1, x2), MARK(sieve(z0))) MARK(if(from(z0), x1, x2)) -> c21(A__IF(a__from(mark(z0)), x1, x2), MARK(from(z0))) MARK(if(head(z0), x1, x2)) -> c21(A__IF(a__head(mark(z0)), x1, x2), MARK(head(z0))) MARK(if(tail(z0), x1, x2)) -> c21(A__IF(a__tail(mark(z0)), x1, x2), MARK(tail(z0))) MARK(if(if(z0, z1, z2), x1, x2)) -> c21(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(filter(z0, z1), x1, x2)) -> c21(A__IF(a__filter(mark(z0), mark(z1)), x1, x2), MARK(filter(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c22(A__FILTER(mark(x0), a__primes), MARK(x0)) MARK(filter(x0, sieve(z0))) -> c22(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, from(z0))) -> c22(A__FILTER(mark(x0), a__from(mark(z0))), MARK(x0)) MARK(filter(x0, head(z0))) -> c22(A__FILTER(mark(x0), a__head(mark(z0))), MARK(x0)) MARK(filter(x0, tail(z0))) -> c22(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(x0)) MARK(filter(x0, if(z0, z1, z2))) -> c22(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0)) MARK(filter(x0, filter(z0, z1))) -> c22(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(x0)) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(x0, s(z0))) -> c22(MARK(x0)) MARK(filter(x0, 0)) -> c22(MARK(x0)) MARK(filter(x0, true)) -> c22(MARK(x0)) MARK(filter(x0, false)) -> c22(MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), x1)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c23(A__FILTER(mark(x0), a__primes), MARK(primes)) MARK(filter(x0, sieve(z0))) -> c23(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, from(z0))) -> c23(A__FILTER(mark(x0), a__from(mark(z0))), MARK(from(z0))) MARK(filter(x0, head(z0))) -> c23(A__FILTER(mark(x0), a__head(mark(z0))), MARK(head(z0))) MARK(filter(x0, tail(z0))) -> c23(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(tail(z0))) MARK(filter(x0, if(z0, z1, z2))) -> c23(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(filter(x0, filter(z0, z1))) -> c23(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(filter(x0, cons(z0, z1))) -> c23(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(primes, x1)) -> c23(A__FILTER(a__primes, mark(x1)), MARK(x1)) MARK(filter(sieve(z0), x1)) -> c23(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(from(z0), x1)) -> c23(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(head(z0), x1)) -> c23(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(tail(z0), x1)) -> c23(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(if(z0, z1, z2), x1)) -> c23(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1)) MARK(filter(filter(z0, z1), x1)) -> c23(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(x1)) MARK(filter(s(z0), x1)) -> c23(A__FILTER(s(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(x0, s(z0))) -> c23(MARK(s(z0))) MARK(filter(x0, divides(z0, z1))) -> c23(MARK(divides(z0, z1))) MARK(filter(0, x1)) -> c23(MARK(x1)) MARK(filter(cons(z0, z1), x1)) -> c23(MARK(x1)) MARK(filter(true, x1)) -> c23(MARK(x1)) MARK(filter(false, x1)) -> c23(MARK(x1)) MARK(filter(divides(z0, z1), x1)) -> c23(MARK(x1)) A__PRIMES -> c(A__SIEVE(cons(s(mark(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(sieve(primes)) -> c17(MARK(primes)) MARK(sieve(sieve(primes))) -> c17(A__SIEVE(a__sieve(a__primes)), MARK(sieve(primes))) MARK(sieve(sieve(sieve(z0)))) -> c17(A__SIEVE(a__sieve(a__sieve(mark(z0)))), MARK(sieve(sieve(z0)))) MARK(sieve(sieve(from(z0)))) -> c17(A__SIEVE(a__sieve(a__from(mark(z0)))), MARK(sieve(from(z0)))) MARK(sieve(sieve(head(z0)))) -> c17(A__SIEVE(a__sieve(a__head(mark(z0)))), MARK(sieve(head(z0)))) MARK(sieve(sieve(tail(z0)))) -> c17(A__SIEVE(a__sieve(a__tail(mark(z0)))), MARK(sieve(tail(z0)))) MARK(sieve(sieve(if(z0, z1, z2)))) -> c17(A__SIEVE(a__sieve(a__if(mark(z0), z1, z2))), MARK(sieve(if(z0, z1, z2)))) MARK(sieve(sieve(filter(z0, z1)))) -> c17(A__SIEVE(a__sieve(a__filter(mark(z0), mark(z1)))), MARK(sieve(filter(z0, z1)))) MARK(sieve(sieve(s(z0)))) -> c17(A__SIEVE(a__sieve(s(mark(z0)))), MARK(sieve(s(z0)))) MARK(sieve(sieve(cons(z0, z1)))) -> c17(A__SIEVE(a__sieve(cons(mark(z0), z1))), MARK(sieve(cons(z0, z1)))) MARK(sieve(sieve(divides(z0, z1)))) -> c17(A__SIEVE(a__sieve(divides(mark(z0), mark(z1)))), MARK(sieve(divides(z0, z1)))) MARK(sieve(sieve(x0))) -> c17(MARK(sieve(x0))) MARK(sieve(sieve(0))) -> c17(A__SIEVE(a__sieve(0))) MARK(sieve(sieve(true))) -> c17(A__SIEVE(a__sieve(true))) MARK(sieve(sieve(false))) -> c17(A__SIEVE(a__sieve(false))) MARK(sieve(from(x0))) -> c17(A__SIEVE(cons(mark(mark(x0)), from(s(mark(x0))))), MARK(from(x0))) MARK(sieve(from(primes))) -> c17(A__SIEVE(a__from(a__primes)), MARK(from(primes))) MARK(sieve(from(sieve(z0)))) -> c17(A__SIEVE(a__from(a__sieve(mark(z0)))), MARK(from(sieve(z0)))) MARK(sieve(from(from(z0)))) -> c17(A__SIEVE(a__from(a__from(mark(z0)))), MARK(from(from(z0)))) MARK(sieve(from(head(z0)))) -> c17(A__SIEVE(a__from(a__head(mark(z0)))), MARK(from(head(z0)))) MARK(sieve(from(tail(z0)))) -> c17(A__SIEVE(a__from(a__tail(mark(z0)))), MARK(from(tail(z0)))) MARK(sieve(from(if(z0, z1, z2)))) -> c17(A__SIEVE(a__from(a__if(mark(z0), z1, z2))), MARK(from(if(z0, z1, z2)))) MARK(sieve(from(filter(z0, z1)))) -> c17(A__SIEVE(a__from(a__filter(mark(z0), mark(z1)))), MARK(from(filter(z0, z1)))) MARK(sieve(from(s(z0)))) -> c17(A__SIEVE(a__from(s(mark(z0)))), MARK(from(s(z0)))) MARK(sieve(from(0))) -> c17(A__SIEVE(a__from(0)), MARK(from(0))) MARK(sieve(from(cons(z0, z1)))) -> c17(A__SIEVE(a__from(cons(mark(z0), z1))), MARK(from(cons(z0, z1)))) MARK(sieve(from(true))) -> c17(A__SIEVE(a__from(true)), MARK(from(true))) MARK(sieve(from(false))) -> c17(A__SIEVE(a__from(false)), MARK(from(false))) MARK(sieve(from(divides(z0, z1)))) -> c17(A__SIEVE(a__from(divides(mark(z0), mark(z1)))), MARK(from(divides(z0, z1)))) MARK(sieve(from(x0))) -> c17(MARK(from(x0))) MARK(sieve(head(primes))) -> c17(A__SIEVE(a__head(a__primes)), MARK(head(primes))) MARK(sieve(head(sieve(z0)))) -> c17(A__SIEVE(a__head(a__sieve(mark(z0)))), MARK(head(sieve(z0)))) MARK(sieve(head(from(z0)))) -> c17(A__SIEVE(a__head(a__from(mark(z0)))), MARK(head(from(z0)))) MARK(sieve(head(head(z0)))) -> c17(A__SIEVE(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(sieve(head(tail(z0)))) -> c17(A__SIEVE(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(sieve(head(if(z0, z1, z2)))) -> c17(A__SIEVE(a__head(a__if(mark(z0), z1, z2))), MARK(head(if(z0, z1, z2)))) MARK(sieve(head(filter(z0, z1)))) -> c17(A__SIEVE(a__head(a__filter(mark(z0), mark(z1)))), MARK(head(filter(z0, z1)))) MARK(sieve(head(s(z0)))) -> 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MARK(filter(0, x1))) MARK(head(filter(cons(z0, z1), x1))) -> c19(A__HEAD(a__filter(cons(mark(z0), z1), mark(x1))), MARK(filter(cons(z0, z1), x1))) MARK(head(filter(true, x1))) -> c19(A__HEAD(a__filter(true, mark(x1))), MARK(filter(true, x1))) MARK(head(filter(false, x1))) -> c19(A__HEAD(a__filter(false, mark(x1))), MARK(filter(false, x1))) MARK(head(filter(divides(z0, z1), x1))) -> c19(A__HEAD(a__filter(divides(mark(z0), mark(z1)), mark(x1))), MARK(filter(divides(z0, z1), x1))) MARK(head(filter(x0, x1))) -> c19(MARK(filter(x0, x1))) MARK(tail(primes)) -> c20(A__TAIL(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(tail(primes)) -> c20(MARK(primes)) MARK(tail(sieve(primes))) -> c20(A__TAIL(a__sieve(a__primes)), MARK(sieve(primes))) MARK(tail(sieve(sieve(z0)))) -> c20(A__TAIL(a__sieve(a__sieve(mark(z0)))), MARK(sieve(sieve(z0)))) MARK(tail(sieve(from(z0)))) -> c20(A__TAIL(a__sieve(a__from(mark(z0)))), MARK(sieve(from(z0)))) MARK(tail(sieve(head(z0)))) -> c20(A__TAIL(a__sieve(a__head(mark(z0)))), MARK(sieve(head(z0)))) MARK(tail(sieve(tail(z0)))) -> c20(A__TAIL(a__sieve(a__tail(mark(z0)))), MARK(sieve(tail(z0)))) MARK(tail(sieve(if(z0, z1, z2)))) -> c20(A__TAIL(a__sieve(a__if(mark(z0), z1, z2))), MARK(sieve(if(z0, z1, z2)))) MARK(tail(sieve(filter(z0, z1)))) -> c20(A__TAIL(a__sieve(a__filter(mark(z0), mark(z1)))), MARK(sieve(filter(z0, z1)))) MARK(tail(sieve(s(z0)))) -> c20(A__TAIL(a__sieve(s(mark(z0)))), MARK(sieve(s(z0)))) MARK(tail(sieve(cons(z0, z1)))) -> c20(A__TAIL(a__sieve(cons(mark(z0), z1))), MARK(sieve(cons(z0, z1)))) MARK(tail(sieve(divides(z0, z1)))) -> c20(A__TAIL(a__sieve(divides(mark(z0), mark(z1)))), MARK(sieve(divides(z0, z1)))) MARK(tail(sieve(x0))) -> c20(MARK(sieve(x0))) MARK(tail(sieve(0))) -> c20(A__TAIL(a__sieve(0))) MARK(tail(sieve(true))) -> c20(A__TAIL(a__sieve(true))) MARK(tail(sieve(false))) -> c20(A__TAIL(a__sieve(false))) MARK(tail(from(x0))) -> c20(A__TAIL(cons(mark(mark(x0)), from(s(mark(x0))))), MARK(from(x0))) MARK(tail(from(primes))) -> c20(A__TAIL(a__from(a__primes)), MARK(from(primes))) MARK(tail(from(sieve(z0)))) -> c20(A__TAIL(a__from(a__sieve(mark(z0)))), MARK(from(sieve(z0)))) MARK(tail(from(from(z0)))) -> c20(A__TAIL(a__from(a__from(mark(z0)))), MARK(from(from(z0)))) MARK(tail(from(head(z0)))) -> c20(A__TAIL(a__from(a__head(mark(z0)))), MARK(from(head(z0)))) MARK(tail(from(tail(z0)))) -> c20(A__TAIL(a__from(a__tail(mark(z0)))), MARK(from(tail(z0)))) MARK(tail(from(if(z0, z1, z2)))) -> c20(A__TAIL(a__from(a__if(mark(z0), z1, z2))), MARK(from(if(z0, z1, z2)))) MARK(tail(from(filter(z0, z1)))) -> c20(A__TAIL(a__from(a__filter(mark(z0), mark(z1)))), MARK(from(filter(z0, z1)))) MARK(tail(from(s(z0)))) -> c20(A__TAIL(a__from(s(mark(z0)))), MARK(from(s(z0)))) MARK(tail(from(0))) -> c20(A__TAIL(a__from(0)), MARK(from(0))) MARK(tail(from(cons(z0, z1)))) -> c20(A__TAIL(a__from(cons(mark(z0), z1))), MARK(from(cons(z0, z1)))) MARK(tail(from(true))) -> c20(A__TAIL(a__from(true)), MARK(from(true))) MARK(tail(from(false))) -> c20(A__TAIL(a__from(false)), MARK(from(false))) MARK(tail(from(divides(z0, z1)))) -> c20(A__TAIL(a__from(divides(mark(z0), mark(z1)))), MARK(from(divides(z0, z1)))) MARK(tail(from(x0))) -> c20(MARK(from(x0))) MARK(tail(head(primes))) -> c20(A__TAIL(a__head(a__primes)), MARK(head(primes))) MARK(tail(head(sieve(z0)))) -> c20(A__TAIL(a__head(a__sieve(mark(z0)))), MARK(head(sieve(z0)))) MARK(tail(head(from(z0)))) -> c20(A__TAIL(a__head(a__from(mark(z0)))), MARK(head(from(z0)))) MARK(tail(head(head(z0)))) -> c20(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c20(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(if(z0, z1, z2)))) -> c20(A__TAIL(a__head(a__if(mark(z0), z1, z2))), MARK(head(if(z0, z1, z2)))) MARK(tail(head(filter(z0, z1)))) -> c20(A__TAIL(a__head(a__filter(mark(z0), mark(z1)))), MARK(head(filter(z0, z1)))) MARK(tail(head(s(z0)))) -> c20(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(cons(z0, z1)))) -> c20(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(divides(z0, z1)))) -> c20(A__TAIL(a__head(divides(mark(z0), mark(z1)))), MARK(head(divides(z0, z1)))) MARK(tail(head(x0))) -> c20(MARK(head(x0))) MARK(tail(head(0))) -> c20(A__TAIL(a__head(0))) MARK(tail(head(true))) -> c20(A__TAIL(a__head(true))) MARK(tail(head(false))) -> c20(A__TAIL(a__head(false))) MARK(tail(tail(primes))) -> c20(A__TAIL(a__tail(a__primes)), MARK(tail(primes))) MARK(tail(tail(sieve(z0)))) -> c20(A__TAIL(a__tail(a__sieve(mark(z0)))), MARK(tail(sieve(z0)))) MARK(tail(tail(from(z0)))) -> c20(A__TAIL(a__tail(a__from(mark(z0)))), MARK(tail(from(z0)))) MARK(tail(tail(head(z0)))) -> c20(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c20(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(if(z0, z1, z2)))) -> c20(A__TAIL(a__tail(a__if(mark(z0), z1, z2))), MARK(tail(if(z0, z1, z2)))) MARK(tail(tail(filter(z0, z1)))) -> c20(A__TAIL(a__tail(a__filter(mark(z0), mark(z1)))), MARK(tail(filter(z0, z1)))) MARK(tail(tail(s(z0)))) -> c20(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c20(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(divides(z0, z1)))) -> c20(A__TAIL(a__tail(divides(mark(z0), mark(z1)))), MARK(tail(divides(z0, z1)))) MARK(tail(tail(x0))) -> c20(MARK(tail(x0))) MARK(tail(tail(0))) -> c20(A__TAIL(a__tail(0))) MARK(tail(tail(true))) -> c20(A__TAIL(a__tail(true))) MARK(tail(tail(false))) -> c20(A__TAIL(a__tail(false))) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c18_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_1, c17_2, c17_1, c19_2, c19_1, c20_2, c20_1, c21_2, c21_1, c22_2, c22_1, c23_2, c23_1, c_2 ---------------------------------------- (175) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(tail(if(z0, z1, z2))) -> c20(A__TAIL(a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) by MARK(tail(if(x0, z1, z2))) -> c20(A__TAIL(if(mark(x0), z1, z2)), MARK(if(x0, z1, z2))) MARK(tail(if(primes, x1, x2))) -> c20(A__TAIL(a__if(a__primes, x1, x2)), MARK(if(primes, x1, x2))) MARK(tail(if(sieve(z0), x1, x2))) -> c20(A__TAIL(a__if(a__sieve(mark(z0)), x1, x2)), MARK(if(sieve(z0), x1, x2))) MARK(tail(if(from(z0), x1, x2))) -> c20(A__TAIL(a__if(a__from(mark(z0)), x1, x2)), MARK(if(from(z0), x1, x2))) MARK(tail(if(head(z0), x1, x2))) -> c20(A__TAIL(a__if(a__head(mark(z0)), x1, x2)), MARK(if(head(z0), x1, x2))) MARK(tail(if(tail(z0), x1, x2))) -> c20(A__TAIL(a__if(a__tail(mark(z0)), x1, x2)), MARK(if(tail(z0), x1, x2))) MARK(tail(if(if(z0, z1, z2), x1, x2))) -> c20(A__TAIL(a__if(a__if(mark(z0), z1, z2), x1, x2)), MARK(if(if(z0, z1, z2), x1, x2))) MARK(tail(if(filter(z0, z1), x1, x2))) -> c20(A__TAIL(a__if(a__filter(mark(z0), mark(z1)), x1, x2)), MARK(if(filter(z0, z1), x1, x2))) MARK(tail(if(s(z0), x1, x2))) -> c20(A__TAIL(a__if(s(mark(z0)), x1, x2)), MARK(if(s(z0), x1, x2))) MARK(tail(if(0, x1, x2))) -> c20(A__TAIL(a__if(0, x1, x2)), MARK(if(0, x1, x2))) MARK(tail(if(cons(z0, z1), x1, x2))) -> c20(A__TAIL(a__if(cons(mark(z0), z1), x1, x2)), MARK(if(cons(z0, z1), x1, x2))) MARK(tail(if(true, x1, x2))) -> c20(A__TAIL(a__if(true, x1, x2)), MARK(if(true, x1, x2))) MARK(tail(if(false, x1, x2))) -> c20(A__TAIL(a__if(false, x1, x2)), MARK(if(false, x1, x2))) MARK(tail(if(divides(z0, z1), x1, x2))) -> c20(A__TAIL(a__if(divides(mark(z0), mark(z1)), x1, x2)), MARK(if(divides(z0, z1), x1, x2))) ---------------------------------------- (176) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(filter(z0, z1))) -> c20(A__TAIL(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(primes, x1, x2)) -> c21(A__IF(a__primes, x1, x2), MARK(primes)) MARK(if(sieve(z0), x1, x2)) -> c21(A__IF(a__sieve(mark(z0)), x1, x2), MARK(sieve(z0))) MARK(if(from(z0), x1, x2)) -> c21(A__IF(a__from(mark(z0)), x1, x2), MARK(from(z0))) MARK(if(head(z0), x1, x2)) -> c21(A__IF(a__head(mark(z0)), x1, x2), MARK(head(z0))) MARK(if(tail(z0), x1, x2)) -> c21(A__IF(a__tail(mark(z0)), x1, x2), MARK(tail(z0))) MARK(if(if(z0, z1, z2), x1, x2)) -> c21(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(filter(z0, z1), x1, x2)) -> c21(A__IF(a__filter(mark(z0), mark(z1)), x1, x2), MARK(filter(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c22(A__FILTER(mark(x0), a__primes), MARK(x0)) MARK(filter(x0, sieve(z0))) -> c22(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, from(z0))) -> c22(A__FILTER(mark(x0), a__from(mark(z0))), MARK(x0)) MARK(filter(x0, head(z0))) -> c22(A__FILTER(mark(x0), a__head(mark(z0))), MARK(x0)) MARK(filter(x0, tail(z0))) -> c22(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(x0)) MARK(filter(x0, if(z0, z1, z2))) -> c22(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0)) MARK(filter(x0, filter(z0, z1))) -> c22(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(x0)) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> 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MARK(sieve(if(primes, x1, x2))) -> c17(A__SIEVE(a__if(a__primes, x1, x2)), MARK(if(primes, x1, x2))) MARK(sieve(if(sieve(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__sieve(mark(z0)), x1, x2)), MARK(if(sieve(z0), x1, x2))) MARK(sieve(if(from(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__from(mark(z0)), x1, x2)), MARK(if(from(z0), x1, x2))) MARK(sieve(if(head(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__head(mark(z0)), x1, x2)), MARK(if(head(z0), x1, x2))) MARK(sieve(if(tail(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__tail(mark(z0)), x1, x2)), MARK(if(tail(z0), x1, x2))) MARK(sieve(if(if(z0, z1, z2), x1, x2))) -> c17(A__SIEVE(a__if(a__if(mark(z0), z1, z2), x1, x2)), MARK(if(if(z0, z1, z2), x1, x2))) MARK(sieve(if(filter(z0, z1), x1, x2))) -> c17(A__SIEVE(a__if(a__filter(mark(z0), mark(z1)), x1, x2)), MARK(if(filter(z0, z1), x1, x2))) MARK(sieve(if(s(z0), x1, x2))) -> c17(A__SIEVE(a__if(s(mark(z0)), x1, x2)), MARK(if(s(z0), x1, x2))) MARK(sieve(if(cons(z0, z1), x1, x2))) -> c17(A__SIEVE(a__if(cons(mark(z0), 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MARK(tail(if(tail(z0), x1, x2))) -> c20(A__TAIL(a__if(a__tail(mark(z0)), x1, x2)), MARK(if(tail(z0), x1, x2))) MARK(tail(if(if(z0, z1, z2), x1, x2))) -> c20(A__TAIL(a__if(a__if(mark(z0), z1, z2), x1, x2)), MARK(if(if(z0, z1, z2), x1, x2))) MARK(tail(if(filter(z0, z1), x1, x2))) -> c20(A__TAIL(a__if(a__filter(mark(z0), mark(z1)), x1, x2)), MARK(if(filter(z0, z1), x1, x2))) MARK(tail(if(s(z0), x1, x2))) -> c20(A__TAIL(a__if(s(mark(z0)), x1, x2)), MARK(if(s(z0), x1, x2))) MARK(tail(if(0, x1, x2))) -> c20(A__TAIL(a__if(0, x1, x2)), MARK(if(0, x1, x2))) MARK(tail(if(cons(z0, z1), x1, x2))) -> c20(A__TAIL(a__if(cons(mark(z0), z1), x1, x2)), MARK(if(cons(z0, z1), x1, x2))) MARK(tail(if(true, x1, x2))) -> c20(A__TAIL(a__if(true, x1, x2)), MARK(if(true, x1, x2))) MARK(tail(if(false, x1, x2))) -> c20(A__TAIL(a__if(false, x1, x2)), MARK(if(false, x1, x2))) MARK(tail(if(divides(z0, z1), x1, x2))) -> c20(A__TAIL(a__if(divides(mark(z0), mark(z1)), x1, x2)), MARK(if(divides(z0, z1), x1, x2))) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c18_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_1, c17_2, c17_1, c19_2, c19_1, c20_2, c20_1, c21_2, c21_1, c22_2, c22_1, c23_2, c23_1, c_2 ---------------------------------------- (177) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (178) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(filter(z0, z1))) -> c20(A__TAIL(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(primes, x1, x2)) -> c21(A__IF(a__primes, x1, x2), MARK(primes)) MARK(if(sieve(z0), x1, x2)) -> c21(A__IF(a__sieve(mark(z0)), x1, x2), MARK(sieve(z0))) MARK(if(from(z0), x1, x2)) -> c21(A__IF(a__from(mark(z0)), x1, x2), MARK(from(z0))) MARK(if(head(z0), x1, x2)) -> c21(A__IF(a__head(mark(z0)), x1, x2), MARK(head(z0))) MARK(if(tail(z0), x1, x2)) -> c21(A__IF(a__tail(mark(z0)), x1, x2), MARK(tail(z0))) MARK(if(if(z0, z1, z2), x1, x2)) -> c21(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(filter(z0, z1), x1, x2)) -> c21(A__IF(a__filter(mark(z0), mark(z1)), x1, x2), MARK(filter(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c22(A__FILTER(mark(x0), a__primes), MARK(x0)) MARK(filter(x0, sieve(z0))) -> c22(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, from(z0))) -> c22(A__FILTER(mark(x0), a__from(mark(z0))), MARK(x0)) MARK(filter(x0, head(z0))) -> c22(A__FILTER(mark(x0), a__head(mark(z0))), MARK(x0)) MARK(filter(x0, tail(z0))) -> c22(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(x0)) MARK(filter(x0, if(z0, z1, z2))) -> c22(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0)) MARK(filter(x0, filter(z0, z1))) -> c22(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(x0)) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(x0, s(z0))) -> c22(MARK(x0)) MARK(filter(x0, 0)) -> c22(MARK(x0)) MARK(filter(x0, true)) -> c22(MARK(x0)) MARK(filter(x0, false)) -> c22(MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), x1)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c23(A__FILTER(mark(x0), a__primes), MARK(primes)) MARK(filter(x0, sieve(z0))) -> c23(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, from(z0))) -> c23(A__FILTER(mark(x0), a__from(mark(z0))), MARK(from(z0))) MARK(filter(x0, head(z0))) -> c23(A__FILTER(mark(x0), a__head(mark(z0))), MARK(head(z0))) MARK(filter(x0, tail(z0))) -> c23(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(tail(z0))) MARK(filter(x0, if(z0, z1, z2))) -> c23(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(filter(x0, filter(z0, z1))) -> c23(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(filter(x0, cons(z0, z1))) -> c23(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(primes, x1)) -> c23(A__FILTER(a__primes, mark(x1)), MARK(x1)) MARK(filter(sieve(z0), x1)) -> c23(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(from(z0), x1)) -> c23(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(head(z0), x1)) -> c23(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(tail(z0), x1)) -> c23(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(if(z0, z1, z2), x1)) -> c23(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1)) MARK(filter(filter(z0, z1), x1)) -> c23(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(x1)) MARK(filter(s(z0), x1)) -> c23(A__FILTER(s(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(x0, s(z0))) -> c23(MARK(s(z0))) MARK(filter(x0, divides(z0, z1))) -> c23(MARK(divides(z0, z1))) MARK(filter(0, x1)) -> c23(MARK(x1)) MARK(filter(cons(z0, z1), x1)) -> c23(MARK(x1)) MARK(filter(true, x1)) -> c23(MARK(x1)) MARK(filter(false, x1)) -> c23(MARK(x1)) MARK(filter(divides(z0, z1), x1)) -> c23(MARK(x1)) A__PRIMES -> c(A__SIEVE(cons(s(mark(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(sieve(primes)) -> c17(MARK(primes)) MARK(sieve(sieve(primes))) -> c17(A__SIEVE(a__sieve(a__primes)), MARK(sieve(primes))) MARK(sieve(sieve(sieve(z0)))) -> c17(A__SIEVE(a__sieve(a__sieve(mark(z0)))), MARK(sieve(sieve(z0)))) MARK(sieve(sieve(from(z0)))) -> c17(A__SIEVE(a__sieve(a__from(mark(z0)))), MARK(sieve(from(z0)))) MARK(sieve(sieve(head(z0)))) -> c17(A__SIEVE(a__sieve(a__head(mark(z0)))), MARK(sieve(head(z0)))) MARK(sieve(sieve(tail(z0)))) -> c17(A__SIEVE(a__sieve(a__tail(mark(z0)))), MARK(sieve(tail(z0)))) MARK(sieve(sieve(if(z0, z1, z2)))) -> c17(A__SIEVE(a__sieve(a__if(mark(z0), z1, z2))), MARK(sieve(if(z0, z1, z2)))) MARK(sieve(sieve(filter(z0, z1)))) -> c17(A__SIEVE(a__sieve(a__filter(mark(z0), mark(z1)))), MARK(sieve(filter(z0, z1)))) MARK(sieve(sieve(s(z0)))) -> c17(A__SIEVE(a__sieve(s(mark(z0)))), MARK(sieve(s(z0)))) MARK(sieve(sieve(cons(z0, z1)))) -> c17(A__SIEVE(a__sieve(cons(mark(z0), z1))), MARK(sieve(cons(z0, z1)))) MARK(sieve(sieve(divides(z0, z1)))) -> c17(A__SIEVE(a__sieve(divides(mark(z0), mark(z1)))), MARK(sieve(divides(z0, z1)))) MARK(sieve(sieve(x0))) -> c17(MARK(sieve(x0))) MARK(sieve(sieve(0))) -> c17(A__SIEVE(a__sieve(0))) MARK(sieve(sieve(true))) -> c17(A__SIEVE(a__sieve(true))) MARK(sieve(sieve(false))) -> c17(A__SIEVE(a__sieve(false))) MARK(sieve(from(x0))) -> c17(A__SIEVE(cons(mark(mark(x0)), from(s(mark(x0))))), MARK(from(x0))) MARK(sieve(from(primes))) -> c17(A__SIEVE(a__from(a__primes)), MARK(from(primes))) MARK(sieve(from(sieve(z0)))) -> c17(A__SIEVE(a__from(a__sieve(mark(z0)))), MARK(from(sieve(z0)))) MARK(sieve(from(from(z0)))) -> c17(A__SIEVE(a__from(a__from(mark(z0)))), MARK(from(from(z0)))) MARK(sieve(from(head(z0)))) -> c17(A__SIEVE(a__from(a__head(mark(z0)))), MARK(from(head(z0)))) MARK(sieve(from(tail(z0)))) -> c17(A__SIEVE(a__from(a__tail(mark(z0)))), MARK(from(tail(z0)))) MARK(sieve(from(if(z0, z1, z2)))) -> c17(A__SIEVE(a__from(a__if(mark(z0), z1, z2))), MARK(from(if(z0, z1, z2)))) MARK(sieve(from(filter(z0, z1)))) -> c17(A__SIEVE(a__from(a__filter(mark(z0), mark(z1)))), MARK(from(filter(z0, z1)))) MARK(sieve(from(s(z0)))) -> c17(A__SIEVE(a__from(s(mark(z0)))), MARK(from(s(z0)))) MARK(sieve(from(0))) -> c17(A__SIEVE(a__from(0)), MARK(from(0))) MARK(sieve(from(cons(z0, z1)))) -> c17(A__SIEVE(a__from(cons(mark(z0), z1))), MARK(from(cons(z0, z1)))) MARK(sieve(from(true))) -> c17(A__SIEVE(a__from(true)), MARK(from(true))) MARK(sieve(from(false))) -> c17(A__SIEVE(a__from(false)), MARK(from(false))) MARK(sieve(from(divides(z0, z1)))) -> c17(A__SIEVE(a__from(divides(mark(z0), mark(z1)))), MARK(from(divides(z0, z1)))) MARK(sieve(from(x0))) -> c17(MARK(from(x0))) MARK(sieve(head(primes))) -> c17(A__SIEVE(a__head(a__primes)), MARK(head(primes))) MARK(sieve(head(sieve(z0)))) -> c17(A__SIEVE(a__head(a__sieve(mark(z0)))), MARK(head(sieve(z0)))) MARK(sieve(head(from(z0)))) -> c17(A__SIEVE(a__head(a__from(mark(z0)))), MARK(head(from(z0)))) MARK(sieve(head(head(z0)))) -> c17(A__SIEVE(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(sieve(head(tail(z0)))) -> c17(A__SIEVE(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(sieve(head(if(z0, z1, z2)))) -> c17(A__SIEVE(a__head(a__if(mark(z0), z1, z2))), MARK(head(if(z0, z1, z2)))) MARK(sieve(head(filter(z0, z1)))) -> c17(A__SIEVE(a__head(a__filter(mark(z0), mark(z1)))), MARK(head(filter(z0, z1)))) MARK(sieve(head(s(z0)))) -> 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MARK(head(filter(x0, true))) -> c19(A__HEAD(a__filter(mark(x0), true)), MARK(filter(x0, true))) MARK(head(filter(x0, false))) -> c19(A__HEAD(a__filter(mark(x0), false)), MARK(filter(x0, false))) MARK(head(filter(x0, divides(z0, z1)))) -> c19(A__HEAD(a__filter(mark(x0), divides(mark(z0), mark(z1)))), MARK(filter(x0, divides(z0, z1)))) MARK(head(filter(primes, x1))) -> c19(A__HEAD(a__filter(a__primes, mark(x1))), MARK(filter(primes, x1))) MARK(head(filter(sieve(z0), x1))) -> c19(A__HEAD(a__filter(a__sieve(mark(z0)), mark(x1))), MARK(filter(sieve(z0), x1))) MARK(head(filter(from(z0), x1))) -> c19(A__HEAD(a__filter(a__from(mark(z0)), mark(x1))), MARK(filter(from(z0), x1))) MARK(head(filter(head(z0), x1))) -> c19(A__HEAD(a__filter(a__head(mark(z0)), mark(x1))), MARK(filter(head(z0), x1))) MARK(head(filter(tail(z0), x1))) -> c19(A__HEAD(a__filter(a__tail(mark(z0)), mark(x1))), MARK(filter(tail(z0), x1))) MARK(head(filter(if(z0, z1, z2), x1))) -> c19(A__HEAD(a__filter(a__if(mark(z0), z1, z2), mark(x1))), MARK(filter(if(z0, z1, z2), x1))) MARK(head(filter(filter(z0, z1), x1))) -> c19(A__HEAD(a__filter(a__filter(mark(z0), mark(z1)), mark(x1))), MARK(filter(filter(z0, z1), x1))) MARK(head(filter(s(z0), x1))) -> c19(A__HEAD(a__filter(s(mark(z0)), mark(x1))), MARK(filter(s(z0), x1))) MARK(head(filter(0, x1))) -> c19(A__HEAD(a__filter(0, mark(x1))), MARK(filter(0, x1))) MARK(head(filter(cons(z0, z1), x1))) -> c19(A__HEAD(a__filter(cons(mark(z0), z1), mark(x1))), MARK(filter(cons(z0, z1), x1))) MARK(head(filter(true, x1))) -> c19(A__HEAD(a__filter(true, mark(x1))), MARK(filter(true, x1))) MARK(head(filter(false, x1))) -> c19(A__HEAD(a__filter(false, mark(x1))), MARK(filter(false, x1))) MARK(head(filter(divides(z0, z1), x1))) -> c19(A__HEAD(a__filter(divides(mark(z0), mark(z1)), mark(x1))), MARK(filter(divides(z0, z1), x1))) MARK(head(filter(x0, x1))) -> c19(MARK(filter(x0, x1))) MARK(tail(primes)) -> c20(A__TAIL(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(tail(primes)) -> c20(MARK(primes)) MARK(tail(sieve(primes))) -> c20(A__TAIL(a__sieve(a__primes)), MARK(sieve(primes))) MARK(tail(sieve(sieve(z0)))) -> c20(A__TAIL(a__sieve(a__sieve(mark(z0)))), MARK(sieve(sieve(z0)))) MARK(tail(sieve(from(z0)))) -> c20(A__TAIL(a__sieve(a__from(mark(z0)))), MARK(sieve(from(z0)))) MARK(tail(sieve(head(z0)))) -> c20(A__TAIL(a__sieve(a__head(mark(z0)))), MARK(sieve(head(z0)))) MARK(tail(sieve(tail(z0)))) -> c20(A__TAIL(a__sieve(a__tail(mark(z0)))), MARK(sieve(tail(z0)))) MARK(tail(sieve(if(z0, z1, z2)))) -> c20(A__TAIL(a__sieve(a__if(mark(z0), z1, z2))), MARK(sieve(if(z0, z1, z2)))) MARK(tail(sieve(filter(z0, z1)))) -> c20(A__TAIL(a__sieve(a__filter(mark(z0), mark(z1)))), MARK(sieve(filter(z0, z1)))) MARK(tail(sieve(s(z0)))) -> c20(A__TAIL(a__sieve(s(mark(z0)))), MARK(sieve(s(z0)))) MARK(tail(sieve(cons(z0, z1)))) -> c20(A__TAIL(a__sieve(cons(mark(z0), z1))), MARK(sieve(cons(z0, z1)))) MARK(tail(sieve(divides(z0, z1)))) -> c20(A__TAIL(a__sieve(divides(mark(z0), mark(z1)))), MARK(sieve(divides(z0, z1)))) MARK(tail(sieve(x0))) -> c20(MARK(sieve(x0))) MARK(tail(sieve(0))) -> c20(A__TAIL(a__sieve(0))) MARK(tail(sieve(true))) -> c20(A__TAIL(a__sieve(true))) MARK(tail(sieve(false))) -> c20(A__TAIL(a__sieve(false))) MARK(tail(from(x0))) -> c20(A__TAIL(cons(mark(mark(x0)), from(s(mark(x0))))), MARK(from(x0))) MARK(tail(from(primes))) -> c20(A__TAIL(a__from(a__primes)), MARK(from(primes))) MARK(tail(from(sieve(z0)))) -> c20(A__TAIL(a__from(a__sieve(mark(z0)))), MARK(from(sieve(z0)))) MARK(tail(from(from(z0)))) -> c20(A__TAIL(a__from(a__from(mark(z0)))), MARK(from(from(z0)))) MARK(tail(from(head(z0)))) -> c20(A__TAIL(a__from(a__head(mark(z0)))), MARK(from(head(z0)))) MARK(tail(from(tail(z0)))) -> c20(A__TAIL(a__from(a__tail(mark(z0)))), MARK(from(tail(z0)))) MARK(tail(from(if(z0, z1, z2)))) -> c20(A__TAIL(a__from(a__if(mark(z0), z1, z2))), MARK(from(if(z0, z1, z2)))) MARK(tail(from(filter(z0, z1)))) -> c20(A__TAIL(a__from(a__filter(mark(z0), mark(z1)))), MARK(from(filter(z0, z1)))) MARK(tail(from(s(z0)))) -> c20(A__TAIL(a__from(s(mark(z0)))), MARK(from(s(z0)))) MARK(tail(from(0))) -> c20(A__TAIL(a__from(0)), MARK(from(0))) MARK(tail(from(cons(z0, z1)))) -> c20(A__TAIL(a__from(cons(mark(z0), z1))), MARK(from(cons(z0, z1)))) MARK(tail(from(true))) -> c20(A__TAIL(a__from(true)), MARK(from(true))) MARK(tail(from(false))) -> c20(A__TAIL(a__from(false)), MARK(from(false))) MARK(tail(from(divides(z0, z1)))) -> c20(A__TAIL(a__from(divides(mark(z0), mark(z1)))), MARK(from(divides(z0, z1)))) MARK(tail(from(x0))) -> c20(MARK(from(x0))) MARK(tail(head(primes))) -> c20(A__TAIL(a__head(a__primes)), MARK(head(primes))) MARK(tail(head(sieve(z0)))) -> c20(A__TAIL(a__head(a__sieve(mark(z0)))), MARK(head(sieve(z0)))) MARK(tail(head(from(z0)))) -> c20(A__TAIL(a__head(a__from(mark(z0)))), MARK(head(from(z0)))) MARK(tail(head(head(z0)))) -> c20(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c20(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(if(z0, z1, z2)))) -> c20(A__TAIL(a__head(a__if(mark(z0), z1, z2))), MARK(head(if(z0, z1, z2)))) MARK(tail(head(filter(z0, z1)))) -> c20(A__TAIL(a__head(a__filter(mark(z0), mark(z1)))), MARK(head(filter(z0, z1)))) MARK(tail(head(s(z0)))) -> c20(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(cons(z0, z1)))) -> c20(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(divides(z0, z1)))) -> c20(A__TAIL(a__head(divides(mark(z0), mark(z1)))), MARK(head(divides(z0, z1)))) MARK(tail(head(x0))) -> c20(MARK(head(x0))) MARK(tail(head(0))) -> c20(A__TAIL(a__head(0))) MARK(tail(head(true))) -> c20(A__TAIL(a__head(true))) MARK(tail(head(false))) -> c20(A__TAIL(a__head(false))) MARK(tail(tail(primes))) -> c20(A__TAIL(a__tail(a__primes)), MARK(tail(primes))) MARK(tail(tail(sieve(z0)))) -> c20(A__TAIL(a__tail(a__sieve(mark(z0)))), MARK(tail(sieve(z0)))) MARK(tail(tail(from(z0)))) -> c20(A__TAIL(a__tail(a__from(mark(z0)))), MARK(tail(from(z0)))) MARK(tail(tail(head(z0)))) -> c20(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c20(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(if(z0, z1, z2)))) -> c20(A__TAIL(a__tail(a__if(mark(z0), z1, z2))), MARK(tail(if(z0, z1, z2)))) MARK(tail(tail(filter(z0, z1)))) -> c20(A__TAIL(a__tail(a__filter(mark(z0), mark(z1)))), MARK(tail(filter(z0, z1)))) MARK(tail(tail(s(z0)))) -> c20(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c20(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(divides(z0, z1)))) -> c20(A__TAIL(a__tail(divides(mark(z0), mark(z1)))), MARK(tail(divides(z0, z1)))) MARK(tail(tail(x0))) -> c20(MARK(tail(x0))) MARK(tail(tail(0))) -> c20(A__TAIL(a__tail(0))) MARK(tail(tail(true))) -> c20(A__TAIL(a__tail(true))) MARK(tail(tail(false))) -> c20(A__TAIL(a__tail(false))) MARK(tail(if(primes, x1, x2))) -> c20(A__TAIL(a__if(a__primes, x1, x2)), MARK(if(primes, x1, x2))) MARK(tail(if(sieve(z0), x1, x2))) -> c20(A__TAIL(a__if(a__sieve(mark(z0)), x1, x2)), MARK(if(sieve(z0), x1, x2))) MARK(tail(if(from(z0), x1, x2))) -> c20(A__TAIL(a__if(a__from(mark(z0)), x1, x2)), MARK(if(from(z0), x1, x2))) MARK(tail(if(head(z0), x1, x2))) -> c20(A__TAIL(a__if(a__head(mark(z0)), x1, x2)), MARK(if(head(z0), x1, x2))) MARK(tail(if(tail(z0), x1, x2))) -> c20(A__TAIL(a__if(a__tail(mark(z0)), x1, x2)), MARK(if(tail(z0), x1, x2))) MARK(tail(if(if(z0, z1, z2), x1, x2))) -> c20(A__TAIL(a__if(a__if(mark(z0), z1, z2), x1, x2)), MARK(if(if(z0, z1, z2), x1, x2))) MARK(tail(if(filter(z0, z1), x1, x2))) -> c20(A__TAIL(a__if(a__filter(mark(z0), mark(z1)), x1, x2)), MARK(if(filter(z0, z1), x1, x2))) MARK(tail(if(s(z0), x1, x2))) -> c20(A__TAIL(a__if(s(mark(z0)), x1, x2)), MARK(if(s(z0), x1, x2))) MARK(tail(if(cons(z0, z1), x1, x2))) -> c20(A__TAIL(a__if(cons(mark(z0), z1), x1, x2)), MARK(if(cons(z0, z1), x1, x2))) MARK(tail(if(true, x1, x2))) -> c20(A__TAIL(a__if(true, x1, x2)), MARK(if(true, x1, x2))) MARK(tail(if(false, x1, x2))) -> c20(A__TAIL(a__if(false, x1, x2)), MARK(if(false, x1, x2))) MARK(tail(if(divides(z0, z1), x1, x2))) -> c20(A__TAIL(a__if(divides(mark(z0), mark(z1)), x1, x2)), MARK(if(divides(z0, z1), x1, x2))) MARK(tail(if(x0, z1, z2))) -> c20(MARK(if(x0, z1, z2))) MARK(tail(if(0, x1, x2))) -> c20(A__TAIL(a__if(0, x1, x2))) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c18_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_1, c17_2, c17_1, c19_2, c19_1, c20_2, c20_1, c21_2, c21_1, c22_2, c22_1, c23_2, c23_1, c_2 ---------------------------------------- (179) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(tail(filter(z0, z1))) -> c20(A__TAIL(a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) by MARK(tail(filter(x0, x1))) -> c20(A__TAIL(filter(mark(x0), mark(x1))), MARK(filter(x0, x1))) MARK(tail(filter(x0, primes))) -> c20(A__TAIL(a__filter(mark(x0), a__primes)), MARK(filter(x0, primes))) MARK(tail(filter(x0, sieve(z0)))) -> c20(A__TAIL(a__filter(mark(x0), a__sieve(mark(z0)))), MARK(filter(x0, sieve(z0)))) MARK(tail(filter(x0, from(z0)))) -> c20(A__TAIL(a__filter(mark(x0), a__from(mark(z0)))), MARK(filter(x0, from(z0)))) MARK(tail(filter(x0, head(z0)))) -> c20(A__TAIL(a__filter(mark(x0), a__head(mark(z0)))), MARK(filter(x0, head(z0)))) MARK(tail(filter(x0, tail(z0)))) -> c20(A__TAIL(a__filter(mark(x0), a__tail(mark(z0)))), MARK(filter(x0, tail(z0)))) MARK(tail(filter(x0, if(z0, z1, z2)))) -> c20(A__TAIL(a__filter(mark(x0), a__if(mark(z0), z1, z2))), MARK(filter(x0, if(z0, z1, z2)))) MARK(tail(filter(x0, filter(z0, z1)))) -> c20(A__TAIL(a__filter(mark(x0), a__filter(mark(z0), mark(z1)))), MARK(filter(x0, filter(z0, z1)))) MARK(tail(filter(x0, s(z0)))) -> c20(A__TAIL(a__filter(mark(x0), s(mark(z0)))), MARK(filter(x0, s(z0)))) MARK(tail(filter(x0, 0))) -> c20(A__TAIL(a__filter(mark(x0), 0)), MARK(filter(x0, 0))) MARK(tail(filter(x0, cons(z0, z1)))) -> c20(A__TAIL(a__filter(mark(x0), cons(mark(z0), z1))), MARK(filter(x0, cons(z0, z1)))) MARK(tail(filter(x0, true))) -> c20(A__TAIL(a__filter(mark(x0), true)), MARK(filter(x0, true))) MARK(tail(filter(x0, false))) -> c20(A__TAIL(a__filter(mark(x0), false)), MARK(filter(x0, false))) MARK(tail(filter(x0, divides(z0, z1)))) -> c20(A__TAIL(a__filter(mark(x0), divides(mark(z0), mark(z1)))), MARK(filter(x0, divides(z0, z1)))) MARK(tail(filter(primes, x1))) -> c20(A__TAIL(a__filter(a__primes, mark(x1))), MARK(filter(primes, x1))) MARK(tail(filter(sieve(z0), x1))) -> c20(A__TAIL(a__filter(a__sieve(mark(z0)), mark(x1))), MARK(filter(sieve(z0), x1))) MARK(tail(filter(from(z0), x1))) -> c20(A__TAIL(a__filter(a__from(mark(z0)), mark(x1))), MARK(filter(from(z0), x1))) MARK(tail(filter(head(z0), x1))) -> c20(A__TAIL(a__filter(a__head(mark(z0)), mark(x1))), MARK(filter(head(z0), x1))) MARK(tail(filter(tail(z0), x1))) -> c20(A__TAIL(a__filter(a__tail(mark(z0)), mark(x1))), MARK(filter(tail(z0), x1))) MARK(tail(filter(if(z0, z1, z2), x1))) -> c20(A__TAIL(a__filter(a__if(mark(z0), z1, z2), mark(x1))), MARK(filter(if(z0, z1, z2), x1))) MARK(tail(filter(filter(z0, z1), x1))) -> c20(A__TAIL(a__filter(a__filter(mark(z0), mark(z1)), mark(x1))), MARK(filter(filter(z0, z1), x1))) MARK(tail(filter(s(z0), x1))) -> c20(A__TAIL(a__filter(s(mark(z0)), mark(x1))), MARK(filter(s(z0), x1))) MARK(tail(filter(0, x1))) -> c20(A__TAIL(a__filter(0, mark(x1))), MARK(filter(0, x1))) MARK(tail(filter(cons(z0, z1), x1))) -> c20(A__TAIL(a__filter(cons(mark(z0), z1), mark(x1))), MARK(filter(cons(z0, z1), x1))) MARK(tail(filter(true, x1))) -> c20(A__TAIL(a__filter(true, mark(x1))), MARK(filter(true, x1))) MARK(tail(filter(false, x1))) -> c20(A__TAIL(a__filter(false, mark(x1))), MARK(filter(false, x1))) MARK(tail(filter(divides(z0, z1), x1))) -> c20(A__TAIL(a__filter(divides(mark(z0), mark(z1)), mark(x1))), MARK(filter(divides(z0, z1), x1))) ---------------------------------------- (180) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(primes, x1, x2)) -> c21(A__IF(a__primes, x1, x2), MARK(primes)) MARK(if(sieve(z0), x1, x2)) -> c21(A__IF(a__sieve(mark(z0)), x1, x2), MARK(sieve(z0))) MARK(if(from(z0), x1, x2)) -> c21(A__IF(a__from(mark(z0)), x1, x2), MARK(from(z0))) MARK(if(head(z0), x1, x2)) -> c21(A__IF(a__head(mark(z0)), x1, x2), MARK(head(z0))) MARK(if(tail(z0), x1, x2)) -> c21(A__IF(a__tail(mark(z0)), x1, x2), MARK(tail(z0))) MARK(if(if(z0, z1, z2), x1, x2)) -> c21(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(filter(z0, z1), x1, x2)) -> c21(A__IF(a__filter(mark(z0), mark(z1)), x1, x2), MARK(filter(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c22(A__FILTER(mark(x0), a__primes), MARK(x0)) MARK(filter(x0, sieve(z0))) -> c22(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, from(z0))) -> c22(A__FILTER(mark(x0), a__from(mark(z0))), MARK(x0)) MARK(filter(x0, head(z0))) -> c22(A__FILTER(mark(x0), a__head(mark(z0))), MARK(x0)) MARK(filter(x0, tail(z0))) -> c22(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(x0)) MARK(filter(x0, if(z0, z1, z2))) -> c22(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0)) MARK(filter(x0, filter(z0, z1))) -> c22(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(x0)) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(x0, s(z0))) -> c22(MARK(x0)) MARK(filter(x0, 0)) -> c22(MARK(x0)) MARK(filter(x0, true)) -> c22(MARK(x0)) MARK(filter(x0, false)) -> c22(MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), x1)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c23(A__FILTER(mark(x0), a__primes), MARK(primes)) MARK(filter(x0, sieve(z0))) -> c23(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, from(z0))) -> c23(A__FILTER(mark(x0), a__from(mark(z0))), MARK(from(z0))) MARK(filter(x0, head(z0))) -> c23(A__FILTER(mark(x0), a__head(mark(z0))), MARK(head(z0))) MARK(filter(x0, tail(z0))) -> c23(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(tail(z0))) MARK(filter(x0, if(z0, z1, z2))) -> c23(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(filter(x0, filter(z0, z1))) -> c23(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(filter(x0, cons(z0, z1))) -> c23(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(primes, x1)) -> c23(A__FILTER(a__primes, mark(x1)), MARK(x1)) MARK(filter(sieve(z0), x1)) -> c23(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(from(z0), x1)) -> c23(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(head(z0), x1)) -> c23(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(tail(z0), x1)) -> c23(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(if(z0, z1, z2), x1)) -> c23(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1)) MARK(filter(filter(z0, z1), x1)) -> c23(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(x1)) MARK(filter(s(z0), x1)) -> c23(A__FILTER(s(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(x0, s(z0))) -> c23(MARK(s(z0))) MARK(filter(x0, divides(z0, z1))) -> c23(MARK(divides(z0, z1))) MARK(filter(0, x1)) -> c23(MARK(x1)) MARK(filter(cons(z0, z1), x1)) -> c23(MARK(x1)) MARK(filter(true, x1)) -> c23(MARK(x1)) MARK(filter(false, x1)) -> c23(MARK(x1)) MARK(filter(divides(z0, z1), x1)) -> c23(MARK(x1)) A__PRIMES -> c(A__SIEVE(cons(s(mark(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(sieve(primes)) -> c17(MARK(primes)) MARK(sieve(sieve(primes))) -> c17(A__SIEVE(a__sieve(a__primes)), MARK(sieve(primes))) MARK(sieve(sieve(sieve(z0)))) -> c17(A__SIEVE(a__sieve(a__sieve(mark(z0)))), MARK(sieve(sieve(z0)))) MARK(sieve(sieve(from(z0)))) -> c17(A__SIEVE(a__sieve(a__from(mark(z0)))), MARK(sieve(from(z0)))) MARK(sieve(sieve(head(z0)))) -> c17(A__SIEVE(a__sieve(a__head(mark(z0)))), MARK(sieve(head(z0)))) MARK(sieve(sieve(tail(z0)))) -> c17(A__SIEVE(a__sieve(a__tail(mark(z0)))), MARK(sieve(tail(z0)))) MARK(sieve(sieve(if(z0, z1, z2)))) -> c17(A__SIEVE(a__sieve(a__if(mark(z0), z1, z2))), MARK(sieve(if(z0, z1, z2)))) MARK(sieve(sieve(filter(z0, z1)))) -> c17(A__SIEVE(a__sieve(a__filter(mark(z0), mark(z1)))), MARK(sieve(filter(z0, z1)))) MARK(sieve(sieve(s(z0)))) -> c17(A__SIEVE(a__sieve(s(mark(z0)))), MARK(sieve(s(z0)))) MARK(sieve(sieve(cons(z0, z1)))) -> c17(A__SIEVE(a__sieve(cons(mark(z0), z1))), MARK(sieve(cons(z0, z1)))) MARK(sieve(sieve(divides(z0, z1)))) -> c17(A__SIEVE(a__sieve(divides(mark(z0), mark(z1)))), MARK(sieve(divides(z0, z1)))) MARK(sieve(sieve(x0))) -> c17(MARK(sieve(x0))) MARK(sieve(sieve(0))) -> c17(A__SIEVE(a__sieve(0))) MARK(sieve(sieve(true))) -> c17(A__SIEVE(a__sieve(true))) MARK(sieve(sieve(false))) -> c17(A__SIEVE(a__sieve(false))) MARK(sieve(from(x0))) -> c17(A__SIEVE(cons(mark(mark(x0)), from(s(mark(x0))))), MARK(from(x0))) MARK(sieve(from(primes))) -> c17(A__SIEVE(a__from(a__primes)), MARK(from(primes))) MARK(sieve(from(sieve(z0)))) -> c17(A__SIEVE(a__from(a__sieve(mark(z0)))), MARK(from(sieve(z0)))) MARK(sieve(from(from(z0)))) -> c17(A__SIEVE(a__from(a__from(mark(z0)))), MARK(from(from(z0)))) MARK(sieve(from(head(z0)))) -> c17(A__SIEVE(a__from(a__head(mark(z0)))), MARK(from(head(z0)))) MARK(sieve(from(tail(z0)))) -> c17(A__SIEVE(a__from(a__tail(mark(z0)))), MARK(from(tail(z0)))) MARK(sieve(from(if(z0, z1, z2)))) -> c17(A__SIEVE(a__from(a__if(mark(z0), z1, z2))), MARK(from(if(z0, z1, z2)))) MARK(sieve(from(filter(z0, z1)))) -> c17(A__SIEVE(a__from(a__filter(mark(z0), mark(z1)))), MARK(from(filter(z0, z1)))) MARK(sieve(from(s(z0)))) -> c17(A__SIEVE(a__from(s(mark(z0)))), MARK(from(s(z0)))) MARK(sieve(from(0))) -> c17(A__SIEVE(a__from(0)), MARK(from(0))) MARK(sieve(from(cons(z0, z1)))) -> c17(A__SIEVE(a__from(cons(mark(z0), z1))), MARK(from(cons(z0, z1)))) MARK(sieve(from(true))) -> c17(A__SIEVE(a__from(true)), MARK(from(true))) MARK(sieve(from(false))) -> c17(A__SIEVE(a__from(false)), MARK(from(false))) MARK(sieve(from(divides(z0, z1)))) -> c17(A__SIEVE(a__from(divides(mark(z0), mark(z1)))), MARK(from(divides(z0, z1)))) MARK(sieve(from(x0))) -> c17(MARK(from(x0))) MARK(sieve(head(primes))) -> c17(A__SIEVE(a__head(a__primes)), MARK(head(primes))) MARK(sieve(head(sieve(z0)))) -> c17(A__SIEVE(a__head(a__sieve(mark(z0)))), MARK(head(sieve(z0)))) MARK(sieve(head(from(z0)))) -> c17(A__SIEVE(a__head(a__from(mark(z0)))), MARK(head(from(z0)))) MARK(sieve(head(head(z0)))) -> c17(A__SIEVE(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(sieve(head(tail(z0)))) -> c17(A__SIEVE(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(sieve(head(if(z0, z1, z2)))) -> c17(A__SIEVE(a__head(a__if(mark(z0), z1, z2))), MARK(head(if(z0, z1, z2)))) MARK(sieve(head(filter(z0, z1)))) -> c17(A__SIEVE(a__head(a__filter(mark(z0), mark(z1)))), MARK(head(filter(z0, z1)))) MARK(sieve(head(s(z0)))) -> c17(A__SIEVE(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(sieve(head(cons(z0, z1)))) -> c17(A__SIEVE(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(sieve(head(divides(z0, z1)))) -> c17(A__SIEVE(a__head(divides(mark(z0), mark(z1)))), MARK(head(divides(z0, z1)))) MARK(sieve(head(x0))) -> c17(MARK(head(x0))) MARK(sieve(head(0))) -> c17(A__SIEVE(a__head(0))) MARK(sieve(head(true))) -> c17(A__SIEVE(a__head(true))) MARK(sieve(head(false))) -> c17(A__SIEVE(a__head(false))) MARK(sieve(tail(primes))) -> c17(A__SIEVE(a__tail(a__primes)), MARK(tail(primes))) MARK(sieve(tail(sieve(z0)))) -> c17(A__SIEVE(a__tail(a__sieve(mark(z0)))), MARK(tail(sieve(z0)))) MARK(sieve(tail(from(z0)))) -> c17(A__SIEVE(a__tail(a__from(mark(z0)))), MARK(tail(from(z0)))) MARK(sieve(tail(head(z0)))) -> c17(A__SIEVE(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(sieve(tail(tail(z0)))) -> c17(A__SIEVE(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(sieve(tail(if(z0, z1, z2)))) -> c17(A__SIEVE(a__tail(a__if(mark(z0), z1, z2))), MARK(tail(if(z0, z1, z2)))) MARK(sieve(tail(filter(z0, z1)))) -> c17(A__SIEVE(a__tail(a__filter(mark(z0), mark(z1)))), MARK(tail(filter(z0, z1)))) MARK(sieve(tail(s(z0)))) -> c17(A__SIEVE(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(sieve(tail(cons(z0, z1)))) -> c17(A__SIEVE(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(sieve(tail(divides(z0, z1)))) -> c17(A__SIEVE(a__tail(divides(mark(z0), mark(z1)))), MARK(tail(divides(z0, z1)))) MARK(sieve(tail(x0))) -> c17(MARK(tail(x0))) MARK(sieve(tail(0))) -> c17(A__SIEVE(a__tail(0))) MARK(sieve(tail(true))) -> c17(A__SIEVE(a__tail(true))) MARK(sieve(tail(false))) -> c17(A__SIEVE(a__tail(false))) MARK(sieve(if(primes, x1, x2))) -> c17(A__SIEVE(a__if(a__primes, x1, x2)), MARK(if(primes, x1, x2))) MARK(sieve(if(sieve(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__sieve(mark(z0)), x1, x2)), MARK(if(sieve(z0), x1, x2))) MARK(sieve(if(from(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__from(mark(z0)), x1, x2)), MARK(if(from(z0), x1, x2))) MARK(sieve(if(head(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__head(mark(z0)), x1, x2)), MARK(if(head(z0), x1, x2))) MARK(sieve(if(tail(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__tail(mark(z0)), x1, x2)), MARK(if(tail(z0), x1, x2))) MARK(sieve(if(if(z0, z1, z2), x1, x2))) -> c17(A__SIEVE(a__if(a__if(mark(z0), z1, z2), x1, x2)), MARK(if(if(z0, z1, 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c20(A__TAIL(a__filter(a__from(mark(z0)), mark(x1))), MARK(filter(from(z0), x1))) MARK(tail(filter(head(z0), x1))) -> c20(A__TAIL(a__filter(a__head(mark(z0)), mark(x1))), MARK(filter(head(z0), x1))) MARK(tail(filter(tail(z0), x1))) -> c20(A__TAIL(a__filter(a__tail(mark(z0)), mark(x1))), MARK(filter(tail(z0), x1))) MARK(tail(filter(if(z0, z1, z2), x1))) -> c20(A__TAIL(a__filter(a__if(mark(z0), z1, z2), mark(x1))), MARK(filter(if(z0, z1, z2), x1))) MARK(tail(filter(filter(z0, z1), x1))) -> c20(A__TAIL(a__filter(a__filter(mark(z0), mark(z1)), mark(x1))), MARK(filter(filter(z0, z1), x1))) MARK(tail(filter(s(z0), x1))) -> c20(A__TAIL(a__filter(s(mark(z0)), mark(x1))), MARK(filter(s(z0), x1))) MARK(tail(filter(0, x1))) -> c20(A__TAIL(a__filter(0, mark(x1))), MARK(filter(0, x1))) MARK(tail(filter(cons(z0, z1), x1))) -> c20(A__TAIL(a__filter(cons(mark(z0), z1), mark(x1))), MARK(filter(cons(z0, z1), x1))) MARK(tail(filter(true, x1))) -> c20(A__TAIL(a__filter(true, mark(x1))), MARK(filter(true, x1))) MARK(tail(filter(false, x1))) -> c20(A__TAIL(a__filter(false, mark(x1))), MARK(filter(false, x1))) MARK(tail(filter(divides(z0, z1), x1))) -> c20(A__TAIL(a__filter(divides(mark(z0), mark(z1)), mark(x1))), MARK(filter(divides(z0, z1), x1))) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c18_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_1, c17_2, c17_1, c19_2, c19_1, c20_2, c20_1, c21_2, c21_1, c22_2, c22_1, c23_2, c23_1, c_2 ---------------------------------------- (181) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (182) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(primes, x1, x2)) -> c21(A__IF(a__primes, x1, x2), MARK(primes)) MARK(if(sieve(z0), x1, x2)) -> c21(A__IF(a__sieve(mark(z0)), x1, x2), MARK(sieve(z0))) MARK(if(from(z0), x1, x2)) -> c21(A__IF(a__from(mark(z0)), x1, x2), MARK(from(z0))) MARK(if(head(z0), x1, x2)) -> c21(A__IF(a__head(mark(z0)), x1, x2), MARK(head(z0))) MARK(if(tail(z0), x1, x2)) -> c21(A__IF(a__tail(mark(z0)), x1, x2), MARK(tail(z0))) MARK(if(if(z0, z1, z2), x1, x2)) -> c21(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(filter(z0, z1), x1, x2)) -> c21(A__IF(a__filter(mark(z0), mark(z1)), x1, x2), MARK(filter(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c22(A__FILTER(mark(x0), a__primes), MARK(x0)) MARK(filter(x0, sieve(z0))) -> c22(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, from(z0))) -> c22(A__FILTER(mark(x0), a__from(mark(z0))), MARK(x0)) MARK(filter(x0, head(z0))) -> c22(A__FILTER(mark(x0), a__head(mark(z0))), MARK(x0)) MARK(filter(x0, tail(z0))) -> c22(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(x0)) MARK(filter(x0, if(z0, z1, z2))) -> c22(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0)) MARK(filter(x0, filter(z0, z1))) -> c22(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(x0)) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(x0, s(z0))) -> c22(MARK(x0)) MARK(filter(x0, 0)) -> c22(MARK(x0)) MARK(filter(x0, true)) -> c22(MARK(x0)) MARK(filter(x0, false)) -> c22(MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), x1)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c23(A__FILTER(mark(x0), a__primes), MARK(primes)) MARK(filter(x0, sieve(z0))) -> c23(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, from(z0))) -> c23(A__FILTER(mark(x0), a__from(mark(z0))), MARK(from(z0))) MARK(filter(x0, head(z0))) -> c23(A__FILTER(mark(x0), a__head(mark(z0))), MARK(head(z0))) MARK(filter(x0, tail(z0))) -> c23(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(tail(z0))) MARK(filter(x0, if(z0, z1, z2))) -> c23(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(filter(x0, filter(z0, z1))) -> c23(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(filter(x0, cons(z0, z1))) -> c23(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(primes, x1)) -> c23(A__FILTER(a__primes, mark(x1)), MARK(x1)) MARK(filter(sieve(z0), x1)) -> c23(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(from(z0), x1)) -> c23(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(head(z0), x1)) -> c23(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(tail(z0), x1)) -> c23(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(if(z0, z1, z2), x1)) -> c23(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1)) MARK(filter(filter(z0, z1), x1)) -> c23(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(x1)) MARK(filter(s(z0), x1)) -> c23(A__FILTER(s(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(x0, s(z0))) -> c23(MARK(s(z0))) MARK(filter(x0, divides(z0, z1))) -> c23(MARK(divides(z0, z1))) MARK(filter(0, x1)) -> c23(MARK(x1)) MARK(filter(cons(z0, z1), x1)) -> c23(MARK(x1)) MARK(filter(true, x1)) -> c23(MARK(x1)) MARK(filter(false, x1)) -> c23(MARK(x1)) MARK(filter(divides(z0, z1), x1)) -> c23(MARK(x1)) A__PRIMES -> c(A__SIEVE(cons(s(mark(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(sieve(primes)) -> c17(MARK(primes)) MARK(sieve(sieve(primes))) -> c17(A__SIEVE(a__sieve(a__primes)), MARK(sieve(primes))) MARK(sieve(sieve(sieve(z0)))) -> c17(A__SIEVE(a__sieve(a__sieve(mark(z0)))), MARK(sieve(sieve(z0)))) MARK(sieve(sieve(from(z0)))) -> c17(A__SIEVE(a__sieve(a__from(mark(z0)))), MARK(sieve(from(z0)))) MARK(sieve(sieve(head(z0)))) -> c17(A__SIEVE(a__sieve(a__head(mark(z0)))), MARK(sieve(head(z0)))) MARK(sieve(sieve(tail(z0)))) -> c17(A__SIEVE(a__sieve(a__tail(mark(z0)))), MARK(sieve(tail(z0)))) MARK(sieve(sieve(if(z0, z1, z2)))) -> c17(A__SIEVE(a__sieve(a__if(mark(z0), z1, z2))), MARK(sieve(if(z0, z1, z2)))) MARK(sieve(sieve(filter(z0, z1)))) -> c17(A__SIEVE(a__sieve(a__filter(mark(z0), mark(z1)))), MARK(sieve(filter(z0, z1)))) MARK(sieve(sieve(s(z0)))) -> c17(A__SIEVE(a__sieve(s(mark(z0)))), MARK(sieve(s(z0)))) MARK(sieve(sieve(cons(z0, z1)))) -> c17(A__SIEVE(a__sieve(cons(mark(z0), z1))), MARK(sieve(cons(z0, z1)))) MARK(sieve(sieve(divides(z0, z1)))) -> c17(A__SIEVE(a__sieve(divides(mark(z0), mark(z1)))), MARK(sieve(divides(z0, z1)))) MARK(sieve(sieve(x0))) -> c17(MARK(sieve(x0))) MARK(sieve(sieve(0))) -> c17(A__SIEVE(a__sieve(0))) MARK(sieve(sieve(true))) -> c17(A__SIEVE(a__sieve(true))) MARK(sieve(sieve(false))) -> c17(A__SIEVE(a__sieve(false))) MARK(sieve(from(x0))) -> c17(A__SIEVE(cons(mark(mark(x0)), from(s(mark(x0))))), MARK(from(x0))) MARK(sieve(from(primes))) -> c17(A__SIEVE(a__from(a__primes)), MARK(from(primes))) MARK(sieve(from(sieve(z0)))) -> c17(A__SIEVE(a__from(a__sieve(mark(z0)))), MARK(from(sieve(z0)))) MARK(sieve(from(from(z0)))) -> c17(A__SIEVE(a__from(a__from(mark(z0)))), MARK(from(from(z0)))) MARK(sieve(from(head(z0)))) -> c17(A__SIEVE(a__from(a__head(mark(z0)))), MARK(from(head(z0)))) MARK(sieve(from(tail(z0)))) -> c17(A__SIEVE(a__from(a__tail(mark(z0)))), MARK(from(tail(z0)))) MARK(sieve(from(if(z0, z1, z2)))) -> c17(A__SIEVE(a__from(a__if(mark(z0), z1, z2))), MARK(from(if(z0, z1, z2)))) MARK(sieve(from(filter(z0, z1)))) -> c17(A__SIEVE(a__from(a__filter(mark(z0), mark(z1)))), MARK(from(filter(z0, z1)))) MARK(sieve(from(s(z0)))) -> c17(A__SIEVE(a__from(s(mark(z0)))), MARK(from(s(z0)))) MARK(sieve(from(0))) -> c17(A__SIEVE(a__from(0)), MARK(from(0))) MARK(sieve(from(cons(z0, z1)))) -> c17(A__SIEVE(a__from(cons(mark(z0), z1))), MARK(from(cons(z0, z1)))) MARK(sieve(from(true))) -> c17(A__SIEVE(a__from(true)), MARK(from(true))) MARK(sieve(from(false))) -> c17(A__SIEVE(a__from(false)), MARK(from(false))) MARK(sieve(from(divides(z0, z1)))) -> c17(A__SIEVE(a__from(divides(mark(z0), mark(z1)))), MARK(from(divides(z0, z1)))) MARK(sieve(from(x0))) -> c17(MARK(from(x0))) MARK(sieve(head(primes))) -> c17(A__SIEVE(a__head(a__primes)), MARK(head(primes))) MARK(sieve(head(sieve(z0)))) -> c17(A__SIEVE(a__head(a__sieve(mark(z0)))), MARK(head(sieve(z0)))) MARK(sieve(head(from(z0)))) -> c17(A__SIEVE(a__head(a__from(mark(z0)))), MARK(head(from(z0)))) MARK(sieve(head(head(z0)))) -> c17(A__SIEVE(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(sieve(head(tail(z0)))) -> c17(A__SIEVE(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(sieve(head(if(z0, z1, z2)))) -> c17(A__SIEVE(a__head(a__if(mark(z0), z1, z2))), MARK(head(if(z0, z1, z2)))) MARK(sieve(head(filter(z0, z1)))) -> c17(A__SIEVE(a__head(a__filter(mark(z0), mark(z1)))), MARK(head(filter(z0, z1)))) MARK(sieve(head(s(z0)))) -> c17(A__SIEVE(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(sieve(head(cons(z0, z1)))) -> c17(A__SIEVE(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(sieve(head(divides(z0, z1)))) -> c17(A__SIEVE(a__head(divides(mark(z0), mark(z1)))), MARK(head(divides(z0, z1)))) MARK(sieve(head(x0))) -> c17(MARK(head(x0))) MARK(sieve(head(0))) -> c17(A__SIEVE(a__head(0))) MARK(sieve(head(true))) -> c17(A__SIEVE(a__head(true))) MARK(sieve(head(false))) -> c17(A__SIEVE(a__head(false))) MARK(sieve(tail(primes))) -> c17(A__SIEVE(a__tail(a__primes)), MARK(tail(primes))) MARK(sieve(tail(sieve(z0)))) -> c17(A__SIEVE(a__tail(a__sieve(mark(z0)))), MARK(tail(sieve(z0)))) MARK(sieve(tail(from(z0)))) -> c17(A__SIEVE(a__tail(a__from(mark(z0)))), MARK(tail(from(z0)))) MARK(sieve(tail(head(z0)))) -> c17(A__SIEVE(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(sieve(tail(tail(z0)))) -> c17(A__SIEVE(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(sieve(tail(if(z0, z1, z2)))) -> c17(A__SIEVE(a__tail(a__if(mark(z0), z1, z2))), MARK(tail(if(z0, z1, z2)))) MARK(sieve(tail(filter(z0, z1)))) -> c17(A__SIEVE(a__tail(a__filter(mark(z0), mark(z1)))), MARK(tail(filter(z0, z1)))) MARK(sieve(tail(s(z0)))) -> c17(A__SIEVE(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(sieve(tail(cons(z0, z1)))) -> c17(A__SIEVE(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(sieve(tail(divides(z0, z1)))) -> c17(A__SIEVE(a__tail(divides(mark(z0), mark(z1)))), MARK(tail(divides(z0, z1)))) MARK(sieve(tail(x0))) -> c17(MARK(tail(x0))) MARK(sieve(tail(0))) -> c17(A__SIEVE(a__tail(0))) MARK(sieve(tail(true))) -> c17(A__SIEVE(a__tail(true))) MARK(sieve(tail(false))) -> c17(A__SIEVE(a__tail(false))) MARK(sieve(if(primes, x1, x2))) -> c17(A__SIEVE(a__if(a__primes, x1, x2)), MARK(if(primes, x1, x2))) MARK(sieve(if(sieve(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__sieve(mark(z0)), x1, x2)), MARK(if(sieve(z0), x1, x2))) MARK(sieve(if(from(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__from(mark(z0)), x1, x2)), MARK(if(from(z0), x1, x2))) MARK(sieve(if(head(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__head(mark(z0)), x1, x2)), MARK(if(head(z0), x1, x2))) MARK(sieve(if(tail(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__tail(mark(z0)), x1, x2)), MARK(if(tail(z0), x1, x2))) MARK(sieve(if(if(z0, z1, z2), x1, x2))) -> c17(A__SIEVE(a__if(a__if(mark(z0), z1, z2), x1, x2)), MARK(if(if(z0, z1, z2), x1, x2))) MARK(sieve(if(filter(z0, z1), x1, x2))) -> c17(A__SIEVE(a__if(a__filter(mark(z0), mark(z1)), x1, x2)), MARK(if(filter(z0, z1), x1, x2))) MARK(sieve(if(s(z0), x1, x2))) -> c17(A__SIEVE(a__if(s(mark(z0)), x1, x2)), MARK(if(s(z0), x1, x2))) MARK(sieve(if(cons(z0, z1), x1, x2))) -> c17(A__SIEVE(a__if(cons(mark(z0), z1), x1, x2)), MARK(if(cons(z0, z1), x1, x2))) MARK(sieve(if(true, x1, x2))) -> c17(A__SIEVE(a__if(true, x1, x2)), MARK(if(true, x1, x2))) 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MARK(filter(s(z0), x1)) -> c23(A__FILTER(s(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(x0, s(z0))) -> c23(MARK(s(z0))) MARK(filter(x0, divides(z0, z1))) -> c23(MARK(divides(z0, z1))) MARK(filter(0, x1)) -> c23(MARK(x1)) MARK(filter(cons(z0, z1), x1)) -> c23(MARK(x1)) MARK(filter(true, x1)) -> c23(MARK(x1)) MARK(filter(false, x1)) -> c23(MARK(x1)) MARK(filter(divides(z0, z1), x1)) -> c23(MARK(x1)) A__PRIMES -> c(A__SIEVE(cons(s(mark(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(sieve(primes)) -> c17(MARK(primes)) MARK(sieve(sieve(primes))) -> c17(A__SIEVE(a__sieve(a__primes)), MARK(sieve(primes))) MARK(sieve(sieve(sieve(z0)))) -> c17(A__SIEVE(a__sieve(a__sieve(mark(z0)))), MARK(sieve(sieve(z0)))) MARK(sieve(sieve(from(z0)))) -> c17(A__SIEVE(a__sieve(a__from(mark(z0)))), MARK(sieve(from(z0)))) MARK(sieve(sieve(head(z0)))) -> c17(A__SIEVE(a__sieve(a__head(mark(z0)))), MARK(sieve(head(z0)))) 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MARK(tail(primes))) MARK(tail(tail(sieve(z0)))) -> c20(A__TAIL(a__tail(a__sieve(mark(z0)))), MARK(tail(sieve(z0)))) MARK(tail(tail(from(z0)))) -> c20(A__TAIL(a__tail(a__from(mark(z0)))), MARK(tail(from(z0)))) MARK(tail(tail(head(z0)))) -> c20(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c20(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(if(z0, z1, z2)))) -> c20(A__TAIL(a__tail(a__if(mark(z0), z1, z2))), MARK(tail(if(z0, z1, z2)))) MARK(tail(tail(filter(z0, z1)))) -> c20(A__TAIL(a__tail(a__filter(mark(z0), mark(z1)))), MARK(tail(filter(z0, z1)))) MARK(tail(tail(s(z0)))) -> c20(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c20(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(divides(z0, z1)))) -> c20(A__TAIL(a__tail(divides(mark(z0), mark(z1)))), MARK(tail(divides(z0, z1)))) MARK(tail(tail(x0))) -> c20(MARK(tail(x0))) MARK(tail(tail(0))) -> 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c20(A__TAIL(a__if(s(mark(z0)), x1, x2)), MARK(if(s(z0), x1, x2))) MARK(tail(if(cons(z0, z1), x1, x2))) -> c20(A__TAIL(a__if(cons(mark(z0), z1), x1, x2)), MARK(if(cons(z0, z1), x1, x2))) MARK(tail(if(true, x1, x2))) -> c20(A__TAIL(a__if(true, x1, x2)), MARK(if(true, x1, x2))) MARK(tail(if(false, x1, x2))) -> c20(A__TAIL(a__if(false, x1, x2)), MARK(if(false, x1, x2))) MARK(tail(if(divides(z0, z1), x1, x2))) -> c20(A__TAIL(a__if(divides(mark(z0), mark(z1)), x1, x2)), MARK(if(divides(z0, z1), x1, x2))) MARK(tail(if(x0, z1, z2))) -> c20(MARK(if(x0, z1, z2))) MARK(tail(if(0, x1, x2))) -> c20(A__TAIL(a__if(0, x1, x2))) MARK(tail(filter(x0, primes))) -> c20(A__TAIL(a__filter(mark(x0), a__primes)), MARK(filter(x0, primes))) MARK(tail(filter(x0, sieve(z0)))) -> c20(A__TAIL(a__filter(mark(x0), a__sieve(mark(z0)))), MARK(filter(x0, sieve(z0)))) MARK(tail(filter(x0, from(z0)))) -> c20(A__TAIL(a__filter(mark(x0), a__from(mark(z0)))), MARK(filter(x0, from(z0)))) MARK(tail(filter(x0, head(z0)))) -> 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MARK(tail(filter(x0, divides(z0, z1)))) -> c20(A__TAIL(a__filter(mark(x0), divides(mark(z0), mark(z1)))), MARK(filter(x0, divides(z0, z1)))) MARK(tail(filter(primes, x1))) -> c20(A__TAIL(a__filter(a__primes, mark(x1))), MARK(filter(primes, x1))) MARK(tail(filter(sieve(z0), x1))) -> c20(A__TAIL(a__filter(a__sieve(mark(z0)), mark(x1))), MARK(filter(sieve(z0), x1))) MARK(tail(filter(from(z0), x1))) -> c20(A__TAIL(a__filter(a__from(mark(z0)), mark(x1))), MARK(filter(from(z0), x1))) MARK(tail(filter(head(z0), x1))) -> c20(A__TAIL(a__filter(a__head(mark(z0)), mark(x1))), MARK(filter(head(z0), x1))) MARK(tail(filter(tail(z0), x1))) -> c20(A__TAIL(a__filter(a__tail(mark(z0)), mark(x1))), MARK(filter(tail(z0), x1))) MARK(tail(filter(if(z0, z1, z2), x1))) -> c20(A__TAIL(a__filter(a__if(mark(z0), z1, z2), mark(x1))), MARK(filter(if(z0, z1, z2), x1))) MARK(tail(filter(filter(z0, z1), x1))) -> c20(A__TAIL(a__filter(a__filter(mark(z0), mark(z1)), mark(x1))), MARK(filter(filter(z0, z1), x1))) MARK(tail(filter(s(z0), x1))) -> c20(A__TAIL(a__filter(s(mark(z0)), mark(x1))), MARK(filter(s(z0), x1))) MARK(tail(filter(0, x1))) -> c20(A__TAIL(a__filter(0, mark(x1))), MARK(filter(0, x1))) MARK(tail(filter(cons(z0, z1), x1))) -> c20(A__TAIL(a__filter(cons(mark(z0), z1), mark(x1))), MARK(filter(cons(z0, z1), x1))) MARK(tail(filter(true, x1))) -> c20(A__TAIL(a__filter(true, mark(x1))), MARK(filter(true, x1))) MARK(tail(filter(false, x1))) -> c20(A__TAIL(a__filter(false, mark(x1))), MARK(filter(false, x1))) MARK(tail(filter(divides(z0, z1), x1))) -> c20(A__TAIL(a__filter(divides(mark(z0), mark(z1)), mark(x1))), MARK(filter(divides(z0, z1), x1))) MARK(tail(filter(x0, x1))) -> c20(MARK(filter(x0, x1))) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c18_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_1, c17_2, c17_1, c19_2, c19_1, c20_2, c20_1, c21_2, c21_1, c22_2, c22_1, c23_2, c23_1, c_2 ---------------------------------------- (183) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(if(primes, x1, x2)) -> c21(A__IF(a__primes, x1, x2), MARK(primes)) by MARK(if(primes, x0, x1)) -> c21(A__IF(a__sieve(a__from(s(s(0)))), x0, x1), MARK(primes)) MARK(if(primes, x0, x1)) -> c21(A__IF(primes, x0, x1), MARK(primes)) ---------------------------------------- (184) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(sieve(z0), x1, x2)) -> c21(A__IF(a__sieve(mark(z0)), x1, x2), MARK(sieve(z0))) MARK(if(from(z0), x1, x2)) -> c21(A__IF(a__from(mark(z0)), x1, x2), MARK(from(z0))) MARK(if(head(z0), x1, x2)) -> c21(A__IF(a__head(mark(z0)), x1, x2), MARK(head(z0))) MARK(if(tail(z0), x1, x2)) -> c21(A__IF(a__tail(mark(z0)), x1, x2), MARK(tail(z0))) MARK(if(if(z0, z1, z2), x1, x2)) -> c21(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(filter(z0, z1), x1, x2)) -> c21(A__IF(a__filter(mark(z0), mark(z1)), x1, x2), MARK(filter(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c22(A__FILTER(mark(x0), a__primes), MARK(x0)) MARK(filter(x0, sieve(z0))) -> c22(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, from(z0))) -> c22(A__FILTER(mark(x0), a__from(mark(z0))), MARK(x0)) MARK(filter(x0, head(z0))) -> c22(A__FILTER(mark(x0), a__head(mark(z0))), MARK(x0)) MARK(filter(x0, tail(z0))) -> c22(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(x0)) MARK(filter(x0, if(z0, z1, z2))) -> c22(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0)) MARK(filter(x0, filter(z0, z1))) -> c22(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(x0)) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(x0, s(z0))) -> c22(MARK(x0)) MARK(filter(x0, 0)) -> c22(MARK(x0)) MARK(filter(x0, true)) -> c22(MARK(x0)) MARK(filter(x0, false)) -> c22(MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), x1)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c23(A__FILTER(mark(x0), a__primes), MARK(primes)) MARK(filter(x0, sieve(z0))) -> c23(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, from(z0))) -> c23(A__FILTER(mark(x0), a__from(mark(z0))), MARK(from(z0))) MARK(filter(x0, head(z0))) -> c23(A__FILTER(mark(x0), a__head(mark(z0))), MARK(head(z0))) MARK(filter(x0, tail(z0))) -> c23(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(tail(z0))) MARK(filter(x0, if(z0, z1, z2))) -> c23(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(filter(x0, filter(z0, z1))) -> c23(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(filter(x0, cons(z0, z1))) -> c23(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(primes, x1)) -> c23(A__FILTER(a__primes, mark(x1)), MARK(x1)) MARK(filter(sieve(z0), x1)) -> c23(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(from(z0), x1)) -> c23(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(head(z0), x1)) -> c23(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(tail(z0), x1)) -> c23(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(if(z0, z1, z2), x1)) -> c23(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1)) MARK(filter(filter(z0, z1), x1)) -> c23(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(x1)) MARK(filter(s(z0), x1)) -> c23(A__FILTER(s(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(x0, s(z0))) -> c23(MARK(s(z0))) MARK(filter(x0, divides(z0, z1))) -> c23(MARK(divides(z0, z1))) MARK(filter(0, x1)) -> c23(MARK(x1)) MARK(filter(cons(z0, z1), x1)) -> c23(MARK(x1)) MARK(filter(true, x1)) -> c23(MARK(x1)) MARK(filter(false, x1)) -> c23(MARK(x1)) MARK(filter(divides(z0, z1), x1)) -> c23(MARK(x1)) A__PRIMES -> c(A__SIEVE(cons(s(mark(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(sieve(primes)) -> c17(MARK(primes)) MARK(sieve(sieve(primes))) -> c17(A__SIEVE(a__sieve(a__primes)), MARK(sieve(primes))) MARK(sieve(sieve(sieve(z0)))) -> c17(A__SIEVE(a__sieve(a__sieve(mark(z0)))), MARK(sieve(sieve(z0)))) MARK(sieve(sieve(from(z0)))) -> c17(A__SIEVE(a__sieve(a__from(mark(z0)))), MARK(sieve(from(z0)))) MARK(sieve(sieve(head(z0)))) -> c17(A__SIEVE(a__sieve(a__head(mark(z0)))), MARK(sieve(head(z0)))) MARK(sieve(sieve(tail(z0)))) -> c17(A__SIEVE(a__sieve(a__tail(mark(z0)))), MARK(sieve(tail(z0)))) MARK(sieve(sieve(if(z0, z1, z2)))) -> c17(A__SIEVE(a__sieve(a__if(mark(z0), z1, z2))), MARK(sieve(if(z0, z1, z2)))) MARK(sieve(sieve(filter(z0, z1)))) -> c17(A__SIEVE(a__sieve(a__filter(mark(z0), mark(z1)))), MARK(sieve(filter(z0, z1)))) MARK(sieve(sieve(s(z0)))) -> c17(A__SIEVE(a__sieve(s(mark(z0)))), MARK(sieve(s(z0)))) MARK(sieve(sieve(cons(z0, z1)))) -> c17(A__SIEVE(a__sieve(cons(mark(z0), z1))), MARK(sieve(cons(z0, z1)))) MARK(sieve(sieve(divides(z0, z1)))) -> c17(A__SIEVE(a__sieve(divides(mark(z0), mark(z1)))), MARK(sieve(divides(z0, z1)))) MARK(sieve(sieve(x0))) -> c17(MARK(sieve(x0))) MARK(sieve(sieve(0))) -> c17(A__SIEVE(a__sieve(0))) MARK(sieve(sieve(true))) -> c17(A__SIEVE(a__sieve(true))) MARK(sieve(sieve(false))) -> c17(A__SIEVE(a__sieve(false))) MARK(sieve(from(x0))) -> c17(A__SIEVE(cons(mark(mark(x0)), from(s(mark(x0))))), MARK(from(x0))) MARK(sieve(from(primes))) -> c17(A__SIEVE(a__from(a__primes)), MARK(from(primes))) MARK(sieve(from(sieve(z0)))) -> c17(A__SIEVE(a__from(a__sieve(mark(z0)))), MARK(from(sieve(z0)))) MARK(sieve(from(from(z0)))) -> c17(A__SIEVE(a__from(a__from(mark(z0)))), MARK(from(from(z0)))) MARK(sieve(from(head(z0)))) -> c17(A__SIEVE(a__from(a__head(mark(z0)))), MARK(from(head(z0)))) MARK(sieve(from(tail(z0)))) -> c17(A__SIEVE(a__from(a__tail(mark(z0)))), MARK(from(tail(z0)))) MARK(sieve(from(if(z0, z1, z2)))) -> c17(A__SIEVE(a__from(a__if(mark(z0), z1, z2))), MARK(from(if(z0, z1, z2)))) MARK(sieve(from(filter(z0, z1)))) -> c17(A__SIEVE(a__from(a__filter(mark(z0), mark(z1)))), MARK(from(filter(z0, z1)))) MARK(sieve(from(s(z0)))) -> c17(A__SIEVE(a__from(s(mark(z0)))), MARK(from(s(z0)))) MARK(sieve(from(0))) -> c17(A__SIEVE(a__from(0)), MARK(from(0))) MARK(sieve(from(cons(z0, z1)))) -> c17(A__SIEVE(a__from(cons(mark(z0), z1))), MARK(from(cons(z0, z1)))) MARK(sieve(from(true))) -> c17(A__SIEVE(a__from(true)), MARK(from(true))) MARK(sieve(from(false))) -> c17(A__SIEVE(a__from(false)), MARK(from(false))) MARK(sieve(from(divides(z0, z1)))) -> c17(A__SIEVE(a__from(divides(mark(z0), mark(z1)))), MARK(from(divides(z0, z1)))) MARK(sieve(from(x0))) -> c17(MARK(from(x0))) MARK(sieve(head(primes))) -> c17(A__SIEVE(a__head(a__primes)), MARK(head(primes))) MARK(sieve(head(sieve(z0)))) -> c17(A__SIEVE(a__head(a__sieve(mark(z0)))), MARK(head(sieve(z0)))) MARK(sieve(head(from(z0)))) -> c17(A__SIEVE(a__head(a__from(mark(z0)))), MARK(head(from(z0)))) MARK(sieve(head(head(z0)))) -> c17(A__SIEVE(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(sieve(head(tail(z0)))) -> c17(A__SIEVE(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(sieve(head(if(z0, z1, z2)))) -> c17(A__SIEVE(a__head(a__if(mark(z0), z1, z2))), MARK(head(if(z0, z1, z2)))) MARK(sieve(head(filter(z0, z1)))) -> c17(A__SIEVE(a__head(a__filter(mark(z0), mark(z1)))), MARK(head(filter(z0, z1)))) MARK(sieve(head(s(z0)))) -> c17(A__SIEVE(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(sieve(head(cons(z0, z1)))) -> c17(A__SIEVE(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(sieve(head(divides(z0, z1)))) -> c17(A__SIEVE(a__head(divides(mark(z0), mark(z1)))), MARK(head(divides(z0, z1)))) MARK(sieve(head(x0))) -> c17(MARK(head(x0))) MARK(sieve(head(0))) -> c17(A__SIEVE(a__head(0))) MARK(sieve(head(true))) -> c17(A__SIEVE(a__head(true))) MARK(sieve(head(false))) -> c17(A__SIEVE(a__head(false))) MARK(sieve(tail(primes))) -> c17(A__SIEVE(a__tail(a__primes)), MARK(tail(primes))) MARK(sieve(tail(sieve(z0)))) -> c17(A__SIEVE(a__tail(a__sieve(mark(z0)))), MARK(tail(sieve(z0)))) MARK(sieve(tail(from(z0)))) -> c17(A__SIEVE(a__tail(a__from(mark(z0)))), MARK(tail(from(z0)))) MARK(sieve(tail(head(z0)))) -> c17(A__SIEVE(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(sieve(tail(tail(z0)))) -> c17(A__SIEVE(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(sieve(tail(if(z0, z1, z2)))) -> c17(A__SIEVE(a__tail(a__if(mark(z0), z1, z2))), MARK(tail(if(z0, z1, z2)))) MARK(sieve(tail(filter(z0, z1)))) -> c17(A__SIEVE(a__tail(a__filter(mark(z0), mark(z1)))), MARK(tail(filter(z0, z1)))) MARK(sieve(tail(s(z0)))) -> c17(A__SIEVE(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(sieve(tail(cons(z0, z1)))) -> c17(A__SIEVE(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(sieve(tail(divides(z0, z1)))) -> c17(A__SIEVE(a__tail(divides(mark(z0), mark(z1)))), MARK(tail(divides(z0, z1)))) MARK(sieve(tail(x0))) -> c17(MARK(tail(x0))) MARK(sieve(tail(0))) -> c17(A__SIEVE(a__tail(0))) MARK(sieve(tail(true))) -> c17(A__SIEVE(a__tail(true))) MARK(sieve(tail(false))) -> c17(A__SIEVE(a__tail(false))) MARK(sieve(if(primes, x1, x2))) -> c17(A__SIEVE(a__if(a__primes, x1, x2)), MARK(if(primes, x1, x2))) MARK(sieve(if(sieve(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__sieve(mark(z0)), x1, x2)), MARK(if(sieve(z0), x1, x2))) MARK(sieve(if(from(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__from(mark(z0)), x1, x2)), MARK(if(from(z0), x1, x2))) MARK(sieve(if(head(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__head(mark(z0)), x1, x2)), MARK(if(head(z0), x1, x2))) MARK(sieve(if(tail(z0), x1, x2))) -> 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x1))) -> c20(A__TAIL(a__filter(cons(mark(z0), z1), mark(x1))), MARK(filter(cons(z0, z1), x1))) MARK(tail(filter(true, x1))) -> c20(A__TAIL(a__filter(true, mark(x1))), MARK(filter(true, x1))) MARK(tail(filter(false, x1))) -> c20(A__TAIL(a__filter(false, mark(x1))), MARK(filter(false, x1))) MARK(tail(filter(divides(z0, z1), x1))) -> c20(A__TAIL(a__filter(divides(mark(z0), mark(z1)), mark(x1))), MARK(filter(divides(z0, z1), x1))) MARK(tail(filter(x0, x1))) -> c20(MARK(filter(x0, x1))) MARK(if(primes, x0, x1)) -> c21(A__IF(a__sieve(a__from(s(s(0)))), x0, x1), MARK(primes)) MARK(if(primes, x0, x1)) -> c21(A__IF(primes, x0, x1), MARK(primes)) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c18_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_1, c17_2, c17_1, c19_2, c19_1, c20_2, c20_1, c21_2, c21_1, c22_2, c22_1, c23_2, c23_1, c_2 ---------------------------------------- (185) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (186) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(sieve(z0), x1, x2)) -> c21(A__IF(a__sieve(mark(z0)), x1, x2), MARK(sieve(z0))) MARK(if(from(z0), x1, x2)) -> c21(A__IF(a__from(mark(z0)), x1, x2), MARK(from(z0))) MARK(if(head(z0), x1, x2)) -> c21(A__IF(a__head(mark(z0)), x1, x2), MARK(head(z0))) MARK(if(tail(z0), x1, x2)) -> c21(A__IF(a__tail(mark(z0)), x1, x2), MARK(tail(z0))) MARK(if(if(z0, z1, z2), x1, x2)) -> c21(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(filter(z0, z1), x1, x2)) -> c21(A__IF(a__filter(mark(z0), mark(z1)), x1, x2), MARK(filter(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c22(A__FILTER(mark(x0), a__primes), MARK(x0)) MARK(filter(x0, sieve(z0))) -> c22(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, from(z0))) -> c22(A__FILTER(mark(x0), a__from(mark(z0))), MARK(x0)) MARK(filter(x0, head(z0))) -> c22(A__FILTER(mark(x0), a__head(mark(z0))), MARK(x0)) MARK(filter(x0, tail(z0))) -> c22(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(x0)) MARK(filter(x0, if(z0, z1, z2))) -> c22(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0)) MARK(filter(x0, filter(z0, z1))) -> c22(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(x0)) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(x0, s(z0))) -> c22(MARK(x0)) MARK(filter(x0, 0)) -> c22(MARK(x0)) MARK(filter(x0, true)) -> c22(MARK(x0)) MARK(filter(x0, false)) -> c22(MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), x1)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c23(A__FILTER(mark(x0), a__primes), MARK(primes)) MARK(filter(x0, sieve(z0))) -> c23(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, from(z0))) -> c23(A__FILTER(mark(x0), a__from(mark(z0))), MARK(from(z0))) MARK(filter(x0, head(z0))) -> c23(A__FILTER(mark(x0), a__head(mark(z0))), MARK(head(z0))) MARK(filter(x0, tail(z0))) -> c23(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(tail(z0))) MARK(filter(x0, if(z0, z1, z2))) -> c23(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(filter(x0, filter(z0, z1))) -> c23(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(filter(x0, cons(z0, z1))) -> c23(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(primes, x1)) -> c23(A__FILTER(a__primes, mark(x1)), MARK(x1)) MARK(filter(sieve(z0), x1)) -> c23(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(from(z0), x1)) -> c23(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(head(z0), x1)) -> c23(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(tail(z0), x1)) -> c23(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(if(z0, z1, z2), x1)) -> c23(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1)) MARK(filter(filter(z0, z1), x1)) -> c23(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(x1)) MARK(filter(s(z0), x1)) -> c23(A__FILTER(s(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(x0, s(z0))) -> c23(MARK(s(z0))) MARK(filter(x0, divides(z0, z1))) -> c23(MARK(divides(z0, z1))) MARK(filter(0, x1)) -> c23(MARK(x1)) MARK(filter(cons(z0, z1), x1)) -> c23(MARK(x1)) MARK(filter(true, x1)) -> c23(MARK(x1)) MARK(filter(false, x1)) -> c23(MARK(x1)) MARK(filter(divides(z0, z1), x1)) -> c23(MARK(x1)) A__PRIMES -> c(A__SIEVE(cons(s(mark(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(sieve(primes)) -> c17(MARK(primes)) MARK(sieve(sieve(primes))) -> c17(A__SIEVE(a__sieve(a__primes)), MARK(sieve(primes))) MARK(sieve(sieve(sieve(z0)))) -> c17(A__SIEVE(a__sieve(a__sieve(mark(z0)))), MARK(sieve(sieve(z0)))) MARK(sieve(sieve(from(z0)))) -> c17(A__SIEVE(a__sieve(a__from(mark(z0)))), MARK(sieve(from(z0)))) MARK(sieve(sieve(head(z0)))) -> c17(A__SIEVE(a__sieve(a__head(mark(z0)))), MARK(sieve(head(z0)))) MARK(sieve(sieve(tail(z0)))) -> c17(A__SIEVE(a__sieve(a__tail(mark(z0)))), MARK(sieve(tail(z0)))) MARK(sieve(sieve(if(z0, z1, z2)))) -> c17(A__SIEVE(a__sieve(a__if(mark(z0), z1, z2))), MARK(sieve(if(z0, z1, z2)))) MARK(sieve(sieve(filter(z0, z1)))) -> c17(A__SIEVE(a__sieve(a__filter(mark(z0), mark(z1)))), MARK(sieve(filter(z0, z1)))) MARK(sieve(sieve(s(z0)))) -> c17(A__SIEVE(a__sieve(s(mark(z0)))), MARK(sieve(s(z0)))) MARK(sieve(sieve(cons(z0, z1)))) -> c17(A__SIEVE(a__sieve(cons(mark(z0), z1))), MARK(sieve(cons(z0, z1)))) MARK(sieve(sieve(divides(z0, z1)))) -> c17(A__SIEVE(a__sieve(divides(mark(z0), mark(z1)))), MARK(sieve(divides(z0, z1)))) MARK(sieve(sieve(x0))) -> c17(MARK(sieve(x0))) MARK(sieve(sieve(0))) -> c17(A__SIEVE(a__sieve(0))) MARK(sieve(sieve(true))) -> c17(A__SIEVE(a__sieve(true))) MARK(sieve(sieve(false))) -> c17(A__SIEVE(a__sieve(false))) MARK(sieve(from(x0))) -> c17(A__SIEVE(cons(mark(mark(x0)), from(s(mark(x0))))), MARK(from(x0))) MARK(sieve(from(primes))) -> c17(A__SIEVE(a__from(a__primes)), MARK(from(primes))) MARK(sieve(from(sieve(z0)))) -> c17(A__SIEVE(a__from(a__sieve(mark(z0)))), MARK(from(sieve(z0)))) MARK(sieve(from(from(z0)))) -> c17(A__SIEVE(a__from(a__from(mark(z0)))), MARK(from(from(z0)))) MARK(sieve(from(head(z0)))) -> c17(A__SIEVE(a__from(a__head(mark(z0)))), MARK(from(head(z0)))) MARK(sieve(from(tail(z0)))) -> c17(A__SIEVE(a__from(a__tail(mark(z0)))), MARK(from(tail(z0)))) MARK(sieve(from(if(z0, z1, z2)))) -> c17(A__SIEVE(a__from(a__if(mark(z0), z1, z2))), MARK(from(if(z0, z1, z2)))) MARK(sieve(from(filter(z0, z1)))) -> c17(A__SIEVE(a__from(a__filter(mark(z0), mark(z1)))), MARK(from(filter(z0, z1)))) MARK(sieve(from(s(z0)))) -> c17(A__SIEVE(a__from(s(mark(z0)))), MARK(from(s(z0)))) MARK(sieve(from(0))) -> c17(A__SIEVE(a__from(0)), MARK(from(0))) MARK(sieve(from(cons(z0, z1)))) -> c17(A__SIEVE(a__from(cons(mark(z0), z1))), MARK(from(cons(z0, z1)))) MARK(sieve(from(true))) -> c17(A__SIEVE(a__from(true)), MARK(from(true))) MARK(sieve(from(false))) -> c17(A__SIEVE(a__from(false)), MARK(from(false))) MARK(sieve(from(divides(z0, z1)))) -> c17(A__SIEVE(a__from(divides(mark(z0), mark(z1)))), MARK(from(divides(z0, z1)))) MARK(sieve(from(x0))) -> c17(MARK(from(x0))) MARK(sieve(head(primes))) -> c17(A__SIEVE(a__head(a__primes)), MARK(head(primes))) MARK(sieve(head(sieve(z0)))) -> c17(A__SIEVE(a__head(a__sieve(mark(z0)))), MARK(head(sieve(z0)))) MARK(sieve(head(from(z0)))) -> c17(A__SIEVE(a__head(a__from(mark(z0)))), MARK(head(from(z0)))) MARK(sieve(head(head(z0)))) -> c17(A__SIEVE(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(sieve(head(tail(z0)))) -> c17(A__SIEVE(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(sieve(head(if(z0, z1, z2)))) -> c17(A__SIEVE(a__head(a__if(mark(z0), z1, z2))), MARK(head(if(z0, z1, z2)))) MARK(sieve(head(filter(z0, z1)))) -> c17(A__SIEVE(a__head(a__filter(mark(z0), mark(z1)))), MARK(head(filter(z0, z1)))) MARK(sieve(head(s(z0)))) -> c17(A__SIEVE(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(sieve(head(cons(z0, z1)))) -> c17(A__SIEVE(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(sieve(head(divides(z0, z1)))) -> c17(A__SIEVE(a__head(divides(mark(z0), mark(z1)))), MARK(head(divides(z0, z1)))) MARK(sieve(head(x0))) -> c17(MARK(head(x0))) MARK(sieve(head(0))) -> c17(A__SIEVE(a__head(0))) MARK(sieve(head(true))) -> c17(A__SIEVE(a__head(true))) MARK(sieve(head(false))) -> c17(A__SIEVE(a__head(false))) MARK(sieve(tail(primes))) -> c17(A__SIEVE(a__tail(a__primes)), MARK(tail(primes))) MARK(sieve(tail(sieve(z0)))) -> c17(A__SIEVE(a__tail(a__sieve(mark(z0)))), MARK(tail(sieve(z0)))) MARK(sieve(tail(from(z0)))) -> c17(A__SIEVE(a__tail(a__from(mark(z0)))), MARK(tail(from(z0)))) MARK(sieve(tail(head(z0)))) -> c17(A__SIEVE(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(sieve(tail(tail(z0)))) -> c17(A__SIEVE(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(sieve(tail(if(z0, z1, z2)))) -> c17(A__SIEVE(a__tail(a__if(mark(z0), z1, z2))), MARK(tail(if(z0, z1, z2)))) MARK(sieve(tail(filter(z0, z1)))) -> c17(A__SIEVE(a__tail(a__filter(mark(z0), mark(z1)))), MARK(tail(filter(z0, z1)))) MARK(sieve(tail(s(z0)))) -> c17(A__SIEVE(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(sieve(tail(cons(z0, z1)))) -> c17(A__SIEVE(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(sieve(tail(divides(z0, z1)))) -> c17(A__SIEVE(a__tail(divides(mark(z0), mark(z1)))), MARK(tail(divides(z0, z1)))) MARK(sieve(tail(x0))) -> c17(MARK(tail(x0))) MARK(sieve(tail(0))) -> c17(A__SIEVE(a__tail(0))) MARK(sieve(tail(true))) -> c17(A__SIEVE(a__tail(true))) MARK(sieve(tail(false))) -> c17(A__SIEVE(a__tail(false))) MARK(sieve(if(primes, x1, x2))) -> c17(A__SIEVE(a__if(a__primes, x1, x2)), MARK(if(primes, x1, x2))) MARK(sieve(if(sieve(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__sieve(mark(z0)), x1, x2)), MARK(if(sieve(z0), x1, x2))) MARK(sieve(if(from(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__from(mark(z0)), x1, x2)), MARK(if(from(z0), x1, x2))) MARK(sieve(if(head(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__head(mark(z0)), x1, x2)), MARK(if(head(z0), x1, x2))) MARK(sieve(if(tail(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__tail(mark(z0)), x1, x2)), MARK(if(tail(z0), x1, x2))) MARK(sieve(if(if(z0, z1, z2), x1, x2))) -> c17(A__SIEVE(a__if(a__if(mark(z0), z1, z2), x1, x2)), MARK(if(if(z0, z1, z2), x1, x2))) MARK(sieve(if(filter(z0, z1), x1, x2))) -> c17(A__SIEVE(a__if(a__filter(mark(z0), mark(z1)), x1, 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c20(A__TAIL(a__from(a__primes)), MARK(from(primes))) MARK(tail(from(sieve(z0)))) -> c20(A__TAIL(a__from(a__sieve(mark(z0)))), MARK(from(sieve(z0)))) MARK(tail(from(from(z0)))) -> c20(A__TAIL(a__from(a__from(mark(z0)))), MARK(from(from(z0)))) MARK(tail(from(head(z0)))) -> c20(A__TAIL(a__from(a__head(mark(z0)))), MARK(from(head(z0)))) MARK(tail(from(tail(z0)))) -> c20(A__TAIL(a__from(a__tail(mark(z0)))), MARK(from(tail(z0)))) MARK(tail(from(if(z0, z1, z2)))) -> c20(A__TAIL(a__from(a__if(mark(z0), z1, z2))), MARK(from(if(z0, z1, z2)))) MARK(tail(from(filter(z0, z1)))) -> c20(A__TAIL(a__from(a__filter(mark(z0), mark(z1)))), MARK(from(filter(z0, z1)))) MARK(tail(from(s(z0)))) -> c20(A__TAIL(a__from(s(mark(z0)))), MARK(from(s(z0)))) MARK(tail(from(0))) -> c20(A__TAIL(a__from(0)), MARK(from(0))) MARK(tail(from(cons(z0, z1)))) -> c20(A__TAIL(a__from(cons(mark(z0), z1))), MARK(from(cons(z0, z1)))) MARK(tail(from(true))) -> c20(A__TAIL(a__from(true)), MARK(from(true))) MARK(tail(from(false))) -> 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c20(A__TAIL(a__tail(a__filter(mark(z0), mark(z1)))), MARK(tail(filter(z0, z1)))) MARK(tail(tail(s(z0)))) -> c20(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c20(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(divides(z0, z1)))) -> c20(A__TAIL(a__tail(divides(mark(z0), mark(z1)))), MARK(tail(divides(z0, z1)))) MARK(tail(tail(x0))) -> c20(MARK(tail(x0))) MARK(tail(tail(0))) -> c20(A__TAIL(a__tail(0))) MARK(tail(tail(true))) -> c20(A__TAIL(a__tail(true))) MARK(tail(tail(false))) -> c20(A__TAIL(a__tail(false))) MARK(tail(if(primes, x1, x2))) -> c20(A__TAIL(a__if(a__primes, x1, x2)), MARK(if(primes, x1, x2))) MARK(tail(if(sieve(z0), x1, x2))) -> c20(A__TAIL(a__if(a__sieve(mark(z0)), x1, x2)), MARK(if(sieve(z0), x1, x2))) MARK(tail(if(from(z0), x1, x2))) -> c20(A__TAIL(a__if(a__from(mark(z0)), x1, x2)), MARK(if(from(z0), x1, x2))) MARK(tail(if(head(z0), x1, x2))) -> c20(A__TAIL(a__if(a__head(mark(z0)), x1, x2)), MARK(if(head(z0), x1, x2))) MARK(tail(if(tail(z0), x1, x2))) -> c20(A__TAIL(a__if(a__tail(mark(z0)), x1, x2)), MARK(if(tail(z0), x1, x2))) MARK(tail(if(if(z0, z1, z2), x1, x2))) -> c20(A__TAIL(a__if(a__if(mark(z0), z1, z2), x1, x2)), MARK(if(if(z0, z1, z2), x1, x2))) MARK(tail(if(filter(z0, z1), x1, x2))) -> c20(A__TAIL(a__if(a__filter(mark(z0), mark(z1)), x1, x2)), MARK(if(filter(z0, z1), x1, x2))) MARK(tail(if(s(z0), x1, x2))) -> c20(A__TAIL(a__if(s(mark(z0)), x1, x2)), MARK(if(s(z0), x1, x2))) MARK(tail(if(cons(z0, z1), x1, x2))) -> c20(A__TAIL(a__if(cons(mark(z0), z1), x1, x2)), MARK(if(cons(z0, z1), x1, x2))) MARK(tail(if(true, x1, x2))) -> c20(A__TAIL(a__if(true, x1, x2)), MARK(if(true, x1, x2))) MARK(tail(if(false, x1, x2))) -> c20(A__TAIL(a__if(false, x1, x2)), MARK(if(false, x1, x2))) MARK(tail(if(divides(z0, z1), x1, x2))) -> c20(A__TAIL(a__if(divides(mark(z0), mark(z1)), x1, x2)), MARK(if(divides(z0, z1), x1, x2))) MARK(tail(if(x0, z1, z2))) -> c20(MARK(if(x0, z1, z2))) MARK(tail(if(0, x1, x2))) -> c20(A__TAIL(a__if(0, x1, x2))) MARK(tail(filter(x0, primes))) -> c20(A__TAIL(a__filter(mark(x0), a__primes)), MARK(filter(x0, primes))) MARK(tail(filter(x0, sieve(z0)))) -> c20(A__TAIL(a__filter(mark(x0), a__sieve(mark(z0)))), MARK(filter(x0, sieve(z0)))) MARK(tail(filter(x0, from(z0)))) -> c20(A__TAIL(a__filter(mark(x0), a__from(mark(z0)))), MARK(filter(x0, from(z0)))) MARK(tail(filter(x0, head(z0)))) -> c20(A__TAIL(a__filter(mark(x0), a__head(mark(z0)))), MARK(filter(x0, head(z0)))) MARK(tail(filter(x0, tail(z0)))) -> c20(A__TAIL(a__filter(mark(x0), a__tail(mark(z0)))), MARK(filter(x0, tail(z0)))) MARK(tail(filter(x0, if(z0, z1, z2)))) -> c20(A__TAIL(a__filter(mark(x0), a__if(mark(z0), z1, z2))), MARK(filter(x0, if(z0, z1, z2)))) MARK(tail(filter(x0, filter(z0, z1)))) -> c20(A__TAIL(a__filter(mark(x0), a__filter(mark(z0), mark(z1)))), MARK(filter(x0, filter(z0, z1)))) MARK(tail(filter(x0, s(z0)))) -> c20(A__TAIL(a__filter(mark(x0), s(mark(z0)))), 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mark(x1))), MARK(filter(head(z0), x1))) MARK(tail(filter(tail(z0), x1))) -> c20(A__TAIL(a__filter(a__tail(mark(z0)), mark(x1))), MARK(filter(tail(z0), x1))) MARK(tail(filter(if(z0, z1, z2), x1))) -> c20(A__TAIL(a__filter(a__if(mark(z0), z1, z2), mark(x1))), MARK(filter(if(z0, z1, z2), x1))) MARK(tail(filter(filter(z0, z1), x1))) -> c20(A__TAIL(a__filter(a__filter(mark(z0), mark(z1)), mark(x1))), MARK(filter(filter(z0, z1), x1))) MARK(tail(filter(s(z0), x1))) -> c20(A__TAIL(a__filter(s(mark(z0)), mark(x1))), MARK(filter(s(z0), x1))) MARK(tail(filter(0, x1))) -> c20(A__TAIL(a__filter(0, mark(x1))), MARK(filter(0, x1))) MARK(tail(filter(cons(z0, z1), x1))) -> c20(A__TAIL(a__filter(cons(mark(z0), z1), mark(x1))), MARK(filter(cons(z0, z1), x1))) MARK(tail(filter(true, x1))) -> c20(A__TAIL(a__filter(true, mark(x1))), MARK(filter(true, x1))) MARK(tail(filter(false, x1))) -> c20(A__TAIL(a__filter(false, mark(x1))), MARK(filter(false, x1))) MARK(tail(filter(divides(z0, z1), x1))) -> c20(A__TAIL(a__filter(divides(mark(z0), mark(z1)), mark(x1))), MARK(filter(divides(z0, z1), x1))) MARK(tail(filter(x0, x1))) -> c20(MARK(filter(x0, x1))) MARK(if(primes, x0, x1)) -> c21(A__IF(a__sieve(a__from(s(s(0)))), x0, x1), MARK(primes)) MARK(if(primes, x0, x1)) -> c21(MARK(primes)) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c18_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_1, c17_2, c17_1, c19_2, c19_1, c20_2, c20_1, c21_2, c21_1, c22_2, c22_1, c23_2, c23_1, c_2 ---------------------------------------- (187) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(if(sieve(z0), x1, x2)) -> c21(A__IF(a__sieve(mark(z0)), x1, x2), MARK(sieve(z0))) by MARK(if(sieve(x0), x1, x2)) -> c21(A__IF(sieve(mark(x0)), x1, x2), MARK(sieve(x0))) MARK(if(sieve(primes), x1, x2)) -> c21(A__IF(a__sieve(a__primes), x1, x2), MARK(sieve(primes))) MARK(if(sieve(sieve(z0)), x1, x2)) -> c21(A__IF(a__sieve(a__sieve(mark(z0))), x1, x2), MARK(sieve(sieve(z0)))) MARK(if(sieve(from(z0)), x1, x2)) -> c21(A__IF(a__sieve(a__from(mark(z0))), x1, x2), MARK(sieve(from(z0)))) MARK(if(sieve(head(z0)), x1, x2)) -> c21(A__IF(a__sieve(a__head(mark(z0))), x1, x2), MARK(sieve(head(z0)))) MARK(if(sieve(tail(z0)), x1, x2)) -> c21(A__IF(a__sieve(a__tail(mark(z0))), x1, x2), MARK(sieve(tail(z0)))) MARK(if(sieve(if(z0, z1, z2)), x1, x2)) -> c21(A__IF(a__sieve(a__if(mark(z0), z1, z2)), x1, x2), MARK(sieve(if(z0, z1, z2)))) MARK(if(sieve(filter(z0, z1)), x1, x2)) -> c21(A__IF(a__sieve(a__filter(mark(z0), mark(z1))), x1, x2), MARK(sieve(filter(z0, z1)))) MARK(if(sieve(s(z0)), x1, x2)) -> c21(A__IF(a__sieve(s(mark(z0))), x1, x2), MARK(sieve(s(z0)))) MARK(if(sieve(0), x1, x2)) -> c21(A__IF(a__sieve(0), x1, x2), MARK(sieve(0))) MARK(if(sieve(cons(z0, z1)), x1, x2)) -> c21(A__IF(a__sieve(cons(mark(z0), z1)), x1, x2), MARK(sieve(cons(z0, z1)))) MARK(if(sieve(true), x1, x2)) -> c21(A__IF(a__sieve(true), x1, x2), MARK(sieve(true))) MARK(if(sieve(false), x1, x2)) -> c21(A__IF(a__sieve(false), x1, x2), MARK(sieve(false))) MARK(if(sieve(divides(z0, z1)), x1, x2)) -> c21(A__IF(a__sieve(divides(mark(z0), mark(z1))), x1, x2), MARK(sieve(divides(z0, z1)))) ---------------------------------------- (188) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(from(z0), x1, x2)) -> c21(A__IF(a__from(mark(z0)), x1, x2), MARK(from(z0))) MARK(if(head(z0), x1, x2)) -> c21(A__IF(a__head(mark(z0)), x1, x2), MARK(head(z0))) MARK(if(tail(z0), x1, x2)) -> c21(A__IF(a__tail(mark(z0)), x1, x2), MARK(tail(z0))) MARK(if(if(z0, z1, z2), x1, x2)) -> c21(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(filter(z0, z1), x1, x2)) -> c21(A__IF(a__filter(mark(z0), mark(z1)), x1, x2), MARK(filter(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c22(A__FILTER(mark(x0), a__primes), MARK(x0)) MARK(filter(x0, sieve(z0))) -> c22(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, from(z0))) -> c22(A__FILTER(mark(x0), a__from(mark(z0))), MARK(x0)) MARK(filter(x0, head(z0))) -> c22(A__FILTER(mark(x0), a__head(mark(z0))), MARK(x0)) MARK(filter(x0, tail(z0))) -> c22(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(x0)) MARK(filter(x0, if(z0, z1, z2))) -> c22(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0)) MARK(filter(x0, filter(z0, z1))) -> c22(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(x0)) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(x0, s(z0))) -> c22(MARK(x0)) MARK(filter(x0, 0)) -> c22(MARK(x0)) MARK(filter(x0, true)) -> c22(MARK(x0)) MARK(filter(x0, false)) -> c22(MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), x1)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c23(A__FILTER(mark(x0), a__primes), MARK(primes)) MARK(filter(x0, sieve(z0))) -> c23(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, from(z0))) -> c23(A__FILTER(mark(x0), a__from(mark(z0))), MARK(from(z0))) MARK(filter(x0, head(z0))) -> c23(A__FILTER(mark(x0), a__head(mark(z0))), MARK(head(z0))) MARK(filter(x0, tail(z0))) -> c23(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(tail(z0))) MARK(filter(x0, if(z0, z1, z2))) -> c23(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(filter(x0, filter(z0, z1))) -> c23(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(filter(x0, cons(z0, z1))) -> c23(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(primes, x1)) -> c23(A__FILTER(a__primes, mark(x1)), MARK(x1)) MARK(filter(sieve(z0), x1)) -> c23(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(from(z0), x1)) -> c23(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(head(z0), x1)) -> c23(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(tail(z0), x1)) -> c23(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(if(z0, z1, z2), x1)) -> c23(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1)) MARK(filter(filter(z0, z1), x1)) -> c23(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(x1)) MARK(filter(s(z0), x1)) -> c23(A__FILTER(s(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(x0, s(z0))) -> c23(MARK(s(z0))) MARK(filter(x0, divides(z0, z1))) -> c23(MARK(divides(z0, z1))) MARK(filter(0, x1)) -> c23(MARK(x1)) MARK(filter(cons(z0, z1), x1)) -> c23(MARK(x1)) MARK(filter(true, x1)) -> c23(MARK(x1)) MARK(filter(false, x1)) -> c23(MARK(x1)) MARK(filter(divides(z0, z1), x1)) -> c23(MARK(x1)) A__PRIMES -> c(A__SIEVE(cons(s(mark(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(sieve(primes)) -> c17(MARK(primes)) MARK(sieve(sieve(primes))) -> c17(A__SIEVE(a__sieve(a__primes)), MARK(sieve(primes))) MARK(sieve(sieve(sieve(z0)))) -> c17(A__SIEVE(a__sieve(a__sieve(mark(z0)))), MARK(sieve(sieve(z0)))) MARK(sieve(sieve(from(z0)))) -> c17(A__SIEVE(a__sieve(a__from(mark(z0)))), MARK(sieve(from(z0)))) MARK(sieve(sieve(head(z0)))) -> c17(A__SIEVE(a__sieve(a__head(mark(z0)))), MARK(sieve(head(z0)))) MARK(sieve(sieve(tail(z0)))) -> c17(A__SIEVE(a__sieve(a__tail(mark(z0)))), MARK(sieve(tail(z0)))) MARK(sieve(sieve(if(z0, z1, z2)))) -> c17(A__SIEVE(a__sieve(a__if(mark(z0), z1, z2))), MARK(sieve(if(z0, z1, z2)))) MARK(sieve(sieve(filter(z0, z1)))) -> c17(A__SIEVE(a__sieve(a__filter(mark(z0), mark(z1)))), MARK(sieve(filter(z0, z1)))) MARK(sieve(sieve(s(z0)))) -> c17(A__SIEVE(a__sieve(s(mark(z0)))), MARK(sieve(s(z0)))) MARK(sieve(sieve(cons(z0, z1)))) -> c17(A__SIEVE(a__sieve(cons(mark(z0), z1))), MARK(sieve(cons(z0, z1)))) MARK(sieve(sieve(divides(z0, z1)))) -> c17(A__SIEVE(a__sieve(divides(mark(z0), mark(z1)))), MARK(sieve(divides(z0, z1)))) MARK(sieve(sieve(x0))) -> c17(MARK(sieve(x0))) MARK(sieve(sieve(0))) -> c17(A__SIEVE(a__sieve(0))) MARK(sieve(sieve(true))) -> c17(A__SIEVE(a__sieve(true))) MARK(sieve(sieve(false))) -> c17(A__SIEVE(a__sieve(false))) MARK(sieve(from(x0))) -> c17(A__SIEVE(cons(mark(mark(x0)), from(s(mark(x0))))), MARK(from(x0))) MARK(sieve(from(primes))) -> c17(A__SIEVE(a__from(a__primes)), MARK(from(primes))) MARK(sieve(from(sieve(z0)))) -> c17(A__SIEVE(a__from(a__sieve(mark(z0)))), MARK(from(sieve(z0)))) MARK(sieve(from(from(z0)))) -> c17(A__SIEVE(a__from(a__from(mark(z0)))), MARK(from(from(z0)))) MARK(sieve(from(head(z0)))) -> c17(A__SIEVE(a__from(a__head(mark(z0)))), MARK(from(head(z0)))) MARK(sieve(from(tail(z0)))) -> c17(A__SIEVE(a__from(a__tail(mark(z0)))), MARK(from(tail(z0)))) MARK(sieve(from(if(z0, z1, z2)))) -> c17(A__SIEVE(a__from(a__if(mark(z0), z1, z2))), MARK(from(if(z0, z1, z2)))) MARK(sieve(from(filter(z0, z1)))) -> c17(A__SIEVE(a__from(a__filter(mark(z0), mark(z1)))), MARK(from(filter(z0, z1)))) MARK(sieve(from(s(z0)))) -> c17(A__SIEVE(a__from(s(mark(z0)))), MARK(from(s(z0)))) MARK(sieve(from(0))) -> c17(A__SIEVE(a__from(0)), MARK(from(0))) MARK(sieve(from(cons(z0, z1)))) -> c17(A__SIEVE(a__from(cons(mark(z0), z1))), MARK(from(cons(z0, z1)))) MARK(sieve(from(true))) -> c17(A__SIEVE(a__from(true)), MARK(from(true))) MARK(sieve(from(false))) -> c17(A__SIEVE(a__from(false)), MARK(from(false))) MARK(sieve(from(divides(z0, z1)))) -> c17(A__SIEVE(a__from(divides(mark(z0), mark(z1)))), MARK(from(divides(z0, z1)))) MARK(sieve(from(x0))) -> c17(MARK(from(x0))) MARK(sieve(head(primes))) -> c17(A__SIEVE(a__head(a__primes)), MARK(head(primes))) MARK(sieve(head(sieve(z0)))) -> c17(A__SIEVE(a__head(a__sieve(mark(z0)))), MARK(head(sieve(z0)))) MARK(sieve(head(from(z0)))) -> c17(A__SIEVE(a__head(a__from(mark(z0)))), MARK(head(from(z0)))) MARK(sieve(head(head(z0)))) -> c17(A__SIEVE(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(sieve(head(tail(z0)))) -> c17(A__SIEVE(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(sieve(head(if(z0, z1, z2)))) -> c17(A__SIEVE(a__head(a__if(mark(z0), z1, z2))), MARK(head(if(z0, z1, z2)))) MARK(sieve(head(filter(z0, z1)))) -> c17(A__SIEVE(a__head(a__filter(mark(z0), mark(z1)))), MARK(head(filter(z0, z1)))) MARK(sieve(head(s(z0)))) -> c17(A__SIEVE(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(sieve(head(cons(z0, z1)))) -> c17(A__SIEVE(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(sieve(head(divides(z0, z1)))) -> c17(A__SIEVE(a__head(divides(mark(z0), mark(z1)))), MARK(head(divides(z0, z1)))) MARK(sieve(head(x0))) -> c17(MARK(head(x0))) MARK(sieve(head(0))) -> c17(A__SIEVE(a__head(0))) 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c20(A__TAIL(a__sieve(false))) MARK(tail(from(x0))) -> c20(A__TAIL(cons(mark(mark(x0)), from(s(mark(x0))))), MARK(from(x0))) MARK(tail(from(primes))) -> c20(A__TAIL(a__from(a__primes)), MARK(from(primes))) MARK(tail(from(sieve(z0)))) -> c20(A__TAIL(a__from(a__sieve(mark(z0)))), MARK(from(sieve(z0)))) MARK(tail(from(from(z0)))) -> c20(A__TAIL(a__from(a__from(mark(z0)))), MARK(from(from(z0)))) MARK(tail(from(head(z0)))) -> c20(A__TAIL(a__from(a__head(mark(z0)))), MARK(from(head(z0)))) MARK(tail(from(tail(z0)))) -> c20(A__TAIL(a__from(a__tail(mark(z0)))), MARK(from(tail(z0)))) MARK(tail(from(if(z0, z1, z2)))) -> c20(A__TAIL(a__from(a__if(mark(z0), z1, z2))), MARK(from(if(z0, z1, z2)))) MARK(tail(from(filter(z0, z1)))) -> c20(A__TAIL(a__from(a__filter(mark(z0), mark(z1)))), MARK(from(filter(z0, z1)))) MARK(tail(from(s(z0)))) -> c20(A__TAIL(a__from(s(mark(z0)))), MARK(from(s(z0)))) MARK(tail(from(0))) -> c20(A__TAIL(a__from(0)), MARK(from(0))) MARK(tail(from(cons(z0, z1)))) -> c20(A__TAIL(a__from(cons(mark(z0), z1))), MARK(from(cons(z0, z1)))) MARK(tail(from(true))) -> c20(A__TAIL(a__from(true)), MARK(from(true))) MARK(tail(from(false))) -> c20(A__TAIL(a__from(false)), MARK(from(false))) MARK(tail(from(divides(z0, z1)))) -> c20(A__TAIL(a__from(divides(mark(z0), mark(z1)))), MARK(from(divides(z0, z1)))) MARK(tail(from(x0))) -> c20(MARK(from(x0))) MARK(tail(head(primes))) -> c20(A__TAIL(a__head(a__primes)), MARK(head(primes))) MARK(tail(head(sieve(z0)))) -> c20(A__TAIL(a__head(a__sieve(mark(z0)))), MARK(head(sieve(z0)))) MARK(tail(head(from(z0)))) -> c20(A__TAIL(a__head(a__from(mark(z0)))), MARK(head(from(z0)))) MARK(tail(head(head(z0)))) -> c20(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c20(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(if(z0, z1, z2)))) -> c20(A__TAIL(a__head(a__if(mark(z0), z1, z2))), MARK(head(if(z0, z1, z2)))) MARK(tail(head(filter(z0, z1)))) -> c20(A__TAIL(a__head(a__filter(mark(z0), mark(z1)))), MARK(head(filter(z0, z1)))) MARK(tail(head(s(z0)))) -> c20(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(cons(z0, z1)))) -> c20(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(divides(z0, z1)))) -> c20(A__TAIL(a__head(divides(mark(z0), mark(z1)))), MARK(head(divides(z0, z1)))) MARK(tail(head(x0))) -> c20(MARK(head(x0))) MARK(tail(head(0))) -> c20(A__TAIL(a__head(0))) MARK(tail(head(true))) -> c20(A__TAIL(a__head(true))) MARK(tail(head(false))) -> c20(A__TAIL(a__head(false))) MARK(tail(tail(primes))) -> c20(A__TAIL(a__tail(a__primes)), MARK(tail(primes))) MARK(tail(tail(sieve(z0)))) -> c20(A__TAIL(a__tail(a__sieve(mark(z0)))), MARK(tail(sieve(z0)))) MARK(tail(tail(from(z0)))) -> c20(A__TAIL(a__tail(a__from(mark(z0)))), MARK(tail(from(z0)))) MARK(tail(tail(head(z0)))) -> c20(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c20(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(if(z0, z1, z2)))) -> c20(A__TAIL(a__tail(a__if(mark(z0), z1, z2))), MARK(tail(if(z0, z1, z2)))) MARK(tail(tail(filter(z0, z1)))) -> c20(A__TAIL(a__tail(a__filter(mark(z0), mark(z1)))), MARK(tail(filter(z0, z1)))) MARK(tail(tail(s(z0)))) -> c20(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c20(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(divides(z0, z1)))) -> c20(A__TAIL(a__tail(divides(mark(z0), mark(z1)))), MARK(tail(divides(z0, z1)))) MARK(tail(tail(x0))) -> c20(MARK(tail(x0))) MARK(tail(tail(0))) -> c20(A__TAIL(a__tail(0))) MARK(tail(tail(true))) -> c20(A__TAIL(a__tail(true))) MARK(tail(tail(false))) -> c20(A__TAIL(a__tail(false))) MARK(tail(if(primes, x1, x2))) -> c20(A__TAIL(a__if(a__primes, x1, x2)), MARK(if(primes, x1, x2))) MARK(tail(if(sieve(z0), x1, x2))) -> c20(A__TAIL(a__if(a__sieve(mark(z0)), x1, x2)), MARK(if(sieve(z0), x1, x2))) MARK(tail(if(from(z0), x1, x2))) -> c20(A__TAIL(a__if(a__from(mark(z0)), x1, x2)), MARK(if(from(z0), x1, x2))) MARK(tail(if(head(z0), x1, x2))) -> c20(A__TAIL(a__if(a__head(mark(z0)), x1, x2)), MARK(if(head(z0), x1, x2))) MARK(tail(if(tail(z0), x1, x2))) -> c20(A__TAIL(a__if(a__tail(mark(z0)), x1, x2)), MARK(if(tail(z0), x1, x2))) MARK(tail(if(if(z0, z1, z2), x1, x2))) -> c20(A__TAIL(a__if(a__if(mark(z0), z1, z2), x1, x2)), MARK(if(if(z0, z1, z2), x1, x2))) MARK(tail(if(filter(z0, z1), x1, x2))) -> c20(A__TAIL(a__if(a__filter(mark(z0), mark(z1)), x1, x2)), MARK(if(filter(z0, z1), x1, x2))) MARK(tail(if(s(z0), x1, x2))) -> c20(A__TAIL(a__if(s(mark(z0)), x1, x2)), MARK(if(s(z0), x1, x2))) MARK(tail(if(cons(z0, z1), x1, x2))) -> c20(A__TAIL(a__if(cons(mark(z0), z1), x1, x2)), MARK(if(cons(z0, z1), x1, x2))) MARK(tail(if(true, x1, x2))) -> c20(A__TAIL(a__if(true, x1, x2)), MARK(if(true, x1, x2))) MARK(tail(if(false, x1, x2))) -> c20(A__TAIL(a__if(false, x1, x2)), MARK(if(false, x1, 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z1)))) -> c20(A__TAIL(a__filter(mark(x0), a__filter(mark(z0), mark(z1)))), MARK(filter(x0, filter(z0, z1)))) MARK(tail(filter(x0, s(z0)))) -> c20(A__TAIL(a__filter(mark(x0), s(mark(z0)))), MARK(filter(x0, s(z0)))) MARK(tail(filter(x0, 0))) -> c20(A__TAIL(a__filter(mark(x0), 0)), MARK(filter(x0, 0))) MARK(tail(filter(x0, cons(z0, z1)))) -> c20(A__TAIL(a__filter(mark(x0), cons(mark(z0), z1))), MARK(filter(x0, cons(z0, z1)))) MARK(tail(filter(x0, true))) -> c20(A__TAIL(a__filter(mark(x0), true)), MARK(filter(x0, true))) MARK(tail(filter(x0, false))) -> c20(A__TAIL(a__filter(mark(x0), false)), MARK(filter(x0, false))) MARK(tail(filter(x0, divides(z0, z1)))) -> c20(A__TAIL(a__filter(mark(x0), divides(mark(z0), mark(z1)))), MARK(filter(x0, divides(z0, z1)))) MARK(tail(filter(primes, x1))) -> c20(A__TAIL(a__filter(a__primes, mark(x1))), MARK(filter(primes, x1))) MARK(tail(filter(sieve(z0), x1))) -> c20(A__TAIL(a__filter(a__sieve(mark(z0)), mark(x1))), MARK(filter(sieve(z0), x1))) MARK(tail(filter(from(z0), x1))) -> c20(A__TAIL(a__filter(a__from(mark(z0)), mark(x1))), MARK(filter(from(z0), x1))) MARK(tail(filter(head(z0), x1))) -> c20(A__TAIL(a__filter(a__head(mark(z0)), mark(x1))), MARK(filter(head(z0), x1))) MARK(tail(filter(tail(z0), x1))) -> c20(A__TAIL(a__filter(a__tail(mark(z0)), mark(x1))), MARK(filter(tail(z0), x1))) MARK(tail(filter(if(z0, z1, z2), x1))) -> c20(A__TAIL(a__filter(a__if(mark(z0), z1, z2), mark(x1))), MARK(filter(if(z0, z1, z2), x1))) MARK(tail(filter(filter(z0, z1), x1))) -> c20(A__TAIL(a__filter(a__filter(mark(z0), mark(z1)), mark(x1))), MARK(filter(filter(z0, z1), x1))) MARK(tail(filter(s(z0), x1))) -> c20(A__TAIL(a__filter(s(mark(z0)), mark(x1))), MARK(filter(s(z0), x1))) MARK(tail(filter(0, x1))) -> c20(A__TAIL(a__filter(0, mark(x1))), MARK(filter(0, x1))) MARK(tail(filter(cons(z0, z1), x1))) -> c20(A__TAIL(a__filter(cons(mark(z0), z1), mark(x1))), MARK(filter(cons(z0, z1), x1))) MARK(tail(filter(true, x1))) -> c20(A__TAIL(a__filter(true, mark(x1))), MARK(filter(true, x1))) MARK(tail(filter(false, x1))) -> c20(A__TAIL(a__filter(false, mark(x1))), MARK(filter(false, x1))) MARK(tail(filter(divides(z0, z1), x1))) -> c20(A__TAIL(a__filter(divides(mark(z0), mark(z1)), mark(x1))), MARK(filter(divides(z0, z1), x1))) MARK(tail(filter(x0, x1))) -> c20(MARK(filter(x0, x1))) MARK(if(primes, x0, x1)) -> c21(A__IF(a__sieve(a__from(s(s(0)))), x0, x1), MARK(primes)) MARK(if(primes, x0, x1)) -> c21(MARK(primes)) MARK(if(sieve(x0), x1, x2)) -> c21(A__IF(sieve(mark(x0)), x1, x2), MARK(sieve(x0))) MARK(if(sieve(primes), x1, x2)) -> c21(A__IF(a__sieve(a__primes), x1, x2), MARK(sieve(primes))) MARK(if(sieve(sieve(z0)), x1, x2)) -> c21(A__IF(a__sieve(a__sieve(mark(z0))), x1, x2), MARK(sieve(sieve(z0)))) MARK(if(sieve(from(z0)), x1, x2)) -> c21(A__IF(a__sieve(a__from(mark(z0))), x1, x2), MARK(sieve(from(z0)))) MARK(if(sieve(head(z0)), x1, x2)) -> c21(A__IF(a__sieve(a__head(mark(z0))), x1, x2), MARK(sieve(head(z0)))) MARK(if(sieve(tail(z0)), x1, x2)) -> c21(A__IF(a__sieve(a__tail(mark(z0))), x1, x2), MARK(sieve(tail(z0)))) MARK(if(sieve(if(z0, z1, z2)), x1, x2)) -> c21(A__IF(a__sieve(a__if(mark(z0), z1, z2)), x1, x2), MARK(sieve(if(z0, z1, z2)))) MARK(if(sieve(filter(z0, z1)), x1, x2)) -> c21(A__IF(a__sieve(a__filter(mark(z0), mark(z1))), x1, x2), MARK(sieve(filter(z0, z1)))) MARK(if(sieve(s(z0)), x1, x2)) -> c21(A__IF(a__sieve(s(mark(z0))), x1, x2), MARK(sieve(s(z0)))) MARK(if(sieve(0), x1, x2)) -> c21(A__IF(a__sieve(0), x1, x2), MARK(sieve(0))) MARK(if(sieve(cons(z0, z1)), x1, x2)) -> c21(A__IF(a__sieve(cons(mark(z0), z1)), x1, x2), MARK(sieve(cons(z0, z1)))) MARK(if(sieve(true), x1, x2)) -> c21(A__IF(a__sieve(true), x1, x2), MARK(sieve(true))) MARK(if(sieve(false), x1, x2)) -> c21(A__IF(a__sieve(false), x1, x2), MARK(sieve(false))) MARK(if(sieve(divides(z0, z1)), x1, x2)) -> c21(A__IF(a__sieve(divides(mark(z0), mark(z1))), x1, x2), MARK(sieve(divides(z0, z1)))) S tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(from(z0), x1, x2)) -> c21(A__IF(a__from(mark(z0)), x1, x2), MARK(from(z0))) MARK(if(head(z0), x1, x2)) -> c21(A__IF(a__head(mark(z0)), x1, x2), MARK(head(z0))) MARK(if(tail(z0), x1, x2)) -> c21(A__IF(a__tail(mark(z0)), x1, x2), MARK(tail(z0))) MARK(if(if(z0, z1, z2), x1, x2)) -> c21(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(filter(z0, z1), x1, x2)) -> c21(A__IF(a__filter(mark(z0), mark(z1)), x1, x2), MARK(filter(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c22(A__FILTER(mark(x0), a__primes), MARK(x0)) MARK(filter(x0, sieve(z0))) -> c22(A__FILTER(mark(x0), a__sieve(mark(z0))), 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MARK(tail(filter(s(z0), x1))) -> c20(A__TAIL(a__filter(s(mark(z0)), mark(x1))), MARK(filter(s(z0), x1))) MARK(tail(filter(0, x1))) -> c20(A__TAIL(a__filter(0, mark(x1))), MARK(filter(0, x1))) MARK(tail(filter(cons(z0, z1), x1))) -> c20(A__TAIL(a__filter(cons(mark(z0), z1), mark(x1))), MARK(filter(cons(z0, z1), x1))) MARK(tail(filter(true, x1))) -> c20(A__TAIL(a__filter(true, mark(x1))), MARK(filter(true, x1))) MARK(tail(filter(false, x1))) -> c20(A__TAIL(a__filter(false, mark(x1))), MARK(filter(false, x1))) MARK(tail(filter(divides(z0, z1), x1))) -> c20(A__TAIL(a__filter(divides(mark(z0), mark(z1)), mark(x1))), MARK(filter(divides(z0, z1), x1))) MARK(tail(filter(x0, x1))) -> c20(MARK(filter(x0, x1))) MARK(if(primes, x0, x1)) -> c21(A__IF(a__sieve(a__from(s(s(0)))), x0, x1), MARK(primes)) MARK(if(primes, x0, x1)) -> c21(MARK(primes)) MARK(if(sieve(x0), x1, x2)) -> c21(A__IF(sieve(mark(x0)), x1, x2), MARK(sieve(x0))) MARK(if(sieve(primes), x1, x2)) -> c21(A__IF(a__sieve(a__primes), x1, x2), MARK(sieve(primes))) MARK(if(sieve(sieve(z0)), x1, x2)) -> c21(A__IF(a__sieve(a__sieve(mark(z0))), x1, x2), MARK(sieve(sieve(z0)))) MARK(if(sieve(from(z0)), x1, x2)) -> c21(A__IF(a__sieve(a__from(mark(z0))), x1, x2), MARK(sieve(from(z0)))) MARK(if(sieve(head(z0)), x1, x2)) -> c21(A__IF(a__sieve(a__head(mark(z0))), x1, x2), MARK(sieve(head(z0)))) MARK(if(sieve(tail(z0)), x1, x2)) -> c21(A__IF(a__sieve(a__tail(mark(z0))), x1, x2), MARK(sieve(tail(z0)))) MARK(if(sieve(if(z0, z1, z2)), x1, x2)) -> c21(A__IF(a__sieve(a__if(mark(z0), z1, z2)), x1, x2), MARK(sieve(if(z0, z1, z2)))) MARK(if(sieve(filter(z0, z1)), x1, x2)) -> c21(A__IF(a__sieve(a__filter(mark(z0), mark(z1))), x1, x2), MARK(sieve(filter(z0, z1)))) MARK(if(sieve(s(z0)), x1, x2)) -> c21(A__IF(a__sieve(s(mark(z0))), x1, x2), MARK(sieve(s(z0)))) MARK(if(sieve(0), x1, x2)) -> c21(A__IF(a__sieve(0), x1, x2), MARK(sieve(0))) MARK(if(sieve(cons(z0, z1)), x1, x2)) -> c21(A__IF(a__sieve(cons(mark(z0), z1)), x1, x2), MARK(sieve(cons(z0, z1)))) MARK(if(sieve(true), x1, x2)) -> c21(A__IF(a__sieve(true), x1, x2), MARK(sieve(true))) MARK(if(sieve(false), x1, x2)) -> c21(A__IF(a__sieve(false), x1, x2), MARK(sieve(false))) MARK(if(sieve(divides(z0, z1)), x1, x2)) -> c21(A__IF(a__sieve(divides(mark(z0), mark(z1))), x1, x2), MARK(sieve(divides(z0, z1)))) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c18_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_1, c17_2, c17_1, c19_2, c19_1, c20_2, c20_1, c21_2, c21_1, c22_2, c22_1, c23_2, c23_1, c_2 ---------------------------------------- (189) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing tuple parts ---------------------------------------- (190) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(from(z0), x1, x2)) -> c21(A__IF(a__from(mark(z0)), x1, x2), MARK(from(z0))) MARK(if(head(z0), x1, x2)) -> c21(A__IF(a__head(mark(z0)), x1, x2), MARK(head(z0))) MARK(if(tail(z0), x1, x2)) -> c21(A__IF(a__tail(mark(z0)), x1, x2), MARK(tail(z0))) MARK(if(if(z0, z1, z2), x1, x2)) -> c21(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(filter(z0, z1), x1, x2)) -> c21(A__IF(a__filter(mark(z0), mark(z1)), x1, x2), MARK(filter(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c22(A__FILTER(mark(x0), a__primes), MARK(x0)) MARK(filter(x0, sieve(z0))) -> c22(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, from(z0))) -> c22(A__FILTER(mark(x0), a__from(mark(z0))), MARK(x0)) MARK(filter(x0, head(z0))) -> c22(A__FILTER(mark(x0), a__head(mark(z0))), MARK(x0)) MARK(filter(x0, tail(z0))) -> c22(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(x0)) MARK(filter(x0, if(z0, z1, z2))) -> c22(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0)) MARK(filter(x0, filter(z0, z1))) -> c22(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(x0)) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(x0, s(z0))) -> c22(MARK(x0)) MARK(filter(x0, 0)) -> c22(MARK(x0)) MARK(filter(x0, true)) -> c22(MARK(x0)) MARK(filter(x0, false)) -> c22(MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), x1)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c23(A__FILTER(mark(x0), a__primes), MARK(primes)) MARK(filter(x0, sieve(z0))) -> c23(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, from(z0))) -> c23(A__FILTER(mark(x0), a__from(mark(z0))), MARK(from(z0))) MARK(filter(x0, head(z0))) -> c23(A__FILTER(mark(x0), a__head(mark(z0))), MARK(head(z0))) MARK(filter(x0, tail(z0))) -> c23(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(tail(z0))) MARK(filter(x0, if(z0, z1, z2))) -> c23(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(filter(x0, filter(z0, z1))) -> c23(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(filter(x0, cons(z0, z1))) -> c23(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(primes, x1)) -> c23(A__FILTER(a__primes, mark(x1)), MARK(x1)) MARK(filter(sieve(z0), x1)) -> c23(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(from(z0), x1)) -> c23(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(head(z0), x1)) -> c23(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(tail(z0), x1)) -> c23(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(if(z0, z1, z2), x1)) -> c23(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1)) MARK(filter(filter(z0, z1), x1)) -> c23(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(x1)) MARK(filter(s(z0), x1)) -> c23(A__FILTER(s(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(x0, s(z0))) -> c23(MARK(s(z0))) MARK(filter(x0, divides(z0, z1))) -> c23(MARK(divides(z0, z1))) MARK(filter(0, x1)) -> c23(MARK(x1)) MARK(filter(cons(z0, z1), x1)) -> c23(MARK(x1)) MARK(filter(true, x1)) -> c23(MARK(x1)) MARK(filter(false, x1)) -> c23(MARK(x1)) MARK(filter(divides(z0, z1), x1)) -> c23(MARK(x1)) A__PRIMES -> c(A__SIEVE(cons(s(mark(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(sieve(primes)) -> c17(MARK(primes)) MARK(sieve(sieve(primes))) -> c17(A__SIEVE(a__sieve(a__primes)), MARK(sieve(primes))) MARK(sieve(sieve(sieve(z0)))) -> c17(A__SIEVE(a__sieve(a__sieve(mark(z0)))), MARK(sieve(sieve(z0)))) MARK(sieve(sieve(from(z0)))) -> c17(A__SIEVE(a__sieve(a__from(mark(z0)))), MARK(sieve(from(z0)))) MARK(sieve(sieve(head(z0)))) -> c17(A__SIEVE(a__sieve(a__head(mark(z0)))), MARK(sieve(head(z0)))) MARK(sieve(sieve(tail(z0)))) -> c17(A__SIEVE(a__sieve(a__tail(mark(z0)))), MARK(sieve(tail(z0)))) MARK(sieve(sieve(if(z0, z1, z2)))) -> c17(A__SIEVE(a__sieve(a__if(mark(z0), z1, z2))), MARK(sieve(if(z0, z1, z2)))) MARK(sieve(sieve(filter(z0, z1)))) -> c17(A__SIEVE(a__sieve(a__filter(mark(z0), mark(z1)))), MARK(sieve(filter(z0, z1)))) MARK(sieve(sieve(s(z0)))) -> c17(A__SIEVE(a__sieve(s(mark(z0)))), MARK(sieve(s(z0)))) MARK(sieve(sieve(cons(z0, z1)))) -> c17(A__SIEVE(a__sieve(cons(mark(z0), z1))), MARK(sieve(cons(z0, z1)))) MARK(sieve(sieve(divides(z0, z1)))) -> c17(A__SIEVE(a__sieve(divides(mark(z0), mark(z1)))), MARK(sieve(divides(z0, z1)))) MARK(sieve(sieve(x0))) -> c17(MARK(sieve(x0))) MARK(sieve(sieve(0))) -> c17(A__SIEVE(a__sieve(0))) MARK(sieve(sieve(true))) -> c17(A__SIEVE(a__sieve(true))) MARK(sieve(sieve(false))) -> c17(A__SIEVE(a__sieve(false))) MARK(sieve(from(x0))) -> c17(A__SIEVE(cons(mark(mark(x0)), from(s(mark(x0))))), MARK(from(x0))) MARK(sieve(from(primes))) -> c17(A__SIEVE(a__from(a__primes)), MARK(from(primes))) MARK(sieve(from(sieve(z0)))) -> c17(A__SIEVE(a__from(a__sieve(mark(z0)))), MARK(from(sieve(z0)))) MARK(sieve(from(from(z0)))) -> c17(A__SIEVE(a__from(a__from(mark(z0)))), MARK(from(from(z0)))) MARK(sieve(from(head(z0)))) -> c17(A__SIEVE(a__from(a__head(mark(z0)))), MARK(from(head(z0)))) MARK(sieve(from(tail(z0)))) -> c17(A__SIEVE(a__from(a__tail(mark(z0)))), MARK(from(tail(z0)))) MARK(sieve(from(if(z0, z1, z2)))) -> c17(A__SIEVE(a__from(a__if(mark(z0), z1, z2))), MARK(from(if(z0, z1, z2)))) MARK(sieve(from(filter(z0, z1)))) -> c17(A__SIEVE(a__from(a__filter(mark(z0), mark(z1)))), MARK(from(filter(z0, z1)))) MARK(sieve(from(s(z0)))) -> c17(A__SIEVE(a__from(s(mark(z0)))), MARK(from(s(z0)))) MARK(sieve(from(0))) -> c17(A__SIEVE(a__from(0)), MARK(from(0))) MARK(sieve(from(cons(z0, z1)))) -> c17(A__SIEVE(a__from(cons(mark(z0), z1))), 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MARK(tail(if(0, x1, x2))) -> c20(A__TAIL(a__if(0, x1, x2))) MARK(tail(filter(x0, primes))) -> c20(A__TAIL(a__filter(mark(x0), a__primes)), MARK(filter(x0, primes))) MARK(tail(filter(x0, sieve(z0)))) -> c20(A__TAIL(a__filter(mark(x0), a__sieve(mark(z0)))), MARK(filter(x0, sieve(z0)))) MARK(tail(filter(x0, from(z0)))) -> c20(A__TAIL(a__filter(mark(x0), a__from(mark(z0)))), MARK(filter(x0, from(z0)))) MARK(tail(filter(x0, head(z0)))) -> c20(A__TAIL(a__filter(mark(x0), a__head(mark(z0)))), MARK(filter(x0, head(z0)))) MARK(tail(filter(x0, tail(z0)))) -> c20(A__TAIL(a__filter(mark(x0), a__tail(mark(z0)))), MARK(filter(x0, tail(z0)))) MARK(tail(filter(x0, if(z0, z1, z2)))) -> c20(A__TAIL(a__filter(mark(x0), a__if(mark(z0), z1, z2))), MARK(filter(x0, if(z0, z1, z2)))) MARK(tail(filter(x0, filter(z0, z1)))) -> c20(A__TAIL(a__filter(mark(x0), a__filter(mark(z0), mark(z1)))), MARK(filter(x0, filter(z0, z1)))) MARK(tail(filter(x0, s(z0)))) -> c20(A__TAIL(a__filter(mark(x0), s(mark(z0)))), MARK(filter(x0, s(z0)))) MARK(tail(filter(x0, 0))) -> c20(A__TAIL(a__filter(mark(x0), 0)), MARK(filter(x0, 0))) MARK(tail(filter(x0, cons(z0, z1)))) -> c20(A__TAIL(a__filter(mark(x0), cons(mark(z0), z1))), MARK(filter(x0, cons(z0, z1)))) MARK(tail(filter(x0, true))) -> c20(A__TAIL(a__filter(mark(x0), true)), MARK(filter(x0, true))) MARK(tail(filter(x0, false))) -> c20(A__TAIL(a__filter(mark(x0), false)), MARK(filter(x0, false))) MARK(tail(filter(x0, divides(z0, z1)))) -> c20(A__TAIL(a__filter(mark(x0), divides(mark(z0), mark(z1)))), MARK(filter(x0, divides(z0, z1)))) MARK(tail(filter(primes, x1))) -> c20(A__TAIL(a__filter(a__primes, mark(x1))), MARK(filter(primes, x1))) MARK(tail(filter(sieve(z0), x1))) -> c20(A__TAIL(a__filter(a__sieve(mark(z0)), mark(x1))), MARK(filter(sieve(z0), x1))) MARK(tail(filter(from(z0), x1))) -> c20(A__TAIL(a__filter(a__from(mark(z0)), mark(x1))), MARK(filter(from(z0), x1))) MARK(tail(filter(head(z0), x1))) -> c20(A__TAIL(a__filter(a__head(mark(z0)), mark(x1))), MARK(filter(head(z0), x1))) MARK(tail(filter(tail(z0), x1))) -> c20(A__TAIL(a__filter(a__tail(mark(z0)), mark(x1))), MARK(filter(tail(z0), x1))) MARK(tail(filter(if(z0, z1, z2), x1))) -> c20(A__TAIL(a__filter(a__if(mark(z0), z1, z2), mark(x1))), MARK(filter(if(z0, z1, z2), x1))) MARK(tail(filter(filter(z0, z1), x1))) -> c20(A__TAIL(a__filter(a__filter(mark(z0), mark(z1)), mark(x1))), MARK(filter(filter(z0, z1), x1))) MARK(tail(filter(s(z0), x1))) -> c20(A__TAIL(a__filter(s(mark(z0)), mark(x1))), MARK(filter(s(z0), x1))) MARK(tail(filter(0, x1))) -> c20(A__TAIL(a__filter(0, mark(x1))), MARK(filter(0, x1))) MARK(tail(filter(cons(z0, z1), x1))) -> c20(A__TAIL(a__filter(cons(mark(z0), z1), mark(x1))), MARK(filter(cons(z0, z1), x1))) MARK(tail(filter(true, x1))) -> c20(A__TAIL(a__filter(true, mark(x1))), MARK(filter(true, x1))) MARK(tail(filter(false, x1))) -> c20(A__TAIL(a__filter(false, mark(x1))), MARK(filter(false, x1))) MARK(tail(filter(divides(z0, z1), x1))) -> c20(A__TAIL(a__filter(divides(mark(z0), mark(z1)), mark(x1))), MARK(filter(divides(z0, z1), x1))) MARK(tail(filter(x0, x1))) -> c20(MARK(filter(x0, x1))) MARK(if(primes, x0, x1)) -> c21(A__IF(a__sieve(a__from(s(s(0)))), x0, x1), MARK(primes)) MARK(if(primes, x0, x1)) -> c21(MARK(primes)) MARK(if(sieve(primes), x1, x2)) -> c21(A__IF(a__sieve(a__primes), x1, x2), MARK(sieve(primes))) MARK(if(sieve(sieve(z0)), x1, x2)) -> c21(A__IF(a__sieve(a__sieve(mark(z0))), x1, x2), MARK(sieve(sieve(z0)))) MARK(if(sieve(from(z0)), x1, x2)) -> c21(A__IF(a__sieve(a__from(mark(z0))), x1, x2), MARK(sieve(from(z0)))) MARK(if(sieve(head(z0)), x1, x2)) -> c21(A__IF(a__sieve(a__head(mark(z0))), x1, x2), MARK(sieve(head(z0)))) MARK(if(sieve(tail(z0)), x1, x2)) -> c21(A__IF(a__sieve(a__tail(mark(z0))), x1, x2), MARK(sieve(tail(z0)))) MARK(if(sieve(if(z0, z1, z2)), x1, x2)) -> c21(A__IF(a__sieve(a__if(mark(z0), z1, z2)), x1, x2), MARK(sieve(if(z0, z1, z2)))) MARK(if(sieve(filter(z0, z1)), x1, x2)) -> c21(A__IF(a__sieve(a__filter(mark(z0), mark(z1))), x1, x2), MARK(sieve(filter(z0, z1)))) MARK(if(sieve(s(z0)), x1, x2)) -> c21(A__IF(a__sieve(s(mark(z0))), x1, x2), MARK(sieve(s(z0)))) MARK(if(sieve(cons(z0, z1)), x1, x2)) -> c21(A__IF(a__sieve(cons(mark(z0), z1)), x1, x2), MARK(sieve(cons(z0, z1)))) MARK(if(sieve(divides(z0, z1)), x1, x2)) -> c21(A__IF(a__sieve(divides(mark(z0), mark(z1))), x1, x2), MARK(sieve(divides(z0, z1)))) MARK(if(sieve(x0), x1, x2)) -> c21(MARK(sieve(x0))) MARK(if(sieve(0), x1, x2)) -> c21(A__IF(a__sieve(0), x1, x2)) MARK(if(sieve(true), x1, x2)) -> c21(A__IF(a__sieve(true), x1, x2)) MARK(if(sieve(false), x1, x2)) -> c21(A__IF(a__sieve(false), x1, x2)) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c18_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_1, c17_2, c17_1, c19_2, c19_1, c20_2, c20_1, c21_2, c21_1, c22_2, c22_1, c23_2, c23_1, c_2 ---------------------------------------- (191) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(if(from(z0), x1, x2)) -> c21(A__IF(a__from(mark(z0)), x1, x2), MARK(from(z0))) by MARK(if(from(x0), x1, x2)) -> c21(A__IF(cons(mark(mark(x0)), from(s(mark(x0)))), x1, x2), MARK(from(x0))) MARK(if(from(x0), x1, x2)) -> c21(A__IF(from(mark(x0)), x1, x2), MARK(from(x0))) MARK(if(from(primes), x1, x2)) -> c21(A__IF(a__from(a__primes), x1, x2), MARK(from(primes))) MARK(if(from(sieve(z0)), x1, x2)) -> c21(A__IF(a__from(a__sieve(mark(z0))), x1, x2), MARK(from(sieve(z0)))) MARK(if(from(from(z0)), x1, x2)) -> c21(A__IF(a__from(a__from(mark(z0))), x1, x2), MARK(from(from(z0)))) MARK(if(from(head(z0)), x1, x2)) -> c21(A__IF(a__from(a__head(mark(z0))), x1, x2), MARK(from(head(z0)))) MARK(if(from(tail(z0)), x1, x2)) -> c21(A__IF(a__from(a__tail(mark(z0))), x1, x2), MARK(from(tail(z0)))) MARK(if(from(if(z0, z1, z2)), x1, x2)) -> c21(A__IF(a__from(a__if(mark(z0), z1, z2)), x1, x2), MARK(from(if(z0, z1, z2)))) MARK(if(from(filter(z0, z1)), x1, x2)) -> c21(A__IF(a__from(a__filter(mark(z0), mark(z1))), x1, x2), MARK(from(filter(z0, z1)))) MARK(if(from(s(z0)), x1, x2)) -> c21(A__IF(a__from(s(mark(z0))), x1, x2), MARK(from(s(z0)))) MARK(if(from(0), x1, x2)) -> c21(A__IF(a__from(0), x1, x2), MARK(from(0))) MARK(if(from(cons(z0, z1)), x1, x2)) -> c21(A__IF(a__from(cons(mark(z0), z1)), x1, x2), MARK(from(cons(z0, z1)))) MARK(if(from(true), x1, x2)) -> c21(A__IF(a__from(true), x1, x2), MARK(from(true))) MARK(if(from(false), x1, x2)) -> c21(A__IF(a__from(false), x1, x2), MARK(from(false))) MARK(if(from(divides(z0, z1)), x1, x2)) -> c21(A__IF(a__from(divides(mark(z0), mark(z1))), x1, x2), MARK(from(divides(z0, z1)))) MARK(if(from(x0), x1, x2)) -> c21(MARK(from(x0))) ---------------------------------------- (192) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(head(z0), x1, x2)) -> c21(A__IF(a__head(mark(z0)), x1, x2), MARK(head(z0))) MARK(if(tail(z0), x1, x2)) -> c21(A__IF(a__tail(mark(z0)), x1, x2), MARK(tail(z0))) MARK(if(if(z0, z1, z2), x1, x2)) -> c21(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(filter(z0, z1), x1, x2)) -> c21(A__IF(a__filter(mark(z0), mark(z1)), x1, x2), MARK(filter(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c22(A__FILTER(mark(x0), a__primes), MARK(x0)) MARK(filter(x0, sieve(z0))) -> c22(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, from(z0))) -> c22(A__FILTER(mark(x0), a__from(mark(z0))), MARK(x0)) MARK(filter(x0, head(z0))) -> c22(A__FILTER(mark(x0), a__head(mark(z0))), MARK(x0)) MARK(filter(x0, tail(z0))) -> c22(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(x0)) MARK(filter(x0, if(z0, z1, z2))) -> c22(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0)) MARK(filter(x0, filter(z0, z1))) -> c22(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(x0)) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(x0, s(z0))) -> c22(MARK(x0)) MARK(filter(x0, 0)) -> c22(MARK(x0)) MARK(filter(x0, true)) -> c22(MARK(x0)) MARK(filter(x0, false)) -> c22(MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), x1)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c23(A__FILTER(mark(x0), a__primes), MARK(primes)) MARK(filter(x0, sieve(z0))) -> c23(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, from(z0))) -> c23(A__FILTER(mark(x0), a__from(mark(z0))), MARK(from(z0))) MARK(filter(x0, head(z0))) -> c23(A__FILTER(mark(x0), a__head(mark(z0))), MARK(head(z0))) 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c17(A__SIEVE(a__from(a__from(mark(z0)))), MARK(from(from(z0)))) MARK(sieve(from(head(z0)))) -> c17(A__SIEVE(a__from(a__head(mark(z0)))), MARK(from(head(z0)))) MARK(sieve(from(tail(z0)))) -> c17(A__SIEVE(a__from(a__tail(mark(z0)))), MARK(from(tail(z0)))) MARK(sieve(from(if(z0, z1, z2)))) -> c17(A__SIEVE(a__from(a__if(mark(z0), z1, z2))), MARK(from(if(z0, z1, z2)))) MARK(sieve(from(filter(z0, z1)))) -> c17(A__SIEVE(a__from(a__filter(mark(z0), mark(z1)))), MARK(from(filter(z0, z1)))) MARK(sieve(from(s(z0)))) -> c17(A__SIEVE(a__from(s(mark(z0)))), MARK(from(s(z0)))) MARK(sieve(from(0))) -> c17(A__SIEVE(a__from(0)), MARK(from(0))) MARK(sieve(from(cons(z0, z1)))) -> c17(A__SIEVE(a__from(cons(mark(z0), z1))), MARK(from(cons(z0, z1)))) MARK(sieve(from(true))) -> c17(A__SIEVE(a__from(true)), MARK(from(true))) MARK(sieve(from(false))) -> c17(A__SIEVE(a__from(false)), MARK(from(false))) MARK(sieve(from(divides(z0, z1)))) -> c17(A__SIEVE(a__from(divides(mark(z0), mark(z1)))), MARK(from(divides(z0, 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MARK(tail(from(s(z0)))) -> c20(A__TAIL(a__from(s(mark(z0)))), MARK(from(s(z0)))) MARK(tail(from(0))) -> c20(A__TAIL(a__from(0)), MARK(from(0))) MARK(tail(from(cons(z0, z1)))) -> c20(A__TAIL(a__from(cons(mark(z0), z1))), MARK(from(cons(z0, z1)))) MARK(tail(from(true))) -> c20(A__TAIL(a__from(true)), MARK(from(true))) MARK(tail(from(false))) -> c20(A__TAIL(a__from(false)), MARK(from(false))) MARK(tail(from(divides(z0, z1)))) -> c20(A__TAIL(a__from(divides(mark(z0), mark(z1)))), MARK(from(divides(z0, z1)))) MARK(tail(from(x0))) -> c20(MARK(from(x0))) MARK(tail(head(primes))) -> c20(A__TAIL(a__head(a__primes)), MARK(head(primes))) MARK(tail(head(sieve(z0)))) -> c20(A__TAIL(a__head(a__sieve(mark(z0)))), MARK(head(sieve(z0)))) MARK(tail(head(from(z0)))) -> c20(A__TAIL(a__head(a__from(mark(z0)))), MARK(head(from(z0)))) MARK(tail(head(head(z0)))) -> c20(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c20(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(if(z0, z1, z2)))) -> c20(A__TAIL(a__head(a__if(mark(z0), z1, z2))), MARK(head(if(z0, z1, z2)))) MARK(tail(head(filter(z0, z1)))) -> c20(A__TAIL(a__head(a__filter(mark(z0), mark(z1)))), MARK(head(filter(z0, z1)))) MARK(tail(head(s(z0)))) -> c20(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(cons(z0, z1)))) -> c20(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(divides(z0, z1)))) -> c20(A__TAIL(a__head(divides(mark(z0), mark(z1)))), MARK(head(divides(z0, z1)))) MARK(tail(head(x0))) -> c20(MARK(head(x0))) MARK(tail(head(0))) -> c20(A__TAIL(a__head(0))) MARK(tail(head(true))) -> c20(A__TAIL(a__head(true))) MARK(tail(head(false))) -> c20(A__TAIL(a__head(false))) MARK(tail(tail(primes))) -> c20(A__TAIL(a__tail(a__primes)), MARK(tail(primes))) MARK(tail(tail(sieve(z0)))) -> c20(A__TAIL(a__tail(a__sieve(mark(z0)))), MARK(tail(sieve(z0)))) MARK(tail(tail(from(z0)))) -> c20(A__TAIL(a__tail(a__from(mark(z0)))), MARK(tail(from(z0)))) MARK(tail(tail(head(z0)))) -> c20(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c20(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(if(z0, z1, z2)))) -> c20(A__TAIL(a__tail(a__if(mark(z0), z1, z2))), MARK(tail(if(z0, z1, z2)))) MARK(tail(tail(filter(z0, z1)))) -> c20(A__TAIL(a__tail(a__filter(mark(z0), mark(z1)))), MARK(tail(filter(z0, z1)))) MARK(tail(tail(s(z0)))) -> c20(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c20(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(divides(z0, z1)))) -> c20(A__TAIL(a__tail(divides(mark(z0), mark(z1)))), MARK(tail(divides(z0, z1)))) MARK(tail(tail(x0))) -> c20(MARK(tail(x0))) MARK(tail(tail(0))) -> c20(A__TAIL(a__tail(0))) MARK(tail(tail(true))) -> c20(A__TAIL(a__tail(true))) MARK(tail(tail(false))) -> c20(A__TAIL(a__tail(false))) MARK(tail(if(primes, x1, x2))) -> c20(A__TAIL(a__if(a__primes, x1, x2)), MARK(if(primes, x1, x2))) MARK(tail(if(sieve(z0), x1, x2))) -> c20(A__TAIL(a__if(a__sieve(mark(z0)), x1, x2)), MARK(if(sieve(z0), x1, x2))) MARK(tail(if(from(z0), x1, x2))) -> c20(A__TAIL(a__if(a__from(mark(z0)), x1, x2)), MARK(if(from(z0), x1, x2))) MARK(tail(if(head(z0), x1, x2))) -> c20(A__TAIL(a__if(a__head(mark(z0)), x1, x2)), MARK(if(head(z0), x1, x2))) MARK(tail(if(tail(z0), x1, x2))) -> c20(A__TAIL(a__if(a__tail(mark(z0)), x1, x2)), MARK(if(tail(z0), x1, x2))) MARK(tail(if(if(z0, z1, z2), x1, x2))) -> c20(A__TAIL(a__if(a__if(mark(z0), z1, z2), x1, x2)), MARK(if(if(z0, z1, z2), x1, x2))) MARK(tail(if(filter(z0, z1), x1, x2))) -> c20(A__TAIL(a__if(a__filter(mark(z0), mark(z1)), x1, x2)), MARK(if(filter(z0, z1), x1, x2))) MARK(tail(if(s(z0), x1, x2))) -> c20(A__TAIL(a__if(s(mark(z0)), x1, x2)), MARK(if(s(z0), x1, x2))) MARK(tail(if(cons(z0, z1), x1, x2))) -> c20(A__TAIL(a__if(cons(mark(z0), z1), x1, x2)), MARK(if(cons(z0, z1), x1, x2))) MARK(tail(if(true, x1, x2))) -> c20(A__TAIL(a__if(true, x1, x2)), MARK(if(true, x1, x2))) MARK(tail(if(false, x1, x2))) -> c20(A__TAIL(a__if(false, x1, x2)), MARK(if(false, x1, x2))) MARK(tail(if(divides(z0, z1), x1, x2))) -> c20(A__TAIL(a__if(divides(mark(z0), mark(z1)), x1, x2)), MARK(if(divides(z0, z1), x1, x2))) MARK(tail(if(x0, z1, z2))) -> c20(MARK(if(x0, z1, z2))) MARK(tail(if(0, x1, x2))) -> c20(A__TAIL(a__if(0, x1, x2))) MARK(tail(filter(x0, primes))) -> c20(A__TAIL(a__filter(mark(x0), a__primes)), MARK(filter(x0, primes))) MARK(tail(filter(x0, sieve(z0)))) -> c20(A__TAIL(a__filter(mark(x0), a__sieve(mark(z0)))), MARK(filter(x0, sieve(z0)))) MARK(tail(filter(x0, from(z0)))) -> c20(A__TAIL(a__filter(mark(x0), a__from(mark(z0)))), MARK(filter(x0, from(z0)))) MARK(tail(filter(x0, head(z0)))) -> c20(A__TAIL(a__filter(mark(x0), a__head(mark(z0)))), MARK(filter(x0, head(z0)))) MARK(tail(filter(x0, tail(z0)))) -> c20(A__TAIL(a__filter(mark(x0), a__tail(mark(z0)))), MARK(filter(x0, tail(z0)))) MARK(tail(filter(x0, if(z0, z1, z2)))) -> c20(A__TAIL(a__filter(mark(x0), a__if(mark(z0), z1, z2))), MARK(filter(x0, if(z0, z1, z2)))) MARK(tail(filter(x0, filter(z0, z1)))) -> c20(A__TAIL(a__filter(mark(x0), a__filter(mark(z0), mark(z1)))), MARK(filter(x0, filter(z0, z1)))) MARK(tail(filter(x0, s(z0)))) -> c20(A__TAIL(a__filter(mark(x0), s(mark(z0)))), MARK(filter(x0, s(z0)))) MARK(tail(filter(x0, 0))) -> c20(A__TAIL(a__filter(mark(x0), 0)), MARK(filter(x0, 0))) MARK(tail(filter(x0, cons(z0, z1)))) -> c20(A__TAIL(a__filter(mark(x0), cons(mark(z0), z1))), MARK(filter(x0, cons(z0, z1)))) MARK(tail(filter(x0, true))) -> c20(A__TAIL(a__filter(mark(x0), true)), MARK(filter(x0, true))) MARK(tail(filter(x0, false))) -> c20(A__TAIL(a__filter(mark(x0), false)), MARK(filter(x0, false))) MARK(tail(filter(x0, divides(z0, z1)))) -> c20(A__TAIL(a__filter(mark(x0), divides(mark(z0), mark(z1)))), MARK(filter(x0, divides(z0, z1)))) MARK(tail(filter(primes, x1))) -> c20(A__TAIL(a__filter(a__primes, mark(x1))), MARK(filter(primes, x1))) MARK(tail(filter(sieve(z0), x1))) -> c20(A__TAIL(a__filter(a__sieve(mark(z0)), mark(x1))), MARK(filter(sieve(z0), x1))) MARK(tail(filter(from(z0), x1))) -> c20(A__TAIL(a__filter(a__from(mark(z0)), mark(x1))), MARK(filter(from(z0), x1))) MARK(tail(filter(head(z0), x1))) -> c20(A__TAIL(a__filter(a__head(mark(z0)), mark(x1))), MARK(filter(head(z0), x1))) MARK(tail(filter(tail(z0), x1))) -> c20(A__TAIL(a__filter(a__tail(mark(z0)), mark(x1))), MARK(filter(tail(z0), x1))) MARK(tail(filter(if(z0, z1, z2), x1))) -> c20(A__TAIL(a__filter(a__if(mark(z0), z1, z2), mark(x1))), MARK(filter(if(z0, z1, z2), x1))) MARK(tail(filter(filter(z0, z1), x1))) -> c20(A__TAIL(a__filter(a__filter(mark(z0), mark(z1)), mark(x1))), MARK(filter(filter(z0, z1), x1))) MARK(tail(filter(s(z0), x1))) -> c20(A__TAIL(a__filter(s(mark(z0)), mark(x1))), MARK(filter(s(z0), x1))) MARK(tail(filter(0, x1))) -> c20(A__TAIL(a__filter(0, mark(x1))), MARK(filter(0, x1))) MARK(tail(filter(cons(z0, z1), x1))) -> c20(A__TAIL(a__filter(cons(mark(z0), z1), mark(x1))), MARK(filter(cons(z0, z1), x1))) MARK(tail(filter(true, x1))) -> c20(A__TAIL(a__filter(true, mark(x1))), MARK(filter(true, x1))) MARK(tail(filter(false, x1))) -> c20(A__TAIL(a__filter(false, mark(x1))), MARK(filter(false, x1))) MARK(tail(filter(divides(z0, z1), x1))) -> c20(A__TAIL(a__filter(divides(mark(z0), mark(z1)), mark(x1))), MARK(filter(divides(z0, z1), x1))) MARK(tail(filter(x0, x1))) -> c20(MARK(filter(x0, x1))) MARK(if(primes, x0, x1)) -> c21(A__IF(a__sieve(a__from(s(s(0)))), x0, x1), MARK(primes)) MARK(if(primes, x0, x1)) -> c21(MARK(primes)) MARK(if(sieve(primes), x1, x2)) -> c21(A__IF(a__sieve(a__primes), x1, x2), MARK(sieve(primes))) MARK(if(sieve(sieve(z0)), x1, x2)) -> c21(A__IF(a__sieve(a__sieve(mark(z0))), x1, x2), MARK(sieve(sieve(z0)))) MARK(if(sieve(from(z0)), x1, x2)) -> c21(A__IF(a__sieve(a__from(mark(z0))), x1, x2), MARK(sieve(from(z0)))) MARK(if(sieve(head(z0)), x1, x2)) -> c21(A__IF(a__sieve(a__head(mark(z0))), x1, x2), MARK(sieve(head(z0)))) MARK(if(sieve(tail(z0)), x1, x2)) -> c21(A__IF(a__sieve(a__tail(mark(z0))), x1, x2), MARK(sieve(tail(z0)))) MARK(if(sieve(if(z0, z1, z2)), x1, x2)) -> c21(A__IF(a__sieve(a__if(mark(z0), z1, z2)), x1, x2), MARK(sieve(if(z0, z1, z2)))) MARK(if(sieve(filter(z0, z1)), x1, x2)) -> c21(A__IF(a__sieve(a__filter(mark(z0), mark(z1))), x1, x2), MARK(sieve(filter(z0, z1)))) MARK(if(sieve(s(z0)), x1, x2)) -> c21(A__IF(a__sieve(s(mark(z0))), x1, x2), MARK(sieve(s(z0)))) MARK(if(sieve(cons(z0, z1)), x1, x2)) -> c21(A__IF(a__sieve(cons(mark(z0), z1)), x1, x2), MARK(sieve(cons(z0, z1)))) MARK(if(sieve(divides(z0, z1)), x1, x2)) -> c21(A__IF(a__sieve(divides(mark(z0), mark(z1))), x1, x2), MARK(sieve(divides(z0, z1)))) MARK(if(sieve(x0), x1, x2)) -> c21(MARK(sieve(x0))) MARK(if(sieve(0), x1, x2)) -> c21(A__IF(a__sieve(0), x1, x2)) MARK(if(sieve(true), x1, x2)) -> c21(A__IF(a__sieve(true), x1, x2)) MARK(if(sieve(false), x1, x2)) -> c21(A__IF(a__sieve(false), x1, x2)) MARK(if(from(x0), x1, x2)) -> c21(A__IF(cons(mark(mark(x0)), from(s(mark(x0)))), x1, x2), MARK(from(x0))) MARK(if(from(x0), x1, x2)) -> c21(A__IF(from(mark(x0)), x1, x2), MARK(from(x0))) MARK(if(from(primes), x1, x2)) -> c21(A__IF(a__from(a__primes), x1, x2), MARK(from(primes))) MARK(if(from(sieve(z0)), x1, x2)) -> c21(A__IF(a__from(a__sieve(mark(z0))), x1, x2), MARK(from(sieve(z0)))) MARK(if(from(from(z0)), x1, x2)) -> c21(A__IF(a__from(a__from(mark(z0))), x1, x2), MARK(from(from(z0)))) MARK(if(from(head(z0)), x1, x2)) -> c21(A__IF(a__from(a__head(mark(z0))), x1, x2), MARK(from(head(z0)))) MARK(if(from(tail(z0)), x1, x2)) -> c21(A__IF(a__from(a__tail(mark(z0))), x1, x2), MARK(from(tail(z0)))) MARK(if(from(if(z0, z1, z2)), x1, x2)) -> c21(A__IF(a__from(a__if(mark(z0), z1, z2)), x1, x2), MARK(from(if(z0, z1, z2)))) MARK(if(from(filter(z0, z1)), x1, x2)) -> c21(A__IF(a__from(a__filter(mark(z0), mark(z1))), x1, x2), MARK(from(filter(z0, z1)))) MARK(if(from(s(z0)), x1, x2)) -> c21(A__IF(a__from(s(mark(z0))), x1, x2), MARK(from(s(z0)))) MARK(if(from(0), x1, x2)) -> c21(A__IF(a__from(0), x1, x2), MARK(from(0))) MARK(if(from(cons(z0, z1)), x1, x2)) -> c21(A__IF(a__from(cons(mark(z0), z1)), x1, x2), MARK(from(cons(z0, z1)))) MARK(if(from(true), x1, x2)) -> c21(A__IF(a__from(true), x1, x2), MARK(from(true))) MARK(if(from(false), x1, x2)) -> c21(A__IF(a__from(false), x1, x2), MARK(from(false))) MARK(if(from(divides(z0, z1)), x1, x2)) -> c21(A__IF(a__from(divides(mark(z0), mark(z1))), x1, x2), MARK(from(divides(z0, z1)))) MARK(if(from(x0), x1, x2)) -> c21(MARK(from(x0))) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c18_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_1, c17_2, c17_1, c19_2, c19_1, c20_2, c20_1, c21_2, c21_1, c22_2, c22_1, c23_2, c23_1, c_2 ---------------------------------------- (193) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (194) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(head(z0), x1, x2)) -> c21(A__IF(a__head(mark(z0)), x1, x2), MARK(head(z0))) MARK(if(tail(z0), x1, x2)) -> c21(A__IF(a__tail(mark(z0)), x1, x2), MARK(tail(z0))) MARK(if(if(z0, z1, z2), x1, x2)) -> c21(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(filter(z0, z1), x1, x2)) -> c21(A__IF(a__filter(mark(z0), mark(z1)), x1, x2), MARK(filter(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c22(A__FILTER(mark(x0), a__primes), MARK(x0)) MARK(filter(x0, sieve(z0))) -> c22(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, from(z0))) -> c22(A__FILTER(mark(x0), a__from(mark(z0))), MARK(x0)) MARK(filter(x0, head(z0))) -> c22(A__FILTER(mark(x0), a__head(mark(z0))), MARK(x0)) MARK(filter(x0, tail(z0))) -> c22(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(x0)) MARK(filter(x0, if(z0, z1, z2))) -> c22(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0)) MARK(filter(x0, filter(z0, z1))) -> c22(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(x0)) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(x0, s(z0))) -> c22(MARK(x0)) MARK(filter(x0, 0)) -> c22(MARK(x0)) MARK(filter(x0, true)) -> c22(MARK(x0)) MARK(filter(x0, false)) -> c22(MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), x1)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c23(A__FILTER(mark(x0), a__primes), MARK(primes)) MARK(filter(x0, sieve(z0))) -> c23(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, from(z0))) -> c23(A__FILTER(mark(x0), a__from(mark(z0))), MARK(from(z0))) MARK(filter(x0, head(z0))) -> c23(A__FILTER(mark(x0), a__head(mark(z0))), MARK(head(z0))) MARK(filter(x0, tail(z0))) -> c23(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(tail(z0))) MARK(filter(x0, if(z0, z1, z2))) -> c23(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(filter(x0, filter(z0, z1))) -> c23(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(filter(x0, cons(z0, z1))) -> c23(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(primes, x1)) -> c23(A__FILTER(a__primes, mark(x1)), MARK(x1)) MARK(filter(sieve(z0), x1)) -> c23(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(from(z0), x1)) -> c23(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(head(z0), x1)) -> c23(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(tail(z0), x1)) -> c23(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(if(z0, z1, z2), x1)) -> c23(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1)) MARK(filter(filter(z0, z1), x1)) -> c23(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(x1)) MARK(filter(s(z0), x1)) -> c23(A__FILTER(s(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(x0, s(z0))) -> c23(MARK(s(z0))) MARK(filter(x0, divides(z0, z1))) -> c23(MARK(divides(z0, z1))) MARK(filter(0, x1)) -> c23(MARK(x1)) MARK(filter(cons(z0, z1), x1)) -> c23(MARK(x1)) MARK(filter(true, x1)) -> c23(MARK(x1)) MARK(filter(false, x1)) -> c23(MARK(x1)) MARK(filter(divides(z0, z1), x1)) -> c23(MARK(x1)) A__PRIMES -> c(A__SIEVE(cons(s(mark(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(sieve(primes)) -> c17(MARK(primes)) MARK(sieve(sieve(primes))) -> c17(A__SIEVE(a__sieve(a__primes)), MARK(sieve(primes))) MARK(sieve(sieve(sieve(z0)))) -> c17(A__SIEVE(a__sieve(a__sieve(mark(z0)))), MARK(sieve(sieve(z0)))) MARK(sieve(sieve(from(z0)))) -> c17(A__SIEVE(a__sieve(a__from(mark(z0)))), MARK(sieve(from(z0)))) MARK(sieve(sieve(head(z0)))) -> c17(A__SIEVE(a__sieve(a__head(mark(z0)))), MARK(sieve(head(z0)))) MARK(sieve(sieve(tail(z0)))) -> c17(A__SIEVE(a__sieve(a__tail(mark(z0)))), MARK(sieve(tail(z0)))) MARK(sieve(sieve(if(z0, z1, z2)))) -> c17(A__SIEVE(a__sieve(a__if(mark(z0), z1, z2))), MARK(sieve(if(z0, z1, z2)))) MARK(sieve(sieve(filter(z0, z1)))) -> c17(A__SIEVE(a__sieve(a__filter(mark(z0), mark(z1)))), MARK(sieve(filter(z0, z1)))) MARK(sieve(sieve(s(z0)))) -> c17(A__SIEVE(a__sieve(s(mark(z0)))), MARK(sieve(s(z0)))) MARK(sieve(sieve(cons(z0, z1)))) -> c17(A__SIEVE(a__sieve(cons(mark(z0), z1))), MARK(sieve(cons(z0, z1)))) MARK(sieve(sieve(divides(z0, z1)))) -> c17(A__SIEVE(a__sieve(divides(mark(z0), mark(z1)))), MARK(sieve(divides(z0, z1)))) MARK(sieve(sieve(x0))) -> c17(MARK(sieve(x0))) MARK(sieve(sieve(0))) -> c17(A__SIEVE(a__sieve(0))) MARK(sieve(sieve(true))) -> c17(A__SIEVE(a__sieve(true))) MARK(sieve(sieve(false))) -> c17(A__SIEVE(a__sieve(false))) MARK(sieve(from(x0))) -> c17(A__SIEVE(cons(mark(mark(x0)), from(s(mark(x0))))), MARK(from(x0))) MARK(sieve(from(primes))) -> c17(A__SIEVE(a__from(a__primes)), MARK(from(primes))) MARK(sieve(from(sieve(z0)))) -> c17(A__SIEVE(a__from(a__sieve(mark(z0)))), MARK(from(sieve(z0)))) MARK(sieve(from(from(z0)))) -> c17(A__SIEVE(a__from(a__from(mark(z0)))), MARK(from(from(z0)))) MARK(sieve(from(head(z0)))) -> c17(A__SIEVE(a__from(a__head(mark(z0)))), MARK(from(head(z0)))) MARK(sieve(from(tail(z0)))) -> c17(A__SIEVE(a__from(a__tail(mark(z0)))), MARK(from(tail(z0)))) MARK(sieve(from(if(z0, z1, z2)))) -> c17(A__SIEVE(a__from(a__if(mark(z0), z1, z2))), MARK(from(if(z0, z1, z2)))) MARK(sieve(from(filter(z0, z1)))) -> c17(A__SIEVE(a__from(a__filter(mark(z0), mark(z1)))), MARK(from(filter(z0, z1)))) MARK(sieve(from(s(z0)))) -> c17(A__SIEVE(a__from(s(mark(z0)))), MARK(from(s(z0)))) MARK(sieve(from(0))) -> c17(A__SIEVE(a__from(0)), MARK(from(0))) MARK(sieve(from(cons(z0, z1)))) -> c17(A__SIEVE(a__from(cons(mark(z0), z1))), MARK(from(cons(z0, z1)))) MARK(sieve(from(true))) -> c17(A__SIEVE(a__from(true)), MARK(from(true))) MARK(sieve(from(false))) -> c17(A__SIEVE(a__from(false)), MARK(from(false))) MARK(sieve(from(divides(z0, z1)))) -> c17(A__SIEVE(a__from(divides(mark(z0), mark(z1)))), MARK(from(divides(z0, z1)))) MARK(sieve(from(x0))) -> c17(MARK(from(x0))) MARK(sieve(head(primes))) -> c17(A__SIEVE(a__head(a__primes)), MARK(head(primes))) MARK(sieve(head(sieve(z0)))) -> c17(A__SIEVE(a__head(a__sieve(mark(z0)))), MARK(head(sieve(z0)))) MARK(sieve(head(from(z0)))) -> c17(A__SIEVE(a__head(a__from(mark(z0)))), MARK(head(from(z0)))) MARK(sieve(head(head(z0)))) -> c17(A__SIEVE(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(sieve(head(tail(z0)))) -> c17(A__SIEVE(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(sieve(head(if(z0, z1, z2)))) -> c17(A__SIEVE(a__head(a__if(mark(z0), z1, z2))), MARK(head(if(z0, z1, z2)))) MARK(sieve(head(filter(z0, z1)))) -> c17(A__SIEVE(a__head(a__filter(mark(z0), mark(z1)))), MARK(head(filter(z0, z1)))) MARK(sieve(head(s(z0)))) -> c17(A__SIEVE(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(sieve(head(cons(z0, z1)))) -> c17(A__SIEVE(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(sieve(head(divides(z0, z1)))) -> c17(A__SIEVE(a__head(divides(mark(z0), mark(z1)))), MARK(head(divides(z0, z1)))) MARK(sieve(head(x0))) -> c17(MARK(head(x0))) MARK(sieve(head(0))) -> c17(A__SIEVE(a__head(0))) MARK(sieve(head(true))) -> c17(A__SIEVE(a__head(true))) MARK(sieve(head(false))) -> c17(A__SIEVE(a__head(false))) MARK(sieve(tail(primes))) -> c17(A__SIEVE(a__tail(a__primes)), MARK(tail(primes))) MARK(sieve(tail(sieve(z0)))) -> c17(A__SIEVE(a__tail(a__sieve(mark(z0)))), MARK(tail(sieve(z0)))) MARK(sieve(tail(from(z0)))) -> c17(A__SIEVE(a__tail(a__from(mark(z0)))), MARK(tail(from(z0)))) MARK(sieve(tail(head(z0)))) -> c17(A__SIEVE(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(sieve(tail(tail(z0)))) -> c17(A__SIEVE(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(sieve(tail(if(z0, z1, z2)))) -> c17(A__SIEVE(a__tail(a__if(mark(z0), z1, z2))), MARK(tail(if(z0, z1, z2)))) MARK(sieve(tail(filter(z0, z1)))) -> c17(A__SIEVE(a__tail(a__filter(mark(z0), mark(z1)))), MARK(tail(filter(z0, z1)))) MARK(sieve(tail(s(z0)))) -> c17(A__SIEVE(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(sieve(tail(cons(z0, z1)))) -> c17(A__SIEVE(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(sieve(tail(divides(z0, z1)))) -> c17(A__SIEVE(a__tail(divides(mark(z0), mark(z1)))), MARK(tail(divides(z0, z1)))) MARK(sieve(tail(x0))) -> c17(MARK(tail(x0))) MARK(sieve(tail(0))) -> c17(A__SIEVE(a__tail(0))) MARK(sieve(tail(true))) -> c17(A__SIEVE(a__tail(true))) MARK(sieve(tail(false))) -> c17(A__SIEVE(a__tail(false))) MARK(sieve(if(primes, x1, x2))) -> c17(A__SIEVE(a__if(a__primes, x1, x2)), MARK(if(primes, x1, x2))) MARK(sieve(if(sieve(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__sieve(mark(z0)), x1, x2)), MARK(if(sieve(z0), x1, x2))) MARK(sieve(if(from(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__from(mark(z0)), x1, x2)), MARK(if(from(z0), x1, x2))) MARK(sieve(if(head(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__head(mark(z0)), x1, x2)), MARK(if(head(z0), x1, x2))) MARK(sieve(if(tail(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__tail(mark(z0)), x1, x2)), MARK(if(tail(z0), x1, x2))) MARK(sieve(if(if(z0, z1, z2), x1, x2))) -> c17(A__SIEVE(a__if(a__if(mark(z0), z1, z2), x1, x2)), MARK(if(if(z0, z1, z2), x1, x2))) MARK(sieve(if(filter(z0, z1), x1, x2))) -> c17(A__SIEVE(a__if(a__filter(mark(z0), mark(z1)), x1, x2)), MARK(if(filter(z0, z1), x1, x2))) MARK(sieve(if(s(z0), x1, x2))) -> c17(A__SIEVE(a__if(s(mark(z0)), x1, x2)), MARK(if(s(z0), x1, x2))) MARK(sieve(if(cons(z0, z1), x1, x2))) -> c17(A__SIEVE(a__if(cons(mark(z0), z1), x1, x2)), MARK(if(cons(z0, z1), x1, x2))) MARK(sieve(if(true, x1, x2))) -> c17(A__SIEVE(a__if(true, x1, x2)), MARK(if(true, x1, x2))) MARK(sieve(if(false, x1, x2))) -> c17(A__SIEVE(a__if(false, x1, x2)), MARK(if(false, x1, x2))) MARK(sieve(if(divides(z0, z1), x1, x2))) -> c17(A__SIEVE(a__if(divides(mark(z0), mark(z1)), x1, x2)), MARK(if(divides(z0, z1), x1, x2))) MARK(sieve(if(x0, z1, z2))) -> c17(MARK(if(x0, z1, z2))) MARK(sieve(if(0, x1, x2))) -> c17(A__SIEVE(a__if(0, x1, x2))) MARK(sieve(filter(x0, primes))) -> c17(A__SIEVE(a__filter(mark(x0), a__primes)), MARK(filter(x0, primes))) MARK(sieve(filter(x0, sieve(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), a__sieve(mark(z0)))), MARK(filter(x0, sieve(z0)))) MARK(sieve(filter(x0, from(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), a__from(mark(z0)))), MARK(filter(x0, from(z0)))) MARK(sieve(filter(x0, head(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), a__head(mark(z0)))), MARK(filter(x0, head(z0)))) MARK(sieve(filter(x0, tail(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), a__tail(mark(z0)))), MARK(filter(x0, tail(z0)))) MARK(sieve(filter(x0, if(z0, z1, z2)))) -> c17(A__SIEVE(a__filter(mark(x0), a__if(mark(z0), z1, z2))), MARK(filter(x0, if(z0, z1, z2)))) MARK(sieve(filter(x0, filter(z0, z1)))) -> c17(A__SIEVE(a__filter(mark(x0), a__filter(mark(z0), mark(z1)))), MARK(filter(x0, filter(z0, z1)))) MARK(sieve(filter(x0, s(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), s(mark(z0)))), MARK(filter(x0, s(z0)))) MARK(sieve(filter(x0, 0))) -> c17(A__SIEVE(a__filter(mark(x0), 0)), MARK(filter(x0, 0))) MARK(sieve(filter(x0, cons(z0, z1)))) -> c17(A__SIEVE(a__filter(mark(x0), cons(mark(z0), z1))), MARK(filter(x0, cons(z0, z1)))) MARK(sieve(filter(x0, true))) -> c17(A__SIEVE(a__filter(mark(x0), true)), MARK(filter(x0, true))) MARK(sieve(filter(x0, false))) -> c17(A__SIEVE(a__filter(mark(x0), false)), MARK(filter(x0, false))) MARK(sieve(filter(x0, divides(z0, z1)))) -> c17(A__SIEVE(a__filter(mark(x0), 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c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(x0, s(z0))) -> c22(MARK(x0)) MARK(filter(x0, 0)) -> c22(MARK(x0)) MARK(filter(x0, true)) -> c22(MARK(x0)) MARK(filter(x0, false)) -> c22(MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), x1)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c23(A__FILTER(mark(x0), a__primes), MARK(primes)) MARK(filter(x0, sieve(z0))) -> c23(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, from(z0))) -> c23(A__FILTER(mark(x0), a__from(mark(z0))), MARK(from(z0))) MARK(filter(x0, head(z0))) -> c23(A__FILTER(mark(x0), a__head(mark(z0))), MARK(head(z0))) MARK(filter(x0, tail(z0))) -> c23(A__FILTER(mark(x0), 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x2)) -> c21(A__IF(a__sieve(a__from(mark(z0))), x1, x2), MARK(sieve(from(z0)))) MARK(if(sieve(head(z0)), x1, x2)) -> c21(A__IF(a__sieve(a__head(mark(z0))), x1, x2), MARK(sieve(head(z0)))) MARK(if(sieve(tail(z0)), x1, x2)) -> c21(A__IF(a__sieve(a__tail(mark(z0))), x1, x2), MARK(sieve(tail(z0)))) MARK(if(sieve(if(z0, z1, z2)), x1, x2)) -> c21(A__IF(a__sieve(a__if(mark(z0), z1, z2)), x1, x2), MARK(sieve(if(z0, z1, z2)))) MARK(if(sieve(filter(z0, z1)), x1, x2)) -> c21(A__IF(a__sieve(a__filter(mark(z0), mark(z1))), x1, x2), MARK(sieve(filter(z0, z1)))) MARK(if(sieve(s(z0)), x1, x2)) -> c21(A__IF(a__sieve(s(mark(z0))), x1, x2), MARK(sieve(s(z0)))) MARK(if(sieve(cons(z0, z1)), x1, x2)) -> c21(A__IF(a__sieve(cons(mark(z0), z1)), x1, x2), MARK(sieve(cons(z0, z1)))) MARK(if(sieve(divides(z0, z1)), x1, x2)) -> c21(A__IF(a__sieve(divides(mark(z0), mark(z1))), x1, x2), MARK(sieve(divides(z0, z1)))) MARK(if(sieve(x0), x1, x2)) -> c21(MARK(sieve(x0))) MARK(if(sieve(0), x1, x2)) -> c21(A__IF(a__sieve(0), x1, x2)) MARK(if(sieve(true), x1, x2)) -> c21(A__IF(a__sieve(true), x1, x2)) MARK(if(sieve(false), x1, x2)) -> c21(A__IF(a__sieve(false), x1, x2)) MARK(if(from(primes), x1, x2)) -> c21(A__IF(a__from(a__primes), x1, x2), MARK(from(primes))) MARK(if(from(sieve(z0)), x1, x2)) -> c21(A__IF(a__from(a__sieve(mark(z0))), x1, x2), MARK(from(sieve(z0)))) MARK(if(from(from(z0)), x1, x2)) -> c21(A__IF(a__from(a__from(mark(z0))), x1, x2), MARK(from(from(z0)))) MARK(if(from(head(z0)), x1, x2)) -> c21(A__IF(a__from(a__head(mark(z0))), x1, x2), MARK(from(head(z0)))) MARK(if(from(tail(z0)), x1, x2)) -> c21(A__IF(a__from(a__tail(mark(z0))), x1, x2), MARK(from(tail(z0)))) MARK(if(from(if(z0, z1, z2)), x1, x2)) -> c21(A__IF(a__from(a__if(mark(z0), z1, z2)), x1, x2), MARK(from(if(z0, z1, z2)))) MARK(if(from(filter(z0, z1)), x1, x2)) -> c21(A__IF(a__from(a__filter(mark(z0), mark(z1))), x1, x2), MARK(from(filter(z0, z1)))) MARK(if(from(s(z0)), x1, x2)) -> c21(A__IF(a__from(s(mark(z0))), x1, x2), MARK(from(s(z0)))) MARK(if(from(0), x1, x2)) -> c21(A__IF(a__from(0), x1, x2), MARK(from(0))) MARK(if(from(cons(z0, z1)), x1, x2)) -> c21(A__IF(a__from(cons(mark(z0), z1)), x1, x2), MARK(from(cons(z0, z1)))) MARK(if(from(true), x1, x2)) -> c21(A__IF(a__from(true), x1, x2), MARK(from(true))) MARK(if(from(false), x1, x2)) -> c21(A__IF(a__from(false), x1, x2), MARK(from(false))) MARK(if(from(divides(z0, z1)), x1, x2)) -> c21(A__IF(a__from(divides(mark(z0), mark(z1))), x1, x2), MARK(from(divides(z0, z1)))) MARK(if(from(x0), x1, x2)) -> c21(MARK(from(x0))) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c18_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_1, c17_2, c17_1, c19_2, c19_1, c20_2, c20_1, c21_2, c21_1, c22_2, c22_1, c23_2, c23_1, c_2 ---------------------------------------- (195) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(if(head(z0), x1, x2)) -> c21(A__IF(a__head(mark(z0)), x1, x2), MARK(head(z0))) by MARK(if(head(x0), x1, x2)) -> c21(A__IF(head(mark(x0)), x1, x2), MARK(head(x0))) MARK(if(head(primes), x1, x2)) -> c21(A__IF(a__head(a__primes), x1, x2), MARK(head(primes))) MARK(if(head(sieve(z0)), x1, x2)) -> c21(A__IF(a__head(a__sieve(mark(z0))), x1, x2), MARK(head(sieve(z0)))) MARK(if(head(from(z0)), x1, x2)) -> c21(A__IF(a__head(a__from(mark(z0))), x1, x2), MARK(head(from(z0)))) MARK(if(head(head(z0)), x1, x2)) -> c21(A__IF(a__head(a__head(mark(z0))), x1, x2), MARK(head(head(z0)))) MARK(if(head(tail(z0)), x1, x2)) -> c21(A__IF(a__head(a__tail(mark(z0))), x1, x2), MARK(head(tail(z0)))) MARK(if(head(if(z0, z1, z2)), x1, x2)) -> c21(A__IF(a__head(a__if(mark(z0), z1, z2)), x1, x2), MARK(head(if(z0, z1, z2)))) MARK(if(head(filter(z0, z1)), x1, x2)) -> c21(A__IF(a__head(a__filter(mark(z0), mark(z1))), x1, x2), MARK(head(filter(z0, z1)))) MARK(if(head(s(z0)), x1, x2)) -> c21(A__IF(a__head(s(mark(z0))), x1, x2), MARK(head(s(z0)))) MARK(if(head(0), x1, x2)) -> c21(A__IF(a__head(0), x1, x2), MARK(head(0))) MARK(if(head(cons(z0, z1)), x1, x2)) -> c21(A__IF(a__head(cons(mark(z0), z1)), x1, x2), MARK(head(cons(z0, z1)))) MARK(if(head(true), x1, x2)) -> c21(A__IF(a__head(true), x1, x2), MARK(head(true))) MARK(if(head(false), x1, x2)) -> c21(A__IF(a__head(false), x1, x2), MARK(head(false))) MARK(if(head(divides(z0, z1)), x1, x2)) -> c21(A__IF(a__head(divides(mark(z0), mark(z1))), x1, x2), MARK(head(divides(z0, z1)))) ---------------------------------------- (196) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(tail(z0), x1, x2)) -> c21(A__IF(a__tail(mark(z0)), x1, x2), MARK(tail(z0))) MARK(if(if(z0, z1, z2), x1, x2)) -> c21(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(filter(z0, z1), x1, x2)) -> c21(A__IF(a__filter(mark(z0), mark(z1)), x1, x2), MARK(filter(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c22(A__FILTER(mark(x0), a__primes), MARK(x0)) MARK(filter(x0, sieve(z0))) -> c22(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, from(z0))) -> c22(A__FILTER(mark(x0), a__from(mark(z0))), MARK(x0)) MARK(filter(x0, head(z0))) -> c22(A__FILTER(mark(x0), a__head(mark(z0))), MARK(x0)) MARK(filter(x0, tail(z0))) -> c22(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(x0)) MARK(filter(x0, if(z0, z1, z2))) -> c22(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0)) MARK(filter(x0, filter(z0, z1))) -> c22(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(x0)) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(x0, s(z0))) -> c22(MARK(x0)) MARK(filter(x0, 0)) -> c22(MARK(x0)) MARK(filter(x0, true)) -> c22(MARK(x0)) MARK(filter(x0, false)) -> c22(MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), x1)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c23(A__FILTER(mark(x0), a__primes), MARK(primes)) MARK(filter(x0, sieve(z0))) -> c23(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, from(z0))) -> c23(A__FILTER(mark(x0), a__from(mark(z0))), MARK(from(z0))) MARK(filter(x0, head(z0))) -> c23(A__FILTER(mark(x0), a__head(mark(z0))), MARK(head(z0))) MARK(filter(x0, tail(z0))) -> c23(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(tail(z0))) MARK(filter(x0, if(z0, z1, z2))) -> c23(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(filter(x0, filter(z0, z1))) -> c23(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(filter(x0, cons(z0, z1))) -> c23(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(primes, x1)) -> c23(A__FILTER(a__primes, mark(x1)), MARK(x1)) MARK(filter(sieve(z0), x1)) -> c23(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(from(z0), x1)) -> c23(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(head(z0), x1)) -> c23(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(tail(z0), x1)) -> c23(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(if(z0, z1, z2), x1)) -> c23(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1)) MARK(filter(filter(z0, z1), x1)) -> c23(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(x1)) MARK(filter(s(z0), x1)) -> c23(A__FILTER(s(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(x0, s(z0))) -> c23(MARK(s(z0))) MARK(filter(x0, divides(z0, z1))) -> c23(MARK(divides(z0, z1))) MARK(filter(0, x1)) -> c23(MARK(x1)) MARK(filter(cons(z0, z1), x1)) -> c23(MARK(x1)) MARK(filter(true, x1)) -> c23(MARK(x1)) MARK(filter(false, x1)) -> c23(MARK(x1)) MARK(filter(divides(z0, z1), x1)) -> c23(MARK(x1)) A__PRIMES -> c(A__SIEVE(cons(s(mark(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(sieve(primes)) -> c17(MARK(primes)) MARK(sieve(sieve(primes))) -> c17(A__SIEVE(a__sieve(a__primes)), MARK(sieve(primes))) MARK(sieve(sieve(sieve(z0)))) -> c17(A__SIEVE(a__sieve(a__sieve(mark(z0)))), MARK(sieve(sieve(z0)))) MARK(sieve(sieve(from(z0)))) -> c17(A__SIEVE(a__sieve(a__from(mark(z0)))), MARK(sieve(from(z0)))) MARK(sieve(sieve(head(z0)))) -> c17(A__SIEVE(a__sieve(a__head(mark(z0)))), MARK(sieve(head(z0)))) MARK(sieve(sieve(tail(z0)))) -> c17(A__SIEVE(a__sieve(a__tail(mark(z0)))), MARK(sieve(tail(z0)))) MARK(sieve(sieve(if(z0, z1, z2)))) -> c17(A__SIEVE(a__sieve(a__if(mark(z0), z1, z2))), MARK(sieve(if(z0, z1, z2)))) MARK(sieve(sieve(filter(z0, z1)))) -> c17(A__SIEVE(a__sieve(a__filter(mark(z0), mark(z1)))), MARK(sieve(filter(z0, z1)))) MARK(sieve(sieve(s(z0)))) -> c17(A__SIEVE(a__sieve(s(mark(z0)))), MARK(sieve(s(z0)))) MARK(sieve(sieve(cons(z0, z1)))) -> c17(A__SIEVE(a__sieve(cons(mark(z0), z1))), MARK(sieve(cons(z0, z1)))) MARK(sieve(sieve(divides(z0, z1)))) -> c17(A__SIEVE(a__sieve(divides(mark(z0), mark(z1)))), MARK(sieve(divides(z0, z1)))) MARK(sieve(sieve(x0))) -> c17(MARK(sieve(x0))) MARK(sieve(sieve(0))) -> c17(A__SIEVE(a__sieve(0))) MARK(sieve(sieve(true))) -> c17(A__SIEVE(a__sieve(true))) MARK(sieve(sieve(false))) -> c17(A__SIEVE(a__sieve(false))) MARK(sieve(from(x0))) -> c17(A__SIEVE(cons(mark(mark(x0)), from(s(mark(x0))))), MARK(from(x0))) MARK(sieve(from(primes))) -> c17(A__SIEVE(a__from(a__primes)), MARK(from(primes))) MARK(sieve(from(sieve(z0)))) -> c17(A__SIEVE(a__from(a__sieve(mark(z0)))), MARK(from(sieve(z0)))) MARK(sieve(from(from(z0)))) -> c17(A__SIEVE(a__from(a__from(mark(z0)))), MARK(from(from(z0)))) MARK(sieve(from(head(z0)))) -> c17(A__SIEVE(a__from(a__head(mark(z0)))), MARK(from(head(z0)))) MARK(sieve(from(tail(z0)))) -> c17(A__SIEVE(a__from(a__tail(mark(z0)))), MARK(from(tail(z0)))) MARK(sieve(from(if(z0, z1, z2)))) -> c17(A__SIEVE(a__from(a__if(mark(z0), z1, z2))), MARK(from(if(z0, z1, z2)))) MARK(sieve(from(filter(z0, z1)))) -> c17(A__SIEVE(a__from(a__filter(mark(z0), mark(z1)))), MARK(from(filter(z0, z1)))) MARK(sieve(from(s(z0)))) -> c17(A__SIEVE(a__from(s(mark(z0)))), MARK(from(s(z0)))) MARK(sieve(from(0))) -> c17(A__SIEVE(a__from(0)), MARK(from(0))) MARK(sieve(from(cons(z0, z1)))) -> c17(A__SIEVE(a__from(cons(mark(z0), z1))), MARK(from(cons(z0, z1)))) MARK(sieve(from(true))) -> c17(A__SIEVE(a__from(true)), MARK(from(true))) MARK(sieve(from(false))) -> c17(A__SIEVE(a__from(false)), MARK(from(false))) MARK(sieve(from(divides(z0, z1)))) -> c17(A__SIEVE(a__from(divides(mark(z0), mark(z1)))), MARK(from(divides(z0, z1)))) MARK(sieve(from(x0))) -> c17(MARK(from(x0))) MARK(sieve(head(primes))) -> 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c22(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(x0)) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(x0, s(z0))) -> c22(MARK(x0)) MARK(filter(x0, 0)) -> c22(MARK(x0)) MARK(filter(x0, true)) -> c22(MARK(x0)) MARK(filter(x0, false)) -> c22(MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), x1)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c23(A__FILTER(mark(x0), a__primes), MARK(primes)) MARK(filter(x0, sieve(z0))) -> c23(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, from(z0))) -> c23(A__FILTER(mark(x0), a__from(mark(z0))), MARK(from(z0))) MARK(filter(x0, head(z0))) -> c23(A__FILTER(mark(x0), a__head(mark(z0))), MARK(head(z0))) MARK(filter(x0, tail(z0))) -> c23(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(tail(z0))) MARK(filter(x0, if(z0, z1, z2))) -> c23(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(filter(x0, filter(z0, z1))) -> c23(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(filter(x0, cons(z0, z1))) -> c23(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(primes, x1)) -> c23(A__FILTER(a__primes, mark(x1)), MARK(x1)) MARK(filter(sieve(z0), x1)) -> c23(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(from(z0), x1)) -> c23(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(head(z0), x1)) -> c23(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(tail(z0), x1)) -> c23(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(if(z0, z1, z2), x1)) -> c23(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1)) MARK(filter(filter(z0, z1), x1)) -> c23(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(x1)) MARK(filter(s(z0), x1)) -> c23(A__FILTER(s(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(x0, s(z0))) -> c23(MARK(s(z0))) MARK(filter(x0, divides(z0, z1))) -> c23(MARK(divides(z0, z1))) MARK(filter(0, x1)) -> c23(MARK(x1)) MARK(filter(cons(z0, z1), x1)) -> c23(MARK(x1)) MARK(filter(true, x1)) -> c23(MARK(x1)) MARK(filter(false, x1)) -> c23(MARK(x1)) MARK(filter(divides(z0, z1), x1)) -> c23(MARK(x1)) A__PRIMES -> c(A__SIEVE(cons(s(mark(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(sieve(primes)) -> c17(MARK(primes)) MARK(sieve(sieve(primes))) -> c17(A__SIEVE(a__sieve(a__primes)), MARK(sieve(primes))) MARK(sieve(sieve(sieve(z0)))) -> c17(A__SIEVE(a__sieve(a__sieve(mark(z0)))), MARK(sieve(sieve(z0)))) MARK(sieve(sieve(from(z0)))) -> c17(A__SIEVE(a__sieve(a__from(mark(z0)))), MARK(sieve(from(z0)))) MARK(sieve(sieve(head(z0)))) -> c17(A__SIEVE(a__sieve(a__head(mark(z0)))), MARK(sieve(head(z0)))) MARK(sieve(sieve(tail(z0)))) -> c17(A__SIEVE(a__sieve(a__tail(mark(z0)))), MARK(sieve(tail(z0)))) MARK(sieve(sieve(if(z0, z1, z2)))) -> c17(A__SIEVE(a__sieve(a__if(mark(z0), z1, z2))), MARK(sieve(if(z0, z1, z2)))) MARK(sieve(sieve(filter(z0, z1)))) -> c17(A__SIEVE(a__sieve(a__filter(mark(z0), mark(z1)))), MARK(sieve(filter(z0, z1)))) MARK(sieve(sieve(s(z0)))) -> c17(A__SIEVE(a__sieve(s(mark(z0)))), MARK(sieve(s(z0)))) MARK(sieve(sieve(cons(z0, z1)))) -> c17(A__SIEVE(a__sieve(cons(mark(z0), z1))), MARK(sieve(cons(z0, z1)))) MARK(sieve(sieve(divides(z0, z1)))) -> c17(A__SIEVE(a__sieve(divides(mark(z0), mark(z1)))), MARK(sieve(divides(z0, z1)))) MARK(sieve(sieve(x0))) -> c17(MARK(sieve(x0))) MARK(sieve(sieve(0))) -> c17(A__SIEVE(a__sieve(0))) MARK(sieve(sieve(true))) -> c17(A__SIEVE(a__sieve(true))) MARK(sieve(sieve(false))) -> c17(A__SIEVE(a__sieve(false))) MARK(sieve(from(x0))) -> c17(A__SIEVE(cons(mark(mark(x0)), from(s(mark(x0))))), MARK(from(x0))) MARK(sieve(from(primes))) -> c17(A__SIEVE(a__from(a__primes)), MARK(from(primes))) MARK(sieve(from(sieve(z0)))) -> c17(A__SIEVE(a__from(a__sieve(mark(z0)))), MARK(from(sieve(z0)))) MARK(sieve(from(from(z0)))) -> c17(A__SIEVE(a__from(a__from(mark(z0)))), MARK(from(from(z0)))) MARK(sieve(from(head(z0)))) -> c17(A__SIEVE(a__from(a__head(mark(z0)))), MARK(from(head(z0)))) MARK(sieve(from(tail(z0)))) -> c17(A__SIEVE(a__from(a__tail(mark(z0)))), MARK(from(tail(z0)))) MARK(sieve(from(if(z0, z1, z2)))) -> c17(A__SIEVE(a__from(a__if(mark(z0), z1, z2))), MARK(from(if(z0, z1, z2)))) MARK(sieve(from(filter(z0, z1)))) -> c17(A__SIEVE(a__from(a__filter(mark(z0), mark(z1)))), MARK(from(filter(z0, z1)))) MARK(sieve(from(s(z0)))) -> c17(A__SIEVE(a__from(s(mark(z0)))), MARK(from(s(z0)))) MARK(sieve(from(0))) -> c17(A__SIEVE(a__from(0)), MARK(from(0))) MARK(sieve(from(cons(z0, z1)))) -> c17(A__SIEVE(a__from(cons(mark(z0), z1))), MARK(from(cons(z0, z1)))) MARK(sieve(from(true))) -> c17(A__SIEVE(a__from(true)), MARK(from(true))) MARK(sieve(from(false))) -> c17(A__SIEVE(a__from(false)), MARK(from(false))) MARK(sieve(from(divides(z0, z1)))) -> c17(A__SIEVE(a__from(divides(mark(z0), mark(z1)))), MARK(from(divides(z0, z1)))) MARK(sieve(from(x0))) -> c17(MARK(from(x0))) MARK(sieve(head(primes))) -> c17(A__SIEVE(a__head(a__primes)), MARK(head(primes))) MARK(sieve(head(sieve(z0)))) -> c17(A__SIEVE(a__head(a__sieve(mark(z0)))), MARK(head(sieve(z0)))) MARK(sieve(head(from(z0)))) -> c17(A__SIEVE(a__head(a__from(mark(z0)))), MARK(head(from(z0)))) MARK(sieve(head(head(z0)))) -> c17(A__SIEVE(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(sieve(head(tail(z0)))) -> c17(A__SIEVE(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(sieve(head(if(z0, z1, z2)))) -> c17(A__SIEVE(a__head(a__if(mark(z0), z1, z2))), MARK(head(if(z0, z1, z2)))) MARK(sieve(head(filter(z0, z1)))) -> c17(A__SIEVE(a__head(a__filter(mark(z0), mark(z1)))), MARK(head(filter(z0, z1)))) MARK(sieve(head(s(z0)))) -> c17(A__SIEVE(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(sieve(head(cons(z0, z1)))) -> c17(A__SIEVE(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(sieve(head(divides(z0, z1)))) -> c17(A__SIEVE(a__head(divides(mark(z0), mark(z1)))), MARK(head(divides(z0, z1)))) MARK(sieve(head(x0))) -> c17(MARK(head(x0))) MARK(sieve(head(0))) -> c17(A__SIEVE(a__head(0))) MARK(sieve(head(true))) -> c17(A__SIEVE(a__head(true))) MARK(sieve(head(false))) -> c17(A__SIEVE(a__head(false))) MARK(sieve(tail(primes))) -> c17(A__SIEVE(a__tail(a__primes)), MARK(tail(primes))) MARK(sieve(tail(sieve(z0)))) -> c17(A__SIEVE(a__tail(a__sieve(mark(z0)))), MARK(tail(sieve(z0)))) MARK(sieve(tail(from(z0)))) -> c17(A__SIEVE(a__tail(a__from(mark(z0)))), MARK(tail(from(z0)))) MARK(sieve(tail(head(z0)))) -> c17(A__SIEVE(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(sieve(tail(tail(z0)))) -> c17(A__SIEVE(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(sieve(tail(if(z0, z1, z2)))) -> c17(A__SIEVE(a__tail(a__if(mark(z0), z1, z2))), MARK(tail(if(z0, z1, z2)))) MARK(sieve(tail(filter(z0, z1)))) -> c17(A__SIEVE(a__tail(a__filter(mark(z0), mark(z1)))), MARK(tail(filter(z0, z1)))) MARK(sieve(tail(s(z0)))) -> c17(A__SIEVE(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(sieve(tail(cons(z0, z1)))) -> c17(A__SIEVE(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) 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c20(A__TAIL(a__from(cons(mark(z0), z1))), MARK(from(cons(z0, z1)))) MARK(tail(from(true))) -> c20(A__TAIL(a__from(true)), MARK(from(true))) MARK(tail(from(false))) -> c20(A__TAIL(a__from(false)), MARK(from(false))) MARK(tail(from(divides(z0, z1)))) -> c20(A__TAIL(a__from(divides(mark(z0), mark(z1)))), MARK(from(divides(z0, z1)))) MARK(tail(from(x0))) -> c20(MARK(from(x0))) MARK(tail(head(primes))) -> c20(A__TAIL(a__head(a__primes)), MARK(head(primes))) MARK(tail(head(sieve(z0)))) -> c20(A__TAIL(a__head(a__sieve(mark(z0)))), MARK(head(sieve(z0)))) MARK(tail(head(from(z0)))) -> c20(A__TAIL(a__head(a__from(mark(z0)))), MARK(head(from(z0)))) MARK(tail(head(head(z0)))) -> c20(A__TAIL(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(tail(head(tail(z0)))) -> c20(A__TAIL(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(tail(head(if(z0, z1, z2)))) -> c20(A__TAIL(a__head(a__if(mark(z0), z1, z2))), MARK(head(if(z0, z1, z2)))) MARK(tail(head(filter(z0, z1)))) -> c20(A__TAIL(a__head(a__filter(mark(z0), mark(z1)))), MARK(head(filter(z0, z1)))) MARK(tail(head(s(z0)))) -> c20(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(cons(z0, z1)))) -> c20(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(divides(z0, z1)))) -> c20(A__TAIL(a__head(divides(mark(z0), mark(z1)))), MARK(head(divides(z0, z1)))) MARK(tail(head(x0))) -> c20(MARK(head(x0))) MARK(tail(head(0))) -> c20(A__TAIL(a__head(0))) MARK(tail(head(true))) -> c20(A__TAIL(a__head(true))) MARK(tail(head(false))) -> c20(A__TAIL(a__head(false))) MARK(tail(tail(primes))) -> c20(A__TAIL(a__tail(a__primes)), MARK(tail(primes))) MARK(tail(tail(sieve(z0)))) -> c20(A__TAIL(a__tail(a__sieve(mark(z0)))), MARK(tail(sieve(z0)))) MARK(tail(tail(from(z0)))) -> c20(A__TAIL(a__tail(a__from(mark(z0)))), MARK(tail(from(z0)))) MARK(tail(tail(head(z0)))) -> c20(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c20(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(if(z0, z1, z2)))) -> c20(A__TAIL(a__tail(a__if(mark(z0), z1, z2))), MARK(tail(if(z0, z1, z2)))) MARK(tail(tail(filter(z0, z1)))) -> c20(A__TAIL(a__tail(a__filter(mark(z0), mark(z1)))), MARK(tail(filter(z0, z1)))) MARK(tail(tail(s(z0)))) -> c20(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c20(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(divides(z0, z1)))) -> c20(A__TAIL(a__tail(divides(mark(z0), mark(z1)))), MARK(tail(divides(z0, z1)))) MARK(tail(tail(x0))) -> c20(MARK(tail(x0))) MARK(tail(tail(0))) -> c20(A__TAIL(a__tail(0))) MARK(tail(tail(true))) -> c20(A__TAIL(a__tail(true))) MARK(tail(tail(false))) -> c20(A__TAIL(a__tail(false))) MARK(tail(if(primes, x1, x2))) -> c20(A__TAIL(a__if(a__primes, x1, x2)), MARK(if(primes, x1, x2))) MARK(tail(if(sieve(z0), x1, x2))) -> c20(A__TAIL(a__if(a__sieve(mark(z0)), x1, x2)), MARK(if(sieve(z0), x1, x2))) MARK(tail(if(from(z0), x1, x2))) -> c20(A__TAIL(a__if(a__from(mark(z0)), x1, x2)), MARK(if(from(z0), x1, x2))) MARK(tail(if(head(z0), x1, x2))) -> c20(A__TAIL(a__if(a__head(mark(z0)), x1, x2)), MARK(if(head(z0), x1, x2))) MARK(tail(if(tail(z0), x1, x2))) -> c20(A__TAIL(a__if(a__tail(mark(z0)), x1, x2)), MARK(if(tail(z0), x1, x2))) MARK(tail(if(if(z0, z1, z2), x1, x2))) -> c20(A__TAIL(a__if(a__if(mark(z0), z1, z2), x1, x2)), MARK(if(if(z0, z1, z2), x1, x2))) MARK(tail(if(filter(z0, z1), x1, x2))) -> c20(A__TAIL(a__if(a__filter(mark(z0), mark(z1)), x1, x2)), MARK(if(filter(z0, z1), x1, x2))) MARK(tail(if(s(z0), x1, x2))) -> c20(A__TAIL(a__if(s(mark(z0)), x1, x2)), MARK(if(s(z0), x1, x2))) MARK(tail(if(cons(z0, z1), x1, x2))) -> c20(A__TAIL(a__if(cons(mark(z0), z1), x1, x2)), MARK(if(cons(z0, z1), x1, x2))) MARK(tail(if(true, x1, x2))) -> c20(A__TAIL(a__if(true, x1, x2)), MARK(if(true, x1, x2))) MARK(tail(if(false, x1, x2))) -> c20(A__TAIL(a__if(false, x1, x2)), MARK(if(false, x1, x2))) MARK(tail(if(divides(z0, z1), x1, x2))) -> c20(A__TAIL(a__if(divides(mark(z0), mark(z1)), x1, x2)), MARK(if(divides(z0, z1), x1, x2))) MARK(tail(if(x0, z1, z2))) -> c20(MARK(if(x0, z1, z2))) MARK(tail(if(0, x1, x2))) -> c20(A__TAIL(a__if(0, x1, x2))) MARK(tail(filter(x0, primes))) -> c20(A__TAIL(a__filter(mark(x0), a__primes)), MARK(filter(x0, primes))) MARK(tail(filter(x0, sieve(z0)))) -> c20(A__TAIL(a__filter(mark(x0), a__sieve(mark(z0)))), MARK(filter(x0, sieve(z0)))) MARK(tail(filter(x0, from(z0)))) -> c20(A__TAIL(a__filter(mark(x0), a__from(mark(z0)))), MARK(filter(x0, from(z0)))) MARK(tail(filter(x0, head(z0)))) -> c20(A__TAIL(a__filter(mark(x0), a__head(mark(z0)))), MARK(filter(x0, head(z0)))) MARK(tail(filter(x0, tail(z0)))) -> c20(A__TAIL(a__filter(mark(x0), a__tail(mark(z0)))), MARK(filter(x0, tail(z0)))) MARK(tail(filter(x0, if(z0, z1, z2)))) -> c20(A__TAIL(a__filter(mark(x0), a__if(mark(z0), z1, z2))), MARK(filter(x0, if(z0, z1, z2)))) MARK(tail(filter(x0, filter(z0, z1)))) -> c20(A__TAIL(a__filter(mark(x0), a__filter(mark(z0), mark(z1)))), MARK(filter(x0, filter(z0, z1)))) MARK(tail(filter(x0, s(z0)))) -> c20(A__TAIL(a__filter(mark(x0), s(mark(z0)))), MARK(filter(x0, s(z0)))) MARK(tail(filter(x0, 0))) -> c20(A__TAIL(a__filter(mark(x0), 0)), MARK(filter(x0, 0))) MARK(tail(filter(x0, cons(z0, z1)))) -> c20(A__TAIL(a__filter(mark(x0), cons(mark(z0), z1))), MARK(filter(x0, cons(z0, z1)))) MARK(tail(filter(x0, true))) -> c20(A__TAIL(a__filter(mark(x0), true)), MARK(filter(x0, true))) MARK(tail(filter(x0, false))) -> c20(A__TAIL(a__filter(mark(x0), false)), MARK(filter(x0, false))) MARK(tail(filter(x0, divides(z0, z1)))) -> c20(A__TAIL(a__filter(mark(x0), divides(mark(z0), mark(z1)))), MARK(filter(x0, divides(z0, z1)))) MARK(tail(filter(primes, x1))) -> c20(A__TAIL(a__filter(a__primes, mark(x1))), MARK(filter(primes, x1))) MARK(tail(filter(sieve(z0), x1))) -> c20(A__TAIL(a__filter(a__sieve(mark(z0)), mark(x1))), MARK(filter(sieve(z0), x1))) MARK(tail(filter(from(z0), x1))) -> c20(A__TAIL(a__filter(a__from(mark(z0)), mark(x1))), MARK(filter(from(z0), x1))) MARK(tail(filter(head(z0), x1))) -> c20(A__TAIL(a__filter(a__head(mark(z0)), mark(x1))), MARK(filter(head(z0), x1))) MARK(tail(filter(tail(z0), x1))) -> c20(A__TAIL(a__filter(a__tail(mark(z0)), mark(x1))), MARK(filter(tail(z0), x1))) MARK(tail(filter(if(z0, z1, z2), x1))) -> c20(A__TAIL(a__filter(a__if(mark(z0), z1, z2), mark(x1))), MARK(filter(if(z0, z1, z2), x1))) MARK(tail(filter(filter(z0, z1), x1))) -> c20(A__TAIL(a__filter(a__filter(mark(z0), mark(z1)), mark(x1))), MARK(filter(filter(z0, z1), x1))) MARK(tail(filter(s(z0), x1))) -> c20(A__TAIL(a__filter(s(mark(z0)), mark(x1))), MARK(filter(s(z0), x1))) MARK(tail(filter(0, x1))) -> c20(A__TAIL(a__filter(0, mark(x1))), MARK(filter(0, x1))) MARK(tail(filter(cons(z0, z1), x1))) -> c20(A__TAIL(a__filter(cons(mark(z0), z1), mark(x1))), MARK(filter(cons(z0, z1), x1))) MARK(tail(filter(true, x1))) -> c20(A__TAIL(a__filter(true, mark(x1))), MARK(filter(true, x1))) MARK(tail(filter(false, x1))) -> c20(A__TAIL(a__filter(false, mark(x1))), MARK(filter(false, x1))) MARK(tail(filter(divides(z0, z1), x1))) -> c20(A__TAIL(a__filter(divides(mark(z0), mark(z1)), mark(x1))), MARK(filter(divides(z0, z1), x1))) MARK(tail(filter(x0, x1))) -> c20(MARK(filter(x0, x1))) MARK(if(primes, x0, x1)) -> c21(A__IF(a__sieve(a__from(s(s(0)))), x0, x1), MARK(primes)) MARK(if(primes, x0, x1)) -> c21(MARK(primes)) MARK(if(sieve(primes), x1, x2)) -> c21(A__IF(a__sieve(a__primes), x1, x2), MARK(sieve(primes))) MARK(if(sieve(sieve(z0)), x1, x2)) -> c21(A__IF(a__sieve(a__sieve(mark(z0))), x1, x2), MARK(sieve(sieve(z0)))) MARK(if(sieve(from(z0)), x1, x2)) -> c21(A__IF(a__sieve(a__from(mark(z0))), x1, x2), MARK(sieve(from(z0)))) MARK(if(sieve(head(z0)), x1, x2)) -> c21(A__IF(a__sieve(a__head(mark(z0))), x1, x2), MARK(sieve(head(z0)))) MARK(if(sieve(tail(z0)), x1, x2)) -> c21(A__IF(a__sieve(a__tail(mark(z0))), x1, x2), MARK(sieve(tail(z0)))) MARK(if(sieve(if(z0, z1, z2)), x1, x2)) -> c21(A__IF(a__sieve(a__if(mark(z0), z1, z2)), x1, x2), MARK(sieve(if(z0, z1, z2)))) MARK(if(sieve(filter(z0, z1)), x1, x2)) -> c21(A__IF(a__sieve(a__filter(mark(z0), mark(z1))), x1, x2), MARK(sieve(filter(z0, z1)))) MARK(if(sieve(s(z0)), x1, x2)) -> c21(A__IF(a__sieve(s(mark(z0))), x1, x2), MARK(sieve(s(z0)))) MARK(if(sieve(cons(z0, z1)), x1, x2)) -> c21(A__IF(a__sieve(cons(mark(z0), z1)), x1, x2), MARK(sieve(cons(z0, z1)))) MARK(if(sieve(divides(z0, z1)), x1, x2)) -> c21(A__IF(a__sieve(divides(mark(z0), mark(z1))), x1, x2), MARK(sieve(divides(z0, z1)))) MARK(if(sieve(x0), x1, x2)) -> c21(MARK(sieve(x0))) MARK(if(sieve(0), x1, x2)) -> c21(A__IF(a__sieve(0), x1, x2)) MARK(if(sieve(true), x1, x2)) -> c21(A__IF(a__sieve(true), x1, x2)) MARK(if(sieve(false), x1, x2)) -> c21(A__IF(a__sieve(false), x1, x2)) MARK(if(from(primes), x1, x2)) -> c21(A__IF(a__from(a__primes), x1, x2), MARK(from(primes))) MARK(if(from(sieve(z0)), x1, x2)) -> c21(A__IF(a__from(a__sieve(mark(z0))), x1, x2), MARK(from(sieve(z0)))) MARK(if(from(from(z0)), x1, x2)) -> c21(A__IF(a__from(a__from(mark(z0))), x1, x2), MARK(from(from(z0)))) MARK(if(from(head(z0)), x1, x2)) -> c21(A__IF(a__from(a__head(mark(z0))), x1, x2), MARK(from(head(z0)))) MARK(if(from(tail(z0)), x1, x2)) -> c21(A__IF(a__from(a__tail(mark(z0))), x1, x2), MARK(from(tail(z0)))) MARK(if(from(if(z0, z1, z2)), x1, x2)) -> c21(A__IF(a__from(a__if(mark(z0), z1, z2)), x1, x2), MARK(from(if(z0, z1, z2)))) MARK(if(from(filter(z0, z1)), x1, x2)) -> c21(A__IF(a__from(a__filter(mark(z0), mark(z1))), x1, x2), MARK(from(filter(z0, z1)))) MARK(if(from(s(z0)), x1, x2)) -> c21(A__IF(a__from(s(mark(z0))), x1, x2), MARK(from(s(z0)))) MARK(if(from(0), x1, x2)) -> c21(A__IF(a__from(0), x1, x2), MARK(from(0))) MARK(if(from(cons(z0, z1)), x1, x2)) -> c21(A__IF(a__from(cons(mark(z0), z1)), x1, x2), MARK(from(cons(z0, z1)))) MARK(if(from(true), x1, x2)) -> c21(A__IF(a__from(true), x1, x2), MARK(from(true))) MARK(if(from(false), x1, x2)) -> c21(A__IF(a__from(false), x1, x2), MARK(from(false))) MARK(if(from(divides(z0, z1)), x1, x2)) -> c21(A__IF(a__from(divides(mark(z0), mark(z1))), x1, x2), MARK(from(divides(z0, z1)))) MARK(if(from(x0), x1, x2)) -> c21(MARK(from(x0))) MARK(if(head(x0), x1, x2)) -> c21(A__IF(head(mark(x0)), x1, x2), MARK(head(x0))) MARK(if(head(primes), x1, x2)) -> c21(A__IF(a__head(a__primes), x1, x2), MARK(head(primes))) MARK(if(head(sieve(z0)), x1, x2)) -> c21(A__IF(a__head(a__sieve(mark(z0))), x1, x2), MARK(head(sieve(z0)))) MARK(if(head(from(z0)), x1, x2)) -> c21(A__IF(a__head(a__from(mark(z0))), x1, x2), MARK(head(from(z0)))) MARK(if(head(head(z0)), x1, x2)) -> c21(A__IF(a__head(a__head(mark(z0))), x1, x2), MARK(head(head(z0)))) MARK(if(head(tail(z0)), x1, x2)) -> c21(A__IF(a__head(a__tail(mark(z0))), x1, x2), MARK(head(tail(z0)))) MARK(if(head(if(z0, z1, z2)), x1, x2)) -> c21(A__IF(a__head(a__if(mark(z0), z1, z2)), x1, x2), MARK(head(if(z0, z1, z2)))) MARK(if(head(filter(z0, z1)), x1, x2)) -> c21(A__IF(a__head(a__filter(mark(z0), mark(z1))), x1, x2), MARK(head(filter(z0, z1)))) MARK(if(head(s(z0)), x1, x2)) -> c21(A__IF(a__head(s(mark(z0))), x1, x2), MARK(head(s(z0)))) MARK(if(head(0), x1, x2)) -> c21(A__IF(a__head(0), x1, x2), MARK(head(0))) MARK(if(head(cons(z0, z1)), x1, x2)) -> c21(A__IF(a__head(cons(mark(z0), z1)), x1, x2), MARK(head(cons(z0, z1)))) MARK(if(head(true), x1, x2)) -> c21(A__IF(a__head(true), x1, x2), MARK(head(true))) MARK(if(head(false), x1, x2)) -> c21(A__IF(a__head(false), x1, x2), MARK(head(false))) MARK(if(head(divides(z0, z1)), x1, x2)) -> c21(A__IF(a__head(divides(mark(z0), mark(z1))), x1, x2), MARK(head(divides(z0, z1)))) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c18_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_1, c17_2, c17_1, c19_2, c19_1, c20_2, c20_1, c21_2, c21_1, c22_2, c22_1, c23_2, c23_1, c_2 ---------------------------------------- (197) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing tuple parts ---------------------------------------- (198) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(tail(z0), x1, x2)) -> c21(A__IF(a__tail(mark(z0)), x1, x2), MARK(tail(z0))) MARK(if(if(z0, z1, z2), x1, x2)) -> c21(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(filter(z0, z1), x1, x2)) -> c21(A__IF(a__filter(mark(z0), mark(z1)), x1, x2), MARK(filter(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c22(A__FILTER(mark(x0), a__primes), MARK(x0)) MARK(filter(x0, sieve(z0))) -> c22(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, from(z0))) -> c22(A__FILTER(mark(x0), a__from(mark(z0))), MARK(x0)) MARK(filter(x0, head(z0))) -> c22(A__FILTER(mark(x0), a__head(mark(z0))), MARK(x0)) MARK(filter(x0, tail(z0))) -> c22(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(x0)) MARK(filter(x0, if(z0, z1, z2))) -> c22(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0)) MARK(filter(x0, filter(z0, z1))) -> c22(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(x0)) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(x0, s(z0))) -> c22(MARK(x0)) MARK(filter(x0, 0)) -> c22(MARK(x0)) MARK(filter(x0, true)) -> c22(MARK(x0)) MARK(filter(x0, false)) -> c22(MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), x1)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c23(A__FILTER(mark(x0), a__primes), MARK(primes)) MARK(filter(x0, sieve(z0))) -> c23(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, from(z0))) -> c23(A__FILTER(mark(x0), a__from(mark(z0))), MARK(from(z0))) MARK(filter(x0, head(z0))) -> c23(A__FILTER(mark(x0), a__head(mark(z0))), MARK(head(z0))) MARK(filter(x0, tail(z0))) -> c23(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(tail(z0))) MARK(filter(x0, if(z0, z1, z2))) -> c23(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(filter(x0, filter(z0, z1))) -> c23(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(filter(x0, cons(z0, z1))) -> c23(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(primes, x1)) -> c23(A__FILTER(a__primes, mark(x1)), MARK(x1)) MARK(filter(sieve(z0), x1)) -> c23(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(from(z0), x1)) -> c23(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(head(z0), x1)) -> c23(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(tail(z0), x1)) -> c23(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(if(z0, z1, z2), x1)) -> c23(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1)) MARK(filter(filter(z0, z1), x1)) -> c23(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(x1)) MARK(filter(s(z0), x1)) -> c23(A__FILTER(s(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(x0, s(z0))) -> c23(MARK(s(z0))) MARK(filter(x0, divides(z0, z1))) -> c23(MARK(divides(z0, z1))) MARK(filter(0, x1)) -> c23(MARK(x1)) MARK(filter(cons(z0, z1), x1)) -> c23(MARK(x1)) MARK(filter(true, x1)) -> c23(MARK(x1)) MARK(filter(false, x1)) -> c23(MARK(x1)) MARK(filter(divides(z0, z1), x1)) -> c23(MARK(x1)) A__PRIMES -> c(A__SIEVE(cons(s(mark(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(sieve(primes)) -> c17(MARK(primes)) MARK(sieve(sieve(primes))) -> c17(A__SIEVE(a__sieve(a__primes)), MARK(sieve(primes))) MARK(sieve(sieve(sieve(z0)))) -> c17(A__SIEVE(a__sieve(a__sieve(mark(z0)))), MARK(sieve(sieve(z0)))) MARK(sieve(sieve(from(z0)))) -> c17(A__SIEVE(a__sieve(a__from(mark(z0)))), MARK(sieve(from(z0)))) MARK(sieve(sieve(head(z0)))) -> c17(A__SIEVE(a__sieve(a__head(mark(z0)))), MARK(sieve(head(z0)))) MARK(sieve(sieve(tail(z0)))) -> c17(A__SIEVE(a__sieve(a__tail(mark(z0)))), MARK(sieve(tail(z0)))) MARK(sieve(sieve(if(z0, z1, z2)))) -> c17(A__SIEVE(a__sieve(a__if(mark(z0), z1, z2))), MARK(sieve(if(z0, z1, z2)))) MARK(sieve(sieve(filter(z0, z1)))) -> c17(A__SIEVE(a__sieve(a__filter(mark(z0), mark(z1)))), MARK(sieve(filter(z0, z1)))) MARK(sieve(sieve(s(z0)))) -> c17(A__SIEVE(a__sieve(s(mark(z0)))), MARK(sieve(s(z0)))) MARK(sieve(sieve(cons(z0, z1)))) -> c17(A__SIEVE(a__sieve(cons(mark(z0), z1))), MARK(sieve(cons(z0, z1)))) MARK(sieve(sieve(divides(z0, z1)))) -> c17(A__SIEVE(a__sieve(divides(mark(z0), mark(z1)))), MARK(sieve(divides(z0, z1)))) MARK(sieve(sieve(x0))) -> c17(MARK(sieve(x0))) MARK(sieve(sieve(0))) -> c17(A__SIEVE(a__sieve(0))) MARK(sieve(sieve(true))) -> c17(A__SIEVE(a__sieve(true))) MARK(sieve(sieve(false))) -> c17(A__SIEVE(a__sieve(false))) MARK(sieve(from(x0))) -> c17(A__SIEVE(cons(mark(mark(x0)), from(s(mark(x0))))), MARK(from(x0))) MARK(sieve(from(primes))) -> c17(A__SIEVE(a__from(a__primes)), MARK(from(primes))) MARK(sieve(from(sieve(z0)))) -> c17(A__SIEVE(a__from(a__sieve(mark(z0)))), MARK(from(sieve(z0)))) MARK(sieve(from(from(z0)))) -> c17(A__SIEVE(a__from(a__from(mark(z0)))), MARK(from(from(z0)))) MARK(sieve(from(head(z0)))) -> c17(A__SIEVE(a__from(a__head(mark(z0)))), MARK(from(head(z0)))) MARK(sieve(from(tail(z0)))) -> c17(A__SIEVE(a__from(a__tail(mark(z0)))), MARK(from(tail(z0)))) MARK(sieve(from(if(z0, z1, z2)))) -> c17(A__SIEVE(a__from(a__if(mark(z0), z1, z2))), MARK(from(if(z0, z1, z2)))) MARK(sieve(from(filter(z0, z1)))) -> c17(A__SIEVE(a__from(a__filter(mark(z0), mark(z1)))), MARK(from(filter(z0, z1)))) MARK(sieve(from(s(z0)))) -> c17(A__SIEVE(a__from(s(mark(z0)))), MARK(from(s(z0)))) MARK(sieve(from(0))) -> c17(A__SIEVE(a__from(0)), MARK(from(0))) MARK(sieve(from(cons(z0, z1)))) -> c17(A__SIEVE(a__from(cons(mark(z0), z1))), MARK(from(cons(z0, z1)))) MARK(sieve(from(true))) -> c17(A__SIEVE(a__from(true)), MARK(from(true))) MARK(sieve(from(false))) -> c17(A__SIEVE(a__from(false)), MARK(from(false))) MARK(sieve(from(divides(z0, z1)))) -> c17(A__SIEVE(a__from(divides(mark(z0), mark(z1)))), MARK(from(divides(z0, z1)))) MARK(sieve(from(x0))) -> c17(MARK(from(x0))) MARK(sieve(head(primes))) -> c17(A__SIEVE(a__head(a__primes)), MARK(head(primes))) MARK(sieve(head(sieve(z0)))) -> c17(A__SIEVE(a__head(a__sieve(mark(z0)))), MARK(head(sieve(z0)))) MARK(sieve(head(from(z0)))) -> c17(A__SIEVE(a__head(a__from(mark(z0)))), MARK(head(from(z0)))) MARK(sieve(head(head(z0)))) -> c17(A__SIEVE(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(sieve(head(tail(z0)))) -> c17(A__SIEVE(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(sieve(head(if(z0, z1, z2)))) -> c17(A__SIEVE(a__head(a__if(mark(z0), z1, z2))), MARK(head(if(z0, z1, z2)))) MARK(sieve(head(filter(z0, z1)))) -> c17(A__SIEVE(a__head(a__filter(mark(z0), mark(z1)))), MARK(head(filter(z0, z1)))) MARK(sieve(head(s(z0)))) -> c17(A__SIEVE(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(sieve(head(cons(z0, z1)))) -> c17(A__SIEVE(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(sieve(head(divides(z0, z1)))) -> c17(A__SIEVE(a__head(divides(mark(z0), mark(z1)))), MARK(head(divides(z0, z1)))) MARK(sieve(head(x0))) -> c17(MARK(head(x0))) MARK(sieve(head(0))) -> c17(A__SIEVE(a__head(0))) MARK(sieve(head(true))) -> c17(A__SIEVE(a__head(true))) MARK(sieve(head(false))) -> c17(A__SIEVE(a__head(false))) MARK(sieve(tail(primes))) -> c17(A__SIEVE(a__tail(a__primes)), MARK(tail(primes))) MARK(sieve(tail(sieve(z0)))) -> c17(A__SIEVE(a__tail(a__sieve(mark(z0)))), MARK(tail(sieve(z0)))) MARK(sieve(tail(from(z0)))) -> c17(A__SIEVE(a__tail(a__from(mark(z0)))), MARK(tail(from(z0)))) MARK(sieve(tail(head(z0)))) -> c17(A__SIEVE(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(sieve(tail(tail(z0)))) -> c17(A__SIEVE(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(sieve(tail(if(z0, z1, z2)))) -> c17(A__SIEVE(a__tail(a__if(mark(z0), z1, z2))), MARK(tail(if(z0, z1, z2)))) MARK(sieve(tail(filter(z0, z1)))) -> c17(A__SIEVE(a__tail(a__filter(mark(z0), mark(z1)))), MARK(tail(filter(z0, z1)))) MARK(sieve(tail(s(z0)))) -> c17(A__SIEVE(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(sieve(tail(cons(z0, z1)))) -> c17(A__SIEVE(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(sieve(tail(divides(z0, z1)))) -> c17(A__SIEVE(a__tail(divides(mark(z0), mark(z1)))), MARK(tail(divides(z0, z1)))) MARK(sieve(tail(x0))) -> c17(MARK(tail(x0))) MARK(sieve(tail(0))) -> c17(A__SIEVE(a__tail(0))) MARK(sieve(tail(true))) -> c17(A__SIEVE(a__tail(true))) MARK(sieve(tail(false))) -> c17(A__SIEVE(a__tail(false))) MARK(sieve(if(primes, x1, x2))) -> c17(A__SIEVE(a__if(a__primes, x1, x2)), MARK(if(primes, x1, x2))) MARK(sieve(if(sieve(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__sieve(mark(z0)), x1, x2)), MARK(if(sieve(z0), x1, x2))) MARK(sieve(if(from(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__from(mark(z0)), x1, x2)), MARK(if(from(z0), x1, x2))) MARK(sieve(if(head(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__head(mark(z0)), x1, x2)), MARK(if(head(z0), x1, x2))) MARK(sieve(if(tail(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__tail(mark(z0)), x1, x2)), MARK(if(tail(z0), x1, x2))) MARK(sieve(if(if(z0, z1, z2), x1, x2))) -> c17(A__SIEVE(a__if(a__if(mark(z0), z1, z2), x1, x2)), MARK(if(if(z0, z1, z2), x1, x2))) MARK(sieve(if(filter(z0, z1), x1, x2))) -> c17(A__SIEVE(a__if(a__filter(mark(z0), mark(z1)), x1, x2)), MARK(if(filter(z0, z1), x1, x2))) MARK(sieve(if(s(z0), x1, x2))) -> c17(A__SIEVE(a__if(s(mark(z0)), x1, x2)), MARK(if(s(z0), x1, x2))) MARK(sieve(if(cons(z0, z1), x1, x2))) -> c17(A__SIEVE(a__if(cons(mark(z0), z1), x1, x2)), MARK(if(cons(z0, z1), x1, x2))) MARK(sieve(if(true, x1, x2))) -> c17(A__SIEVE(a__if(true, x1, x2)), MARK(if(true, x1, x2))) MARK(sieve(if(false, x1, x2))) -> c17(A__SIEVE(a__if(false, x1, x2)), MARK(if(false, x1, x2))) MARK(sieve(if(divides(z0, z1), x1, x2))) -> c17(A__SIEVE(a__if(divides(mark(z0), mark(z1)), x1, x2)), MARK(if(divides(z0, z1), x1, x2))) MARK(sieve(if(x0, z1, z2))) -> c17(MARK(if(x0, z1, z2))) MARK(sieve(if(0, x1, x2))) -> c17(A__SIEVE(a__if(0, x1, x2))) MARK(sieve(filter(x0, primes))) -> c17(A__SIEVE(a__filter(mark(x0), a__primes)), MARK(filter(x0, primes))) MARK(sieve(filter(x0, sieve(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), a__sieve(mark(z0)))), MARK(filter(x0, sieve(z0)))) MARK(sieve(filter(x0, from(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), a__from(mark(z0)))), MARK(filter(x0, from(z0)))) MARK(sieve(filter(x0, head(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), a__head(mark(z0)))), MARK(filter(x0, head(z0)))) MARK(sieve(filter(x0, tail(z0)))) -> 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z1), x1))) MARK(tail(filter(x0, x1))) -> c20(MARK(filter(x0, x1))) MARK(if(primes, x0, x1)) -> c21(A__IF(a__sieve(a__from(s(s(0)))), x0, x1), MARK(primes)) MARK(if(primes, x0, x1)) -> c21(MARK(primes)) MARK(if(sieve(primes), x1, x2)) -> c21(A__IF(a__sieve(a__primes), x1, x2), MARK(sieve(primes))) MARK(if(sieve(sieve(z0)), x1, x2)) -> c21(A__IF(a__sieve(a__sieve(mark(z0))), x1, x2), MARK(sieve(sieve(z0)))) MARK(if(sieve(from(z0)), x1, x2)) -> c21(A__IF(a__sieve(a__from(mark(z0))), x1, x2), MARK(sieve(from(z0)))) MARK(if(sieve(head(z0)), x1, x2)) -> c21(A__IF(a__sieve(a__head(mark(z0))), x1, x2), MARK(sieve(head(z0)))) MARK(if(sieve(tail(z0)), x1, x2)) -> c21(A__IF(a__sieve(a__tail(mark(z0))), x1, x2), MARK(sieve(tail(z0)))) MARK(if(sieve(if(z0, z1, z2)), x1, x2)) -> c21(A__IF(a__sieve(a__if(mark(z0), z1, z2)), x1, x2), MARK(sieve(if(z0, z1, z2)))) MARK(if(sieve(filter(z0, z1)), x1, x2)) -> c21(A__IF(a__sieve(a__filter(mark(z0), mark(z1))), x1, x2), MARK(sieve(filter(z0, z1)))) MARK(if(sieve(s(z0)), x1, x2)) -> c21(A__IF(a__sieve(s(mark(z0))), x1, x2), MARK(sieve(s(z0)))) MARK(if(sieve(cons(z0, z1)), x1, x2)) -> c21(A__IF(a__sieve(cons(mark(z0), z1)), x1, x2), MARK(sieve(cons(z0, z1)))) MARK(if(sieve(divides(z0, z1)), x1, x2)) -> c21(A__IF(a__sieve(divides(mark(z0), mark(z1))), x1, x2), MARK(sieve(divides(z0, z1)))) MARK(if(sieve(x0), x1, x2)) -> c21(MARK(sieve(x0))) MARK(if(sieve(0), x1, x2)) -> c21(A__IF(a__sieve(0), x1, x2)) MARK(if(sieve(true), x1, x2)) -> c21(A__IF(a__sieve(true), x1, x2)) MARK(if(sieve(false), x1, x2)) -> c21(A__IF(a__sieve(false), x1, x2)) MARK(if(from(primes), x1, x2)) -> c21(A__IF(a__from(a__primes), x1, x2), MARK(from(primes))) MARK(if(from(sieve(z0)), x1, x2)) -> c21(A__IF(a__from(a__sieve(mark(z0))), x1, x2), MARK(from(sieve(z0)))) MARK(if(from(from(z0)), x1, x2)) -> c21(A__IF(a__from(a__from(mark(z0))), x1, x2), MARK(from(from(z0)))) MARK(if(from(head(z0)), x1, x2)) -> c21(A__IF(a__from(a__head(mark(z0))), x1, x2), MARK(from(head(z0)))) MARK(if(from(tail(z0)), x1, x2)) -> c21(A__IF(a__from(a__tail(mark(z0))), x1, x2), MARK(from(tail(z0)))) MARK(if(from(if(z0, z1, z2)), x1, x2)) -> c21(A__IF(a__from(a__if(mark(z0), z1, z2)), x1, x2), MARK(from(if(z0, z1, z2)))) MARK(if(from(filter(z0, z1)), x1, x2)) -> c21(A__IF(a__from(a__filter(mark(z0), mark(z1))), x1, x2), MARK(from(filter(z0, z1)))) MARK(if(from(s(z0)), x1, x2)) -> c21(A__IF(a__from(s(mark(z0))), x1, x2), MARK(from(s(z0)))) MARK(if(from(0), x1, x2)) -> c21(A__IF(a__from(0), x1, x2), MARK(from(0))) MARK(if(from(cons(z0, z1)), x1, x2)) -> c21(A__IF(a__from(cons(mark(z0), z1)), x1, x2), MARK(from(cons(z0, z1)))) MARK(if(from(true), x1, x2)) -> c21(A__IF(a__from(true), x1, x2), MARK(from(true))) MARK(if(from(false), x1, x2)) -> c21(A__IF(a__from(false), x1, x2), MARK(from(false))) MARK(if(from(divides(z0, z1)), x1, x2)) -> c21(A__IF(a__from(divides(mark(z0), mark(z1))), x1, x2), MARK(from(divides(z0, z1)))) MARK(if(from(x0), x1, x2)) -> c21(MARK(from(x0))) MARK(if(head(primes), x1, x2)) -> c21(A__IF(a__head(a__primes), x1, x2), MARK(head(primes))) MARK(if(head(sieve(z0)), x1, x2)) -> c21(A__IF(a__head(a__sieve(mark(z0))), x1, x2), MARK(head(sieve(z0)))) MARK(if(head(from(z0)), x1, x2)) -> c21(A__IF(a__head(a__from(mark(z0))), x1, x2), MARK(head(from(z0)))) MARK(if(head(head(z0)), x1, x2)) -> c21(A__IF(a__head(a__head(mark(z0))), x1, x2), MARK(head(head(z0)))) MARK(if(head(tail(z0)), x1, x2)) -> c21(A__IF(a__head(a__tail(mark(z0))), x1, x2), MARK(head(tail(z0)))) MARK(if(head(if(z0, z1, z2)), x1, x2)) -> c21(A__IF(a__head(a__if(mark(z0), z1, z2)), x1, x2), MARK(head(if(z0, z1, z2)))) MARK(if(head(filter(z0, z1)), x1, x2)) -> c21(A__IF(a__head(a__filter(mark(z0), mark(z1))), x1, x2), MARK(head(filter(z0, z1)))) MARK(if(head(s(z0)), x1, x2)) -> c21(A__IF(a__head(s(mark(z0))), x1, x2), MARK(head(s(z0)))) MARK(if(head(cons(z0, z1)), x1, x2)) -> c21(A__IF(a__head(cons(mark(z0), z1)), x1, x2), MARK(head(cons(z0, z1)))) MARK(if(head(divides(z0, z1)), x1, x2)) -> c21(A__IF(a__head(divides(mark(z0), mark(z1))), x1, x2), MARK(head(divides(z0, z1)))) MARK(if(head(x0), x1, x2)) -> c21(MARK(head(x0))) MARK(if(head(0), x1, x2)) -> c21(A__IF(a__head(0), x1, x2)) MARK(if(head(true), x1, x2)) -> c21(A__IF(a__head(true), x1, x2)) MARK(if(head(false), x1, x2)) -> c21(A__IF(a__head(false), x1, x2)) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c18_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_1, c17_2, c17_1, c19_2, c19_1, c20_2, c20_1, c21_2, c21_1, c22_2, c22_1, c23_2, c23_1, c_2 ---------------------------------------- (199) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(if(tail(z0), x1, x2)) -> c21(A__IF(a__tail(mark(z0)), x1, x2), MARK(tail(z0))) by MARK(if(tail(x0), x1, x2)) -> c21(A__IF(tail(mark(x0)), x1, x2), MARK(tail(x0))) MARK(if(tail(primes), x1, x2)) -> c21(A__IF(a__tail(a__primes), x1, x2), MARK(tail(primes))) MARK(if(tail(sieve(z0)), x1, x2)) -> c21(A__IF(a__tail(a__sieve(mark(z0))), x1, x2), MARK(tail(sieve(z0)))) MARK(if(tail(from(z0)), x1, x2)) -> c21(A__IF(a__tail(a__from(mark(z0))), x1, x2), MARK(tail(from(z0)))) MARK(if(tail(head(z0)), x1, x2)) -> c21(A__IF(a__tail(a__head(mark(z0))), x1, x2), MARK(tail(head(z0)))) MARK(if(tail(tail(z0)), x1, x2)) -> c21(A__IF(a__tail(a__tail(mark(z0))), x1, x2), MARK(tail(tail(z0)))) MARK(if(tail(if(z0, z1, z2)), x1, x2)) -> c21(A__IF(a__tail(a__if(mark(z0), z1, z2)), x1, x2), MARK(tail(if(z0, z1, z2)))) MARK(if(tail(filter(z0, z1)), x1, x2)) -> c21(A__IF(a__tail(a__filter(mark(z0), mark(z1))), x1, x2), MARK(tail(filter(z0, z1)))) MARK(if(tail(s(z0)), x1, x2)) -> c21(A__IF(a__tail(s(mark(z0))), x1, x2), MARK(tail(s(z0)))) MARK(if(tail(0), x1, x2)) -> c21(A__IF(a__tail(0), x1, x2), MARK(tail(0))) MARK(if(tail(cons(z0, z1)), x1, x2)) -> c21(A__IF(a__tail(cons(mark(z0), z1)), x1, x2), MARK(tail(cons(z0, z1)))) MARK(if(tail(true), x1, x2)) -> c21(A__IF(a__tail(true), x1, x2), MARK(tail(true))) MARK(if(tail(false), x1, x2)) -> c21(A__IF(a__tail(false), x1, x2), MARK(tail(false))) MARK(if(tail(divides(z0, z1)), x1, x2)) -> c21(A__IF(a__tail(divides(mark(z0), mark(z1))), x1, x2), MARK(tail(divides(z0, z1)))) ---------------------------------------- (200) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(if(z0, z1, z2), x1, x2)) -> c21(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(filter(z0, z1), x1, x2)) -> c21(A__IF(a__filter(mark(z0), mark(z1)), x1, x2), MARK(filter(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c22(A__FILTER(mark(x0), a__primes), MARK(x0)) MARK(filter(x0, sieve(z0))) -> c22(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, from(z0))) -> c22(A__FILTER(mark(x0), a__from(mark(z0))), MARK(x0)) MARK(filter(x0, head(z0))) -> c22(A__FILTER(mark(x0), a__head(mark(z0))), MARK(x0)) MARK(filter(x0, tail(z0))) -> c22(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(x0)) MARK(filter(x0, if(z0, z1, z2))) -> c22(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0)) MARK(filter(x0, filter(z0, z1))) -> c22(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(x0)) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(x0, s(z0))) -> c22(MARK(x0)) MARK(filter(x0, 0)) -> c22(MARK(x0)) MARK(filter(x0, true)) -> c22(MARK(x0)) MARK(filter(x0, false)) -> c22(MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), x1)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c23(A__FILTER(mark(x0), a__primes), MARK(primes)) MARK(filter(x0, sieve(z0))) -> c23(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, from(z0))) -> c23(A__FILTER(mark(x0), a__from(mark(z0))), MARK(from(z0))) MARK(filter(x0, head(z0))) -> c23(A__FILTER(mark(x0), a__head(mark(z0))), MARK(head(z0))) MARK(filter(x0, tail(z0))) -> c23(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(tail(z0))) MARK(filter(x0, if(z0, z1, z2))) -> c23(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(filter(x0, filter(z0, z1))) -> c23(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(filter(x0, cons(z0, z1))) -> c23(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(primes, x1)) -> c23(A__FILTER(a__primes, mark(x1)), MARK(x1)) MARK(filter(sieve(z0), x1)) -> c23(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(from(z0), x1)) -> c23(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(head(z0), x1)) -> c23(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(tail(z0), x1)) -> c23(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(if(z0, z1, z2), x1)) -> c23(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1)) MARK(filter(filter(z0, z1), x1)) -> c23(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(x1)) MARK(filter(s(z0), x1)) -> c23(A__FILTER(s(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(x0, s(z0))) -> c23(MARK(s(z0))) MARK(filter(x0, divides(z0, z1))) -> c23(MARK(divides(z0, z1))) MARK(filter(0, x1)) -> c23(MARK(x1)) MARK(filter(cons(z0, z1), x1)) -> c23(MARK(x1)) MARK(filter(true, x1)) -> c23(MARK(x1)) MARK(filter(false, x1)) -> c23(MARK(x1)) MARK(filter(divides(z0, z1), x1)) -> c23(MARK(x1)) A__PRIMES -> c(A__SIEVE(cons(s(mark(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(sieve(primes)) -> c17(MARK(primes)) MARK(sieve(sieve(primes))) -> c17(A__SIEVE(a__sieve(a__primes)), MARK(sieve(primes))) MARK(sieve(sieve(sieve(z0)))) -> c17(A__SIEVE(a__sieve(a__sieve(mark(z0)))), MARK(sieve(sieve(z0)))) MARK(sieve(sieve(from(z0)))) -> c17(A__SIEVE(a__sieve(a__from(mark(z0)))), MARK(sieve(from(z0)))) MARK(sieve(sieve(head(z0)))) -> c17(A__SIEVE(a__sieve(a__head(mark(z0)))), MARK(sieve(head(z0)))) MARK(sieve(sieve(tail(z0)))) -> c17(A__SIEVE(a__sieve(a__tail(mark(z0)))), MARK(sieve(tail(z0)))) MARK(sieve(sieve(if(z0, z1, z2)))) -> c17(A__SIEVE(a__sieve(a__if(mark(z0), z1, z2))), MARK(sieve(if(z0, z1, z2)))) MARK(sieve(sieve(filter(z0, z1)))) -> c17(A__SIEVE(a__sieve(a__filter(mark(z0), mark(z1)))), MARK(sieve(filter(z0, z1)))) MARK(sieve(sieve(s(z0)))) -> c17(A__SIEVE(a__sieve(s(mark(z0)))), MARK(sieve(s(z0)))) MARK(sieve(sieve(cons(z0, z1)))) -> c17(A__SIEVE(a__sieve(cons(mark(z0), z1))), MARK(sieve(cons(z0, z1)))) MARK(sieve(sieve(divides(z0, z1)))) -> c17(A__SIEVE(a__sieve(divides(mark(z0), mark(z1)))), MARK(sieve(divides(z0, 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MARK(if(sieve(s(z0)), x1, x2)) -> c21(A__IF(a__sieve(s(mark(z0))), x1, x2), MARK(sieve(s(z0)))) MARK(if(sieve(cons(z0, z1)), x1, x2)) -> c21(A__IF(a__sieve(cons(mark(z0), z1)), x1, x2), MARK(sieve(cons(z0, z1)))) MARK(if(sieve(divides(z0, z1)), x1, x2)) -> c21(A__IF(a__sieve(divides(mark(z0), mark(z1))), x1, x2), MARK(sieve(divides(z0, z1)))) MARK(if(sieve(x0), x1, x2)) -> c21(MARK(sieve(x0))) MARK(if(sieve(0), x1, x2)) -> c21(A__IF(a__sieve(0), x1, x2)) MARK(if(sieve(true), x1, x2)) -> c21(A__IF(a__sieve(true), x1, x2)) MARK(if(sieve(false), x1, x2)) -> c21(A__IF(a__sieve(false), x1, x2)) MARK(if(from(primes), x1, x2)) -> c21(A__IF(a__from(a__primes), x1, x2), MARK(from(primes))) MARK(if(from(sieve(z0)), x1, x2)) -> c21(A__IF(a__from(a__sieve(mark(z0))), x1, x2), MARK(from(sieve(z0)))) MARK(if(from(from(z0)), x1, x2)) -> c21(A__IF(a__from(a__from(mark(z0))), x1, x2), MARK(from(from(z0)))) MARK(if(from(head(z0)), x1, x2)) -> c21(A__IF(a__from(a__head(mark(z0))), x1, x2), 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c21(MARK(from(x0))) MARK(if(head(primes), x1, x2)) -> c21(A__IF(a__head(a__primes), x1, x2), MARK(head(primes))) MARK(if(head(sieve(z0)), x1, x2)) -> c21(A__IF(a__head(a__sieve(mark(z0))), x1, x2), MARK(head(sieve(z0)))) MARK(if(head(from(z0)), x1, x2)) -> c21(A__IF(a__head(a__from(mark(z0))), x1, x2), MARK(head(from(z0)))) MARK(if(head(head(z0)), x1, x2)) -> c21(A__IF(a__head(a__head(mark(z0))), x1, x2), MARK(head(head(z0)))) MARK(if(head(tail(z0)), x1, x2)) -> c21(A__IF(a__head(a__tail(mark(z0))), x1, x2), MARK(head(tail(z0)))) MARK(if(head(if(z0, z1, z2)), x1, x2)) -> c21(A__IF(a__head(a__if(mark(z0), z1, z2)), x1, x2), MARK(head(if(z0, z1, z2)))) MARK(if(head(filter(z0, z1)), x1, x2)) -> c21(A__IF(a__head(a__filter(mark(z0), mark(z1))), x1, x2), MARK(head(filter(z0, z1)))) MARK(if(head(s(z0)), x1, x2)) -> c21(A__IF(a__head(s(mark(z0))), x1, x2), MARK(head(s(z0)))) MARK(if(head(cons(z0, z1)), x1, x2)) -> c21(A__IF(a__head(cons(mark(z0), z1)), x1, x2), MARK(head(cons(z0, z1)))) MARK(if(head(divides(z0, z1)), x1, x2)) -> c21(A__IF(a__head(divides(mark(z0), mark(z1))), x1, x2), MARK(head(divides(z0, z1)))) MARK(if(head(x0), x1, x2)) -> c21(MARK(head(x0))) MARK(if(head(0), x1, x2)) -> c21(A__IF(a__head(0), x1, x2)) MARK(if(head(true), x1, x2)) -> c21(A__IF(a__head(true), x1, x2)) MARK(if(head(false), x1, x2)) -> c21(A__IF(a__head(false), x1, x2)) MARK(if(tail(x0), x1, x2)) -> c21(A__IF(tail(mark(x0)), x1, x2), MARK(tail(x0))) MARK(if(tail(primes), x1, x2)) -> c21(A__IF(a__tail(a__primes), x1, x2), MARK(tail(primes))) MARK(if(tail(sieve(z0)), x1, x2)) -> c21(A__IF(a__tail(a__sieve(mark(z0))), x1, x2), MARK(tail(sieve(z0)))) MARK(if(tail(from(z0)), x1, x2)) -> c21(A__IF(a__tail(a__from(mark(z0))), x1, x2), MARK(tail(from(z0)))) MARK(if(tail(head(z0)), x1, x2)) -> c21(A__IF(a__tail(a__head(mark(z0))), x1, x2), MARK(tail(head(z0)))) MARK(if(tail(tail(z0)), x1, x2)) -> c21(A__IF(a__tail(a__tail(mark(z0))), x1, x2), MARK(tail(tail(z0)))) MARK(if(tail(if(z0, z1, z2)), x1, x2)) -> c21(A__IF(a__tail(a__if(mark(z0), z1, z2)), x1, x2), MARK(tail(if(z0, z1, z2)))) MARK(if(tail(filter(z0, z1)), x1, x2)) -> c21(A__IF(a__tail(a__filter(mark(z0), mark(z1))), x1, x2), MARK(tail(filter(z0, z1)))) MARK(if(tail(s(z0)), x1, x2)) -> c21(A__IF(a__tail(s(mark(z0))), x1, x2), MARK(tail(s(z0)))) MARK(if(tail(0), x1, x2)) -> c21(A__IF(a__tail(0), x1, x2), MARK(tail(0))) MARK(if(tail(cons(z0, z1)), x1, x2)) -> c21(A__IF(a__tail(cons(mark(z0), z1)), x1, x2), MARK(tail(cons(z0, z1)))) MARK(if(tail(true), x1, x2)) -> c21(A__IF(a__tail(true), x1, x2), MARK(tail(true))) MARK(if(tail(false), x1, x2)) -> c21(A__IF(a__tail(false), x1, x2), MARK(tail(false))) MARK(if(tail(divides(z0, z1)), x1, x2)) -> c21(A__IF(a__tail(divides(mark(z0), mark(z1))), x1, x2), MARK(tail(divides(z0, z1)))) S tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(if(z0, z1, z2), x1, x2)) -> c21(A__IF(a__if(mark(z0), z1, 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a__filter(mark(z0), mark(z1))), MARK(x0)) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(x0, s(z0))) -> c22(MARK(x0)) MARK(filter(x0, 0)) -> c22(MARK(x0)) MARK(filter(x0, true)) -> c22(MARK(x0)) MARK(filter(x0, false)) -> c22(MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), x1)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c23(A__FILTER(mark(x0), a__primes), MARK(primes)) MARK(filter(x0, sieve(z0))) -> c23(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, from(z0))) -> c23(A__FILTER(mark(x0), a__from(mark(z0))), MARK(from(z0))) MARK(filter(x0, head(z0))) -> c23(A__FILTER(mark(x0), a__head(mark(z0))), MARK(head(z0))) MARK(filter(x0, tail(z0))) -> c23(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(tail(z0))) MARK(filter(x0, if(z0, z1, z2))) -> c23(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(filter(x0, filter(z0, z1))) -> c23(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(filter(x0, cons(z0, z1))) -> c23(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(primes, x1)) -> c23(A__FILTER(a__primes, mark(x1)), MARK(x1)) MARK(filter(sieve(z0), x1)) -> c23(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(from(z0), x1)) -> c23(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(head(z0), x1)) -> c23(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(tail(z0), x1)) -> c23(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(if(z0, z1, z2), x1)) -> c23(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1)) MARK(filter(filter(z0, z1), x1)) -> c23(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(x1)) MARK(filter(s(z0), x1)) -> c23(A__FILTER(s(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(x0, s(z0))) -> c23(MARK(s(z0))) MARK(filter(x0, divides(z0, z1))) -> c23(MARK(divides(z0, z1))) MARK(filter(0, x1)) -> c23(MARK(x1)) MARK(filter(cons(z0, z1), x1)) -> c23(MARK(x1)) MARK(filter(true, x1)) -> c23(MARK(x1)) MARK(filter(false, x1)) -> c23(MARK(x1)) MARK(filter(divides(z0, z1), x1)) -> c23(MARK(x1)) A__PRIMES -> c(A__SIEVE(cons(s(mark(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(sieve(primes)) -> c17(MARK(primes)) MARK(sieve(sieve(primes))) -> c17(A__SIEVE(a__sieve(a__primes)), MARK(sieve(primes))) MARK(sieve(sieve(sieve(z0)))) -> c17(A__SIEVE(a__sieve(a__sieve(mark(z0)))), MARK(sieve(sieve(z0)))) MARK(sieve(sieve(from(z0)))) -> c17(A__SIEVE(a__sieve(a__from(mark(z0)))), MARK(sieve(from(z0)))) MARK(sieve(sieve(head(z0)))) -> c17(A__SIEVE(a__sieve(a__head(mark(z0)))), MARK(sieve(head(z0)))) MARK(sieve(sieve(tail(z0)))) -> c17(A__SIEVE(a__sieve(a__tail(mark(z0)))), MARK(sieve(tail(z0)))) MARK(sieve(sieve(if(z0, z1, z2)))) -> c17(A__SIEVE(a__sieve(a__if(mark(z0), z1, z2))), MARK(sieve(if(z0, z1, z2)))) MARK(sieve(sieve(filter(z0, z1)))) -> c17(A__SIEVE(a__sieve(a__filter(mark(z0), mark(z1)))), MARK(sieve(filter(z0, z1)))) MARK(sieve(sieve(s(z0)))) -> c17(A__SIEVE(a__sieve(s(mark(z0)))), MARK(sieve(s(z0)))) MARK(sieve(sieve(cons(z0, z1)))) -> c17(A__SIEVE(a__sieve(cons(mark(z0), z1))), MARK(sieve(cons(z0, z1)))) MARK(sieve(sieve(divides(z0, z1)))) -> c17(A__SIEVE(a__sieve(divides(mark(z0), mark(z1)))), MARK(sieve(divides(z0, z1)))) MARK(sieve(sieve(x0))) -> c17(MARK(sieve(x0))) MARK(sieve(sieve(0))) -> c17(A__SIEVE(a__sieve(0))) MARK(sieve(sieve(true))) -> c17(A__SIEVE(a__sieve(true))) MARK(sieve(sieve(false))) -> c17(A__SIEVE(a__sieve(false))) MARK(sieve(from(x0))) -> c17(A__SIEVE(cons(mark(mark(x0)), from(s(mark(x0))))), MARK(from(x0))) MARK(sieve(from(primes))) -> c17(A__SIEVE(a__from(a__primes)), MARK(from(primes))) MARK(sieve(from(sieve(z0)))) -> c17(A__SIEVE(a__from(a__sieve(mark(z0)))), MARK(from(sieve(z0)))) MARK(sieve(from(from(z0)))) -> c17(A__SIEVE(a__from(a__from(mark(z0)))), MARK(from(from(z0)))) MARK(sieve(from(head(z0)))) -> c17(A__SIEVE(a__from(a__head(mark(z0)))), MARK(from(head(z0)))) MARK(sieve(from(tail(z0)))) -> c17(A__SIEVE(a__from(a__tail(mark(z0)))), 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c21(A__IF(a__head(a__filter(mark(z0), mark(z1))), x1, x2), MARK(head(filter(z0, z1)))) MARK(if(head(s(z0)), x1, x2)) -> c21(A__IF(a__head(s(mark(z0))), x1, x2), MARK(head(s(z0)))) MARK(if(head(cons(z0, z1)), x1, x2)) -> c21(A__IF(a__head(cons(mark(z0), z1)), x1, x2), MARK(head(cons(z0, z1)))) MARK(if(head(divides(z0, z1)), x1, x2)) -> c21(A__IF(a__head(divides(mark(z0), mark(z1))), x1, x2), MARK(head(divides(z0, z1)))) MARK(if(head(x0), x1, x2)) -> c21(MARK(head(x0))) MARK(if(head(0), x1, x2)) -> c21(A__IF(a__head(0), x1, x2)) MARK(if(head(true), x1, x2)) -> c21(A__IF(a__head(true), x1, x2)) MARK(if(head(false), x1, x2)) -> c21(A__IF(a__head(false), x1, x2)) MARK(if(tail(x0), x1, x2)) -> c21(A__IF(tail(mark(x0)), x1, x2), MARK(tail(x0))) MARK(if(tail(primes), x1, x2)) -> c21(A__IF(a__tail(a__primes), x1, x2), MARK(tail(primes))) MARK(if(tail(sieve(z0)), x1, x2)) -> c21(A__IF(a__tail(a__sieve(mark(z0))), x1, x2), MARK(tail(sieve(z0)))) MARK(if(tail(from(z0)), x1, x2)) -> c21(A__IF(a__tail(a__from(mark(z0))), x1, x2), MARK(tail(from(z0)))) MARK(if(tail(head(z0)), x1, x2)) -> c21(A__IF(a__tail(a__head(mark(z0))), x1, x2), MARK(tail(head(z0)))) MARK(if(tail(tail(z0)), x1, x2)) -> c21(A__IF(a__tail(a__tail(mark(z0))), x1, x2), MARK(tail(tail(z0)))) MARK(if(tail(if(z0, z1, z2)), x1, x2)) -> c21(A__IF(a__tail(a__if(mark(z0), z1, z2)), x1, x2), MARK(tail(if(z0, z1, z2)))) MARK(if(tail(filter(z0, z1)), x1, x2)) -> c21(A__IF(a__tail(a__filter(mark(z0), mark(z1))), x1, x2), MARK(tail(filter(z0, z1)))) MARK(if(tail(s(z0)), x1, x2)) -> c21(A__IF(a__tail(s(mark(z0))), x1, x2), MARK(tail(s(z0)))) MARK(if(tail(0), x1, x2)) -> c21(A__IF(a__tail(0), x1, x2), MARK(tail(0))) MARK(if(tail(cons(z0, z1)), x1, x2)) -> c21(A__IF(a__tail(cons(mark(z0), z1)), x1, x2), MARK(tail(cons(z0, z1)))) MARK(if(tail(true), x1, x2)) -> c21(A__IF(a__tail(true), x1, x2), MARK(tail(true))) MARK(if(tail(false), x1, x2)) -> c21(A__IF(a__tail(false), x1, x2), MARK(tail(false))) MARK(if(tail(divides(z0, z1)), x1, x2)) -> c21(A__IF(a__tail(divides(mark(z0), mark(z1))), x1, x2), MARK(tail(divides(z0, z1)))) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c18_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_1, c17_2, c17_1, c19_2, c19_1, c20_2, c20_1, c21_2, c21_1, c22_2, c22_1, c23_2, c23_1, c_2 ---------------------------------------- (201) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing tuple parts ---------------------------------------- (202) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(if(z0, z1, z2), x1, x2)) -> c21(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) MARK(if(filter(z0, z1), x1, x2)) -> c21(A__IF(a__filter(mark(z0), mark(z1)), x1, x2), MARK(filter(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c22(A__FILTER(mark(x0), a__primes), MARK(x0)) MARK(filter(x0, sieve(z0))) -> c22(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, from(z0))) -> c22(A__FILTER(mark(x0), a__from(mark(z0))), MARK(x0)) MARK(filter(x0, head(z0))) -> c22(A__FILTER(mark(x0), a__head(mark(z0))), MARK(x0)) MARK(filter(x0, tail(z0))) -> c22(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(x0)) MARK(filter(x0, if(z0, z1, z2))) -> c22(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0)) MARK(filter(x0, filter(z0, z1))) -> c22(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(x0)) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(x0, s(z0))) -> c22(MARK(x0)) MARK(filter(x0, 0)) -> c22(MARK(x0)) MARK(filter(x0, true)) -> c22(MARK(x0)) MARK(filter(x0, false)) -> c22(MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), x1)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c23(A__FILTER(mark(x0), a__primes), MARK(primes)) MARK(filter(x0, sieve(z0))) -> c23(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, from(z0))) -> c23(A__FILTER(mark(x0), a__from(mark(z0))), MARK(from(z0))) MARK(filter(x0, head(z0))) -> c23(A__FILTER(mark(x0), a__head(mark(z0))), MARK(head(z0))) MARK(filter(x0, tail(z0))) -> c23(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(tail(z0))) MARK(filter(x0, if(z0, z1, z2))) -> c23(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(filter(x0, filter(z0, z1))) -> c23(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(filter(x0, cons(z0, z1))) -> c23(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(primes, x1)) -> c23(A__FILTER(a__primes, mark(x1)), MARK(x1)) MARK(filter(sieve(z0), x1)) -> c23(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(from(z0), x1)) -> c23(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(head(z0), x1)) -> c23(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(tail(z0), x1)) -> c23(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(if(z0, z1, z2), x1)) -> c23(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1)) MARK(filter(filter(z0, z1), x1)) -> c23(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(x1)) MARK(filter(s(z0), x1)) -> c23(A__FILTER(s(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(x0, s(z0))) -> c23(MARK(s(z0))) MARK(filter(x0, divides(z0, z1))) -> c23(MARK(divides(z0, z1))) MARK(filter(0, x1)) -> c23(MARK(x1)) MARK(filter(cons(z0, z1), x1)) -> c23(MARK(x1)) MARK(filter(true, x1)) -> c23(MARK(x1)) MARK(filter(false, x1)) -> c23(MARK(x1)) MARK(filter(divides(z0, z1), x1)) -> c23(MARK(x1)) A__PRIMES -> c(A__SIEVE(cons(s(mark(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(sieve(primes)) -> c17(MARK(primes)) MARK(sieve(sieve(primes))) -> c17(A__SIEVE(a__sieve(a__primes)), MARK(sieve(primes))) MARK(sieve(sieve(sieve(z0)))) -> c17(A__SIEVE(a__sieve(a__sieve(mark(z0)))), MARK(sieve(sieve(z0)))) MARK(sieve(sieve(from(z0)))) -> c17(A__SIEVE(a__sieve(a__from(mark(z0)))), MARK(sieve(from(z0)))) MARK(sieve(sieve(head(z0)))) -> c17(A__SIEVE(a__sieve(a__head(mark(z0)))), MARK(sieve(head(z0)))) MARK(sieve(sieve(tail(z0)))) -> c17(A__SIEVE(a__sieve(a__tail(mark(z0)))), MARK(sieve(tail(z0)))) MARK(sieve(sieve(if(z0, z1, z2)))) -> c17(A__SIEVE(a__sieve(a__if(mark(z0), z1, z2))), MARK(sieve(if(z0, z1, z2)))) MARK(sieve(sieve(filter(z0, z1)))) -> c17(A__SIEVE(a__sieve(a__filter(mark(z0), mark(z1)))), MARK(sieve(filter(z0, z1)))) MARK(sieve(sieve(s(z0)))) -> c17(A__SIEVE(a__sieve(s(mark(z0)))), MARK(sieve(s(z0)))) MARK(sieve(sieve(cons(z0, z1)))) -> c17(A__SIEVE(a__sieve(cons(mark(z0), z1))), MARK(sieve(cons(z0, z1)))) MARK(sieve(sieve(divides(z0, z1)))) -> c17(A__SIEVE(a__sieve(divides(mark(z0), mark(z1)))), MARK(sieve(divides(z0, z1)))) MARK(sieve(sieve(x0))) -> c17(MARK(sieve(x0))) MARK(sieve(sieve(0))) -> c17(A__SIEVE(a__sieve(0))) MARK(sieve(sieve(true))) -> c17(A__SIEVE(a__sieve(true))) MARK(sieve(sieve(false))) -> c17(A__SIEVE(a__sieve(false))) MARK(sieve(from(x0))) -> c17(A__SIEVE(cons(mark(mark(x0)), from(s(mark(x0))))), MARK(from(x0))) MARK(sieve(from(primes))) -> c17(A__SIEVE(a__from(a__primes)), MARK(from(primes))) MARK(sieve(from(sieve(z0)))) -> c17(A__SIEVE(a__from(a__sieve(mark(z0)))), MARK(from(sieve(z0)))) MARK(sieve(from(from(z0)))) -> c17(A__SIEVE(a__from(a__from(mark(z0)))), MARK(from(from(z0)))) MARK(sieve(from(head(z0)))) -> c17(A__SIEVE(a__from(a__head(mark(z0)))), MARK(from(head(z0)))) MARK(sieve(from(tail(z0)))) -> c17(A__SIEVE(a__from(a__tail(mark(z0)))), MARK(from(tail(z0)))) MARK(sieve(from(if(z0, z1, z2)))) -> c17(A__SIEVE(a__from(a__if(mark(z0), z1, z2))), MARK(from(if(z0, z1, z2)))) MARK(sieve(from(filter(z0, z1)))) -> c17(A__SIEVE(a__from(a__filter(mark(z0), mark(z1)))), MARK(from(filter(z0, z1)))) MARK(sieve(from(s(z0)))) -> c17(A__SIEVE(a__from(s(mark(z0)))), MARK(from(s(z0)))) MARK(sieve(from(0))) -> c17(A__SIEVE(a__from(0)), MARK(from(0))) MARK(sieve(from(cons(z0, z1)))) -> c17(A__SIEVE(a__from(cons(mark(z0), z1))), MARK(from(cons(z0, z1)))) MARK(sieve(from(true))) -> c17(A__SIEVE(a__from(true)), MARK(from(true))) MARK(sieve(from(false))) -> c17(A__SIEVE(a__from(false)), MARK(from(false))) MARK(sieve(from(divides(z0, z1)))) -> c17(A__SIEVE(a__from(divides(mark(z0), mark(z1)))), MARK(from(divides(z0, z1)))) MARK(sieve(from(x0))) -> c17(MARK(from(x0))) MARK(sieve(head(primes))) -> c17(A__SIEVE(a__head(a__primes)), MARK(head(primes))) MARK(sieve(head(sieve(z0)))) -> 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c17(A__SIEVE(a__from(a__from(mark(z0)))), MARK(from(from(z0)))) MARK(sieve(from(head(z0)))) -> c17(A__SIEVE(a__from(a__head(mark(z0)))), MARK(from(head(z0)))) MARK(sieve(from(tail(z0)))) -> c17(A__SIEVE(a__from(a__tail(mark(z0)))), MARK(from(tail(z0)))) MARK(sieve(from(if(z0, z1, z2)))) -> c17(A__SIEVE(a__from(a__if(mark(z0), z1, z2))), MARK(from(if(z0, z1, z2)))) MARK(sieve(from(filter(z0, z1)))) -> c17(A__SIEVE(a__from(a__filter(mark(z0), mark(z1)))), MARK(from(filter(z0, z1)))) MARK(sieve(from(s(z0)))) -> c17(A__SIEVE(a__from(s(mark(z0)))), MARK(from(s(z0)))) MARK(sieve(from(0))) -> c17(A__SIEVE(a__from(0)), MARK(from(0))) MARK(sieve(from(cons(z0, z1)))) -> c17(A__SIEVE(a__from(cons(mark(z0), z1))), MARK(from(cons(z0, z1)))) MARK(sieve(from(true))) -> c17(A__SIEVE(a__from(true)), MARK(from(true))) MARK(sieve(from(false))) -> c17(A__SIEVE(a__from(false)), MARK(from(false))) MARK(sieve(from(divides(z0, z1)))) -> c17(A__SIEVE(a__from(divides(mark(z0), mark(z1)))), MARK(from(divides(z0, 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MARK(head(tail(z0)))) MARK(tail(head(if(z0, z1, z2)))) -> c20(A__TAIL(a__head(a__if(mark(z0), z1, z2))), MARK(head(if(z0, z1, z2)))) MARK(tail(head(filter(z0, z1)))) -> c20(A__TAIL(a__head(a__filter(mark(z0), mark(z1)))), MARK(head(filter(z0, z1)))) MARK(tail(head(s(z0)))) -> c20(A__TAIL(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(tail(head(cons(z0, z1)))) -> c20(A__TAIL(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(tail(head(divides(z0, z1)))) -> c20(A__TAIL(a__head(divides(mark(z0), mark(z1)))), MARK(head(divides(z0, z1)))) MARK(tail(head(x0))) -> c20(MARK(head(x0))) MARK(tail(head(0))) -> c20(A__TAIL(a__head(0))) MARK(tail(head(true))) -> c20(A__TAIL(a__head(true))) MARK(tail(head(false))) -> c20(A__TAIL(a__head(false))) MARK(tail(tail(primes))) -> c20(A__TAIL(a__tail(a__primes)), MARK(tail(primes))) MARK(tail(tail(sieve(z0)))) -> c20(A__TAIL(a__tail(a__sieve(mark(z0)))), MARK(tail(sieve(z0)))) MARK(tail(tail(from(z0)))) -> c20(A__TAIL(a__tail(a__from(mark(z0)))), MARK(tail(from(z0)))) MARK(tail(tail(head(z0)))) -> c20(A__TAIL(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(tail(tail(tail(z0)))) -> c20(A__TAIL(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(tail(tail(if(z0, z1, z2)))) -> c20(A__TAIL(a__tail(a__if(mark(z0), z1, z2))), MARK(tail(if(z0, z1, z2)))) MARK(tail(tail(filter(z0, z1)))) -> c20(A__TAIL(a__tail(a__filter(mark(z0), mark(z1)))), MARK(tail(filter(z0, z1)))) MARK(tail(tail(s(z0)))) -> c20(A__TAIL(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(tail(tail(cons(z0, z1)))) -> c20(A__TAIL(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(tail(tail(divides(z0, z1)))) -> c20(A__TAIL(a__tail(divides(mark(z0), mark(z1)))), MARK(tail(divides(z0, z1)))) MARK(tail(tail(x0))) -> c20(MARK(tail(x0))) MARK(tail(tail(0))) -> c20(A__TAIL(a__tail(0))) MARK(tail(tail(true))) -> c20(A__TAIL(a__tail(true))) MARK(tail(tail(false))) -> c20(A__TAIL(a__tail(false))) MARK(tail(if(primes, x1, x2))) -> c20(A__TAIL(a__if(a__primes, x1, x2)), MARK(if(primes, x1, x2))) MARK(tail(if(sieve(z0), x1, x2))) -> c20(A__TAIL(a__if(a__sieve(mark(z0)), x1, x2)), MARK(if(sieve(z0), x1, x2))) MARK(tail(if(from(z0), x1, x2))) -> c20(A__TAIL(a__if(a__from(mark(z0)), x1, x2)), MARK(if(from(z0), x1, x2))) MARK(tail(if(head(z0), x1, x2))) -> c20(A__TAIL(a__if(a__head(mark(z0)), x1, x2)), MARK(if(head(z0), x1, x2))) MARK(tail(if(tail(z0), x1, x2))) -> c20(A__TAIL(a__if(a__tail(mark(z0)), x1, x2)), MARK(if(tail(z0), x1, x2))) MARK(tail(if(if(z0, z1, z2), x1, x2))) -> c20(A__TAIL(a__if(a__if(mark(z0), z1, z2), x1, x2)), MARK(if(if(z0, z1, z2), x1, x2))) MARK(tail(if(filter(z0, z1), x1, x2))) -> c20(A__TAIL(a__if(a__filter(mark(z0), mark(z1)), x1, x2)), MARK(if(filter(z0, z1), x1, x2))) MARK(tail(if(s(z0), x1, x2))) -> c20(A__TAIL(a__if(s(mark(z0)), x1, x2)), MARK(if(s(z0), x1, x2))) MARK(tail(if(cons(z0, z1), x1, x2))) -> c20(A__TAIL(a__if(cons(mark(z0), z1), x1, x2)), MARK(if(cons(z0, z1), x1, x2))) MARK(tail(if(true, x1, x2))) -> c20(A__TAIL(a__if(true, x1, x2)), MARK(if(true, x1, x2))) MARK(tail(if(false, x1, x2))) -> c20(A__TAIL(a__if(false, x1, x2)), MARK(if(false, x1, x2))) MARK(tail(if(divides(z0, z1), x1, x2))) -> c20(A__TAIL(a__if(divides(mark(z0), mark(z1)), x1, x2)), MARK(if(divides(z0, z1), x1, x2))) MARK(tail(if(x0, z1, z2))) -> c20(MARK(if(x0, z1, z2))) MARK(tail(if(0, x1, x2))) -> c20(A__TAIL(a__if(0, x1, x2))) MARK(tail(filter(x0, primes))) -> c20(A__TAIL(a__filter(mark(x0), a__primes)), MARK(filter(x0, primes))) MARK(tail(filter(x0, sieve(z0)))) -> c20(A__TAIL(a__filter(mark(x0), a__sieve(mark(z0)))), MARK(filter(x0, sieve(z0)))) MARK(tail(filter(x0, from(z0)))) -> c20(A__TAIL(a__filter(mark(x0), a__from(mark(z0)))), MARK(filter(x0, from(z0)))) MARK(tail(filter(x0, head(z0)))) -> c20(A__TAIL(a__filter(mark(x0), a__head(mark(z0)))), MARK(filter(x0, head(z0)))) MARK(tail(filter(x0, tail(z0)))) -> c20(A__TAIL(a__filter(mark(x0), a__tail(mark(z0)))), MARK(filter(x0, tail(z0)))) MARK(tail(filter(x0, if(z0, z1, z2)))) -> c20(A__TAIL(a__filter(mark(x0), a__if(mark(z0), z1, z2))), MARK(filter(x0, if(z0, z1, z2)))) MARK(tail(filter(x0, filter(z0, z1)))) -> c20(A__TAIL(a__filter(mark(x0), a__filter(mark(z0), mark(z1)))), MARK(filter(x0, filter(z0, z1)))) MARK(tail(filter(x0, s(z0)))) -> c20(A__TAIL(a__filter(mark(x0), s(mark(z0)))), MARK(filter(x0, s(z0)))) MARK(tail(filter(x0, 0))) -> c20(A__TAIL(a__filter(mark(x0), 0)), MARK(filter(x0, 0))) MARK(tail(filter(x0, cons(z0, z1)))) -> c20(A__TAIL(a__filter(mark(x0), cons(mark(z0), z1))), MARK(filter(x0, cons(z0, z1)))) MARK(tail(filter(x0, true))) -> c20(A__TAIL(a__filter(mark(x0), true)), MARK(filter(x0, true))) MARK(tail(filter(x0, false))) -> c20(A__TAIL(a__filter(mark(x0), false)), MARK(filter(x0, false))) MARK(tail(filter(x0, divides(z0, z1)))) -> c20(A__TAIL(a__filter(mark(x0), divides(mark(z0), mark(z1)))), MARK(filter(x0, divides(z0, z1)))) MARK(tail(filter(primes, x1))) -> c20(A__TAIL(a__filter(a__primes, mark(x1))), MARK(filter(primes, x1))) MARK(tail(filter(sieve(z0), x1))) -> c20(A__TAIL(a__filter(a__sieve(mark(z0)), mark(x1))), MARK(filter(sieve(z0), x1))) MARK(tail(filter(from(z0), x1))) -> c20(A__TAIL(a__filter(a__from(mark(z0)), mark(x1))), MARK(filter(from(z0), x1))) MARK(tail(filter(head(z0), x1))) -> c20(A__TAIL(a__filter(a__head(mark(z0)), mark(x1))), MARK(filter(head(z0), x1))) MARK(tail(filter(tail(z0), x1))) -> c20(A__TAIL(a__filter(a__tail(mark(z0)), mark(x1))), MARK(filter(tail(z0), x1))) MARK(tail(filter(if(z0, z1, z2), x1))) -> c20(A__TAIL(a__filter(a__if(mark(z0), z1, z2), mark(x1))), MARK(filter(if(z0, z1, z2), x1))) MARK(tail(filter(filter(z0, z1), x1))) -> c20(A__TAIL(a__filter(a__filter(mark(z0), mark(z1)), mark(x1))), MARK(filter(filter(z0, z1), x1))) MARK(tail(filter(s(z0), x1))) -> c20(A__TAIL(a__filter(s(mark(z0)), mark(x1))), MARK(filter(s(z0), x1))) MARK(tail(filter(0, x1))) -> c20(A__TAIL(a__filter(0, mark(x1))), MARK(filter(0, x1))) MARK(tail(filter(cons(z0, z1), x1))) -> c20(A__TAIL(a__filter(cons(mark(z0), z1), mark(x1))), MARK(filter(cons(z0, z1), x1))) MARK(tail(filter(true, x1))) -> c20(A__TAIL(a__filter(true, mark(x1))), MARK(filter(true, x1))) MARK(tail(filter(false, x1))) -> c20(A__TAIL(a__filter(false, mark(x1))), MARK(filter(false, x1))) MARK(tail(filter(divides(z0, z1), x1))) -> c20(A__TAIL(a__filter(divides(mark(z0), mark(z1)), mark(x1))), MARK(filter(divides(z0, z1), x1))) MARK(tail(filter(x0, x1))) -> c20(MARK(filter(x0, x1))) MARK(if(primes, x0, x1)) -> c21(A__IF(a__sieve(a__from(s(s(0)))), x0, x1), MARK(primes)) MARK(if(primes, x0, x1)) -> c21(MARK(primes)) MARK(if(sieve(primes), x1, x2)) -> c21(A__IF(a__sieve(a__primes), x1, x2), MARK(sieve(primes))) MARK(if(sieve(sieve(z0)), x1, x2)) -> c21(A__IF(a__sieve(a__sieve(mark(z0))), x1, x2), MARK(sieve(sieve(z0)))) MARK(if(sieve(from(z0)), x1, x2)) -> c21(A__IF(a__sieve(a__from(mark(z0))), x1, x2), MARK(sieve(from(z0)))) MARK(if(sieve(head(z0)), x1, x2)) -> c21(A__IF(a__sieve(a__head(mark(z0))), x1, x2), MARK(sieve(head(z0)))) MARK(if(sieve(tail(z0)), x1, x2)) -> c21(A__IF(a__sieve(a__tail(mark(z0))), x1, x2), MARK(sieve(tail(z0)))) MARK(if(sieve(if(z0, z1, z2)), x1, x2)) -> c21(A__IF(a__sieve(a__if(mark(z0), z1, z2)), x1, x2), MARK(sieve(if(z0, z1, z2)))) MARK(if(sieve(filter(z0, z1)), x1, x2)) -> c21(A__IF(a__sieve(a__filter(mark(z0), mark(z1))), x1, x2), MARK(sieve(filter(z0, z1)))) MARK(if(sieve(s(z0)), x1, x2)) -> c21(A__IF(a__sieve(s(mark(z0))), x1, x2), MARK(sieve(s(z0)))) MARK(if(sieve(cons(z0, z1)), x1, x2)) -> c21(A__IF(a__sieve(cons(mark(z0), z1)), x1, x2), MARK(sieve(cons(z0, z1)))) MARK(if(sieve(divides(z0, z1)), x1, x2)) -> c21(A__IF(a__sieve(divides(mark(z0), mark(z1))), x1, x2), MARK(sieve(divides(z0, z1)))) MARK(if(sieve(x0), x1, x2)) -> c21(MARK(sieve(x0))) MARK(if(sieve(0), x1, x2)) -> c21(A__IF(a__sieve(0), x1, x2)) MARK(if(sieve(true), x1, x2)) -> c21(A__IF(a__sieve(true), x1, x2)) MARK(if(sieve(false), x1, x2)) -> c21(A__IF(a__sieve(false), x1, x2)) MARK(if(from(primes), x1, x2)) -> c21(A__IF(a__from(a__primes), x1, x2), MARK(from(primes))) MARK(if(from(sieve(z0)), x1, x2)) -> c21(A__IF(a__from(a__sieve(mark(z0))), x1, x2), MARK(from(sieve(z0)))) MARK(if(from(from(z0)), x1, x2)) -> c21(A__IF(a__from(a__from(mark(z0))), x1, x2), MARK(from(from(z0)))) MARK(if(from(head(z0)), x1, x2)) -> c21(A__IF(a__from(a__head(mark(z0))), x1, x2), MARK(from(head(z0)))) MARK(if(from(tail(z0)), x1, x2)) -> c21(A__IF(a__from(a__tail(mark(z0))), x1, x2), MARK(from(tail(z0)))) MARK(if(from(if(z0, z1, z2)), x1, x2)) -> c21(A__IF(a__from(a__if(mark(z0), z1, z2)), x1, x2), MARK(from(if(z0, z1, z2)))) MARK(if(from(filter(z0, z1)), x1, x2)) -> c21(A__IF(a__from(a__filter(mark(z0), mark(z1))), x1, x2), MARK(from(filter(z0, z1)))) MARK(if(from(s(z0)), x1, x2)) -> c21(A__IF(a__from(s(mark(z0))), x1, x2), MARK(from(s(z0)))) MARK(if(from(0), x1, x2)) -> c21(A__IF(a__from(0), x1, x2), MARK(from(0))) MARK(if(from(cons(z0, z1)), x1, x2)) -> c21(A__IF(a__from(cons(mark(z0), z1)), x1, x2), MARK(from(cons(z0, z1)))) MARK(if(from(true), x1, x2)) -> c21(A__IF(a__from(true), x1, x2), MARK(from(true))) MARK(if(from(false), x1, x2)) -> c21(A__IF(a__from(false), x1, x2), MARK(from(false))) MARK(if(from(divides(z0, z1)), x1, x2)) -> c21(A__IF(a__from(divides(mark(z0), mark(z1))), x1, x2), MARK(from(divides(z0, z1)))) MARK(if(from(x0), x1, x2)) -> c21(MARK(from(x0))) MARK(if(head(primes), x1, x2)) -> c21(A__IF(a__head(a__primes), x1, x2), MARK(head(primes))) MARK(if(head(sieve(z0)), x1, x2)) -> c21(A__IF(a__head(a__sieve(mark(z0))), x1, x2), MARK(head(sieve(z0)))) MARK(if(head(from(z0)), x1, x2)) -> c21(A__IF(a__head(a__from(mark(z0))), x1, x2), MARK(head(from(z0)))) MARK(if(head(head(z0)), x1, x2)) -> c21(A__IF(a__head(a__head(mark(z0))), x1, x2), MARK(head(head(z0)))) MARK(if(head(tail(z0)), x1, x2)) -> c21(A__IF(a__head(a__tail(mark(z0))), x1, x2), MARK(head(tail(z0)))) MARK(if(head(if(z0, z1, z2)), x1, x2)) -> c21(A__IF(a__head(a__if(mark(z0), z1, z2)), x1, x2), MARK(head(if(z0, z1, z2)))) MARK(if(head(filter(z0, z1)), x1, x2)) -> c21(A__IF(a__head(a__filter(mark(z0), mark(z1))), x1, x2), MARK(head(filter(z0, z1)))) MARK(if(head(s(z0)), x1, x2)) -> c21(A__IF(a__head(s(mark(z0))), x1, x2), MARK(head(s(z0)))) MARK(if(head(cons(z0, z1)), x1, x2)) -> c21(A__IF(a__head(cons(mark(z0), z1)), x1, x2), MARK(head(cons(z0, z1)))) MARK(if(head(divides(z0, z1)), x1, x2)) -> c21(A__IF(a__head(divides(mark(z0), mark(z1))), x1, x2), MARK(head(divides(z0, z1)))) MARK(if(head(x0), x1, x2)) -> c21(MARK(head(x0))) MARK(if(head(0), x1, x2)) -> c21(A__IF(a__head(0), x1, x2)) MARK(if(head(true), x1, x2)) -> c21(A__IF(a__head(true), x1, x2)) MARK(if(head(false), x1, x2)) -> c21(A__IF(a__head(false), x1, x2)) MARK(if(tail(primes), x1, x2)) -> c21(A__IF(a__tail(a__primes), x1, x2), MARK(tail(primes))) MARK(if(tail(sieve(z0)), x1, x2)) -> c21(A__IF(a__tail(a__sieve(mark(z0))), x1, x2), MARK(tail(sieve(z0)))) MARK(if(tail(from(z0)), x1, x2)) -> c21(A__IF(a__tail(a__from(mark(z0))), x1, x2), MARK(tail(from(z0)))) MARK(if(tail(head(z0)), x1, x2)) -> c21(A__IF(a__tail(a__head(mark(z0))), x1, x2), MARK(tail(head(z0)))) MARK(if(tail(tail(z0)), x1, x2)) -> c21(A__IF(a__tail(a__tail(mark(z0))), x1, x2), MARK(tail(tail(z0)))) MARK(if(tail(if(z0, z1, z2)), x1, x2)) -> c21(A__IF(a__tail(a__if(mark(z0), z1, z2)), x1, x2), MARK(tail(if(z0, z1, z2)))) MARK(if(tail(filter(z0, z1)), x1, x2)) -> c21(A__IF(a__tail(a__filter(mark(z0), mark(z1))), x1, x2), MARK(tail(filter(z0, z1)))) MARK(if(tail(s(z0)), x1, x2)) -> c21(A__IF(a__tail(s(mark(z0))), x1, x2), MARK(tail(s(z0)))) MARK(if(tail(cons(z0, z1)), x1, x2)) -> c21(A__IF(a__tail(cons(mark(z0), z1)), x1, x2), MARK(tail(cons(z0, z1)))) MARK(if(tail(divides(z0, z1)), x1, x2)) -> c21(A__IF(a__tail(divides(mark(z0), mark(z1))), x1, x2), MARK(tail(divides(z0, z1)))) MARK(if(tail(x0), x1, x2)) -> c21(MARK(tail(x0))) MARK(if(tail(0), x1, x2)) -> c21(A__IF(a__tail(0), x1, x2)) MARK(if(tail(true), x1, x2)) -> c21(A__IF(a__tail(true), x1, x2)) MARK(if(tail(false), x1, x2)) -> c21(A__IF(a__tail(false), x1, x2)) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c18_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_1, c17_2, c17_1, c19_2, c19_1, c20_2, c20_1, c21_2, c21_1, c22_2, c22_1, c23_2, c23_1, c_2 ---------------------------------------- (203) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(if(if(z0, z1, z2), x1, x2)) -> c21(A__IF(a__if(mark(z0), z1, z2), x1, x2), MARK(if(z0, z1, z2))) by MARK(if(if(x0, z1, z2), x3, x4)) -> c21(A__IF(if(mark(x0), z1, z2), x3, x4), MARK(if(x0, z1, z2))) MARK(if(if(primes, x1, x2), x3, x4)) -> c21(A__IF(a__if(a__primes, x1, x2), x3, x4), MARK(if(primes, x1, x2))) MARK(if(if(sieve(z0), x1, x2), x3, x4)) -> c21(A__IF(a__if(a__sieve(mark(z0)), x1, x2), x3, x4), MARK(if(sieve(z0), x1, x2))) MARK(if(if(from(z0), x1, x2), x3, x4)) -> c21(A__IF(a__if(a__from(mark(z0)), x1, x2), x3, x4), MARK(if(from(z0), x1, x2))) MARK(if(if(head(z0), x1, x2), x3, x4)) -> c21(A__IF(a__if(a__head(mark(z0)), x1, x2), x3, x4), MARK(if(head(z0), x1, x2))) MARK(if(if(tail(z0), x1, x2), x3, x4)) -> c21(A__IF(a__if(a__tail(mark(z0)), x1, x2), x3, x4), MARK(if(tail(z0), x1, x2))) MARK(if(if(if(z0, z1, z2), x1, x2), x3, x4)) -> c21(A__IF(a__if(a__if(mark(z0), z1, z2), x1, x2), x3, x4), MARK(if(if(z0, z1, z2), x1, x2))) MARK(if(if(filter(z0, z1), x1, x2), x3, x4)) -> c21(A__IF(a__if(a__filter(mark(z0), mark(z1)), x1, x2), x3, x4), MARK(if(filter(z0, z1), x1, x2))) MARK(if(if(s(z0), x1, x2), x3, x4)) -> c21(A__IF(a__if(s(mark(z0)), x1, x2), x3, x4), MARK(if(s(z0), x1, x2))) MARK(if(if(0, x1, x2), x3, x4)) -> c21(A__IF(a__if(0, x1, x2), x3, x4), MARK(if(0, x1, x2))) MARK(if(if(cons(z0, z1), x1, x2), x3, x4)) -> c21(A__IF(a__if(cons(mark(z0), z1), x1, x2), x3, x4), MARK(if(cons(z0, z1), x1, x2))) MARK(if(if(true, x1, x2), x3, x4)) -> c21(A__IF(a__if(true, x1, x2), x3, x4), MARK(if(true, x1, x2))) MARK(if(if(false, x1, x2), x3, x4)) -> c21(A__IF(a__if(false, x1, x2), x3, x4), MARK(if(false, x1, x2))) MARK(if(if(divides(z0, z1), x1, x2), x3, x4)) -> c21(A__IF(a__if(divides(mark(z0), mark(z1)), x1, x2), x3, x4), MARK(if(divides(z0, z1), x1, x2))) ---------------------------------------- (204) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(filter(z0, z1), x1, x2)) -> c21(A__IF(a__filter(mark(z0), mark(z1)), x1, x2), MARK(filter(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c22(A__FILTER(mark(x0), a__primes), MARK(x0)) MARK(filter(x0, sieve(z0))) -> c22(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, from(z0))) -> c22(A__FILTER(mark(x0), a__from(mark(z0))), MARK(x0)) MARK(filter(x0, head(z0))) -> c22(A__FILTER(mark(x0), a__head(mark(z0))), MARK(x0)) MARK(filter(x0, tail(z0))) -> c22(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(x0)) MARK(filter(x0, if(z0, z1, z2))) -> c22(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0)) MARK(filter(x0, filter(z0, z1))) -> c22(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(x0)) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(x0, s(z0))) -> c22(MARK(x0)) MARK(filter(x0, 0)) -> c22(MARK(x0)) MARK(filter(x0, true)) -> c22(MARK(x0)) MARK(filter(x0, false)) -> c22(MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), x1)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c23(A__FILTER(mark(x0), a__primes), MARK(primes)) MARK(filter(x0, sieve(z0))) -> c23(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, from(z0))) -> c23(A__FILTER(mark(x0), a__from(mark(z0))), MARK(from(z0))) MARK(filter(x0, head(z0))) -> c23(A__FILTER(mark(x0), a__head(mark(z0))), MARK(head(z0))) MARK(filter(x0, tail(z0))) -> c23(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(tail(z0))) MARK(filter(x0, if(z0, z1, z2))) -> c23(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(filter(x0, filter(z0, z1))) -> c23(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(filter(x0, cons(z0, z1))) -> c23(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(primes, x1)) -> c23(A__FILTER(a__primes, mark(x1)), MARK(x1)) MARK(filter(sieve(z0), x1)) -> c23(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(from(z0), x1)) -> c23(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(head(z0), x1)) -> c23(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(tail(z0), x1)) -> c23(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(if(z0, z1, z2), x1)) -> c23(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1)) MARK(filter(filter(z0, z1), x1)) -> c23(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(x1)) MARK(filter(s(z0), x1)) -> c23(A__FILTER(s(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(x0, s(z0))) -> c23(MARK(s(z0))) MARK(filter(x0, divides(z0, z1))) -> c23(MARK(divides(z0, z1))) MARK(filter(0, x1)) -> c23(MARK(x1)) MARK(filter(cons(z0, z1), x1)) -> c23(MARK(x1)) MARK(filter(true, x1)) -> c23(MARK(x1)) MARK(filter(false, x1)) -> c23(MARK(x1)) MARK(filter(divides(z0, z1), x1)) -> c23(MARK(x1)) A__PRIMES -> c(A__SIEVE(cons(s(mark(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(sieve(primes)) -> c17(MARK(primes)) MARK(sieve(sieve(primes))) -> c17(A__SIEVE(a__sieve(a__primes)), MARK(sieve(primes))) MARK(sieve(sieve(sieve(z0)))) -> c17(A__SIEVE(a__sieve(a__sieve(mark(z0)))), MARK(sieve(sieve(z0)))) MARK(sieve(sieve(from(z0)))) -> c17(A__SIEVE(a__sieve(a__from(mark(z0)))), MARK(sieve(from(z0)))) MARK(sieve(sieve(head(z0)))) -> c17(A__SIEVE(a__sieve(a__head(mark(z0)))), MARK(sieve(head(z0)))) MARK(sieve(sieve(tail(z0)))) -> c17(A__SIEVE(a__sieve(a__tail(mark(z0)))), MARK(sieve(tail(z0)))) MARK(sieve(sieve(if(z0, z1, z2)))) -> c17(A__SIEVE(a__sieve(a__if(mark(z0), z1, z2))), MARK(sieve(if(z0, z1, z2)))) MARK(sieve(sieve(filter(z0, z1)))) -> c17(A__SIEVE(a__sieve(a__filter(mark(z0), mark(z1)))), MARK(sieve(filter(z0, z1)))) MARK(sieve(sieve(s(z0)))) -> c17(A__SIEVE(a__sieve(s(mark(z0)))), MARK(sieve(s(z0)))) MARK(sieve(sieve(cons(z0, z1)))) -> c17(A__SIEVE(a__sieve(cons(mark(z0), z1))), MARK(sieve(cons(z0, z1)))) MARK(sieve(sieve(divides(z0, z1)))) -> c17(A__SIEVE(a__sieve(divides(mark(z0), mark(z1)))), MARK(sieve(divides(z0, z1)))) MARK(sieve(sieve(x0))) -> c17(MARK(sieve(x0))) MARK(sieve(sieve(0))) -> c17(A__SIEVE(a__sieve(0))) MARK(sieve(sieve(true))) -> c17(A__SIEVE(a__sieve(true))) MARK(sieve(sieve(false))) -> c17(A__SIEVE(a__sieve(false))) MARK(sieve(from(x0))) -> c17(A__SIEVE(cons(mark(mark(x0)), from(s(mark(x0))))), MARK(from(x0))) MARK(sieve(from(primes))) -> c17(A__SIEVE(a__from(a__primes)), MARK(from(primes))) MARK(sieve(from(sieve(z0)))) -> c17(A__SIEVE(a__from(a__sieve(mark(z0)))), MARK(from(sieve(z0)))) MARK(sieve(from(from(z0)))) -> c17(A__SIEVE(a__from(a__from(mark(z0)))), MARK(from(from(z0)))) MARK(sieve(from(head(z0)))) -> c17(A__SIEVE(a__from(a__head(mark(z0)))), MARK(from(head(z0)))) MARK(sieve(from(tail(z0)))) -> c17(A__SIEVE(a__from(a__tail(mark(z0)))), MARK(from(tail(z0)))) MARK(sieve(from(if(z0, z1, z2)))) -> c17(A__SIEVE(a__from(a__if(mark(z0), z1, z2))), MARK(from(if(z0, z1, z2)))) MARK(sieve(from(filter(z0, z1)))) -> c17(A__SIEVE(a__from(a__filter(mark(z0), mark(z1)))), MARK(from(filter(z0, z1)))) MARK(sieve(from(s(z0)))) -> c17(A__SIEVE(a__from(s(mark(z0)))), MARK(from(s(z0)))) MARK(sieve(from(0))) -> c17(A__SIEVE(a__from(0)), MARK(from(0))) MARK(sieve(from(cons(z0, z1)))) -> c17(A__SIEVE(a__from(cons(mark(z0), z1))), MARK(from(cons(z0, z1)))) MARK(sieve(from(true))) -> c17(A__SIEVE(a__from(true)), MARK(from(true))) MARK(sieve(from(false))) -> c17(A__SIEVE(a__from(false)), MARK(from(false))) MARK(sieve(from(divides(z0, z1)))) -> c17(A__SIEVE(a__from(divides(mark(z0), mark(z1)))), MARK(from(divides(z0, z1)))) MARK(sieve(from(x0))) -> c17(MARK(from(x0))) MARK(sieve(head(primes))) -> c17(A__SIEVE(a__head(a__primes)), MARK(head(primes))) MARK(sieve(head(sieve(z0)))) -> c17(A__SIEVE(a__head(a__sieve(mark(z0)))), MARK(head(sieve(z0)))) MARK(sieve(head(from(z0)))) -> c17(A__SIEVE(a__head(a__from(mark(z0)))), MARK(head(from(z0)))) MARK(sieve(head(head(z0)))) -> c17(A__SIEVE(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(sieve(head(tail(z0)))) -> c17(A__SIEVE(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(sieve(head(if(z0, z1, z2)))) -> c17(A__SIEVE(a__head(a__if(mark(z0), z1, z2))), MARK(head(if(z0, z1, z2)))) MARK(sieve(head(filter(z0, z1)))) -> c17(A__SIEVE(a__head(a__filter(mark(z0), mark(z1)))), MARK(head(filter(z0, z1)))) MARK(sieve(head(s(z0)))) -> c17(A__SIEVE(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(sieve(head(cons(z0, z1)))) -> c17(A__SIEVE(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(sieve(head(divides(z0, z1)))) -> c17(A__SIEVE(a__head(divides(mark(z0), mark(z1)))), MARK(head(divides(z0, z1)))) MARK(sieve(head(x0))) -> c17(MARK(head(x0))) MARK(sieve(head(0))) -> c17(A__SIEVE(a__head(0))) MARK(sieve(head(true))) -> c17(A__SIEVE(a__head(true))) MARK(sieve(head(false))) -> c17(A__SIEVE(a__head(false))) MARK(sieve(tail(primes))) -> c17(A__SIEVE(a__tail(a__primes)), MARK(tail(primes))) MARK(sieve(tail(sieve(z0)))) -> c17(A__SIEVE(a__tail(a__sieve(mark(z0)))), MARK(tail(sieve(z0)))) MARK(sieve(tail(from(z0)))) -> c17(A__SIEVE(a__tail(a__from(mark(z0)))), MARK(tail(from(z0)))) MARK(sieve(tail(head(z0)))) -> c17(A__SIEVE(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(sieve(tail(tail(z0)))) -> c17(A__SIEVE(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(sieve(tail(if(z0, z1, z2)))) -> c17(A__SIEVE(a__tail(a__if(mark(z0), z1, z2))), MARK(tail(if(z0, z1, z2)))) MARK(sieve(tail(filter(z0, z1)))) -> c17(A__SIEVE(a__tail(a__filter(mark(z0), mark(z1)))), MARK(tail(filter(z0, z1)))) MARK(sieve(tail(s(z0)))) -> c17(A__SIEVE(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(sieve(tail(cons(z0, z1)))) -> c17(A__SIEVE(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(sieve(tail(divides(z0, z1)))) -> c17(A__SIEVE(a__tail(divides(mark(z0), mark(z1)))), MARK(tail(divides(z0, z1)))) MARK(sieve(tail(x0))) -> c17(MARK(tail(x0))) MARK(sieve(tail(0))) -> c17(A__SIEVE(a__tail(0))) MARK(sieve(tail(true))) -> c17(A__SIEVE(a__tail(true))) MARK(sieve(tail(false))) -> c17(A__SIEVE(a__tail(false))) MARK(sieve(if(primes, x1, x2))) -> c17(A__SIEVE(a__if(a__primes, x1, x2)), MARK(if(primes, x1, x2))) MARK(sieve(if(sieve(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__sieve(mark(z0)), x1, x2)), MARK(if(sieve(z0), x1, x2))) MARK(sieve(if(from(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__from(mark(z0)), x1, x2)), MARK(if(from(z0), x1, x2))) MARK(sieve(if(head(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__head(mark(z0)), x1, x2)), MARK(if(head(z0), x1, x2))) MARK(sieve(if(tail(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__tail(mark(z0)), x1, x2)), MARK(if(tail(z0), x1, x2))) MARK(sieve(if(if(z0, z1, z2), x1, x2))) -> c17(A__SIEVE(a__if(a__if(mark(z0), z1, z2), x1, x2)), MARK(if(if(z0, z1, z2), x1, x2))) MARK(sieve(if(filter(z0, z1), x1, x2))) -> c17(A__SIEVE(a__if(a__filter(mark(z0), mark(z1)), x1, x2)), MARK(if(filter(z0, z1), x1, x2))) MARK(sieve(if(s(z0), x1, x2))) -> c17(A__SIEVE(a__if(s(mark(z0)), x1, x2)), MARK(if(s(z0), x1, x2))) MARK(sieve(if(cons(z0, z1), x1, x2))) -> c17(A__SIEVE(a__if(cons(mark(z0), z1), x1, x2)), MARK(if(cons(z0, z1), x1, x2))) MARK(sieve(if(true, x1, x2))) -> c17(A__SIEVE(a__if(true, x1, x2)), MARK(if(true, x1, x2))) MARK(sieve(if(false, x1, x2))) -> c17(A__SIEVE(a__if(false, x1, x2)), MARK(if(false, x1, x2))) MARK(sieve(if(divides(z0, z1), x1, x2))) -> c17(A__SIEVE(a__if(divides(mark(z0), mark(z1)), x1, x2)), MARK(if(divides(z0, z1), x1, x2))) MARK(sieve(if(x0, z1, z2))) -> c17(MARK(if(x0, z1, z2))) MARK(sieve(if(0, x1, x2))) -> c17(A__SIEVE(a__if(0, x1, x2))) MARK(sieve(filter(x0, primes))) -> c17(A__SIEVE(a__filter(mark(x0), a__primes)), MARK(filter(x0, primes))) MARK(sieve(filter(x0, sieve(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), a__sieve(mark(z0)))), MARK(filter(x0, sieve(z0)))) MARK(sieve(filter(x0, from(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), a__from(mark(z0)))), MARK(filter(x0, from(z0)))) MARK(sieve(filter(x0, head(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), a__head(mark(z0)))), MARK(filter(x0, head(z0)))) 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x2), x3, x4), MARK(if(head(z0), x1, x2))) MARK(if(if(tail(z0), x1, x2), x3, x4)) -> c21(A__IF(a__if(a__tail(mark(z0)), x1, x2), x3, x4), MARK(if(tail(z0), x1, x2))) MARK(if(if(if(z0, z1, z2), x1, x2), x3, x4)) -> c21(A__IF(a__if(a__if(mark(z0), z1, z2), x1, x2), x3, x4), MARK(if(if(z0, z1, z2), x1, x2))) MARK(if(if(filter(z0, z1), x1, x2), x3, x4)) -> c21(A__IF(a__if(a__filter(mark(z0), mark(z1)), x1, x2), x3, x4), MARK(if(filter(z0, z1), x1, x2))) MARK(if(if(s(z0), x1, x2), x3, x4)) -> c21(A__IF(a__if(s(mark(z0)), x1, x2), x3, x4), MARK(if(s(z0), x1, x2))) MARK(if(if(0, x1, x2), x3, x4)) -> c21(A__IF(a__if(0, x1, x2), x3, x4), MARK(if(0, x1, x2))) MARK(if(if(cons(z0, z1), x1, x2), x3, x4)) -> c21(A__IF(a__if(cons(mark(z0), z1), x1, x2), x3, x4), MARK(if(cons(z0, z1), x1, x2))) MARK(if(if(true, x1, x2), x3, x4)) -> c21(A__IF(a__if(true, x1, x2), x3, x4), MARK(if(true, x1, x2))) MARK(if(if(false, x1, x2), x3, x4)) -> c21(A__IF(a__if(false, x1, x2), x3, x4), MARK(if(false, x1, x2))) MARK(if(if(divides(z0, z1), x1, x2), x3, x4)) -> c21(A__IF(a__if(divides(mark(z0), mark(z1)), x1, x2), x3, x4), MARK(if(divides(z0, z1), x1, x2))) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c18_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_1, c17_2, c17_1, c19_2, c19_1, c20_2, c20_1, c21_2, c21_1, c22_2, c22_1, c23_2, c23_1, c_2 ---------------------------------------- (205) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (206) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(filter(z0, z1), x1, x2)) -> c21(A__IF(a__filter(mark(z0), mark(z1)), x1, x2), MARK(filter(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c22(A__FILTER(mark(x0), a__primes), MARK(x0)) MARK(filter(x0, sieve(z0))) -> c22(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, from(z0))) -> c22(A__FILTER(mark(x0), a__from(mark(z0))), MARK(x0)) MARK(filter(x0, head(z0))) -> c22(A__FILTER(mark(x0), a__head(mark(z0))), MARK(x0)) MARK(filter(x0, tail(z0))) -> c22(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(x0)) MARK(filter(x0, if(z0, z1, z2))) -> c22(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0)) MARK(filter(x0, filter(z0, z1))) -> c22(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(x0)) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(x0, s(z0))) -> c22(MARK(x0)) MARK(filter(x0, 0)) -> c22(MARK(x0)) MARK(filter(x0, true)) -> c22(MARK(x0)) MARK(filter(x0, false)) -> c22(MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), x1)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c23(A__FILTER(mark(x0), a__primes), MARK(primes)) MARK(filter(x0, sieve(z0))) -> c23(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, from(z0))) -> c23(A__FILTER(mark(x0), a__from(mark(z0))), MARK(from(z0))) MARK(filter(x0, head(z0))) -> c23(A__FILTER(mark(x0), a__head(mark(z0))), MARK(head(z0))) MARK(filter(x0, tail(z0))) -> c23(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(tail(z0))) MARK(filter(x0, if(z0, z1, z2))) -> c23(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(filter(x0, filter(z0, z1))) -> c23(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(filter(x0, cons(z0, z1))) -> c23(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(primes, x1)) -> c23(A__FILTER(a__primes, mark(x1)), MARK(x1)) MARK(filter(sieve(z0), x1)) -> c23(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(from(z0), x1)) -> c23(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(head(z0), x1)) -> c23(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(tail(z0), x1)) -> c23(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(if(z0, z1, z2), x1)) -> c23(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1)) MARK(filter(filter(z0, z1), x1)) -> c23(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(x1)) MARK(filter(s(z0), x1)) -> c23(A__FILTER(s(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(x0, s(z0))) -> c23(MARK(s(z0))) MARK(filter(x0, divides(z0, z1))) -> c23(MARK(divides(z0, z1))) MARK(filter(0, x1)) -> c23(MARK(x1)) MARK(filter(cons(z0, z1), x1)) -> c23(MARK(x1)) MARK(filter(true, x1)) -> c23(MARK(x1)) MARK(filter(false, x1)) -> c23(MARK(x1)) MARK(filter(divides(z0, z1), x1)) -> c23(MARK(x1)) A__PRIMES -> c(A__SIEVE(cons(s(mark(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(sieve(primes)) -> c17(MARK(primes)) MARK(sieve(sieve(primes))) -> c17(A__SIEVE(a__sieve(a__primes)), MARK(sieve(primes))) MARK(sieve(sieve(sieve(z0)))) -> c17(A__SIEVE(a__sieve(a__sieve(mark(z0)))), MARK(sieve(sieve(z0)))) MARK(sieve(sieve(from(z0)))) -> c17(A__SIEVE(a__sieve(a__from(mark(z0)))), MARK(sieve(from(z0)))) MARK(sieve(sieve(head(z0)))) -> c17(A__SIEVE(a__sieve(a__head(mark(z0)))), MARK(sieve(head(z0)))) MARK(sieve(sieve(tail(z0)))) -> c17(A__SIEVE(a__sieve(a__tail(mark(z0)))), MARK(sieve(tail(z0)))) MARK(sieve(sieve(if(z0, z1, z2)))) -> c17(A__SIEVE(a__sieve(a__if(mark(z0), z1, z2))), MARK(sieve(if(z0, z1, z2)))) MARK(sieve(sieve(filter(z0, z1)))) -> c17(A__SIEVE(a__sieve(a__filter(mark(z0), mark(z1)))), MARK(sieve(filter(z0, z1)))) MARK(sieve(sieve(s(z0)))) -> c17(A__SIEVE(a__sieve(s(mark(z0)))), MARK(sieve(s(z0)))) MARK(sieve(sieve(cons(z0, z1)))) -> c17(A__SIEVE(a__sieve(cons(mark(z0), z1))), MARK(sieve(cons(z0, z1)))) MARK(sieve(sieve(divides(z0, z1)))) -> c17(A__SIEVE(a__sieve(divides(mark(z0), mark(z1)))), MARK(sieve(divides(z0, z1)))) MARK(sieve(sieve(x0))) -> c17(MARK(sieve(x0))) MARK(sieve(sieve(0))) -> c17(A__SIEVE(a__sieve(0))) MARK(sieve(sieve(true))) -> c17(A__SIEVE(a__sieve(true))) MARK(sieve(sieve(false))) -> c17(A__SIEVE(a__sieve(false))) MARK(sieve(from(x0))) -> c17(A__SIEVE(cons(mark(mark(x0)), from(s(mark(x0))))), MARK(from(x0))) MARK(sieve(from(primes))) -> c17(A__SIEVE(a__from(a__primes)), MARK(from(primes))) MARK(sieve(from(sieve(z0)))) -> c17(A__SIEVE(a__from(a__sieve(mark(z0)))), MARK(from(sieve(z0)))) MARK(sieve(from(from(z0)))) -> c17(A__SIEVE(a__from(a__from(mark(z0)))), MARK(from(from(z0)))) MARK(sieve(from(head(z0)))) -> c17(A__SIEVE(a__from(a__head(mark(z0)))), MARK(from(head(z0)))) MARK(sieve(from(tail(z0)))) -> c17(A__SIEVE(a__from(a__tail(mark(z0)))), MARK(from(tail(z0)))) MARK(sieve(from(if(z0, z1, z2)))) -> c17(A__SIEVE(a__from(a__if(mark(z0), z1, z2))), MARK(from(if(z0, z1, z2)))) MARK(sieve(from(filter(z0, z1)))) -> c17(A__SIEVE(a__from(a__filter(mark(z0), mark(z1)))), MARK(from(filter(z0, z1)))) MARK(sieve(from(s(z0)))) -> c17(A__SIEVE(a__from(s(mark(z0)))), MARK(from(s(z0)))) MARK(sieve(from(0))) -> c17(A__SIEVE(a__from(0)), MARK(from(0))) MARK(sieve(from(cons(z0, z1)))) -> c17(A__SIEVE(a__from(cons(mark(z0), z1))), MARK(from(cons(z0, z1)))) MARK(sieve(from(true))) -> c17(A__SIEVE(a__from(true)), MARK(from(true))) MARK(sieve(from(false))) -> c17(A__SIEVE(a__from(false)), MARK(from(false))) MARK(sieve(from(divides(z0, z1)))) -> c17(A__SIEVE(a__from(divides(mark(z0), mark(z1)))), MARK(from(divides(z0, z1)))) MARK(sieve(from(x0))) -> c17(MARK(from(x0))) MARK(sieve(head(primes))) -> c17(A__SIEVE(a__head(a__primes)), MARK(head(primes))) MARK(sieve(head(sieve(z0)))) -> c17(A__SIEVE(a__head(a__sieve(mark(z0)))), MARK(head(sieve(z0)))) MARK(sieve(head(from(z0)))) -> c17(A__SIEVE(a__head(a__from(mark(z0)))), MARK(head(from(z0)))) MARK(sieve(head(head(z0)))) -> c17(A__SIEVE(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(sieve(head(tail(z0)))) -> c17(A__SIEVE(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(sieve(head(if(z0, z1, z2)))) -> c17(A__SIEVE(a__head(a__if(mark(z0), z1, z2))), MARK(head(if(z0, z1, z2)))) MARK(sieve(head(filter(z0, z1)))) -> c17(A__SIEVE(a__head(a__filter(mark(z0), mark(z1)))), MARK(head(filter(z0, z1)))) MARK(sieve(head(s(z0)))) -> c17(A__SIEVE(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(sieve(head(cons(z0, z1)))) -> c17(A__SIEVE(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(sieve(head(divides(z0, z1)))) -> c17(A__SIEVE(a__head(divides(mark(z0), mark(z1)))), MARK(head(divides(z0, z1)))) MARK(sieve(head(x0))) -> c17(MARK(head(x0))) MARK(sieve(head(0))) -> c17(A__SIEVE(a__head(0))) MARK(sieve(head(true))) -> c17(A__SIEVE(a__head(true))) MARK(sieve(head(false))) -> c17(A__SIEVE(a__head(false))) MARK(sieve(tail(primes))) -> c17(A__SIEVE(a__tail(a__primes)), MARK(tail(primes))) MARK(sieve(tail(sieve(z0)))) -> c17(A__SIEVE(a__tail(a__sieve(mark(z0)))), MARK(tail(sieve(z0)))) MARK(sieve(tail(from(z0)))) -> c17(A__SIEVE(a__tail(a__from(mark(z0)))), MARK(tail(from(z0)))) MARK(sieve(tail(head(z0)))) -> c17(A__SIEVE(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(sieve(tail(tail(z0)))) -> c17(A__SIEVE(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(sieve(tail(if(z0, z1, z2)))) -> c17(A__SIEVE(a__tail(a__if(mark(z0), z1, z2))), MARK(tail(if(z0, z1, z2)))) MARK(sieve(tail(filter(z0, z1)))) -> c17(A__SIEVE(a__tail(a__filter(mark(z0), mark(z1)))), MARK(tail(filter(z0, z1)))) MARK(sieve(tail(s(z0)))) -> c17(A__SIEVE(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(sieve(tail(cons(z0, z1)))) -> c17(A__SIEVE(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(sieve(tail(divides(z0, z1)))) -> c17(A__SIEVE(a__tail(divides(mark(z0), mark(z1)))), MARK(tail(divides(z0, z1)))) MARK(sieve(tail(x0))) -> c17(MARK(tail(x0))) MARK(sieve(tail(0))) -> c17(A__SIEVE(a__tail(0))) MARK(sieve(tail(true))) -> c17(A__SIEVE(a__tail(true))) MARK(sieve(tail(false))) -> c17(A__SIEVE(a__tail(false))) MARK(sieve(if(primes, x1, x2))) -> c17(A__SIEVE(a__if(a__primes, x1, x2)), MARK(if(primes, x1, x2))) MARK(sieve(if(sieve(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__sieve(mark(z0)), x1, x2)), MARK(if(sieve(z0), x1, x2))) MARK(sieve(if(from(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__from(mark(z0)), x1, x2)), MARK(if(from(z0), x1, x2))) MARK(sieve(if(head(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__head(mark(z0)), x1, x2)), MARK(if(head(z0), x1, x2))) MARK(sieve(if(tail(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__tail(mark(z0)), x1, x2)), MARK(if(tail(z0), x1, x2))) MARK(sieve(if(if(z0, z1, z2), x1, x2))) -> c17(A__SIEVE(a__if(a__if(mark(z0), z1, z2), x1, x2)), MARK(if(if(z0, z1, z2), x1, x2))) MARK(sieve(if(filter(z0, z1), x1, x2))) -> c17(A__SIEVE(a__if(a__filter(mark(z0), mark(z1)), x1, 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MARK(tail(tail(divides(z0, z1)))) -> c20(A__TAIL(a__tail(divides(mark(z0), mark(z1)))), MARK(tail(divides(z0, z1)))) MARK(tail(tail(x0))) -> c20(MARK(tail(x0))) MARK(tail(tail(0))) -> c20(A__TAIL(a__tail(0))) MARK(tail(tail(true))) -> c20(A__TAIL(a__tail(true))) MARK(tail(tail(false))) -> c20(A__TAIL(a__tail(false))) MARK(tail(if(primes, x1, x2))) -> c20(A__TAIL(a__if(a__primes, x1, x2)), MARK(if(primes, x1, x2))) MARK(tail(if(sieve(z0), x1, x2))) -> c20(A__TAIL(a__if(a__sieve(mark(z0)), x1, x2)), MARK(if(sieve(z0), x1, x2))) MARK(tail(if(from(z0), x1, x2))) -> c20(A__TAIL(a__if(a__from(mark(z0)), x1, x2)), MARK(if(from(z0), x1, x2))) MARK(tail(if(head(z0), x1, x2))) -> c20(A__TAIL(a__if(a__head(mark(z0)), x1, x2)), MARK(if(head(z0), x1, x2))) MARK(tail(if(tail(z0), x1, x2))) -> c20(A__TAIL(a__if(a__tail(mark(z0)), x1, x2)), MARK(if(tail(z0), x1, x2))) MARK(tail(if(if(z0, z1, z2), x1, x2))) -> c20(A__TAIL(a__if(a__if(mark(z0), z1, z2), x1, x2)), MARK(if(if(z0, z1, z2), x1, x2))) MARK(tail(if(filter(z0, z1), x1, x2))) -> c20(A__TAIL(a__if(a__filter(mark(z0), mark(z1)), x1, x2)), MARK(if(filter(z0, z1), x1, x2))) MARK(tail(if(s(z0), x1, x2))) -> c20(A__TAIL(a__if(s(mark(z0)), x1, x2)), MARK(if(s(z0), x1, x2))) MARK(tail(if(cons(z0, z1), x1, x2))) -> c20(A__TAIL(a__if(cons(mark(z0), z1), x1, x2)), MARK(if(cons(z0, z1), x1, x2))) MARK(tail(if(true, x1, x2))) -> c20(A__TAIL(a__if(true, x1, x2)), MARK(if(true, x1, x2))) MARK(tail(if(false, x1, x2))) -> c20(A__TAIL(a__if(false, x1, x2)), MARK(if(false, x1, x2))) MARK(tail(if(divides(z0, z1), x1, x2))) -> c20(A__TAIL(a__if(divides(mark(z0), mark(z1)), x1, x2)), MARK(if(divides(z0, z1), x1, x2))) MARK(tail(if(x0, z1, z2))) -> c20(MARK(if(x0, z1, z2))) MARK(tail(if(0, x1, x2))) -> c20(A__TAIL(a__if(0, x1, x2))) MARK(tail(filter(x0, primes))) -> c20(A__TAIL(a__filter(mark(x0), a__primes)), MARK(filter(x0, primes))) MARK(tail(filter(x0, sieve(z0)))) -> c20(A__TAIL(a__filter(mark(x0), a__sieve(mark(z0)))), MARK(filter(x0, sieve(z0)))) MARK(tail(filter(x0, from(z0)))) -> c20(A__TAIL(a__filter(mark(x0), a__from(mark(z0)))), MARK(filter(x0, from(z0)))) MARK(tail(filter(x0, head(z0)))) -> c20(A__TAIL(a__filter(mark(x0), a__head(mark(z0)))), MARK(filter(x0, head(z0)))) MARK(tail(filter(x0, tail(z0)))) -> c20(A__TAIL(a__filter(mark(x0), a__tail(mark(z0)))), MARK(filter(x0, tail(z0)))) MARK(tail(filter(x0, if(z0, z1, z2)))) -> c20(A__TAIL(a__filter(mark(x0), a__if(mark(z0), z1, z2))), MARK(filter(x0, if(z0, z1, z2)))) MARK(tail(filter(x0, filter(z0, z1)))) -> c20(A__TAIL(a__filter(mark(x0), a__filter(mark(z0), mark(z1)))), MARK(filter(x0, filter(z0, z1)))) MARK(tail(filter(x0, s(z0)))) -> c20(A__TAIL(a__filter(mark(x0), s(mark(z0)))), MARK(filter(x0, s(z0)))) MARK(tail(filter(x0, 0))) -> c20(A__TAIL(a__filter(mark(x0), 0)), MARK(filter(x0, 0))) MARK(tail(filter(x0, cons(z0, z1)))) -> c20(A__TAIL(a__filter(mark(x0), cons(mark(z0), z1))), MARK(filter(x0, cons(z0, z1)))) MARK(tail(filter(x0, true))) -> c20(A__TAIL(a__filter(mark(x0), true)), MARK(filter(x0, true))) MARK(tail(filter(x0, false))) -> c20(A__TAIL(a__filter(mark(x0), false)), MARK(filter(x0, false))) MARK(tail(filter(x0, divides(z0, z1)))) -> c20(A__TAIL(a__filter(mark(x0), divides(mark(z0), mark(z1)))), MARK(filter(x0, divides(z0, z1)))) MARK(tail(filter(primes, x1))) -> c20(A__TAIL(a__filter(a__primes, mark(x1))), MARK(filter(primes, x1))) MARK(tail(filter(sieve(z0), x1))) -> c20(A__TAIL(a__filter(a__sieve(mark(z0)), mark(x1))), MARK(filter(sieve(z0), x1))) MARK(tail(filter(from(z0), x1))) -> c20(A__TAIL(a__filter(a__from(mark(z0)), mark(x1))), MARK(filter(from(z0), x1))) MARK(tail(filter(head(z0), x1))) -> c20(A__TAIL(a__filter(a__head(mark(z0)), mark(x1))), MARK(filter(head(z0), x1))) MARK(tail(filter(tail(z0), x1))) -> c20(A__TAIL(a__filter(a__tail(mark(z0)), mark(x1))), MARK(filter(tail(z0), x1))) MARK(tail(filter(if(z0, z1, z2), x1))) -> c20(A__TAIL(a__filter(a__if(mark(z0), z1, z2), mark(x1))), MARK(filter(if(z0, z1, z2), x1))) MARK(tail(filter(filter(z0, z1), x1))) -> c20(A__TAIL(a__filter(a__filter(mark(z0), mark(z1)), mark(x1))), MARK(filter(filter(z0, z1), x1))) MARK(tail(filter(s(z0), x1))) -> c20(A__TAIL(a__filter(s(mark(z0)), mark(x1))), MARK(filter(s(z0), x1))) MARK(tail(filter(0, x1))) -> c20(A__TAIL(a__filter(0, mark(x1))), MARK(filter(0, x1))) MARK(tail(filter(cons(z0, z1), x1))) -> c20(A__TAIL(a__filter(cons(mark(z0), z1), mark(x1))), MARK(filter(cons(z0, z1), x1))) MARK(tail(filter(true, x1))) -> c20(A__TAIL(a__filter(true, mark(x1))), MARK(filter(true, x1))) MARK(tail(filter(false, x1))) -> c20(A__TAIL(a__filter(false, mark(x1))), MARK(filter(false, x1))) MARK(tail(filter(divides(z0, z1), x1))) -> c20(A__TAIL(a__filter(divides(mark(z0), mark(z1)), mark(x1))), MARK(filter(divides(z0, z1), x1))) MARK(tail(filter(x0, x1))) -> c20(MARK(filter(x0, x1))) MARK(if(primes, x0, x1)) -> c21(A__IF(a__sieve(a__from(s(s(0)))), x0, x1), MARK(primes)) MARK(if(primes, x0, x1)) -> c21(MARK(primes)) MARK(if(sieve(primes), x1, x2)) -> c21(A__IF(a__sieve(a__primes), x1, x2), MARK(sieve(primes))) MARK(if(sieve(sieve(z0)), x1, x2)) -> c21(A__IF(a__sieve(a__sieve(mark(z0))), x1, x2), MARK(sieve(sieve(z0)))) MARK(if(sieve(from(z0)), x1, x2)) -> c21(A__IF(a__sieve(a__from(mark(z0))), x1, x2), MARK(sieve(from(z0)))) MARK(if(sieve(head(z0)), x1, x2)) -> c21(A__IF(a__sieve(a__head(mark(z0))), x1, x2), MARK(sieve(head(z0)))) MARK(if(sieve(tail(z0)), x1, x2)) -> c21(A__IF(a__sieve(a__tail(mark(z0))), x1, x2), MARK(sieve(tail(z0)))) MARK(if(sieve(if(z0, z1, z2)), x1, x2)) -> c21(A__IF(a__sieve(a__if(mark(z0), z1, z2)), x1, x2), MARK(sieve(if(z0, z1, z2)))) MARK(if(sieve(filter(z0, z1)), x1, x2)) -> c21(A__IF(a__sieve(a__filter(mark(z0), mark(z1))), x1, x2), MARK(sieve(filter(z0, z1)))) MARK(if(sieve(s(z0)), x1, x2)) -> c21(A__IF(a__sieve(s(mark(z0))), x1, x2), MARK(sieve(s(z0)))) MARK(if(sieve(cons(z0, z1)), x1, x2)) -> c21(A__IF(a__sieve(cons(mark(z0), z1)), x1, x2), MARK(sieve(cons(z0, z1)))) MARK(if(sieve(divides(z0, z1)), x1, x2)) -> c21(A__IF(a__sieve(divides(mark(z0), mark(z1))), x1, x2), MARK(sieve(divides(z0, z1)))) MARK(if(sieve(x0), x1, x2)) -> c21(MARK(sieve(x0))) MARK(if(sieve(0), x1, x2)) -> c21(A__IF(a__sieve(0), x1, x2)) MARK(if(sieve(true), x1, x2)) -> c21(A__IF(a__sieve(true), x1, x2)) MARK(if(sieve(false), x1, x2)) -> c21(A__IF(a__sieve(false), x1, x2)) MARK(if(from(primes), x1, x2)) -> c21(A__IF(a__from(a__primes), x1, x2), MARK(from(primes))) MARK(if(from(sieve(z0)), x1, x2)) -> c21(A__IF(a__from(a__sieve(mark(z0))), x1, x2), MARK(from(sieve(z0)))) MARK(if(from(from(z0)), x1, x2)) -> c21(A__IF(a__from(a__from(mark(z0))), x1, x2), MARK(from(from(z0)))) MARK(if(from(head(z0)), x1, x2)) -> c21(A__IF(a__from(a__head(mark(z0))), x1, x2), MARK(from(head(z0)))) MARK(if(from(tail(z0)), x1, x2)) -> c21(A__IF(a__from(a__tail(mark(z0))), x1, x2), MARK(from(tail(z0)))) MARK(if(from(if(z0, z1, z2)), x1, x2)) -> c21(A__IF(a__from(a__if(mark(z0), z1, z2)), x1, x2), MARK(from(if(z0, z1, z2)))) MARK(if(from(filter(z0, z1)), x1, x2)) -> c21(A__IF(a__from(a__filter(mark(z0), mark(z1))), x1, x2), MARK(from(filter(z0, z1)))) MARK(if(from(s(z0)), x1, x2)) -> c21(A__IF(a__from(s(mark(z0))), x1, x2), MARK(from(s(z0)))) MARK(if(from(0), x1, x2)) -> c21(A__IF(a__from(0), x1, x2), MARK(from(0))) MARK(if(from(cons(z0, z1)), x1, x2)) -> c21(A__IF(a__from(cons(mark(z0), z1)), x1, x2), MARK(from(cons(z0, z1)))) MARK(if(from(true), x1, x2)) -> c21(A__IF(a__from(true), x1, x2), MARK(from(true))) MARK(if(from(false), x1, x2)) -> c21(A__IF(a__from(false), x1, x2), MARK(from(false))) MARK(if(from(divides(z0, z1)), x1, x2)) -> c21(A__IF(a__from(divides(mark(z0), mark(z1))), x1, x2), MARK(from(divides(z0, z1)))) MARK(if(from(x0), x1, x2)) -> c21(MARK(from(x0))) MARK(if(head(primes), x1, x2)) -> c21(A__IF(a__head(a__primes), x1, x2), MARK(head(primes))) MARK(if(head(sieve(z0)), x1, x2)) -> c21(A__IF(a__head(a__sieve(mark(z0))), x1, x2), MARK(head(sieve(z0)))) MARK(if(head(from(z0)), x1, x2)) -> c21(A__IF(a__head(a__from(mark(z0))), x1, x2), MARK(head(from(z0)))) MARK(if(head(head(z0)), x1, x2)) -> c21(A__IF(a__head(a__head(mark(z0))), x1, x2), MARK(head(head(z0)))) MARK(if(head(tail(z0)), x1, x2)) -> c21(A__IF(a__head(a__tail(mark(z0))), x1, x2), MARK(head(tail(z0)))) MARK(if(head(if(z0, z1, z2)), x1, x2)) -> c21(A__IF(a__head(a__if(mark(z0), z1, z2)), x1, x2), MARK(head(if(z0, z1, z2)))) MARK(if(head(filter(z0, z1)), x1, x2)) -> c21(A__IF(a__head(a__filter(mark(z0), mark(z1))), x1, x2), MARK(head(filter(z0, z1)))) MARK(if(head(s(z0)), x1, x2)) -> c21(A__IF(a__head(s(mark(z0))), x1, x2), MARK(head(s(z0)))) MARK(if(head(cons(z0, z1)), x1, x2)) -> c21(A__IF(a__head(cons(mark(z0), z1)), x1, x2), MARK(head(cons(z0, z1)))) MARK(if(head(divides(z0, z1)), x1, x2)) -> c21(A__IF(a__head(divides(mark(z0), mark(z1))), x1, x2), MARK(head(divides(z0, z1)))) MARK(if(head(x0), x1, x2)) -> c21(MARK(head(x0))) MARK(if(head(0), x1, x2)) -> c21(A__IF(a__head(0), x1, x2)) MARK(if(head(true), x1, x2)) -> c21(A__IF(a__head(true), x1, x2)) MARK(if(head(false), x1, x2)) -> c21(A__IF(a__head(false), x1, x2)) MARK(if(tail(primes), x1, x2)) -> c21(A__IF(a__tail(a__primes), x1, x2), MARK(tail(primes))) MARK(if(tail(sieve(z0)), x1, x2)) -> c21(A__IF(a__tail(a__sieve(mark(z0))), x1, x2), MARK(tail(sieve(z0)))) MARK(if(tail(from(z0)), x1, x2)) -> c21(A__IF(a__tail(a__from(mark(z0))), x1, x2), MARK(tail(from(z0)))) MARK(if(tail(head(z0)), x1, x2)) -> c21(A__IF(a__tail(a__head(mark(z0))), x1, x2), MARK(tail(head(z0)))) MARK(if(tail(tail(z0)), x1, x2)) -> c21(A__IF(a__tail(a__tail(mark(z0))), x1, x2), MARK(tail(tail(z0)))) MARK(if(tail(if(z0, z1, z2)), x1, x2)) -> c21(A__IF(a__tail(a__if(mark(z0), z1, z2)), x1, x2), MARK(tail(if(z0, z1, z2)))) MARK(if(tail(filter(z0, z1)), x1, x2)) -> c21(A__IF(a__tail(a__filter(mark(z0), mark(z1))), x1, x2), MARK(tail(filter(z0, z1)))) MARK(if(tail(s(z0)), x1, x2)) -> c21(A__IF(a__tail(s(mark(z0))), x1, x2), MARK(tail(s(z0)))) MARK(if(tail(cons(z0, z1)), x1, x2)) -> c21(A__IF(a__tail(cons(mark(z0), z1)), x1, x2), MARK(tail(cons(z0, z1)))) MARK(if(tail(divides(z0, z1)), x1, x2)) -> c21(A__IF(a__tail(divides(mark(z0), mark(z1))), x1, x2), MARK(tail(divides(z0, z1)))) MARK(if(tail(x0), x1, x2)) -> c21(MARK(tail(x0))) MARK(if(tail(0), x1, x2)) -> c21(A__IF(a__tail(0), x1, x2)) MARK(if(tail(true), x1, x2)) -> c21(A__IF(a__tail(true), x1, x2)) MARK(if(tail(false), x1, x2)) -> c21(A__IF(a__tail(false), x1, x2)) MARK(if(if(primes, x1, x2), x3, x4)) -> c21(A__IF(a__if(a__primes, x1, x2), x3, x4), MARK(if(primes, x1, x2))) MARK(if(if(sieve(z0), x1, x2), x3, x4)) -> c21(A__IF(a__if(a__sieve(mark(z0)), x1, x2), x3, x4), MARK(if(sieve(z0), x1, x2))) MARK(if(if(from(z0), x1, x2), x3, x4)) -> c21(A__IF(a__if(a__from(mark(z0)), x1, x2), x3, x4), MARK(if(from(z0), x1, x2))) MARK(if(if(head(z0), x1, x2), x3, x4)) -> c21(A__IF(a__if(a__head(mark(z0)), x1, x2), x3, x4), MARK(if(head(z0), x1, x2))) MARK(if(if(tail(z0), x1, x2), x3, x4)) -> c21(A__IF(a__if(a__tail(mark(z0)), x1, x2), x3, x4), MARK(if(tail(z0), x1, x2))) MARK(if(if(if(z0, z1, z2), x1, x2), x3, x4)) -> c21(A__IF(a__if(a__if(mark(z0), z1, z2), x1, x2), x3, x4), MARK(if(if(z0, z1, z2), x1, x2))) MARK(if(if(filter(z0, z1), x1, x2), x3, x4)) -> c21(A__IF(a__if(a__filter(mark(z0), mark(z1)), x1, x2), x3, x4), MARK(if(filter(z0, z1), x1, x2))) MARK(if(if(s(z0), x1, x2), x3, x4)) -> c21(A__IF(a__if(s(mark(z0)), x1, x2), x3, x4), MARK(if(s(z0), x1, x2))) MARK(if(if(cons(z0, z1), x1, x2), x3, x4)) -> c21(A__IF(a__if(cons(mark(z0), z1), x1, x2), x3, x4), MARK(if(cons(z0, z1), x1, x2))) MARK(if(if(true, x1, x2), x3, x4)) -> c21(A__IF(a__if(true, x1, x2), x3, x4), MARK(if(true, x1, x2))) MARK(if(if(false, x1, x2), x3, x4)) -> c21(A__IF(a__if(false, x1, x2), x3, x4), MARK(if(false, x1, x2))) MARK(if(if(divides(z0, z1), x1, x2), x3, x4)) -> c21(A__IF(a__if(divides(mark(z0), mark(z1)), x1, x2), x3, x4), MARK(if(divides(z0, z1), x1, x2))) MARK(if(if(x0, z1, z2), x3, x4)) -> c21(MARK(if(x0, z1, z2))) MARK(if(if(0, x1, x2), x3, x4)) -> c21(A__IF(a__if(0, x1, x2), x3, x4)) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c18_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_1, c17_2, c17_1, c19_2, c19_1, c20_2, c20_1, c21_2, c21_1, c22_2, c22_1, c23_2, c23_1, c_2 ---------------------------------------- (207) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(if(filter(z0, z1), x1, x2)) -> c21(A__IF(a__filter(mark(z0), mark(z1)), x1, x2), MARK(filter(z0, z1))) by MARK(if(filter(x0, x1), x2, x3)) -> c21(A__IF(filter(mark(x0), mark(x1)), x2, x3), MARK(filter(x0, x1))) MARK(if(filter(x0, primes), x2, x3)) -> c21(A__IF(a__filter(mark(x0), a__primes), x2, x3), MARK(filter(x0, primes))) MARK(if(filter(x0, sieve(z0)), x2, x3)) -> c21(A__IF(a__filter(mark(x0), a__sieve(mark(z0))), x2, x3), MARK(filter(x0, sieve(z0)))) MARK(if(filter(x0, from(z0)), x2, x3)) -> c21(A__IF(a__filter(mark(x0), a__from(mark(z0))), x2, x3), MARK(filter(x0, from(z0)))) MARK(if(filter(x0, head(z0)), x2, x3)) -> c21(A__IF(a__filter(mark(x0), a__head(mark(z0))), x2, x3), MARK(filter(x0, head(z0)))) MARK(if(filter(x0, tail(z0)), x2, x3)) -> c21(A__IF(a__filter(mark(x0), a__tail(mark(z0))), x2, x3), MARK(filter(x0, tail(z0)))) MARK(if(filter(x0, if(z0, z1, z2)), x2, x3)) -> c21(A__IF(a__filter(mark(x0), a__if(mark(z0), z1, z2)), x2, x3), MARK(filter(x0, if(z0, z1, z2)))) MARK(if(filter(x0, filter(z0, z1)), x2, x3)) -> c21(A__IF(a__filter(mark(x0), a__filter(mark(z0), mark(z1))), x2, x3), MARK(filter(x0, filter(z0, z1)))) MARK(if(filter(x0, s(z0)), x2, x3)) -> c21(A__IF(a__filter(mark(x0), s(mark(z0))), x2, x3), MARK(filter(x0, s(z0)))) MARK(if(filter(x0, 0), x2, x3)) -> c21(A__IF(a__filter(mark(x0), 0), x2, x3), MARK(filter(x0, 0))) MARK(if(filter(x0, cons(z0, z1)), x2, x3)) -> c21(A__IF(a__filter(mark(x0), cons(mark(z0), z1)), x2, x3), MARK(filter(x0, cons(z0, z1)))) MARK(if(filter(x0, true), x2, x3)) -> c21(A__IF(a__filter(mark(x0), true), x2, x3), MARK(filter(x0, true))) MARK(if(filter(x0, false), x2, x3)) -> c21(A__IF(a__filter(mark(x0), false), x2, x3), MARK(filter(x0, false))) MARK(if(filter(x0, divides(z0, z1)), x2, x3)) -> c21(A__IF(a__filter(mark(x0), divides(mark(z0), mark(z1))), x2, x3), MARK(filter(x0, divides(z0, z1)))) MARK(if(filter(primes, x1), x2, x3)) -> c21(A__IF(a__filter(a__primes, mark(x1)), x2, x3), MARK(filter(primes, x1))) MARK(if(filter(sieve(z0), x1), x2, x3)) -> c21(A__IF(a__filter(a__sieve(mark(z0)), mark(x1)), x2, x3), MARK(filter(sieve(z0), x1))) MARK(if(filter(from(z0), x1), x2, x3)) -> c21(A__IF(a__filter(a__from(mark(z0)), mark(x1)), x2, x3), MARK(filter(from(z0), x1))) MARK(if(filter(head(z0), x1), x2, x3)) -> c21(A__IF(a__filter(a__head(mark(z0)), mark(x1)), x2, x3), MARK(filter(head(z0), x1))) MARK(if(filter(tail(z0), x1), x2, x3)) -> c21(A__IF(a__filter(a__tail(mark(z0)), mark(x1)), x2, x3), MARK(filter(tail(z0), x1))) MARK(if(filter(if(z0, z1, z2), x1), x2, x3)) -> c21(A__IF(a__filter(a__if(mark(z0), z1, z2), mark(x1)), x2, x3), MARK(filter(if(z0, z1, z2), x1))) MARK(if(filter(filter(z0, z1), x1), x2, x3)) -> c21(A__IF(a__filter(a__filter(mark(z0), mark(z1)), mark(x1)), x2, x3), MARK(filter(filter(z0, z1), x1))) MARK(if(filter(s(z0), x1), x2, x3)) -> c21(A__IF(a__filter(s(mark(z0)), mark(x1)), x2, x3), MARK(filter(s(z0), x1))) MARK(if(filter(0, x1), x2, x3)) -> c21(A__IF(a__filter(0, mark(x1)), x2, x3), MARK(filter(0, x1))) MARK(if(filter(cons(z0, z1), x1), x2, x3)) -> c21(A__IF(a__filter(cons(mark(z0), z1), mark(x1)), x2, x3), MARK(filter(cons(z0, z1), x1))) MARK(if(filter(true, x1), x2, x3)) -> c21(A__IF(a__filter(true, mark(x1)), x2, x3), MARK(filter(true, x1))) MARK(if(filter(false, x1), x2, x3)) -> c21(A__IF(a__filter(false, mark(x1)), x2, x3), MARK(filter(false, x1))) MARK(if(filter(divides(z0, z1), x1), x2, x3)) -> c21(A__IF(a__filter(divides(mark(z0), mark(z1)), mark(x1)), x2, x3), MARK(filter(divides(z0, z1), x1))) ---------------------------------------- (208) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c22(A__FILTER(mark(x0), a__primes), MARK(x0)) MARK(filter(x0, sieve(z0))) -> c22(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, from(z0))) -> c22(A__FILTER(mark(x0), a__from(mark(z0))), MARK(x0)) MARK(filter(x0, head(z0))) -> c22(A__FILTER(mark(x0), a__head(mark(z0))), MARK(x0)) MARK(filter(x0, tail(z0))) -> c22(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(x0)) MARK(filter(x0, if(z0, z1, z2))) -> c22(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0)) MARK(filter(x0, filter(z0, z1))) -> c22(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(x0)) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(x0, s(z0))) -> c22(MARK(x0)) MARK(filter(x0, 0)) -> c22(MARK(x0)) MARK(filter(x0, true)) -> c22(MARK(x0)) MARK(filter(x0, false)) -> c22(MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), x1)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c23(A__FILTER(mark(x0), a__primes), MARK(primes)) MARK(filter(x0, sieve(z0))) -> c23(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, from(z0))) -> c23(A__FILTER(mark(x0), a__from(mark(z0))), MARK(from(z0))) MARK(filter(x0, head(z0))) -> c23(A__FILTER(mark(x0), a__head(mark(z0))), MARK(head(z0))) MARK(filter(x0, tail(z0))) -> c23(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(tail(z0))) MARK(filter(x0, if(z0, z1, z2))) -> c23(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(filter(x0, filter(z0, z1))) -> c23(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(filter(x0, cons(z0, z1))) -> c23(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(primes, x1)) -> c23(A__FILTER(a__primes, mark(x1)), MARK(x1)) MARK(filter(sieve(z0), x1)) -> c23(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(from(z0), x1)) -> c23(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(head(z0), x1)) -> c23(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(tail(z0), x1)) -> c23(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(if(z0, z1, z2), x1)) -> c23(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1)) MARK(filter(filter(z0, z1), x1)) -> c23(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(x1)) MARK(filter(s(z0), x1)) -> c23(A__FILTER(s(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(x0, s(z0))) -> c23(MARK(s(z0))) MARK(filter(x0, divides(z0, z1))) -> c23(MARK(divides(z0, z1))) MARK(filter(0, x1)) -> c23(MARK(x1)) MARK(filter(cons(z0, z1), x1)) -> c23(MARK(x1)) MARK(filter(true, x1)) -> c23(MARK(x1)) MARK(filter(false, x1)) -> c23(MARK(x1)) MARK(filter(divides(z0, z1), x1)) -> c23(MARK(x1)) A__PRIMES -> c(A__SIEVE(cons(s(mark(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(sieve(primes)) -> c17(MARK(primes)) MARK(sieve(sieve(primes))) -> c17(A__SIEVE(a__sieve(a__primes)), MARK(sieve(primes))) MARK(sieve(sieve(sieve(z0)))) -> c17(A__SIEVE(a__sieve(a__sieve(mark(z0)))), MARK(sieve(sieve(z0)))) MARK(sieve(sieve(from(z0)))) -> c17(A__SIEVE(a__sieve(a__from(mark(z0)))), MARK(sieve(from(z0)))) MARK(sieve(sieve(head(z0)))) -> c17(A__SIEVE(a__sieve(a__head(mark(z0)))), MARK(sieve(head(z0)))) MARK(sieve(sieve(tail(z0)))) -> c17(A__SIEVE(a__sieve(a__tail(mark(z0)))), MARK(sieve(tail(z0)))) MARK(sieve(sieve(if(z0, z1, z2)))) -> c17(A__SIEVE(a__sieve(a__if(mark(z0), z1, z2))), MARK(sieve(if(z0, z1, z2)))) MARK(sieve(sieve(filter(z0, z1)))) -> c17(A__SIEVE(a__sieve(a__filter(mark(z0), mark(z1)))), MARK(sieve(filter(z0, z1)))) MARK(sieve(sieve(s(z0)))) -> c17(A__SIEVE(a__sieve(s(mark(z0)))), MARK(sieve(s(z0)))) MARK(sieve(sieve(cons(z0, z1)))) -> c17(A__SIEVE(a__sieve(cons(mark(z0), z1))), MARK(sieve(cons(z0, z1)))) MARK(sieve(sieve(divides(z0, z1)))) -> c17(A__SIEVE(a__sieve(divides(mark(z0), mark(z1)))), MARK(sieve(divides(z0, z1)))) MARK(sieve(sieve(x0))) -> c17(MARK(sieve(x0))) MARK(sieve(sieve(0))) -> c17(A__SIEVE(a__sieve(0))) MARK(sieve(sieve(true))) -> c17(A__SIEVE(a__sieve(true))) MARK(sieve(sieve(false))) -> c17(A__SIEVE(a__sieve(false))) MARK(sieve(from(x0))) -> c17(A__SIEVE(cons(mark(mark(x0)), from(s(mark(x0))))), MARK(from(x0))) MARK(sieve(from(primes))) -> c17(A__SIEVE(a__from(a__primes)), MARK(from(primes))) MARK(sieve(from(sieve(z0)))) -> c17(A__SIEVE(a__from(a__sieve(mark(z0)))), MARK(from(sieve(z0)))) MARK(sieve(from(from(z0)))) -> c17(A__SIEVE(a__from(a__from(mark(z0)))), MARK(from(from(z0)))) MARK(sieve(from(head(z0)))) -> c17(A__SIEVE(a__from(a__head(mark(z0)))), MARK(from(head(z0)))) MARK(sieve(from(tail(z0)))) -> c17(A__SIEVE(a__from(a__tail(mark(z0)))), MARK(from(tail(z0)))) MARK(sieve(from(if(z0, z1, z2)))) -> c17(A__SIEVE(a__from(a__if(mark(z0), z1, z2))), MARK(from(if(z0, z1, z2)))) MARK(sieve(from(filter(z0, z1)))) -> c17(A__SIEVE(a__from(a__filter(mark(z0), mark(z1)))), MARK(from(filter(z0, z1)))) MARK(sieve(from(s(z0)))) -> c17(A__SIEVE(a__from(s(mark(z0)))), MARK(from(s(z0)))) MARK(sieve(from(0))) -> c17(A__SIEVE(a__from(0)), MARK(from(0))) MARK(sieve(from(cons(z0, z1)))) -> c17(A__SIEVE(a__from(cons(mark(z0), z1))), MARK(from(cons(z0, z1)))) MARK(sieve(from(true))) -> c17(A__SIEVE(a__from(true)), MARK(from(true))) MARK(sieve(from(false))) -> c17(A__SIEVE(a__from(false)), MARK(from(false))) MARK(sieve(from(divides(z0, z1)))) -> c17(A__SIEVE(a__from(divides(mark(z0), mark(z1)))), MARK(from(divides(z0, z1)))) MARK(sieve(from(x0))) -> c17(MARK(from(x0))) MARK(sieve(head(primes))) -> c17(A__SIEVE(a__head(a__primes)), MARK(head(primes))) MARK(sieve(head(sieve(z0)))) -> c17(A__SIEVE(a__head(a__sieve(mark(z0)))), MARK(head(sieve(z0)))) MARK(sieve(head(from(z0)))) -> c17(A__SIEVE(a__head(a__from(mark(z0)))), MARK(head(from(z0)))) MARK(sieve(head(head(z0)))) -> c17(A__SIEVE(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(sieve(head(tail(z0)))) -> c17(A__SIEVE(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(sieve(head(if(z0, z1, z2)))) -> c17(A__SIEVE(a__head(a__if(mark(z0), z1, z2))), MARK(head(if(z0, z1, z2)))) MARK(sieve(head(filter(z0, z1)))) -> c17(A__SIEVE(a__head(a__filter(mark(z0), mark(z1)))), MARK(head(filter(z0, z1)))) MARK(sieve(head(s(z0)))) -> c17(A__SIEVE(a__head(s(mark(z0)))), MARK(head(s(z0)))) 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MARK(head(head(z0)))) MARK(if(head(tail(z0)), x1, x2)) -> c21(A__IF(a__head(a__tail(mark(z0))), x1, x2), MARK(head(tail(z0)))) MARK(if(head(if(z0, z1, z2)), x1, x2)) -> c21(A__IF(a__head(a__if(mark(z0), z1, z2)), x1, x2), MARK(head(if(z0, z1, z2)))) MARK(if(head(filter(z0, z1)), x1, x2)) -> c21(A__IF(a__head(a__filter(mark(z0), mark(z1))), x1, x2), MARK(head(filter(z0, z1)))) MARK(if(head(s(z0)), x1, x2)) -> c21(A__IF(a__head(s(mark(z0))), x1, x2), MARK(head(s(z0)))) MARK(if(head(cons(z0, z1)), x1, x2)) -> c21(A__IF(a__head(cons(mark(z0), z1)), x1, x2), MARK(head(cons(z0, z1)))) MARK(if(head(divides(z0, z1)), x1, x2)) -> c21(A__IF(a__head(divides(mark(z0), mark(z1))), x1, x2), MARK(head(divides(z0, z1)))) MARK(if(head(x0), x1, x2)) -> c21(MARK(head(x0))) MARK(if(head(0), x1, x2)) -> c21(A__IF(a__head(0), x1, x2)) MARK(if(head(true), x1, x2)) -> c21(A__IF(a__head(true), x1, x2)) MARK(if(head(false), x1, x2)) -> c21(A__IF(a__head(false), x1, x2)) MARK(if(tail(primes), x1, x2)) -> c21(A__IF(a__tail(a__primes), x1, x2), MARK(tail(primes))) MARK(if(tail(sieve(z0)), x1, x2)) -> c21(A__IF(a__tail(a__sieve(mark(z0))), x1, x2), MARK(tail(sieve(z0)))) MARK(if(tail(from(z0)), x1, x2)) -> c21(A__IF(a__tail(a__from(mark(z0))), x1, x2), MARK(tail(from(z0)))) MARK(if(tail(head(z0)), x1, x2)) -> c21(A__IF(a__tail(a__head(mark(z0))), x1, x2), MARK(tail(head(z0)))) MARK(if(tail(tail(z0)), x1, x2)) -> c21(A__IF(a__tail(a__tail(mark(z0))), x1, x2), MARK(tail(tail(z0)))) MARK(if(tail(if(z0, z1, z2)), x1, x2)) -> c21(A__IF(a__tail(a__if(mark(z0), z1, z2)), x1, x2), MARK(tail(if(z0, z1, z2)))) MARK(if(tail(filter(z0, z1)), x1, x2)) -> c21(A__IF(a__tail(a__filter(mark(z0), mark(z1))), x1, x2), MARK(tail(filter(z0, z1)))) MARK(if(tail(s(z0)), x1, x2)) -> c21(A__IF(a__tail(s(mark(z0))), x1, x2), MARK(tail(s(z0)))) MARK(if(tail(cons(z0, z1)), x1, x2)) -> c21(A__IF(a__tail(cons(mark(z0), z1)), x1, x2), MARK(tail(cons(z0, z1)))) MARK(if(tail(divides(z0, z1)), x1, x2)) -> c21(A__IF(a__tail(divides(mark(z0), mark(z1))), x1, x2), MARK(tail(divides(z0, z1)))) MARK(if(tail(x0), x1, x2)) -> c21(MARK(tail(x0))) MARK(if(tail(0), x1, x2)) -> c21(A__IF(a__tail(0), x1, x2)) MARK(if(tail(true), x1, x2)) -> c21(A__IF(a__tail(true), x1, x2)) MARK(if(tail(false), x1, x2)) -> c21(A__IF(a__tail(false), x1, x2)) MARK(if(if(primes, x1, x2), x3, x4)) -> c21(A__IF(a__if(a__primes, x1, x2), x3, x4), MARK(if(primes, x1, x2))) MARK(if(if(sieve(z0), x1, x2), x3, x4)) -> c21(A__IF(a__if(a__sieve(mark(z0)), x1, x2), x3, x4), MARK(if(sieve(z0), x1, x2))) MARK(if(if(from(z0), x1, x2), x3, x4)) -> c21(A__IF(a__if(a__from(mark(z0)), x1, x2), x3, x4), MARK(if(from(z0), x1, x2))) MARK(if(if(head(z0), x1, x2), x3, x4)) -> c21(A__IF(a__if(a__head(mark(z0)), x1, x2), x3, x4), MARK(if(head(z0), x1, x2))) MARK(if(if(tail(z0), x1, x2), x3, x4)) -> c21(A__IF(a__if(a__tail(mark(z0)), x1, x2), x3, x4), MARK(if(tail(z0), x1, x2))) MARK(if(if(if(z0, z1, z2), x1, x2), x3, x4)) -> c21(A__IF(a__if(a__if(mark(z0), z1, z2), x1, x2), x3, x4), MARK(if(if(z0, z1, z2), x1, x2))) MARK(if(if(filter(z0, z1), x1, x2), x3, x4)) -> c21(A__IF(a__if(a__filter(mark(z0), mark(z1)), x1, x2), x3, x4), MARK(if(filter(z0, z1), x1, x2))) MARK(if(if(s(z0), x1, x2), x3, x4)) -> c21(A__IF(a__if(s(mark(z0)), x1, x2), x3, x4), MARK(if(s(z0), x1, x2))) MARK(if(if(cons(z0, z1), x1, x2), x3, x4)) -> c21(A__IF(a__if(cons(mark(z0), z1), x1, x2), x3, x4), MARK(if(cons(z0, z1), x1, x2))) MARK(if(if(true, x1, x2), x3, x4)) -> c21(A__IF(a__if(true, x1, x2), x3, x4), MARK(if(true, x1, x2))) MARK(if(if(false, x1, x2), x3, x4)) -> c21(A__IF(a__if(false, x1, x2), x3, x4), MARK(if(false, x1, x2))) MARK(if(if(divides(z0, z1), x1, x2), x3, x4)) -> c21(A__IF(a__if(divides(mark(z0), mark(z1)), x1, x2), x3, x4), MARK(if(divides(z0, z1), x1, x2))) MARK(if(if(x0, z1, z2), x3, x4)) -> c21(MARK(if(x0, z1, z2))) MARK(if(if(0, x1, x2), x3, x4)) -> c21(A__IF(a__if(0, x1, x2), x3, x4)) MARK(if(filter(x0, x1), x2, x3)) -> c21(A__IF(filter(mark(x0), mark(x1)), x2, x3), MARK(filter(x0, x1))) MARK(if(filter(x0, primes), x2, x3)) -> c21(A__IF(a__filter(mark(x0), a__primes), x2, x3), MARK(filter(x0, primes))) MARK(if(filter(x0, sieve(z0)), x2, x3)) -> c21(A__IF(a__filter(mark(x0), a__sieve(mark(z0))), x2, x3), MARK(filter(x0, sieve(z0)))) MARK(if(filter(x0, from(z0)), x2, x3)) -> c21(A__IF(a__filter(mark(x0), a__from(mark(z0))), x2, x3), MARK(filter(x0, from(z0)))) MARK(if(filter(x0, head(z0)), x2, x3)) -> c21(A__IF(a__filter(mark(x0), a__head(mark(z0))), x2, x3), MARK(filter(x0, head(z0)))) MARK(if(filter(x0, tail(z0)), x2, x3)) -> c21(A__IF(a__filter(mark(x0), a__tail(mark(z0))), x2, x3), MARK(filter(x0, tail(z0)))) MARK(if(filter(x0, if(z0, z1, z2)), x2, x3)) -> c21(A__IF(a__filter(mark(x0), a__if(mark(z0), z1, z2)), x2, x3), MARK(filter(x0, if(z0, z1, z2)))) MARK(if(filter(x0, filter(z0, z1)), x2, x3)) -> c21(A__IF(a__filter(mark(x0), a__filter(mark(z0), mark(z1))), x2, x3), MARK(filter(x0, filter(z0, z1)))) MARK(if(filter(x0, s(z0)), x2, x3)) -> c21(A__IF(a__filter(mark(x0), s(mark(z0))), x2, x3), MARK(filter(x0, s(z0)))) MARK(if(filter(x0, 0), x2, x3)) -> c21(A__IF(a__filter(mark(x0), 0), x2, x3), MARK(filter(x0, 0))) MARK(if(filter(x0, cons(z0, z1)), x2, x3)) -> c21(A__IF(a__filter(mark(x0), cons(mark(z0), z1)), x2, x3), MARK(filter(x0, cons(z0, z1)))) MARK(if(filter(x0, true), x2, x3)) -> c21(A__IF(a__filter(mark(x0), true), x2, x3), MARK(filter(x0, true))) MARK(if(filter(x0, false), x2, x3)) -> c21(A__IF(a__filter(mark(x0), false), x2, x3), MARK(filter(x0, false))) MARK(if(filter(x0, divides(z0, z1)), x2, x3)) -> c21(A__IF(a__filter(mark(x0), divides(mark(z0), mark(z1))), x2, x3), MARK(filter(x0, divides(z0, z1)))) MARK(if(filter(primes, x1), x2, x3)) -> c21(A__IF(a__filter(a__primes, mark(x1)), x2, x3), MARK(filter(primes, x1))) MARK(if(filter(sieve(z0), x1), x2, x3)) -> c21(A__IF(a__filter(a__sieve(mark(z0)), mark(x1)), x2, x3), MARK(filter(sieve(z0), x1))) MARK(if(filter(from(z0), x1), x2, x3)) -> c21(A__IF(a__filter(a__from(mark(z0)), mark(x1)), x2, x3), MARK(filter(from(z0), x1))) MARK(if(filter(head(z0), x1), x2, x3)) -> c21(A__IF(a__filter(a__head(mark(z0)), mark(x1)), x2, x3), MARK(filter(head(z0), x1))) MARK(if(filter(tail(z0), x1), x2, x3)) -> c21(A__IF(a__filter(a__tail(mark(z0)), mark(x1)), x2, x3), MARK(filter(tail(z0), x1))) MARK(if(filter(if(z0, z1, z2), x1), x2, x3)) -> c21(A__IF(a__filter(a__if(mark(z0), z1, z2), mark(x1)), x2, x3), MARK(filter(if(z0, z1, z2), x1))) MARK(if(filter(filter(z0, z1), x1), x2, x3)) -> c21(A__IF(a__filter(a__filter(mark(z0), mark(z1)), mark(x1)), x2, x3), MARK(filter(filter(z0, z1), x1))) MARK(if(filter(s(z0), x1), x2, x3)) -> c21(A__IF(a__filter(s(mark(z0)), mark(x1)), x2, x3), MARK(filter(s(z0), x1))) MARK(if(filter(0, x1), x2, x3)) -> c21(A__IF(a__filter(0, mark(x1)), x2, x3), MARK(filter(0, x1))) MARK(if(filter(cons(z0, z1), x1), x2, x3)) -> c21(A__IF(a__filter(cons(mark(z0), z1), mark(x1)), x2, x3), MARK(filter(cons(z0, z1), x1))) MARK(if(filter(true, x1), x2, x3)) -> c21(A__IF(a__filter(true, mark(x1)), x2, x3), MARK(filter(true, x1))) MARK(if(filter(false, x1), x2, x3)) -> c21(A__IF(a__filter(false, mark(x1)), x2, x3), MARK(filter(false, x1))) MARK(if(filter(divides(z0, z1), x1), x2, x3)) -> c21(A__IF(a__filter(divides(mark(z0), mark(z1)), mark(x1)), x2, x3), MARK(filter(divides(z0, z1), x1))) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c18_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_1, c17_2, c17_1, c19_2, c19_1, c20_2, c20_1, c21_1, c22_2, c22_1, c23_2, c23_1, c_2, c21_2 ---------------------------------------- (209) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (210) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c22(A__FILTER(mark(x0), a__primes), MARK(x0)) MARK(filter(x0, sieve(z0))) -> c22(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, from(z0))) -> c22(A__FILTER(mark(x0), a__from(mark(z0))), MARK(x0)) MARK(filter(x0, head(z0))) -> c22(A__FILTER(mark(x0), a__head(mark(z0))), MARK(x0)) MARK(filter(x0, tail(z0))) -> c22(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(x0)) MARK(filter(x0, if(z0, z1, z2))) -> c22(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0)) MARK(filter(x0, filter(z0, z1))) -> c22(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(x0)) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) 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c21(A__IF(a__sieve(a__tail(mark(z0))), x1, x2), MARK(sieve(tail(z0)))) MARK(if(sieve(if(z0, z1, z2)), x1, x2)) -> c21(A__IF(a__sieve(a__if(mark(z0), z1, z2)), x1, x2), MARK(sieve(if(z0, z1, z2)))) MARK(if(sieve(filter(z0, z1)), x1, x2)) -> c21(A__IF(a__sieve(a__filter(mark(z0), mark(z1))), x1, x2), MARK(sieve(filter(z0, z1)))) MARK(if(sieve(s(z0)), x1, x2)) -> c21(A__IF(a__sieve(s(mark(z0))), x1, x2), MARK(sieve(s(z0)))) MARK(if(sieve(cons(z0, z1)), x1, x2)) -> c21(A__IF(a__sieve(cons(mark(z0), z1)), x1, x2), MARK(sieve(cons(z0, z1)))) MARK(if(sieve(divides(z0, z1)), x1, x2)) -> c21(A__IF(a__sieve(divides(mark(z0), mark(z1))), x1, x2), MARK(sieve(divides(z0, z1)))) MARK(if(sieve(x0), x1, x2)) -> c21(MARK(sieve(x0))) MARK(if(sieve(0), x1, x2)) -> c21(A__IF(a__sieve(0), x1, x2)) MARK(if(sieve(true), x1, x2)) -> c21(A__IF(a__sieve(true), x1, x2)) MARK(if(sieve(false), x1, x2)) -> c21(A__IF(a__sieve(false), x1, x2)) MARK(if(from(primes), x1, x2)) -> c21(A__IF(a__from(a__primes), x1, x2), MARK(from(primes))) MARK(if(from(sieve(z0)), x1, x2)) -> c21(A__IF(a__from(a__sieve(mark(z0))), x1, x2), MARK(from(sieve(z0)))) MARK(if(from(from(z0)), x1, x2)) -> c21(A__IF(a__from(a__from(mark(z0))), x1, x2), MARK(from(from(z0)))) MARK(if(from(head(z0)), x1, x2)) -> c21(A__IF(a__from(a__head(mark(z0))), x1, x2), MARK(from(head(z0)))) MARK(if(from(tail(z0)), x1, x2)) -> c21(A__IF(a__from(a__tail(mark(z0))), x1, x2), MARK(from(tail(z0)))) MARK(if(from(if(z0, z1, z2)), x1, x2)) -> c21(A__IF(a__from(a__if(mark(z0), z1, z2)), x1, x2), MARK(from(if(z0, z1, z2)))) MARK(if(from(filter(z0, z1)), x1, x2)) -> c21(A__IF(a__from(a__filter(mark(z0), mark(z1))), x1, x2), MARK(from(filter(z0, z1)))) MARK(if(from(s(z0)), x1, x2)) -> c21(A__IF(a__from(s(mark(z0))), x1, x2), MARK(from(s(z0)))) MARK(if(from(0), x1, x2)) -> c21(A__IF(a__from(0), x1, x2), MARK(from(0))) MARK(if(from(cons(z0, z1)), x1, x2)) -> c21(A__IF(a__from(cons(mark(z0), z1)), x1, x2), MARK(from(cons(z0, z1)))) MARK(if(from(true), x1, x2)) -> c21(A__IF(a__from(true), x1, x2), MARK(from(true))) MARK(if(from(false), x1, x2)) -> c21(A__IF(a__from(false), x1, x2), MARK(from(false))) MARK(if(from(divides(z0, z1)), x1, x2)) -> c21(A__IF(a__from(divides(mark(z0), mark(z1))), x1, x2), MARK(from(divides(z0, z1)))) MARK(if(from(x0), x1, x2)) -> c21(MARK(from(x0))) MARK(if(head(primes), x1, x2)) -> c21(A__IF(a__head(a__primes), x1, x2), MARK(head(primes))) MARK(if(head(sieve(z0)), x1, x2)) -> c21(A__IF(a__head(a__sieve(mark(z0))), x1, x2), MARK(head(sieve(z0)))) MARK(if(head(from(z0)), x1, x2)) -> c21(A__IF(a__head(a__from(mark(z0))), x1, x2), MARK(head(from(z0)))) MARK(if(head(head(z0)), x1, x2)) -> c21(A__IF(a__head(a__head(mark(z0))), x1, x2), MARK(head(head(z0)))) MARK(if(head(tail(z0)), x1, x2)) -> c21(A__IF(a__head(a__tail(mark(z0))), x1, x2), MARK(head(tail(z0)))) MARK(if(head(if(z0, z1, z2)), x1, x2)) -> c21(A__IF(a__head(a__if(mark(z0), z1, z2)), x1, x2), MARK(head(if(z0, z1, z2)))) MARK(if(head(filter(z0, z1)), x1, 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MARK(if(tail(head(z0)), x1, x2)) -> c21(A__IF(a__tail(a__head(mark(z0))), x1, x2), MARK(tail(head(z0)))) MARK(if(tail(tail(z0)), x1, x2)) -> c21(A__IF(a__tail(a__tail(mark(z0))), x1, x2), MARK(tail(tail(z0)))) MARK(if(tail(if(z0, z1, z2)), x1, x2)) -> c21(A__IF(a__tail(a__if(mark(z0), z1, z2)), x1, x2), MARK(tail(if(z0, z1, z2)))) MARK(if(tail(filter(z0, z1)), x1, x2)) -> c21(A__IF(a__tail(a__filter(mark(z0), mark(z1))), x1, x2), MARK(tail(filter(z0, z1)))) MARK(if(tail(s(z0)), x1, x2)) -> c21(A__IF(a__tail(s(mark(z0))), x1, x2), MARK(tail(s(z0)))) MARK(if(tail(cons(z0, z1)), x1, x2)) -> c21(A__IF(a__tail(cons(mark(z0), z1)), x1, x2), MARK(tail(cons(z0, z1)))) MARK(if(tail(divides(z0, z1)), x1, x2)) -> c21(A__IF(a__tail(divides(mark(z0), mark(z1))), x1, x2), MARK(tail(divides(z0, z1)))) MARK(if(tail(x0), x1, x2)) -> c21(MARK(tail(x0))) MARK(if(tail(0), x1, x2)) -> c21(A__IF(a__tail(0), x1, x2)) MARK(if(tail(true), x1, x2)) -> c21(A__IF(a__tail(true), x1, x2)) MARK(if(tail(false), x1, x2)) -> c21(A__IF(a__tail(false), x1, x2)) MARK(if(if(primes, x1, x2), x3, x4)) -> c21(A__IF(a__if(a__primes, x1, x2), x3, x4), MARK(if(primes, x1, x2))) MARK(if(if(sieve(z0), x1, x2), x3, x4)) -> c21(A__IF(a__if(a__sieve(mark(z0)), x1, x2), x3, x4), MARK(if(sieve(z0), x1, x2))) MARK(if(if(from(z0), x1, x2), x3, x4)) -> c21(A__IF(a__if(a__from(mark(z0)), x1, x2), x3, x4), MARK(if(from(z0), x1, x2))) MARK(if(if(head(z0), x1, x2), x3, x4)) -> c21(A__IF(a__if(a__head(mark(z0)), x1, x2), x3, x4), MARK(if(head(z0), x1, x2))) MARK(if(if(tail(z0), x1, x2), x3, x4)) -> c21(A__IF(a__if(a__tail(mark(z0)), x1, x2), x3, x4), MARK(if(tail(z0), x1, x2))) MARK(if(if(if(z0, z1, z2), x1, x2), x3, x4)) -> c21(A__IF(a__if(a__if(mark(z0), z1, z2), x1, x2), x3, x4), MARK(if(if(z0, z1, z2), x1, x2))) MARK(if(if(filter(z0, z1), x1, x2), x3, x4)) -> c21(A__IF(a__if(a__filter(mark(z0), mark(z1)), x1, x2), x3, x4), MARK(if(filter(z0, z1), x1, x2))) MARK(if(if(s(z0), x1, x2), x3, x4)) -> 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MARK(filter(tail(z0), x1))) MARK(if(filter(if(z0, z1, z2), x1), x2, x3)) -> c21(A__IF(a__filter(a__if(mark(z0), z1, z2), mark(x1)), x2, x3), MARK(filter(if(z0, z1, z2), x1))) MARK(if(filter(filter(z0, z1), x1), x2, x3)) -> c21(A__IF(a__filter(a__filter(mark(z0), mark(z1)), mark(x1)), x2, x3), MARK(filter(filter(z0, z1), x1))) MARK(if(filter(s(z0), x1), x2, x3)) -> c21(A__IF(a__filter(s(mark(z0)), mark(x1)), x2, x3), MARK(filter(s(z0), x1))) MARK(if(filter(0, x1), x2, x3)) -> c21(A__IF(a__filter(0, mark(x1)), x2, x3), MARK(filter(0, x1))) MARK(if(filter(cons(z0, z1), x1), x2, x3)) -> c21(A__IF(a__filter(cons(mark(z0), z1), mark(x1)), x2, x3), MARK(filter(cons(z0, z1), x1))) MARK(if(filter(true, x1), x2, x3)) -> c21(A__IF(a__filter(true, mark(x1)), x2, x3), MARK(filter(true, x1))) MARK(if(filter(false, x1), x2, x3)) -> c21(A__IF(a__filter(false, mark(x1)), x2, x3), MARK(filter(false, x1))) MARK(if(filter(divides(z0, z1), x1), x2, x3)) -> c21(A__IF(a__filter(divides(mark(z0), mark(z1)), 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c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c22(A__FILTER(mark(x0), a__primes), MARK(x0)) MARK(filter(x0, sieve(z0))) -> c22(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, from(z0))) -> c22(A__FILTER(mark(x0), a__from(mark(z0))), MARK(x0)) MARK(filter(x0, head(z0))) -> c22(A__FILTER(mark(x0), a__head(mark(z0))), MARK(x0)) MARK(filter(x0, tail(z0))) -> c22(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(x0)) MARK(filter(x0, if(z0, z1, z2))) -> c22(A__FILTER(mark(x0), 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x1))) -> c20(A__TAIL(a__filter(true, mark(x1))), MARK(filter(true, x1))) MARK(tail(filter(false, x1))) -> c20(A__TAIL(a__filter(false, mark(x1))), MARK(filter(false, x1))) MARK(tail(filter(divides(z0, z1), x1))) -> c20(A__TAIL(a__filter(divides(mark(z0), mark(z1)), mark(x1))), MARK(filter(divides(z0, z1), x1))) MARK(tail(filter(x0, x1))) -> c20(MARK(filter(x0, x1))) MARK(if(primes, x0, x1)) -> c21(A__IF(a__sieve(a__from(s(s(0)))), x0, x1), MARK(primes)) MARK(if(primes, x0, x1)) -> c21(MARK(primes)) MARK(if(sieve(primes), x1, x2)) -> c21(A__IF(a__sieve(a__primes), x1, x2), MARK(sieve(primes))) MARK(if(sieve(sieve(z0)), x1, x2)) -> c21(A__IF(a__sieve(a__sieve(mark(z0))), x1, x2), MARK(sieve(sieve(z0)))) MARK(if(sieve(from(z0)), x1, x2)) -> c21(A__IF(a__sieve(a__from(mark(z0))), x1, x2), MARK(sieve(from(z0)))) MARK(if(sieve(head(z0)), x1, x2)) -> c21(A__IF(a__sieve(a__head(mark(z0))), x1, x2), MARK(sieve(head(z0)))) MARK(if(sieve(tail(z0)), x1, x2)) -> c21(A__IF(a__sieve(a__tail(mark(z0))), x1, x2), MARK(sieve(tail(z0)))) MARK(if(sieve(if(z0, z1, z2)), x1, x2)) -> c21(A__IF(a__sieve(a__if(mark(z0), z1, z2)), x1, x2), MARK(sieve(if(z0, z1, z2)))) MARK(if(sieve(filter(z0, z1)), x1, x2)) -> c21(A__IF(a__sieve(a__filter(mark(z0), mark(z1))), x1, x2), MARK(sieve(filter(z0, z1)))) MARK(if(sieve(s(z0)), x1, x2)) -> c21(A__IF(a__sieve(s(mark(z0))), x1, x2), MARK(sieve(s(z0)))) MARK(if(sieve(cons(z0, z1)), x1, x2)) -> c21(A__IF(a__sieve(cons(mark(z0), z1)), x1, x2), MARK(sieve(cons(z0, z1)))) MARK(if(sieve(divides(z0, z1)), x1, x2)) -> c21(A__IF(a__sieve(divides(mark(z0), mark(z1))), x1, x2), MARK(sieve(divides(z0, z1)))) MARK(if(sieve(x0), x1, x2)) -> c21(MARK(sieve(x0))) MARK(if(sieve(0), x1, x2)) -> c21(A__IF(a__sieve(0), x1, x2)) MARK(if(sieve(true), x1, x2)) -> c21(A__IF(a__sieve(true), x1, x2)) MARK(if(sieve(false), x1, x2)) -> c21(A__IF(a__sieve(false), x1, x2)) MARK(if(from(primes), x1, x2)) -> c21(A__IF(a__from(a__primes), x1, x2), MARK(from(primes))) MARK(if(from(sieve(z0)), x1, x2)) -> c21(A__IF(a__from(a__sieve(mark(z0))), x1, x2), MARK(from(sieve(z0)))) MARK(if(from(from(z0)), x1, x2)) -> c21(A__IF(a__from(a__from(mark(z0))), x1, x2), MARK(from(from(z0)))) MARK(if(from(head(z0)), x1, x2)) -> c21(A__IF(a__from(a__head(mark(z0))), x1, x2), MARK(from(head(z0)))) MARK(if(from(tail(z0)), x1, x2)) -> c21(A__IF(a__from(a__tail(mark(z0))), x1, x2), MARK(from(tail(z0)))) MARK(if(from(if(z0, z1, z2)), x1, x2)) -> c21(A__IF(a__from(a__if(mark(z0), z1, z2)), x1, x2), MARK(from(if(z0, z1, z2)))) MARK(if(from(filter(z0, z1)), x1, x2)) -> c21(A__IF(a__from(a__filter(mark(z0), mark(z1))), x1, x2), MARK(from(filter(z0, z1)))) MARK(if(from(s(z0)), x1, x2)) -> c21(A__IF(a__from(s(mark(z0))), x1, x2), MARK(from(s(z0)))) MARK(if(from(0), x1, x2)) -> c21(A__IF(a__from(0), x1, x2), MARK(from(0))) MARK(if(from(cons(z0, z1)), x1, x2)) -> c21(A__IF(a__from(cons(mark(z0), z1)), x1, x2), MARK(from(cons(z0, z1)))) MARK(if(from(true), x1, x2)) -> c21(A__IF(a__from(true), x1, x2), MARK(from(true))) MARK(if(from(false), x1, x2)) -> c21(A__IF(a__from(false), x1, x2), MARK(from(false))) MARK(if(from(divides(z0, z1)), x1, x2)) -> c21(A__IF(a__from(divides(mark(z0), mark(z1))), x1, x2), MARK(from(divides(z0, z1)))) MARK(if(from(x0), x1, x2)) -> c21(MARK(from(x0))) MARK(if(head(primes), x1, x2)) -> c21(A__IF(a__head(a__primes), x1, x2), MARK(head(primes))) MARK(if(head(sieve(z0)), x1, x2)) -> c21(A__IF(a__head(a__sieve(mark(z0))), x1, x2), MARK(head(sieve(z0)))) MARK(if(head(from(z0)), x1, x2)) -> c21(A__IF(a__head(a__from(mark(z0))), x1, x2), MARK(head(from(z0)))) MARK(if(head(head(z0)), x1, x2)) -> c21(A__IF(a__head(a__head(mark(z0))), x1, x2), MARK(head(head(z0)))) MARK(if(head(tail(z0)), x1, x2)) -> c21(A__IF(a__head(a__tail(mark(z0))), x1, x2), MARK(head(tail(z0)))) MARK(if(head(if(z0, z1, z2)), x1, x2)) -> c21(A__IF(a__head(a__if(mark(z0), z1, z2)), x1, x2), MARK(head(if(z0, z1, z2)))) MARK(if(head(filter(z0, z1)), x1, x2)) -> c21(A__IF(a__head(a__filter(mark(z0), mark(z1))), x1, x2), MARK(head(filter(z0, z1)))) MARK(if(head(s(z0)), x1, x2)) -> c21(A__IF(a__head(s(mark(z0))), x1, x2), MARK(head(s(z0)))) MARK(if(head(cons(z0, z1)), x1, x2)) -> c21(A__IF(a__head(cons(mark(z0), z1)), x1, x2), MARK(head(cons(z0, z1)))) MARK(if(head(divides(z0, z1)), x1, x2)) -> c21(A__IF(a__head(divides(mark(z0), mark(z1))), x1, x2), MARK(head(divides(z0, z1)))) MARK(if(head(x0), x1, x2)) -> c21(MARK(head(x0))) MARK(if(head(0), x1, x2)) -> c21(A__IF(a__head(0), x1, x2)) MARK(if(head(true), x1, x2)) -> c21(A__IF(a__head(true), x1, x2)) MARK(if(head(false), x1, x2)) -> c21(A__IF(a__head(false), x1, x2)) MARK(if(tail(primes), x1, x2)) -> c21(A__IF(a__tail(a__primes), x1, x2), MARK(tail(primes))) MARK(if(tail(sieve(z0)), x1, x2)) -> c21(A__IF(a__tail(a__sieve(mark(z0))), x1, x2), MARK(tail(sieve(z0)))) MARK(if(tail(from(z0)), x1, x2)) -> c21(A__IF(a__tail(a__from(mark(z0))), x1, x2), MARK(tail(from(z0)))) MARK(if(tail(head(z0)), x1, x2)) -> c21(A__IF(a__tail(a__head(mark(z0))), x1, x2), MARK(tail(head(z0)))) MARK(if(tail(tail(z0)), x1, x2)) -> c21(A__IF(a__tail(a__tail(mark(z0))), x1, x2), MARK(tail(tail(z0)))) MARK(if(tail(if(z0, z1, z2)), x1, x2)) -> c21(A__IF(a__tail(a__if(mark(z0), z1, z2)), x1, x2), MARK(tail(if(z0, z1, z2)))) MARK(if(tail(filter(z0, z1)), x1, x2)) -> c21(A__IF(a__tail(a__filter(mark(z0), mark(z1))), x1, x2), MARK(tail(filter(z0, z1)))) MARK(if(tail(s(z0)), x1, x2)) -> c21(A__IF(a__tail(s(mark(z0))), x1, x2), MARK(tail(s(z0)))) MARK(if(tail(cons(z0, z1)), x1, x2)) -> c21(A__IF(a__tail(cons(mark(z0), z1)), x1, x2), MARK(tail(cons(z0, z1)))) MARK(if(tail(divides(z0, z1)), x1, x2)) -> c21(A__IF(a__tail(divides(mark(z0), mark(z1))), x1, x2), MARK(tail(divides(z0, z1)))) MARK(if(tail(x0), x1, x2)) -> c21(MARK(tail(x0))) MARK(if(tail(0), x1, x2)) -> c21(A__IF(a__tail(0), x1, x2)) MARK(if(tail(true), x1, x2)) -> c21(A__IF(a__tail(true), x1, x2)) MARK(if(tail(false), x1, x2)) -> c21(A__IF(a__tail(false), x1, x2)) MARK(if(if(primes, x1, x2), x3, x4)) -> c21(A__IF(a__if(a__primes, x1, x2), x3, x4), MARK(if(primes, x1, x2))) MARK(if(if(sieve(z0), x1, x2), x3, x4)) -> c21(A__IF(a__if(a__sieve(mark(z0)), x1, x2), x3, x4), MARK(if(sieve(z0), x1, x2))) MARK(if(if(from(z0), x1, x2), x3, x4)) -> c21(A__IF(a__if(a__from(mark(z0)), x1, x2), x3, x4), MARK(if(from(z0), x1, x2))) MARK(if(if(head(z0), x1, x2), x3, x4)) -> c21(A__IF(a__if(a__head(mark(z0)), x1, x2), x3, x4), MARK(if(head(z0), x1, x2))) MARK(if(if(tail(z0), x1, x2), x3, x4)) -> c21(A__IF(a__if(a__tail(mark(z0)), x1, x2), x3, x4), MARK(if(tail(z0), x1, x2))) MARK(if(if(if(z0, z1, z2), x1, x2), x3, x4)) -> c21(A__IF(a__if(a__if(mark(z0), z1, z2), x1, x2), x3, x4), MARK(if(if(z0, z1, z2), x1, x2))) MARK(if(if(filter(z0, z1), x1, x2), x3, x4)) -> c21(A__IF(a__if(a__filter(mark(z0), mark(z1)), x1, x2), x3, x4), MARK(if(filter(z0, z1), x1, x2))) MARK(if(if(s(z0), x1, x2), x3, x4)) -> c21(A__IF(a__if(s(mark(z0)), x1, x2), x3, x4), MARK(if(s(z0), x1, x2))) MARK(if(if(cons(z0, z1), x1, x2), x3, x4)) -> c21(A__IF(a__if(cons(mark(z0), z1), x1, x2), x3, x4), MARK(if(cons(z0, z1), x1, x2))) MARK(if(if(true, x1, x2), x3, x4)) -> c21(A__IF(a__if(true, x1, x2), x3, x4), MARK(if(true, x1, x2))) MARK(if(if(false, x1, x2), x3, x4)) -> c21(A__IF(a__if(false, x1, x2), x3, x4), MARK(if(false, x1, x2))) MARK(if(if(divides(z0, z1), x1, x2), x3, x4)) -> c21(A__IF(a__if(divides(mark(z0), mark(z1)), x1, x2), x3, x4), MARK(if(divides(z0, z1), x1, x2))) MARK(if(if(x0, z1, z2), x3, x4)) -> c21(MARK(if(x0, z1, z2))) MARK(if(if(0, x1, x2), x3, x4)) -> c21(A__IF(a__if(0, x1, x2), x3, x4)) MARK(if(filter(x0, primes), x2, x3)) -> c21(A__IF(a__filter(mark(x0), a__primes), x2, x3), MARK(filter(x0, primes))) MARK(if(filter(x0, sieve(z0)), x2, x3)) -> c21(A__IF(a__filter(mark(x0), a__sieve(mark(z0))), x2, x3), MARK(filter(x0, sieve(z0)))) MARK(if(filter(x0, from(z0)), x2, x3)) -> c21(A__IF(a__filter(mark(x0), a__from(mark(z0))), x2, x3), MARK(filter(x0, from(z0)))) MARK(if(filter(x0, head(z0)), x2, x3)) -> c21(A__IF(a__filter(mark(x0), a__head(mark(z0))), x2, x3), MARK(filter(x0, head(z0)))) MARK(if(filter(x0, tail(z0)), x2, x3)) -> c21(A__IF(a__filter(mark(x0), a__tail(mark(z0))), x2, x3), MARK(filter(x0, tail(z0)))) MARK(if(filter(x0, if(z0, z1, z2)), x2, x3)) -> c21(A__IF(a__filter(mark(x0), a__if(mark(z0), z1, z2)), x2, x3), MARK(filter(x0, if(z0, z1, z2)))) MARK(if(filter(x0, filter(z0, z1)), x2, x3)) -> c21(A__IF(a__filter(mark(x0), a__filter(mark(z0), mark(z1))), x2, x3), MARK(filter(x0, filter(z0, z1)))) MARK(if(filter(x0, s(z0)), x2, x3)) -> c21(A__IF(a__filter(mark(x0), s(mark(z0))), x2, x3), MARK(filter(x0, s(z0)))) MARK(if(filter(x0, 0), x2, x3)) -> c21(A__IF(a__filter(mark(x0), 0), x2, x3), MARK(filter(x0, 0))) MARK(if(filter(x0, cons(z0, z1)), x2, x3)) -> c21(A__IF(a__filter(mark(x0), cons(mark(z0), z1)), x2, x3), MARK(filter(x0, cons(z0, z1)))) MARK(if(filter(x0, true), x2, x3)) -> c21(A__IF(a__filter(mark(x0), true), x2, x3), MARK(filter(x0, true))) MARK(if(filter(x0, false), x2, x3)) -> c21(A__IF(a__filter(mark(x0), false), x2, x3), MARK(filter(x0, false))) MARK(if(filter(x0, divides(z0, z1)), x2, x3)) -> c21(A__IF(a__filter(mark(x0), divides(mark(z0), mark(z1))), x2, x3), MARK(filter(x0, divides(z0, z1)))) MARK(if(filter(primes, x1), x2, x3)) -> c21(A__IF(a__filter(a__primes, mark(x1)), x2, x3), MARK(filter(primes, x1))) MARK(if(filter(sieve(z0), x1), x2, x3)) -> c21(A__IF(a__filter(a__sieve(mark(z0)), mark(x1)), x2, x3), MARK(filter(sieve(z0), x1))) MARK(if(filter(from(z0), x1), x2, x3)) -> c21(A__IF(a__filter(a__from(mark(z0)), mark(x1)), x2, x3), MARK(filter(from(z0), x1))) MARK(if(filter(head(z0), x1), x2, x3)) -> c21(A__IF(a__filter(a__head(mark(z0)), mark(x1)), x2, x3), MARK(filter(head(z0), x1))) MARK(if(filter(tail(z0), x1), x2, x3)) -> c21(A__IF(a__filter(a__tail(mark(z0)), mark(x1)), x2, x3), MARK(filter(tail(z0), x1))) MARK(if(filter(if(z0, z1, z2), x1), x2, x3)) -> c21(A__IF(a__filter(a__if(mark(z0), z1, z2), mark(x1)), x2, x3), MARK(filter(if(z0, z1, z2), x1))) MARK(if(filter(filter(z0, z1), x1), x2, x3)) -> c21(A__IF(a__filter(a__filter(mark(z0), mark(z1)), mark(x1)), x2, x3), MARK(filter(filter(z0, z1), x1))) MARK(if(filter(s(z0), x1), x2, x3)) -> c21(A__IF(a__filter(s(mark(z0)), mark(x1)), x2, x3), MARK(filter(s(z0), x1))) MARK(if(filter(0, x1), x2, x3)) -> c21(A__IF(a__filter(0, mark(x1)), x2, x3), MARK(filter(0, x1))) MARK(if(filter(cons(z0, z1), x1), x2, x3)) -> c21(A__IF(a__filter(cons(mark(z0), z1), mark(x1)), x2, x3), MARK(filter(cons(z0, z1), x1))) MARK(if(filter(true, x1), x2, x3)) -> c21(A__IF(a__filter(true, mark(x1)), x2, x3), MARK(filter(true, x1))) MARK(if(filter(false, x1), x2, x3)) -> c21(A__IF(a__filter(false, mark(x1)), x2, x3), MARK(filter(false, x1))) MARK(if(filter(divides(z0, z1), x1), x2, x3)) -> c21(A__IF(a__filter(divides(mark(z0), mark(z1)), mark(x1)), x2, x3), MARK(filter(divides(z0, z1), x1))) MARK(if(filter(x0, x1), x2, x3)) -> c21(MARK(filter(x0, x1))) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c18_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_1, c17_2, c17_1, c19_2, c19_1, c20_2, c20_1, c21_1, c22_2, c22_1, c23_2, c23_1, c_2, c21_2 ---------------------------------------- (211) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(filter(x0, primes)) -> c22(A__FILTER(mark(x0), a__primes), MARK(x0)) by MARK(filter(x0, primes)) -> c22(A__FILTER(mark(x0), a__sieve(a__from(s(s(0))))), MARK(x0)) MARK(filter(x0, primes)) -> c22(A__FILTER(mark(x0), primes), MARK(x0)) MARK(filter(primes, primes)) -> c22(A__FILTER(a__primes, a__primes), MARK(primes)) MARK(filter(sieve(z0), primes)) -> c22(A__FILTER(a__sieve(mark(z0)), a__primes), MARK(sieve(z0))) MARK(filter(from(z0), primes)) -> c22(A__FILTER(a__from(mark(z0)), a__primes), MARK(from(z0))) MARK(filter(head(z0), primes)) -> c22(A__FILTER(a__head(mark(z0)), a__primes), MARK(head(z0))) MARK(filter(tail(z0), primes)) -> c22(A__FILTER(a__tail(mark(z0)), a__primes), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), primes)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), a__primes), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), primes)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), a__primes), MARK(filter(z0, z1))) MARK(filter(s(z0), primes)) -> c22(A__FILTER(s(mark(z0)), a__primes), MARK(s(z0))) MARK(filter(0, primes)) -> c22(A__FILTER(0, a__primes), MARK(0)) MARK(filter(cons(z0, z1), primes)) -> c22(A__FILTER(cons(mark(z0), z1), a__primes), MARK(cons(z0, z1))) MARK(filter(true, primes)) -> c22(A__FILTER(true, a__primes), MARK(true)) MARK(filter(false, primes)) -> c22(A__FILTER(false, a__primes), MARK(false)) MARK(filter(divides(z0, z1), primes)) -> c22(A__FILTER(divides(mark(z0), mark(z1)), a__primes), MARK(divides(z0, z1))) ---------------------------------------- (212) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, sieve(z0))) -> c22(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, from(z0))) -> c22(A__FILTER(mark(x0), a__from(mark(z0))), MARK(x0)) MARK(filter(x0, head(z0))) -> c22(A__FILTER(mark(x0), a__head(mark(z0))), MARK(x0)) MARK(filter(x0, tail(z0))) -> c22(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(x0)) MARK(filter(x0, if(z0, z1, z2))) -> c22(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0)) MARK(filter(x0, filter(z0, z1))) -> c22(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(x0)) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(x0, s(z0))) -> c22(MARK(x0)) MARK(filter(x0, 0)) -> c22(MARK(x0)) MARK(filter(x0, true)) -> c22(MARK(x0)) MARK(filter(x0, false)) -> c22(MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), x1)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c23(A__FILTER(mark(x0), a__primes), MARK(primes)) MARK(filter(x0, sieve(z0))) -> c23(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, from(z0))) -> c23(A__FILTER(mark(x0), a__from(mark(z0))), MARK(from(z0))) MARK(filter(x0, head(z0))) -> c23(A__FILTER(mark(x0), a__head(mark(z0))), MARK(head(z0))) MARK(filter(x0, tail(z0))) -> c23(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(tail(z0))) MARK(filter(x0, if(z0, z1, z2))) -> c23(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(filter(x0, filter(z0, z1))) -> c23(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(filter(x0, cons(z0, z1))) -> c23(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(primes, x1)) -> c23(A__FILTER(a__primes, mark(x1)), MARK(x1)) MARK(filter(sieve(z0), x1)) -> c23(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(from(z0), x1)) -> c23(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(head(z0), x1)) -> c23(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(tail(z0), x1)) -> c23(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(if(z0, z1, z2), x1)) -> c23(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1)) MARK(filter(filter(z0, z1), x1)) -> c23(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(x1)) MARK(filter(s(z0), x1)) -> c23(A__FILTER(s(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(x0, s(z0))) -> c23(MARK(s(z0))) MARK(filter(x0, divides(z0, z1))) -> c23(MARK(divides(z0, z1))) MARK(filter(0, x1)) -> c23(MARK(x1)) MARK(filter(cons(z0, z1), x1)) -> c23(MARK(x1)) MARK(filter(true, x1)) -> c23(MARK(x1)) MARK(filter(false, x1)) -> c23(MARK(x1)) MARK(filter(divides(z0, z1), x1)) -> c23(MARK(x1)) A__PRIMES -> c(A__SIEVE(cons(s(mark(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(sieve(primes)) -> c17(MARK(primes)) MARK(sieve(sieve(primes))) -> c17(A__SIEVE(a__sieve(a__primes)), MARK(sieve(primes))) MARK(sieve(sieve(sieve(z0)))) -> c17(A__SIEVE(a__sieve(a__sieve(mark(z0)))), MARK(sieve(sieve(z0)))) MARK(sieve(sieve(from(z0)))) -> c17(A__SIEVE(a__sieve(a__from(mark(z0)))), MARK(sieve(from(z0)))) MARK(sieve(sieve(head(z0)))) -> c17(A__SIEVE(a__sieve(a__head(mark(z0)))), MARK(sieve(head(z0)))) MARK(sieve(sieve(tail(z0)))) -> c17(A__SIEVE(a__sieve(a__tail(mark(z0)))), MARK(sieve(tail(z0)))) MARK(sieve(sieve(if(z0, z1, z2)))) -> c17(A__SIEVE(a__sieve(a__if(mark(z0), z1, z2))), MARK(sieve(if(z0, z1, z2)))) MARK(sieve(sieve(filter(z0, z1)))) -> c17(A__SIEVE(a__sieve(a__filter(mark(z0), mark(z1)))), MARK(sieve(filter(z0, z1)))) MARK(sieve(sieve(s(z0)))) -> c17(A__SIEVE(a__sieve(s(mark(z0)))), MARK(sieve(s(z0)))) MARK(sieve(sieve(cons(z0, z1)))) -> c17(A__SIEVE(a__sieve(cons(mark(z0), z1))), MARK(sieve(cons(z0, z1)))) MARK(sieve(sieve(divides(z0, z1)))) -> c17(A__SIEVE(a__sieve(divides(mark(z0), mark(z1)))), MARK(sieve(divides(z0, z1)))) MARK(sieve(sieve(x0))) -> c17(MARK(sieve(x0))) MARK(sieve(sieve(0))) -> c17(A__SIEVE(a__sieve(0))) MARK(sieve(sieve(true))) -> c17(A__SIEVE(a__sieve(true))) MARK(sieve(sieve(false))) -> c17(A__SIEVE(a__sieve(false))) MARK(sieve(from(x0))) -> c17(A__SIEVE(cons(mark(mark(x0)), from(s(mark(x0))))), MARK(from(x0))) MARK(sieve(from(primes))) -> c17(A__SIEVE(a__from(a__primes)), MARK(from(primes))) MARK(sieve(from(sieve(z0)))) -> c17(A__SIEVE(a__from(a__sieve(mark(z0)))), MARK(from(sieve(z0)))) MARK(sieve(from(from(z0)))) -> c17(A__SIEVE(a__from(a__from(mark(z0)))), MARK(from(from(z0)))) MARK(sieve(from(head(z0)))) -> c17(A__SIEVE(a__from(a__head(mark(z0)))), MARK(from(head(z0)))) MARK(sieve(from(tail(z0)))) -> c17(A__SIEVE(a__from(a__tail(mark(z0)))), MARK(from(tail(z0)))) MARK(sieve(from(if(z0, z1, z2)))) -> c17(A__SIEVE(a__from(a__if(mark(z0), z1, z2))), MARK(from(if(z0, z1, z2)))) MARK(sieve(from(filter(z0, z1)))) -> c17(A__SIEVE(a__from(a__filter(mark(z0), mark(z1)))), MARK(from(filter(z0, z1)))) MARK(sieve(from(s(z0)))) -> c17(A__SIEVE(a__from(s(mark(z0)))), MARK(from(s(z0)))) MARK(sieve(from(0))) -> c17(A__SIEVE(a__from(0)), MARK(from(0))) MARK(sieve(from(cons(z0, z1)))) -> c17(A__SIEVE(a__from(cons(mark(z0), z1))), MARK(from(cons(z0, z1)))) MARK(sieve(from(true))) -> c17(A__SIEVE(a__from(true)), MARK(from(true))) MARK(sieve(from(false))) -> c17(A__SIEVE(a__from(false)), MARK(from(false))) MARK(sieve(from(divides(z0, z1)))) -> c17(A__SIEVE(a__from(divides(mark(z0), mark(z1)))), MARK(from(divides(z0, z1)))) MARK(sieve(from(x0))) -> c17(MARK(from(x0))) MARK(sieve(head(primes))) -> c17(A__SIEVE(a__head(a__primes)), MARK(head(primes))) MARK(sieve(head(sieve(z0)))) -> c17(A__SIEVE(a__head(a__sieve(mark(z0)))), MARK(head(sieve(z0)))) MARK(sieve(head(from(z0)))) -> c17(A__SIEVE(a__head(a__from(mark(z0)))), MARK(head(from(z0)))) MARK(sieve(head(head(z0)))) -> c17(A__SIEVE(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(sieve(head(tail(z0)))) -> c17(A__SIEVE(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(sieve(head(if(z0, z1, z2)))) -> c17(A__SIEVE(a__head(a__if(mark(z0), z1, z2))), MARK(head(if(z0, z1, z2)))) MARK(sieve(head(filter(z0, z1)))) -> c17(A__SIEVE(a__head(a__filter(mark(z0), mark(z1)))), MARK(head(filter(z0, z1)))) MARK(sieve(head(s(z0)))) -> c17(A__SIEVE(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(sieve(head(cons(z0, z1)))) -> c17(A__SIEVE(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(sieve(head(divides(z0, z1)))) -> c17(A__SIEVE(a__head(divides(mark(z0), mark(z1)))), MARK(head(divides(z0, z1)))) MARK(sieve(head(x0))) -> c17(MARK(head(x0))) MARK(sieve(head(0))) -> c17(A__SIEVE(a__head(0))) MARK(sieve(head(true))) -> c17(A__SIEVE(a__head(true))) MARK(sieve(head(false))) -> c17(A__SIEVE(a__head(false))) MARK(sieve(tail(primes))) -> c17(A__SIEVE(a__tail(a__primes)), MARK(tail(primes))) MARK(sieve(tail(sieve(z0)))) -> c17(A__SIEVE(a__tail(a__sieve(mark(z0)))), MARK(tail(sieve(z0)))) MARK(sieve(tail(from(z0)))) -> c17(A__SIEVE(a__tail(a__from(mark(z0)))), MARK(tail(from(z0)))) MARK(sieve(tail(head(z0)))) -> c17(A__SIEVE(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(sieve(tail(tail(z0)))) -> c17(A__SIEVE(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(sieve(tail(if(z0, z1, z2)))) -> c17(A__SIEVE(a__tail(a__if(mark(z0), z1, z2))), MARK(tail(if(z0, z1, z2)))) MARK(sieve(tail(filter(z0, z1)))) -> c17(A__SIEVE(a__tail(a__filter(mark(z0), mark(z1)))), MARK(tail(filter(z0, z1)))) MARK(sieve(tail(s(z0)))) -> c17(A__SIEVE(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(sieve(tail(cons(z0, z1)))) -> c17(A__SIEVE(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(sieve(tail(divides(z0, z1)))) -> c17(A__SIEVE(a__tail(divides(mark(z0), mark(z1)))), MARK(tail(divides(z0, z1)))) MARK(sieve(tail(x0))) -> c17(MARK(tail(x0))) MARK(sieve(tail(0))) -> c17(A__SIEVE(a__tail(0))) MARK(sieve(tail(true))) -> c17(A__SIEVE(a__tail(true))) MARK(sieve(tail(false))) -> c17(A__SIEVE(a__tail(false))) MARK(sieve(if(primes, x1, x2))) -> c17(A__SIEVE(a__if(a__primes, x1, x2)), MARK(if(primes, x1, x2))) MARK(sieve(if(sieve(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__sieve(mark(z0)), x1, x2)), MARK(if(sieve(z0), x1, x2))) MARK(sieve(if(from(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__from(mark(z0)), x1, x2)), MARK(if(from(z0), x1, x2))) MARK(sieve(if(head(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__head(mark(z0)), x1, x2)), MARK(if(head(z0), x1, x2))) MARK(sieve(if(tail(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__tail(mark(z0)), x1, x2)), MARK(if(tail(z0), x1, x2))) MARK(sieve(if(if(z0, z1, z2), x1, x2))) -> c17(A__SIEVE(a__if(a__if(mark(z0), z1, z2), x1, x2)), MARK(if(if(z0, z1, z2), x1, x2))) MARK(sieve(if(filter(z0, z1), x1, x2))) -> c17(A__SIEVE(a__if(a__filter(mark(z0), mark(z1)), x1, x2)), MARK(if(filter(z0, z1), x1, x2))) MARK(sieve(if(s(z0), x1, x2))) -> c17(A__SIEVE(a__if(s(mark(z0)), x1, x2)), MARK(if(s(z0), x1, x2))) MARK(sieve(if(cons(z0, z1), x1, x2))) -> c17(A__SIEVE(a__if(cons(mark(z0), z1), x1, x2)), MARK(if(cons(z0, z1), x1, x2))) MARK(sieve(if(true, x1, x2))) -> c17(A__SIEVE(a__if(true, x1, x2)), MARK(if(true, x1, x2))) MARK(sieve(if(false, x1, x2))) -> c17(A__SIEVE(a__if(false, x1, x2)), MARK(if(false, x1, x2))) MARK(sieve(if(divides(z0, z1), x1, x2))) -> c17(A__SIEVE(a__if(divides(mark(z0), mark(z1)), x1, x2)), MARK(if(divides(z0, z1), x1, x2))) MARK(sieve(if(x0, z1, z2))) -> c17(MARK(if(x0, z1, z2))) MARK(sieve(if(0, x1, x2))) -> c17(A__SIEVE(a__if(0, 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x1, x2)) -> c21(A__IF(a__tail(cons(mark(z0), z1)), x1, x2), MARK(tail(cons(z0, z1)))) MARK(if(tail(divides(z0, z1)), x1, x2)) -> c21(A__IF(a__tail(divides(mark(z0), mark(z1))), x1, x2), MARK(tail(divides(z0, z1)))) MARK(if(tail(x0), x1, x2)) -> c21(MARK(tail(x0))) MARK(if(tail(0), x1, x2)) -> c21(A__IF(a__tail(0), x1, x2)) MARK(if(tail(true), x1, x2)) -> c21(A__IF(a__tail(true), x1, x2)) MARK(if(tail(false), x1, x2)) -> c21(A__IF(a__tail(false), x1, x2)) MARK(if(if(primes, x1, x2), x3, x4)) -> c21(A__IF(a__if(a__primes, x1, x2), x3, x4), MARK(if(primes, x1, x2))) MARK(if(if(sieve(z0), x1, x2), x3, x4)) -> c21(A__IF(a__if(a__sieve(mark(z0)), x1, x2), x3, x4), MARK(if(sieve(z0), x1, x2))) MARK(if(if(from(z0), x1, x2), x3, x4)) -> c21(A__IF(a__if(a__from(mark(z0)), x1, x2), x3, x4), MARK(if(from(z0), x1, x2))) MARK(if(if(head(z0), x1, x2), x3, x4)) -> c21(A__IF(a__if(a__head(mark(z0)), x1, x2), x3, x4), MARK(if(head(z0), x1, x2))) MARK(if(if(tail(z0), x1, x2), x3, x4)) -> c21(A__IF(a__if(a__tail(mark(z0)), x1, x2), x3, x4), MARK(if(tail(z0), x1, x2))) MARK(if(if(if(z0, z1, z2), x1, x2), x3, x4)) -> c21(A__IF(a__if(a__if(mark(z0), z1, z2), x1, x2), x3, x4), MARK(if(if(z0, z1, z2), x1, x2))) MARK(if(if(filter(z0, z1), x1, x2), x3, x4)) -> c21(A__IF(a__if(a__filter(mark(z0), mark(z1)), x1, x2), x3, x4), MARK(if(filter(z0, z1), x1, x2))) MARK(if(if(s(z0), x1, x2), x3, x4)) -> c21(A__IF(a__if(s(mark(z0)), x1, x2), x3, x4), MARK(if(s(z0), x1, x2))) MARK(if(if(cons(z0, z1), x1, x2), x3, x4)) -> c21(A__IF(a__if(cons(mark(z0), z1), x1, x2), x3, x4), MARK(if(cons(z0, z1), x1, x2))) MARK(if(if(true, x1, x2), x3, x4)) -> c21(A__IF(a__if(true, x1, x2), x3, x4), MARK(if(true, x1, x2))) MARK(if(if(false, x1, x2), x3, x4)) -> c21(A__IF(a__if(false, x1, x2), x3, x4), MARK(if(false, x1, x2))) MARK(if(if(divides(z0, z1), x1, x2), x3, x4)) -> c21(A__IF(a__if(divides(mark(z0), mark(z1)), x1, x2), x3, x4), MARK(if(divides(z0, z1), x1, x2))) MARK(if(if(x0, z1, z2), x3, x4)) -> c21(MARK(if(x0, z1, z2))) MARK(if(if(0, x1, x2), x3, x4)) -> c21(A__IF(a__if(0, x1, x2), x3, x4)) MARK(if(filter(x0, primes), x2, x3)) -> c21(A__IF(a__filter(mark(x0), a__primes), x2, x3), MARK(filter(x0, primes))) MARK(if(filter(x0, sieve(z0)), x2, x3)) -> c21(A__IF(a__filter(mark(x0), a__sieve(mark(z0))), x2, x3), MARK(filter(x0, sieve(z0)))) MARK(if(filter(x0, from(z0)), x2, x3)) -> c21(A__IF(a__filter(mark(x0), a__from(mark(z0))), x2, x3), MARK(filter(x0, from(z0)))) MARK(if(filter(x0, head(z0)), x2, x3)) -> c21(A__IF(a__filter(mark(x0), a__head(mark(z0))), x2, x3), MARK(filter(x0, head(z0)))) MARK(if(filter(x0, tail(z0)), x2, x3)) -> c21(A__IF(a__filter(mark(x0), a__tail(mark(z0))), x2, x3), MARK(filter(x0, tail(z0)))) MARK(if(filter(x0, if(z0, z1, z2)), x2, x3)) -> c21(A__IF(a__filter(mark(x0), a__if(mark(z0), z1, z2)), x2, x3), MARK(filter(x0, if(z0, z1, z2)))) MARK(if(filter(x0, filter(z0, z1)), x2, x3)) -> c21(A__IF(a__filter(mark(x0), a__filter(mark(z0), mark(z1))), x2, x3), MARK(filter(x0, filter(z0, z1)))) MARK(if(filter(x0, s(z0)), x2, x3)) -> c21(A__IF(a__filter(mark(x0), s(mark(z0))), x2, x3), MARK(filter(x0, s(z0)))) MARK(if(filter(x0, 0), x2, x3)) -> c21(A__IF(a__filter(mark(x0), 0), x2, x3), MARK(filter(x0, 0))) MARK(if(filter(x0, cons(z0, z1)), x2, x3)) -> c21(A__IF(a__filter(mark(x0), cons(mark(z0), z1)), x2, x3), MARK(filter(x0, cons(z0, z1)))) MARK(if(filter(x0, true), x2, x3)) -> c21(A__IF(a__filter(mark(x0), true), x2, x3), MARK(filter(x0, true))) MARK(if(filter(x0, false), x2, x3)) -> c21(A__IF(a__filter(mark(x0), false), x2, x3), MARK(filter(x0, false))) MARK(if(filter(x0, divides(z0, z1)), x2, x3)) -> c21(A__IF(a__filter(mark(x0), divides(mark(z0), mark(z1))), x2, x3), MARK(filter(x0, divides(z0, z1)))) MARK(if(filter(primes, x1), x2, x3)) -> c21(A__IF(a__filter(a__primes, mark(x1)), x2, x3), MARK(filter(primes, x1))) MARK(if(filter(sieve(z0), x1), x2, x3)) -> c21(A__IF(a__filter(a__sieve(mark(z0)), mark(x1)), x2, x3), MARK(filter(sieve(z0), x1))) MARK(if(filter(from(z0), x1), x2, x3)) -> c21(A__IF(a__filter(a__from(mark(z0)), mark(x1)), x2, x3), MARK(filter(from(z0), x1))) MARK(if(filter(head(z0), x1), x2, x3)) -> c21(A__IF(a__filter(a__head(mark(z0)), mark(x1)), x2, x3), MARK(filter(head(z0), x1))) MARK(if(filter(tail(z0), x1), x2, x3)) -> c21(A__IF(a__filter(a__tail(mark(z0)), mark(x1)), x2, x3), MARK(filter(tail(z0), x1))) MARK(if(filter(if(z0, z1, z2), x1), x2, x3)) -> c21(A__IF(a__filter(a__if(mark(z0), z1, z2), mark(x1)), x2, x3), MARK(filter(if(z0, z1, z2), x1))) MARK(if(filter(filter(z0, z1), x1), x2, x3)) -> c21(A__IF(a__filter(a__filter(mark(z0), mark(z1)), mark(x1)), x2, x3), MARK(filter(filter(z0, z1), x1))) MARK(if(filter(s(z0), x1), x2, x3)) -> c21(A__IF(a__filter(s(mark(z0)), mark(x1)), x2, x3), MARK(filter(s(z0), x1))) MARK(if(filter(0, x1), x2, x3)) -> c21(A__IF(a__filter(0, mark(x1)), x2, x3), MARK(filter(0, x1))) MARK(if(filter(cons(z0, z1), x1), x2, x3)) -> c21(A__IF(a__filter(cons(mark(z0), z1), mark(x1)), x2, x3), MARK(filter(cons(z0, z1), x1))) MARK(if(filter(true, x1), x2, x3)) -> c21(A__IF(a__filter(true, mark(x1)), x2, x3), MARK(filter(true, x1))) MARK(if(filter(false, x1), x2, x3)) -> c21(A__IF(a__filter(false, mark(x1)), x2, x3), MARK(filter(false, x1))) MARK(if(filter(divides(z0, z1), x1), x2, x3)) -> c21(A__IF(a__filter(divides(mark(z0), mark(z1)), mark(x1)), x2, x3), MARK(filter(divides(z0, z1), x1))) MARK(if(filter(x0, x1), x2, x3)) -> c21(MARK(filter(x0, x1))) MARK(filter(x0, primes)) -> c22(A__FILTER(mark(x0), a__sieve(a__from(s(s(0))))), MARK(x0)) MARK(filter(x0, primes)) -> c22(A__FILTER(mark(x0), primes), MARK(x0)) MARK(filter(primes, primes)) -> c22(A__FILTER(a__primes, a__primes), MARK(primes)) MARK(filter(sieve(z0), primes)) -> c22(A__FILTER(a__sieve(mark(z0)), a__primes), MARK(sieve(z0))) MARK(filter(from(z0), primes)) -> c22(A__FILTER(a__from(mark(z0)), a__primes), MARK(from(z0))) MARK(filter(head(z0), primes)) -> c22(A__FILTER(a__head(mark(z0)), a__primes), MARK(head(z0))) MARK(filter(tail(z0), primes)) -> c22(A__FILTER(a__tail(mark(z0)), a__primes), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), primes)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), a__primes), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), primes)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), a__primes), MARK(filter(z0, z1))) MARK(filter(s(z0), primes)) -> c22(A__FILTER(s(mark(z0)), a__primes), MARK(s(z0))) MARK(filter(0, primes)) -> c22(A__FILTER(0, a__primes), MARK(0)) MARK(filter(cons(z0, z1), primes)) -> c22(A__FILTER(cons(mark(z0), z1), a__primes), MARK(cons(z0, z1))) MARK(filter(true, primes)) -> c22(A__FILTER(true, a__primes), MARK(true)) MARK(filter(false, primes)) -> c22(A__FILTER(false, a__primes), MARK(false)) MARK(filter(divides(z0, z1), primes)) -> c22(A__FILTER(divides(mark(z0), mark(z1)), a__primes), MARK(divides(z0, z1))) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c18_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_1, c17_2, c17_1, c19_2, c19_1, c20_2, c20_1, c21_1, c22_2, c22_1, c23_2, c23_1, c_2, c21_2 ---------------------------------------- (213) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing nodes: MARK(filter(0, primes)) -> c22(A__FILTER(0, a__primes), MARK(0)) MARK(filter(false, primes)) -> c22(A__FILTER(false, a__primes), MARK(false)) MARK(filter(true, primes)) -> c22(A__FILTER(true, a__primes), MARK(true)) ---------------------------------------- (214) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, sieve(z0))) -> c22(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, from(z0))) -> c22(A__FILTER(mark(x0), a__from(mark(z0))), MARK(x0)) MARK(filter(x0, head(z0))) -> c22(A__FILTER(mark(x0), a__head(mark(z0))), MARK(x0)) MARK(filter(x0, tail(z0))) -> c22(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(x0)) MARK(filter(x0, if(z0, z1, z2))) -> c22(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0)) MARK(filter(x0, filter(z0, z1))) -> c22(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(x0)) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(x0, s(z0))) -> c22(MARK(x0)) MARK(filter(x0, 0)) -> c22(MARK(x0)) MARK(filter(x0, true)) -> c22(MARK(x0)) MARK(filter(x0, false)) -> c22(MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), x1)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c23(A__FILTER(mark(x0), a__primes), MARK(primes)) MARK(filter(x0, sieve(z0))) -> c23(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, from(z0))) -> c23(A__FILTER(mark(x0), a__from(mark(z0))), MARK(from(z0))) MARK(filter(x0, head(z0))) -> c23(A__FILTER(mark(x0), a__head(mark(z0))), MARK(head(z0))) MARK(filter(x0, tail(z0))) -> c23(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(tail(z0))) MARK(filter(x0, if(z0, z1, z2))) -> c23(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(filter(x0, filter(z0, z1))) -> c23(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(filter(x0, cons(z0, z1))) -> c23(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(primes, x1)) -> c23(A__FILTER(a__primes, mark(x1)), MARK(x1)) MARK(filter(sieve(z0), x1)) -> c23(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(from(z0), x1)) -> c23(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(head(z0), x1)) -> c23(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(tail(z0), x1)) -> c23(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(if(z0, z1, z2), x1)) -> c23(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1)) MARK(filter(filter(z0, z1), x1)) -> c23(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(x1)) MARK(filter(s(z0), x1)) -> c23(A__FILTER(s(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(x0, s(z0))) -> c23(MARK(s(z0))) MARK(filter(x0, divides(z0, z1))) -> c23(MARK(divides(z0, z1))) MARK(filter(0, x1)) -> c23(MARK(x1)) MARK(filter(cons(z0, z1), x1)) -> c23(MARK(x1)) MARK(filter(true, x1)) -> c23(MARK(x1)) MARK(filter(false, x1)) -> c23(MARK(x1)) MARK(filter(divides(z0, z1), x1)) -> c23(MARK(x1)) A__PRIMES -> c(A__SIEVE(cons(s(mark(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(sieve(primes)) -> c17(MARK(primes)) MARK(sieve(sieve(primes))) -> 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x4), MARK(if(sieve(z0), x1, x2))) MARK(if(if(from(z0), x1, x2), x3, x4)) -> c21(A__IF(a__if(a__from(mark(z0)), x1, x2), x3, x4), MARK(if(from(z0), x1, x2))) MARK(if(if(head(z0), x1, x2), x3, x4)) -> c21(A__IF(a__if(a__head(mark(z0)), x1, x2), x3, x4), MARK(if(head(z0), x1, x2))) MARK(if(if(tail(z0), x1, x2), x3, x4)) -> c21(A__IF(a__if(a__tail(mark(z0)), x1, x2), x3, x4), MARK(if(tail(z0), x1, x2))) MARK(if(if(if(z0, z1, z2), x1, x2), x3, x4)) -> c21(A__IF(a__if(a__if(mark(z0), z1, z2), x1, x2), x3, x4), MARK(if(if(z0, z1, z2), x1, x2))) MARK(if(if(filter(z0, z1), x1, x2), x3, x4)) -> c21(A__IF(a__if(a__filter(mark(z0), mark(z1)), x1, x2), x3, x4), MARK(if(filter(z0, z1), x1, x2))) MARK(if(if(s(z0), x1, x2), x3, x4)) -> c21(A__IF(a__if(s(mark(z0)), x1, x2), x3, x4), MARK(if(s(z0), x1, x2))) MARK(if(if(cons(z0, z1), x1, x2), x3, x4)) -> c21(A__IF(a__if(cons(mark(z0), z1), x1, x2), x3, x4), MARK(if(cons(z0, z1), x1, x2))) MARK(if(if(true, x1, x2), x3, x4)) -> c21(A__IF(a__if(true, x1, 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mark(x1)), x2, x3), MARK(filter(filter(z0, z1), x1))) MARK(if(filter(s(z0), x1), x2, x3)) -> c21(A__IF(a__filter(s(mark(z0)), mark(x1)), x2, x3), MARK(filter(s(z0), x1))) MARK(if(filter(0, x1), x2, x3)) -> c21(A__IF(a__filter(0, mark(x1)), x2, x3), MARK(filter(0, x1))) MARK(if(filter(cons(z0, z1), x1), x2, x3)) -> c21(A__IF(a__filter(cons(mark(z0), z1), mark(x1)), x2, x3), MARK(filter(cons(z0, z1), x1))) MARK(if(filter(true, x1), x2, x3)) -> c21(A__IF(a__filter(true, mark(x1)), x2, x3), MARK(filter(true, x1))) MARK(if(filter(false, x1), x2, x3)) -> c21(A__IF(a__filter(false, mark(x1)), x2, x3), MARK(filter(false, x1))) MARK(if(filter(divides(z0, z1), x1), x2, x3)) -> c21(A__IF(a__filter(divides(mark(z0), mark(z1)), mark(x1)), x2, x3), MARK(filter(divides(z0, z1), x1))) MARK(if(filter(x0, x1), x2, x3)) -> c21(MARK(filter(x0, x1))) MARK(filter(x0, primes)) -> c22(A__FILTER(mark(x0), a__sieve(a__from(s(s(0))))), MARK(x0)) MARK(filter(x0, primes)) -> c22(A__FILTER(mark(x0), primes), MARK(x0)) MARK(filter(primes, primes)) -> c22(A__FILTER(a__primes, a__primes), MARK(primes)) MARK(filter(sieve(z0), primes)) -> c22(A__FILTER(a__sieve(mark(z0)), a__primes), MARK(sieve(z0))) MARK(filter(from(z0), primes)) -> c22(A__FILTER(a__from(mark(z0)), a__primes), MARK(from(z0))) MARK(filter(head(z0), primes)) -> c22(A__FILTER(a__head(mark(z0)), a__primes), MARK(head(z0))) MARK(filter(tail(z0), primes)) -> c22(A__FILTER(a__tail(mark(z0)), a__primes), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), primes)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), a__primes), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), primes)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), a__primes), MARK(filter(z0, z1))) MARK(filter(s(z0), primes)) -> c22(A__FILTER(s(mark(z0)), a__primes), MARK(s(z0))) MARK(filter(cons(z0, z1), primes)) -> c22(A__FILTER(cons(mark(z0), z1), a__primes), MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), primes)) -> c22(A__FILTER(divides(mark(z0), mark(z1)), a__primes), MARK(divides(z0, z1))) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c18_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_1, c17_2, c17_1, c19_2, c19_1, c20_2, c20_1, c21_1, c22_2, c22_1, c23_2, c23_1, c_2, c21_2 ---------------------------------------- (215) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (216) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, sieve(z0))) -> c22(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(x0)) MARK(filter(x0, from(z0))) -> c22(A__FILTER(mark(x0), a__from(mark(z0))), MARK(x0)) MARK(filter(x0, head(z0))) -> c22(A__FILTER(mark(x0), a__head(mark(z0))), MARK(x0)) MARK(filter(x0, tail(z0))) -> c22(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(x0)) MARK(filter(x0, if(z0, z1, z2))) -> c22(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0)) MARK(filter(x0, filter(z0, z1))) -> c22(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(x0)) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(x0, s(z0))) -> c22(MARK(x0)) MARK(filter(x0, 0)) -> c22(MARK(x0)) MARK(filter(x0, true)) -> c22(MARK(x0)) MARK(filter(x0, false)) -> c22(MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), x1)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c23(A__FILTER(mark(x0), a__primes), MARK(primes)) MARK(filter(x0, sieve(z0))) -> c23(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, from(z0))) -> c23(A__FILTER(mark(x0), a__from(mark(z0))), MARK(from(z0))) MARK(filter(x0, head(z0))) -> c23(A__FILTER(mark(x0), a__head(mark(z0))), MARK(head(z0))) MARK(filter(x0, tail(z0))) -> c23(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(tail(z0))) MARK(filter(x0, if(z0, z1, z2))) -> c23(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(filter(x0, filter(z0, z1))) -> c23(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(filter(x0, cons(z0, z1))) -> c23(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(primes, x1)) -> c23(A__FILTER(a__primes, mark(x1)), MARK(x1)) MARK(filter(sieve(z0), x1)) -> c23(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(from(z0), x1)) -> c23(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(head(z0), x1)) -> c23(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(tail(z0), x1)) -> c23(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(if(z0, z1, z2), x1)) -> c23(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1)) MARK(filter(filter(z0, z1), x1)) -> c23(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(x1)) MARK(filter(s(z0), x1)) -> c23(A__FILTER(s(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(x0, s(z0))) -> c23(MARK(s(z0))) MARK(filter(x0, divides(z0, z1))) -> c23(MARK(divides(z0, z1))) MARK(filter(0, x1)) -> c23(MARK(x1)) MARK(filter(cons(z0, z1), x1)) -> c23(MARK(x1)) MARK(filter(true, x1)) -> c23(MARK(x1)) MARK(filter(false, x1)) -> c23(MARK(x1)) MARK(filter(divides(z0, z1), x1)) -> c23(MARK(x1)) A__PRIMES -> c(A__SIEVE(cons(s(mark(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(sieve(primes)) -> c17(MARK(primes)) MARK(sieve(sieve(primes))) -> c17(A__SIEVE(a__sieve(a__primes)), MARK(sieve(primes))) MARK(sieve(sieve(sieve(z0)))) -> c17(A__SIEVE(a__sieve(a__sieve(mark(z0)))), MARK(sieve(sieve(z0)))) MARK(sieve(sieve(from(z0)))) -> c17(A__SIEVE(a__sieve(a__from(mark(z0)))), MARK(sieve(from(z0)))) MARK(sieve(sieve(head(z0)))) -> c17(A__SIEVE(a__sieve(a__head(mark(z0)))), MARK(sieve(head(z0)))) MARK(sieve(sieve(tail(z0)))) -> c17(A__SIEVE(a__sieve(a__tail(mark(z0)))), MARK(sieve(tail(z0)))) MARK(sieve(sieve(if(z0, z1, z2)))) -> c17(A__SIEVE(a__sieve(a__if(mark(z0), z1, z2))), MARK(sieve(if(z0, z1, z2)))) MARK(sieve(sieve(filter(z0, z1)))) -> c17(A__SIEVE(a__sieve(a__filter(mark(z0), mark(z1)))), MARK(sieve(filter(z0, z1)))) MARK(sieve(sieve(s(z0)))) -> c17(A__SIEVE(a__sieve(s(mark(z0)))), MARK(sieve(s(z0)))) MARK(sieve(sieve(cons(z0, z1)))) -> c17(A__SIEVE(a__sieve(cons(mark(z0), z1))), MARK(sieve(cons(z0, z1)))) MARK(sieve(sieve(divides(z0, z1)))) -> c17(A__SIEVE(a__sieve(divides(mark(z0), mark(z1)))), MARK(sieve(divides(z0, z1)))) MARK(sieve(sieve(x0))) -> c17(MARK(sieve(x0))) MARK(sieve(sieve(0))) -> c17(A__SIEVE(a__sieve(0))) MARK(sieve(sieve(true))) -> c17(A__SIEVE(a__sieve(true))) MARK(sieve(sieve(false))) -> c17(A__SIEVE(a__sieve(false))) MARK(sieve(from(x0))) -> c17(A__SIEVE(cons(mark(mark(x0)), from(s(mark(x0))))), MARK(from(x0))) MARK(sieve(from(primes))) -> c17(A__SIEVE(a__from(a__primes)), MARK(from(primes))) MARK(sieve(from(sieve(z0)))) -> c17(A__SIEVE(a__from(a__sieve(mark(z0)))), MARK(from(sieve(z0)))) MARK(sieve(from(from(z0)))) -> c17(A__SIEVE(a__from(a__from(mark(z0)))), MARK(from(from(z0)))) MARK(sieve(from(head(z0)))) -> c17(A__SIEVE(a__from(a__head(mark(z0)))), MARK(from(head(z0)))) MARK(sieve(from(tail(z0)))) -> c17(A__SIEVE(a__from(a__tail(mark(z0)))), MARK(from(tail(z0)))) MARK(sieve(from(if(z0, z1, z2)))) -> c17(A__SIEVE(a__from(a__if(mark(z0), z1, z2))), MARK(from(if(z0, z1, z2)))) MARK(sieve(from(filter(z0, z1)))) -> c17(A__SIEVE(a__from(a__filter(mark(z0), mark(z1)))), MARK(from(filter(z0, z1)))) MARK(sieve(from(s(z0)))) -> c17(A__SIEVE(a__from(s(mark(z0)))), MARK(from(s(z0)))) MARK(sieve(from(0))) -> c17(A__SIEVE(a__from(0)), MARK(from(0))) MARK(sieve(from(cons(z0, z1)))) -> c17(A__SIEVE(a__from(cons(mark(z0), z1))), MARK(from(cons(z0, z1)))) MARK(sieve(from(true))) -> c17(A__SIEVE(a__from(true)), MARK(from(true))) MARK(sieve(from(false))) -> c17(A__SIEVE(a__from(false)), MARK(from(false))) MARK(sieve(from(divides(z0, z1)))) -> c17(A__SIEVE(a__from(divides(mark(z0), mark(z1)))), MARK(from(divides(z0, z1)))) MARK(sieve(from(x0))) -> c17(MARK(from(x0))) MARK(sieve(head(primes))) -> c17(A__SIEVE(a__head(a__primes)), MARK(head(primes))) MARK(sieve(head(sieve(z0)))) -> c17(A__SIEVE(a__head(a__sieve(mark(z0)))), MARK(head(sieve(z0)))) MARK(sieve(head(from(z0)))) -> c17(A__SIEVE(a__head(a__from(mark(z0)))), MARK(head(from(z0)))) MARK(sieve(head(head(z0)))) -> c17(A__SIEVE(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(sieve(head(tail(z0)))) -> c17(A__SIEVE(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(sieve(head(if(z0, z1, z2)))) -> c17(A__SIEVE(a__head(a__if(mark(z0), z1, z2))), MARK(head(if(z0, z1, z2)))) MARK(sieve(head(filter(z0, z1)))) -> c17(A__SIEVE(a__head(a__filter(mark(z0), mark(z1)))), MARK(head(filter(z0, z1)))) MARK(sieve(head(s(z0)))) -> c17(A__SIEVE(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(sieve(head(cons(z0, z1)))) -> c17(A__SIEVE(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(sieve(head(divides(z0, z1)))) -> c17(A__SIEVE(a__head(divides(mark(z0), mark(z1)))), MARK(head(divides(z0, z1)))) MARK(sieve(head(x0))) -> c17(MARK(head(x0))) MARK(sieve(head(0))) -> c17(A__SIEVE(a__head(0))) MARK(sieve(head(true))) -> c17(A__SIEVE(a__head(true))) MARK(sieve(head(false))) -> c17(A__SIEVE(a__head(false))) MARK(sieve(tail(primes))) -> c17(A__SIEVE(a__tail(a__primes)), MARK(tail(primes))) MARK(sieve(tail(sieve(z0)))) -> c17(A__SIEVE(a__tail(a__sieve(mark(z0)))), MARK(tail(sieve(z0)))) MARK(sieve(tail(from(z0)))) -> c17(A__SIEVE(a__tail(a__from(mark(z0)))), MARK(tail(from(z0)))) MARK(sieve(tail(head(z0)))) -> c17(A__SIEVE(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(sieve(tail(tail(z0)))) -> c17(A__SIEVE(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(sieve(tail(if(z0, z1, z2)))) -> c17(A__SIEVE(a__tail(a__if(mark(z0), z1, z2))), MARK(tail(if(z0, z1, z2)))) MARK(sieve(tail(filter(z0, z1)))) -> c17(A__SIEVE(a__tail(a__filter(mark(z0), mark(z1)))), MARK(tail(filter(z0, z1)))) MARK(sieve(tail(s(z0)))) -> c17(A__SIEVE(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(sieve(tail(cons(z0, z1)))) -> c17(A__SIEVE(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(sieve(tail(divides(z0, z1)))) -> c17(A__SIEVE(a__tail(divides(mark(z0), mark(z1)))), MARK(tail(divides(z0, z1)))) MARK(sieve(tail(x0))) -> c17(MARK(tail(x0))) MARK(sieve(tail(0))) -> c17(A__SIEVE(a__tail(0))) MARK(sieve(tail(true))) -> c17(A__SIEVE(a__tail(true))) MARK(sieve(tail(false))) -> c17(A__SIEVE(a__tail(false))) MARK(sieve(if(primes, x1, x2))) -> c17(A__SIEVE(a__if(a__primes, x1, x2)), MARK(if(primes, x1, x2))) MARK(sieve(if(sieve(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__sieve(mark(z0)), x1, x2)), MARK(if(sieve(z0), x1, x2))) MARK(sieve(if(from(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__from(mark(z0)), x1, x2)), MARK(if(from(z0), x1, x2))) MARK(sieve(if(head(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__head(mark(z0)), x1, x2)), MARK(if(head(z0), x1, x2))) MARK(sieve(if(tail(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__tail(mark(z0)), x1, x2)), MARK(if(tail(z0), x1, x2))) MARK(sieve(if(if(z0, z1, z2), x1, x2))) -> c17(A__SIEVE(a__if(a__if(mark(z0), z1, z2), x1, x2)), MARK(if(if(z0, z1, z2), x1, x2))) MARK(sieve(if(filter(z0, z1), x1, x2))) -> c17(A__SIEVE(a__if(a__filter(mark(z0), mark(z1)), x1, x2)), MARK(if(filter(z0, z1), x1, x2))) MARK(sieve(if(s(z0), x1, x2))) -> c17(A__SIEVE(a__if(s(mark(z0)), x1, x2)), MARK(if(s(z0), x1, x2))) MARK(sieve(if(cons(z0, z1), x1, x2))) -> c17(A__SIEVE(a__if(cons(mark(z0), z1), x1, x2)), MARK(if(cons(z0, z1), x1, x2))) MARK(sieve(if(true, x1, x2))) -> 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c21(A__IF(a__head(a__from(mark(z0))), x1, x2), MARK(head(from(z0)))) MARK(if(head(head(z0)), x1, x2)) -> c21(A__IF(a__head(a__head(mark(z0))), x1, x2), MARK(head(head(z0)))) MARK(if(head(tail(z0)), x1, x2)) -> c21(A__IF(a__head(a__tail(mark(z0))), x1, x2), MARK(head(tail(z0)))) MARK(if(head(if(z0, z1, z2)), x1, x2)) -> c21(A__IF(a__head(a__if(mark(z0), z1, z2)), x1, x2), MARK(head(if(z0, z1, z2)))) MARK(if(head(filter(z0, z1)), x1, x2)) -> c21(A__IF(a__head(a__filter(mark(z0), mark(z1))), x1, x2), MARK(head(filter(z0, z1)))) MARK(if(head(s(z0)), x1, x2)) -> c21(A__IF(a__head(s(mark(z0))), x1, x2), MARK(head(s(z0)))) MARK(if(head(cons(z0, z1)), x1, x2)) -> c21(A__IF(a__head(cons(mark(z0), z1)), x1, x2), MARK(head(cons(z0, z1)))) MARK(if(head(divides(z0, z1)), x1, x2)) -> c21(A__IF(a__head(divides(mark(z0), mark(z1))), x1, x2), MARK(head(divides(z0, z1)))) MARK(if(head(x0), x1, x2)) -> c21(MARK(head(x0))) MARK(if(head(0), x1, x2)) -> c21(A__IF(a__head(0), x1, x2)) MARK(if(head(true), x1, x2)) -> c21(A__IF(a__head(true), x1, x2)) MARK(if(head(false), x1, x2)) -> c21(A__IF(a__head(false), x1, x2)) MARK(if(tail(primes), x1, x2)) -> c21(A__IF(a__tail(a__primes), x1, x2), MARK(tail(primes))) MARK(if(tail(sieve(z0)), x1, x2)) -> c21(A__IF(a__tail(a__sieve(mark(z0))), x1, x2), MARK(tail(sieve(z0)))) MARK(if(tail(from(z0)), x1, x2)) -> c21(A__IF(a__tail(a__from(mark(z0))), x1, x2), MARK(tail(from(z0)))) MARK(if(tail(head(z0)), x1, x2)) -> c21(A__IF(a__tail(a__head(mark(z0))), x1, x2), MARK(tail(head(z0)))) MARK(if(tail(tail(z0)), x1, x2)) -> c21(A__IF(a__tail(a__tail(mark(z0))), x1, x2), MARK(tail(tail(z0)))) MARK(if(tail(if(z0, z1, z2)), x1, x2)) -> c21(A__IF(a__tail(a__if(mark(z0), z1, z2)), x1, x2), MARK(tail(if(z0, z1, z2)))) MARK(if(tail(filter(z0, z1)), x1, x2)) -> c21(A__IF(a__tail(a__filter(mark(z0), mark(z1))), x1, x2), MARK(tail(filter(z0, z1)))) MARK(if(tail(s(z0)), x1, x2)) -> c21(A__IF(a__tail(s(mark(z0))), x1, x2), MARK(tail(s(z0)))) MARK(if(tail(cons(z0, z1)), x1, x2)) -> c21(A__IF(a__tail(cons(mark(z0), z1)), x1, x2), MARK(tail(cons(z0, z1)))) MARK(if(tail(divides(z0, z1)), x1, x2)) -> c21(A__IF(a__tail(divides(mark(z0), mark(z1))), x1, x2), MARK(tail(divides(z0, z1)))) MARK(if(tail(x0), x1, x2)) -> c21(MARK(tail(x0))) MARK(if(tail(0), x1, x2)) -> c21(A__IF(a__tail(0), x1, x2)) MARK(if(tail(true), x1, x2)) -> c21(A__IF(a__tail(true), x1, x2)) MARK(if(tail(false), x1, x2)) -> c21(A__IF(a__tail(false), x1, x2)) MARK(if(if(primes, x1, x2), x3, x4)) -> c21(A__IF(a__if(a__primes, x1, x2), x3, x4), MARK(if(primes, x1, x2))) MARK(if(if(sieve(z0), x1, x2), x3, x4)) -> c21(A__IF(a__if(a__sieve(mark(z0)), x1, x2), x3, x4), MARK(if(sieve(z0), x1, x2))) MARK(if(if(from(z0), x1, x2), x3, x4)) -> c21(A__IF(a__if(a__from(mark(z0)), x1, x2), x3, x4), MARK(if(from(z0), x1, x2))) MARK(if(if(head(z0), x1, x2), x3, x4)) -> c21(A__IF(a__if(a__head(mark(z0)), x1, x2), x3, x4), MARK(if(head(z0), x1, x2))) MARK(if(if(tail(z0), x1, x2), x3, x4)) -> c21(A__IF(a__if(a__tail(mark(z0)), x1, x2), x3, x4), MARK(if(tail(z0), x1, x2))) MARK(if(if(if(z0, z1, z2), x1, x2), x3, x4)) -> c21(A__IF(a__if(a__if(mark(z0), z1, z2), x1, x2), x3, x4), MARK(if(if(z0, z1, z2), x1, x2))) MARK(if(if(filter(z0, z1), x1, x2), x3, x4)) -> c21(A__IF(a__if(a__filter(mark(z0), mark(z1)), x1, x2), x3, x4), MARK(if(filter(z0, z1), x1, x2))) MARK(if(if(s(z0), x1, x2), x3, x4)) -> c21(A__IF(a__if(s(mark(z0)), x1, x2), x3, x4), MARK(if(s(z0), x1, x2))) MARK(if(if(cons(z0, z1), x1, x2), x3, x4)) -> c21(A__IF(a__if(cons(mark(z0), z1), x1, x2), x3, x4), MARK(if(cons(z0, z1), x1, x2))) MARK(if(if(true, x1, x2), x3, x4)) -> c21(A__IF(a__if(true, x1, x2), x3, x4), MARK(if(true, x1, x2))) MARK(if(if(false, x1, x2), x3, x4)) -> c21(A__IF(a__if(false, x1, x2), x3, x4), MARK(if(false, x1, x2))) MARK(if(if(divides(z0, z1), x1, x2), x3, x4)) -> c21(A__IF(a__if(divides(mark(z0), mark(z1)), x1, x2), x3, x4), MARK(if(divides(z0, z1), x1, x2))) MARK(if(if(x0, z1, z2), x3, x4)) -> c21(MARK(if(x0, z1, z2))) MARK(if(if(0, x1, x2), x3, x4)) -> c21(A__IF(a__if(0, x1, x2), x3, x4)) MARK(if(filter(x0, primes), x2, x3)) -> c21(A__IF(a__filter(mark(x0), a__primes), x2, x3), MARK(filter(x0, primes))) MARK(if(filter(x0, sieve(z0)), x2, x3)) -> c21(A__IF(a__filter(mark(x0), a__sieve(mark(z0))), x2, x3), MARK(filter(x0, sieve(z0)))) MARK(if(filter(x0, from(z0)), x2, x3)) -> c21(A__IF(a__filter(mark(x0), a__from(mark(z0))), x2, x3), MARK(filter(x0, from(z0)))) MARK(if(filter(x0, head(z0)), x2, x3)) -> c21(A__IF(a__filter(mark(x0), a__head(mark(z0))), x2, x3), MARK(filter(x0, head(z0)))) MARK(if(filter(x0, tail(z0)), x2, x3)) -> c21(A__IF(a__filter(mark(x0), a__tail(mark(z0))), x2, x3), MARK(filter(x0, tail(z0)))) MARK(if(filter(x0, if(z0, z1, z2)), x2, x3)) -> c21(A__IF(a__filter(mark(x0), a__if(mark(z0), z1, z2)), x2, x3), MARK(filter(x0, if(z0, z1, z2)))) MARK(if(filter(x0, filter(z0, z1)), x2, x3)) -> c21(A__IF(a__filter(mark(x0), a__filter(mark(z0), mark(z1))), x2, x3), MARK(filter(x0, filter(z0, z1)))) MARK(if(filter(x0, s(z0)), x2, x3)) -> c21(A__IF(a__filter(mark(x0), s(mark(z0))), x2, x3), MARK(filter(x0, s(z0)))) MARK(if(filter(x0, 0), x2, x3)) -> c21(A__IF(a__filter(mark(x0), 0), x2, x3), MARK(filter(x0, 0))) MARK(if(filter(x0, cons(z0, z1)), x2, x3)) -> c21(A__IF(a__filter(mark(x0), cons(mark(z0), z1)), x2, x3), MARK(filter(x0, cons(z0, z1)))) MARK(if(filter(x0, true), x2, x3)) -> c21(A__IF(a__filter(mark(x0), true), x2, x3), MARK(filter(x0, true))) MARK(if(filter(x0, false), x2, x3)) -> c21(A__IF(a__filter(mark(x0), false), x2, x3), MARK(filter(x0, false))) MARK(if(filter(x0, divides(z0, z1)), x2, x3)) -> c21(A__IF(a__filter(mark(x0), divides(mark(z0), mark(z1))), x2, x3), MARK(filter(x0, divides(z0, z1)))) MARK(if(filter(primes, x1), x2, x3)) -> c21(A__IF(a__filter(a__primes, mark(x1)), x2, x3), MARK(filter(primes, x1))) MARK(if(filter(sieve(z0), x1), x2, x3)) -> c21(A__IF(a__filter(a__sieve(mark(z0)), mark(x1)), x2, x3), MARK(filter(sieve(z0), x1))) MARK(if(filter(from(z0), x1), x2, x3)) -> c21(A__IF(a__filter(a__from(mark(z0)), mark(x1)), x2, x3), MARK(filter(from(z0), x1))) MARK(if(filter(head(z0), x1), x2, x3)) -> c21(A__IF(a__filter(a__head(mark(z0)), mark(x1)), x2, x3), MARK(filter(head(z0), x1))) MARK(if(filter(tail(z0), x1), x2, x3)) -> c21(A__IF(a__filter(a__tail(mark(z0)), mark(x1)), x2, x3), MARK(filter(tail(z0), x1))) MARK(if(filter(if(z0, z1, z2), x1), x2, x3)) -> c21(A__IF(a__filter(a__if(mark(z0), z1, z2), mark(x1)), x2, x3), MARK(filter(if(z0, z1, z2), x1))) MARK(if(filter(filter(z0, z1), x1), x2, x3)) -> c21(A__IF(a__filter(a__filter(mark(z0), mark(z1)), mark(x1)), x2, x3), MARK(filter(filter(z0, z1), x1))) MARK(if(filter(s(z0), x1), x2, x3)) -> c21(A__IF(a__filter(s(mark(z0)), mark(x1)), x2, x3), MARK(filter(s(z0), x1))) MARK(if(filter(0, x1), x2, x3)) -> c21(A__IF(a__filter(0, mark(x1)), x2, x3), MARK(filter(0, x1))) MARK(if(filter(cons(z0, z1), x1), x2, x3)) -> c21(A__IF(a__filter(cons(mark(z0), z1), mark(x1)), x2, x3), MARK(filter(cons(z0, z1), x1))) MARK(if(filter(true, x1), x2, x3)) -> c21(A__IF(a__filter(true, mark(x1)), x2, x3), MARK(filter(true, x1))) MARK(if(filter(false, x1), x2, x3)) -> c21(A__IF(a__filter(false, mark(x1)), x2, x3), MARK(filter(false, x1))) MARK(if(filter(divides(z0, z1), x1), x2, x3)) -> c21(A__IF(a__filter(divides(mark(z0), mark(z1)), mark(x1)), x2, x3), MARK(filter(divides(z0, z1), x1))) MARK(if(filter(x0, x1), x2, x3)) -> c21(MARK(filter(x0, x1))) MARK(filter(x0, primes)) -> c22(A__FILTER(mark(x0), a__sieve(a__from(s(s(0))))), MARK(x0)) MARK(filter(primes, primes)) -> c22(A__FILTER(a__primes, a__primes), MARK(primes)) MARK(filter(sieve(z0), primes)) -> c22(A__FILTER(a__sieve(mark(z0)), a__primes), MARK(sieve(z0))) MARK(filter(from(z0), primes)) -> c22(A__FILTER(a__from(mark(z0)), a__primes), MARK(from(z0))) MARK(filter(head(z0), primes)) -> c22(A__FILTER(a__head(mark(z0)), a__primes), MARK(head(z0))) MARK(filter(tail(z0), primes)) -> c22(A__FILTER(a__tail(mark(z0)), a__primes), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), primes)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), a__primes), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), primes)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), a__primes), MARK(filter(z0, z1))) MARK(filter(s(z0), primes)) -> c22(A__FILTER(s(mark(z0)), a__primes), MARK(s(z0))) MARK(filter(x0, primes)) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), primes)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), primes)) -> c22(MARK(divides(z0, z1))) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c18_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_1, c17_2, c17_1, c19_2, c19_1, c20_2, c20_1, c21_1, c22_2, c22_1, c23_2, c23_1, c_2, c21_2 ---------------------------------------- (217) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(filter(x0, sieve(z0))) -> c22(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(x0)) by MARK(filter(x0, sieve(x1))) -> c22(A__FILTER(mark(x0), sieve(mark(x1))), MARK(x0)) MARK(filter(x0, sieve(primes))) -> c22(A__FILTER(mark(x0), a__sieve(a__primes)), MARK(x0)) MARK(filter(x0, sieve(sieve(z0)))) -> c22(A__FILTER(mark(x0), a__sieve(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, sieve(from(z0)))) -> c22(A__FILTER(mark(x0), a__sieve(a__from(mark(z0)))), MARK(x0)) MARK(filter(x0, sieve(head(z0)))) -> c22(A__FILTER(mark(x0), a__sieve(a__head(mark(z0)))), MARK(x0)) MARK(filter(x0, sieve(tail(z0)))) -> c22(A__FILTER(mark(x0), a__sieve(a__tail(mark(z0)))), MARK(x0)) MARK(filter(x0, sieve(if(z0, z1, z2)))) -> c22(A__FILTER(mark(x0), a__sieve(a__if(mark(z0), z1, z2))), MARK(x0)) MARK(filter(x0, sieve(filter(z0, z1)))) -> c22(A__FILTER(mark(x0), a__sieve(a__filter(mark(z0), mark(z1)))), MARK(x0)) MARK(filter(x0, sieve(s(z0)))) -> c22(A__FILTER(mark(x0), a__sieve(s(mark(z0)))), MARK(x0)) MARK(filter(x0, sieve(0))) -> c22(A__FILTER(mark(x0), a__sieve(0)), MARK(x0)) MARK(filter(x0, sieve(cons(z0, z1)))) -> c22(A__FILTER(mark(x0), a__sieve(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, sieve(true))) -> c22(A__FILTER(mark(x0), a__sieve(true)), MARK(x0)) MARK(filter(x0, sieve(false))) -> c22(A__FILTER(mark(x0), a__sieve(false)), MARK(x0)) MARK(filter(x0, sieve(divides(z0, z1)))) -> c22(A__FILTER(mark(x0), a__sieve(divides(mark(z0), mark(z1)))), MARK(x0)) MARK(filter(primes, sieve(x1))) -> c22(A__FILTER(a__primes, a__sieve(mark(x1))), MARK(primes)) MARK(filter(sieve(z0), sieve(x1))) -> c22(A__FILTER(a__sieve(mark(z0)), a__sieve(mark(x1))), MARK(sieve(z0))) MARK(filter(from(z0), sieve(x1))) -> c22(A__FILTER(a__from(mark(z0)), a__sieve(mark(x1))), MARK(from(z0))) MARK(filter(head(z0), sieve(x1))) -> c22(A__FILTER(a__head(mark(z0)), a__sieve(mark(x1))), MARK(head(z0))) MARK(filter(tail(z0), sieve(x1))) -> c22(A__FILTER(a__tail(mark(z0)), a__sieve(mark(x1))), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), sieve(x1))) -> c22(A__FILTER(a__if(mark(z0), z1, z2), a__sieve(mark(x1))), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), sieve(x1))) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), a__sieve(mark(x1))), MARK(filter(z0, z1))) MARK(filter(s(z0), sieve(x1))) -> c22(A__FILTER(s(mark(z0)), a__sieve(mark(x1))), MARK(s(z0))) MARK(filter(0, sieve(x1))) -> c22(A__FILTER(0, a__sieve(mark(x1))), MARK(0)) MARK(filter(cons(z0, z1), sieve(x1))) -> c22(A__FILTER(cons(mark(z0), z1), a__sieve(mark(x1))), MARK(cons(z0, z1))) MARK(filter(true, sieve(x1))) -> c22(A__FILTER(true, a__sieve(mark(x1))), MARK(true)) MARK(filter(false, sieve(x1))) -> c22(A__FILTER(false, a__sieve(mark(x1))), MARK(false)) MARK(filter(divides(z0, z1), sieve(x1))) -> c22(A__FILTER(divides(mark(z0), mark(z1)), a__sieve(mark(x1))), MARK(divides(z0, z1))) ---------------------------------------- (218) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, 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MARK(if(if(tail(z0), x1, x2), x3, x4)) -> c21(A__IF(a__if(a__tail(mark(z0)), x1, x2), x3, x4), MARK(if(tail(z0), x1, x2))) MARK(if(if(if(z0, z1, z2), x1, x2), x3, x4)) -> c21(A__IF(a__if(a__if(mark(z0), z1, z2), x1, x2), x3, x4), MARK(if(if(z0, z1, z2), x1, x2))) MARK(if(if(filter(z0, z1), x1, x2), x3, x4)) -> c21(A__IF(a__if(a__filter(mark(z0), mark(z1)), x1, x2), x3, x4), MARK(if(filter(z0, z1), x1, x2))) MARK(if(if(s(z0), x1, x2), x3, x4)) -> c21(A__IF(a__if(s(mark(z0)), x1, x2), x3, x4), MARK(if(s(z0), x1, x2))) MARK(if(if(cons(z0, z1), x1, x2), x3, x4)) -> c21(A__IF(a__if(cons(mark(z0), z1), x1, x2), x3, x4), MARK(if(cons(z0, z1), x1, x2))) MARK(if(if(true, x1, x2), x3, x4)) -> c21(A__IF(a__if(true, x1, x2), x3, x4), MARK(if(true, x1, x2))) MARK(if(if(false, x1, x2), x3, x4)) -> c21(A__IF(a__if(false, x1, x2), x3, x4), MARK(if(false, x1, x2))) MARK(if(if(divides(z0, z1), x1, x2), x3, x4)) -> c21(A__IF(a__if(divides(mark(z0), mark(z1)), x1, x2), x3, x4), MARK(if(divides(z0, z1), x1, x2))) MARK(if(if(x0, z1, z2), x3, x4)) -> c21(MARK(if(x0, z1, z2))) MARK(if(if(0, x1, x2), x3, x4)) -> c21(A__IF(a__if(0, x1, x2), x3, x4)) MARK(if(filter(x0, primes), x2, x3)) -> c21(A__IF(a__filter(mark(x0), a__primes), x2, x3), MARK(filter(x0, primes))) MARK(if(filter(x0, sieve(z0)), x2, x3)) -> c21(A__IF(a__filter(mark(x0), a__sieve(mark(z0))), x2, x3), MARK(filter(x0, sieve(z0)))) MARK(if(filter(x0, from(z0)), x2, x3)) -> c21(A__IF(a__filter(mark(x0), a__from(mark(z0))), x2, x3), MARK(filter(x0, from(z0)))) MARK(if(filter(x0, head(z0)), x2, x3)) -> c21(A__IF(a__filter(mark(x0), a__head(mark(z0))), x2, x3), MARK(filter(x0, head(z0)))) MARK(if(filter(x0, tail(z0)), x2, x3)) -> c21(A__IF(a__filter(mark(x0), a__tail(mark(z0))), x2, x3), MARK(filter(x0, tail(z0)))) MARK(if(filter(x0, if(z0, z1, z2)), x2, x3)) -> c21(A__IF(a__filter(mark(x0), a__if(mark(z0), z1, z2)), x2, x3), MARK(filter(x0, if(z0, z1, z2)))) MARK(if(filter(x0, filter(z0, z1)), x2, x3)) -> c21(A__IF(a__filter(mark(x0), a__filter(mark(z0), mark(z1))), x2, x3), MARK(filter(x0, filter(z0, z1)))) MARK(if(filter(x0, s(z0)), x2, x3)) -> c21(A__IF(a__filter(mark(x0), s(mark(z0))), x2, x3), MARK(filter(x0, s(z0)))) MARK(if(filter(x0, 0), x2, x3)) -> c21(A__IF(a__filter(mark(x0), 0), x2, x3), MARK(filter(x0, 0))) MARK(if(filter(x0, cons(z0, z1)), x2, x3)) -> c21(A__IF(a__filter(mark(x0), cons(mark(z0), z1)), x2, x3), MARK(filter(x0, cons(z0, z1)))) MARK(if(filter(x0, true), x2, x3)) -> c21(A__IF(a__filter(mark(x0), true), x2, x3), MARK(filter(x0, true))) MARK(if(filter(x0, false), x2, x3)) -> c21(A__IF(a__filter(mark(x0), false), x2, x3), MARK(filter(x0, false))) MARK(if(filter(x0, divides(z0, z1)), x2, x3)) -> c21(A__IF(a__filter(mark(x0), divides(mark(z0), mark(z1))), x2, x3), MARK(filter(x0, divides(z0, z1)))) MARK(if(filter(primes, x1), x2, x3)) -> c21(A__IF(a__filter(a__primes, mark(x1)), x2, x3), MARK(filter(primes, x1))) MARK(if(filter(sieve(z0), x1), x2, x3)) -> c21(A__IF(a__filter(a__sieve(mark(z0)), mark(x1)), x2, x3), MARK(filter(sieve(z0), x1))) MARK(if(filter(from(z0), x1), x2, x3)) -> c21(A__IF(a__filter(a__from(mark(z0)), mark(x1)), x2, x3), MARK(filter(from(z0), x1))) MARK(if(filter(head(z0), x1), x2, x3)) -> c21(A__IF(a__filter(a__head(mark(z0)), mark(x1)), x2, x3), MARK(filter(head(z0), x1))) MARK(if(filter(tail(z0), x1), x2, x3)) -> c21(A__IF(a__filter(a__tail(mark(z0)), mark(x1)), x2, x3), MARK(filter(tail(z0), x1))) MARK(if(filter(if(z0, z1, z2), x1), x2, x3)) -> c21(A__IF(a__filter(a__if(mark(z0), z1, z2), mark(x1)), x2, x3), MARK(filter(if(z0, z1, z2), x1))) MARK(if(filter(filter(z0, z1), x1), x2, x3)) -> c21(A__IF(a__filter(a__filter(mark(z0), mark(z1)), mark(x1)), x2, x3), MARK(filter(filter(z0, z1), x1))) MARK(if(filter(s(z0), x1), x2, x3)) -> c21(A__IF(a__filter(s(mark(z0)), mark(x1)), x2, x3), MARK(filter(s(z0), x1))) MARK(if(filter(0, x1), x2, x3)) -> c21(A__IF(a__filter(0, mark(x1)), x2, x3), MARK(filter(0, x1))) MARK(if(filter(cons(z0, z1), x1), x2, x3)) -> c21(A__IF(a__filter(cons(mark(z0), z1), mark(x1)), x2, x3), MARK(filter(cons(z0, z1), x1))) MARK(if(filter(true, x1), x2, x3)) -> c21(A__IF(a__filter(true, mark(x1)), x2, x3), MARK(filter(true, x1))) MARK(if(filter(false, x1), x2, x3)) -> c21(A__IF(a__filter(false, mark(x1)), x2, x3), MARK(filter(false, x1))) MARK(if(filter(divides(z0, z1), x1), x2, x3)) -> c21(A__IF(a__filter(divides(mark(z0), mark(z1)), mark(x1)), x2, x3), MARK(filter(divides(z0, z1), x1))) MARK(if(filter(x0, x1), x2, x3)) -> c21(MARK(filter(x0, x1))) MARK(filter(x0, primes)) -> c22(A__FILTER(mark(x0), a__sieve(a__from(s(s(0))))), MARK(x0)) MARK(filter(primes, primes)) -> c22(A__FILTER(a__primes, a__primes), MARK(primes)) MARK(filter(sieve(z0), primes)) -> c22(A__FILTER(a__sieve(mark(z0)), a__primes), MARK(sieve(z0))) MARK(filter(from(z0), primes)) -> c22(A__FILTER(a__from(mark(z0)), a__primes), MARK(from(z0))) MARK(filter(head(z0), primes)) -> c22(A__FILTER(a__head(mark(z0)), a__primes), MARK(head(z0))) MARK(filter(tail(z0), primes)) -> c22(A__FILTER(a__tail(mark(z0)), a__primes), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), primes)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), a__primes), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), primes)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), a__primes), MARK(filter(z0, z1))) MARK(filter(s(z0), primes)) -> c22(A__FILTER(s(mark(z0)), a__primes), MARK(s(z0))) MARK(filter(x0, primes)) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), primes)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), primes)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, sieve(x1))) -> c22(A__FILTER(mark(x0), sieve(mark(x1))), MARK(x0)) MARK(filter(x0, sieve(primes))) -> c22(A__FILTER(mark(x0), a__sieve(a__primes)), MARK(x0)) MARK(filter(x0, sieve(sieve(z0)))) -> c22(A__FILTER(mark(x0), a__sieve(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, sieve(from(z0)))) -> c22(A__FILTER(mark(x0), a__sieve(a__from(mark(z0)))), MARK(x0)) MARK(filter(x0, sieve(head(z0)))) -> c22(A__FILTER(mark(x0), a__sieve(a__head(mark(z0)))), MARK(x0)) MARK(filter(x0, sieve(tail(z0)))) -> c22(A__FILTER(mark(x0), a__sieve(a__tail(mark(z0)))), MARK(x0)) MARK(filter(x0, sieve(if(z0, z1, z2)))) -> c22(A__FILTER(mark(x0), a__sieve(a__if(mark(z0), z1, z2))), MARK(x0)) MARK(filter(x0, sieve(filter(z0, z1)))) -> c22(A__FILTER(mark(x0), a__sieve(a__filter(mark(z0), mark(z1)))), MARK(x0)) MARK(filter(x0, sieve(s(z0)))) -> c22(A__FILTER(mark(x0), a__sieve(s(mark(z0)))), MARK(x0)) MARK(filter(x0, sieve(0))) -> c22(A__FILTER(mark(x0), a__sieve(0)), MARK(x0)) MARK(filter(x0, sieve(cons(z0, z1)))) -> c22(A__FILTER(mark(x0), a__sieve(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, sieve(true))) -> c22(A__FILTER(mark(x0), a__sieve(true)), MARK(x0)) MARK(filter(x0, sieve(false))) -> c22(A__FILTER(mark(x0), a__sieve(false)), MARK(x0)) MARK(filter(x0, sieve(divides(z0, z1)))) -> c22(A__FILTER(mark(x0), a__sieve(divides(mark(z0), mark(z1)))), MARK(x0)) MARK(filter(primes, sieve(x1))) -> c22(A__FILTER(a__primes, a__sieve(mark(x1))), MARK(primes)) MARK(filter(sieve(z0), sieve(x1))) -> c22(A__FILTER(a__sieve(mark(z0)), a__sieve(mark(x1))), MARK(sieve(z0))) MARK(filter(from(z0), sieve(x1))) -> c22(A__FILTER(a__from(mark(z0)), a__sieve(mark(x1))), MARK(from(z0))) MARK(filter(head(z0), sieve(x1))) -> c22(A__FILTER(a__head(mark(z0)), a__sieve(mark(x1))), MARK(head(z0))) MARK(filter(tail(z0), sieve(x1))) -> c22(A__FILTER(a__tail(mark(z0)), a__sieve(mark(x1))), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), sieve(x1))) -> c22(A__FILTER(a__if(mark(z0), z1, z2), a__sieve(mark(x1))), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), sieve(x1))) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), a__sieve(mark(x1))), MARK(filter(z0, z1))) MARK(filter(s(z0), sieve(x1))) -> c22(A__FILTER(s(mark(z0)), a__sieve(mark(x1))), MARK(s(z0))) MARK(filter(0, sieve(x1))) -> c22(A__FILTER(0, a__sieve(mark(x1))), MARK(0)) MARK(filter(cons(z0, z1), sieve(x1))) -> c22(A__FILTER(cons(mark(z0), z1), a__sieve(mark(x1))), MARK(cons(z0, z1))) MARK(filter(true, sieve(x1))) -> c22(A__FILTER(true, a__sieve(mark(x1))), MARK(true)) MARK(filter(false, sieve(x1))) -> c22(A__FILTER(false, a__sieve(mark(x1))), MARK(false)) MARK(filter(divides(z0, z1), sieve(x1))) -> c22(A__FILTER(divides(mark(z0), mark(z1)), a__sieve(mark(x1))), MARK(divides(z0, z1))) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c18_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_1, c17_2, c17_1, c19_2, c19_1, c20_2, c20_1, c21_1, c22_2, c22_1, c23_2, c23_1, c_2, c21_2 ---------------------------------------- (219) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing nodes: MARK(filter(0, sieve(x1))) -> c22(A__FILTER(0, a__sieve(mark(x1))), MARK(0)) MARK(filter(true, sieve(x1))) -> c22(A__FILTER(true, a__sieve(mark(x1))), MARK(true)) MARK(filter(false, sieve(x1))) -> c22(A__FILTER(false, a__sieve(mark(x1))), MARK(false)) ---------------------------------------- (220) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, from(z0))) -> c22(A__FILTER(mark(x0), a__from(mark(z0))), MARK(x0)) MARK(filter(x0, head(z0))) -> c22(A__FILTER(mark(x0), a__head(mark(z0))), MARK(x0)) MARK(filter(x0, tail(z0))) 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c21(A__IF(a__filter(mark(x0), a__filter(mark(z0), mark(z1))), x2, x3), MARK(filter(x0, filter(z0, z1)))) MARK(if(filter(x0, s(z0)), x2, x3)) -> c21(A__IF(a__filter(mark(x0), s(mark(z0))), x2, x3), MARK(filter(x0, s(z0)))) MARK(if(filter(x0, 0), x2, x3)) -> c21(A__IF(a__filter(mark(x0), 0), x2, x3), MARK(filter(x0, 0))) MARK(if(filter(x0, cons(z0, z1)), x2, x3)) -> c21(A__IF(a__filter(mark(x0), cons(mark(z0), z1)), x2, x3), MARK(filter(x0, cons(z0, z1)))) MARK(if(filter(x0, true), x2, x3)) -> c21(A__IF(a__filter(mark(x0), true), x2, x3), MARK(filter(x0, true))) MARK(if(filter(x0, false), x2, x3)) -> c21(A__IF(a__filter(mark(x0), false), x2, x3), MARK(filter(x0, false))) MARK(if(filter(x0, divides(z0, z1)), x2, x3)) -> c21(A__IF(a__filter(mark(x0), divides(mark(z0), mark(z1))), x2, x3), MARK(filter(x0, divides(z0, z1)))) MARK(if(filter(primes, x1), x2, x3)) -> c21(A__IF(a__filter(a__primes, mark(x1)), x2, x3), MARK(filter(primes, x1))) MARK(if(filter(sieve(z0), x1), x2, x3)) -> c21(A__IF(a__filter(a__sieve(mark(z0)), mark(x1)), x2, x3), MARK(filter(sieve(z0), x1))) MARK(if(filter(from(z0), x1), x2, x3)) -> c21(A__IF(a__filter(a__from(mark(z0)), mark(x1)), x2, x3), MARK(filter(from(z0), x1))) MARK(if(filter(head(z0), x1), x2, x3)) -> c21(A__IF(a__filter(a__head(mark(z0)), mark(x1)), x2, x3), MARK(filter(head(z0), x1))) MARK(if(filter(tail(z0), x1), x2, x3)) -> c21(A__IF(a__filter(a__tail(mark(z0)), mark(x1)), x2, x3), MARK(filter(tail(z0), x1))) MARK(if(filter(if(z0, z1, z2), x1), x2, x3)) -> c21(A__IF(a__filter(a__if(mark(z0), z1, z2), mark(x1)), x2, x3), MARK(filter(if(z0, z1, z2), x1))) MARK(if(filter(filter(z0, z1), x1), x2, x3)) -> c21(A__IF(a__filter(a__filter(mark(z0), mark(z1)), mark(x1)), x2, x3), MARK(filter(filter(z0, z1), x1))) MARK(if(filter(s(z0), x1), x2, x3)) -> c21(A__IF(a__filter(s(mark(z0)), mark(x1)), x2, x3), MARK(filter(s(z0), x1))) MARK(if(filter(0, x1), x2, x3)) -> c21(A__IF(a__filter(0, mark(x1)), x2, x3), MARK(filter(0, x1))) MARK(if(filter(cons(z0, z1), x1), x2, x3)) -> c21(A__IF(a__filter(cons(mark(z0), z1), mark(x1)), x2, x3), MARK(filter(cons(z0, z1), x1))) MARK(if(filter(true, x1), x2, x3)) -> c21(A__IF(a__filter(true, mark(x1)), x2, x3), MARK(filter(true, x1))) MARK(if(filter(false, x1), x2, x3)) -> c21(A__IF(a__filter(false, mark(x1)), x2, x3), MARK(filter(false, x1))) MARK(if(filter(divides(z0, z1), x1), x2, x3)) -> c21(A__IF(a__filter(divides(mark(z0), mark(z1)), mark(x1)), x2, x3), MARK(filter(divides(z0, z1), x1))) MARK(if(filter(x0, x1), x2, x3)) -> c21(MARK(filter(x0, x1))) MARK(filter(x0, primes)) -> c22(A__FILTER(mark(x0), a__sieve(a__from(s(s(0))))), MARK(x0)) MARK(filter(primes, primes)) -> c22(A__FILTER(a__primes, a__primes), MARK(primes)) MARK(filter(sieve(z0), primes)) -> c22(A__FILTER(a__sieve(mark(z0)), a__primes), MARK(sieve(z0))) MARK(filter(from(z0), primes)) -> c22(A__FILTER(a__from(mark(z0)), a__primes), MARK(from(z0))) MARK(filter(head(z0), primes)) -> c22(A__FILTER(a__head(mark(z0)), a__primes), MARK(head(z0))) MARK(filter(tail(z0), primes)) -> c22(A__FILTER(a__tail(mark(z0)), a__primes), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), primes)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), a__primes), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), primes)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), a__primes), MARK(filter(z0, z1))) MARK(filter(s(z0), primes)) -> c22(A__FILTER(s(mark(z0)), a__primes), MARK(s(z0))) MARK(filter(x0, primes)) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), primes)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), primes)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, sieve(x1))) -> c22(A__FILTER(mark(x0), sieve(mark(x1))), MARK(x0)) MARK(filter(x0, sieve(primes))) -> c22(A__FILTER(mark(x0), a__sieve(a__primes)), MARK(x0)) MARK(filter(x0, sieve(sieve(z0)))) -> c22(A__FILTER(mark(x0), a__sieve(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, sieve(from(z0)))) -> c22(A__FILTER(mark(x0), a__sieve(a__from(mark(z0)))), MARK(x0)) MARK(filter(x0, sieve(head(z0)))) -> c22(A__FILTER(mark(x0), a__sieve(a__head(mark(z0)))), MARK(x0)) MARK(filter(x0, sieve(tail(z0)))) -> c22(A__FILTER(mark(x0), a__sieve(a__tail(mark(z0)))), MARK(x0)) MARK(filter(x0, sieve(if(z0, z1, z2)))) -> c22(A__FILTER(mark(x0), a__sieve(a__if(mark(z0), z1, z2))), MARK(x0)) MARK(filter(x0, sieve(filter(z0, z1)))) -> c22(A__FILTER(mark(x0), a__sieve(a__filter(mark(z0), mark(z1)))), MARK(x0)) MARK(filter(x0, sieve(s(z0)))) -> c22(A__FILTER(mark(x0), a__sieve(s(mark(z0)))), MARK(x0)) MARK(filter(x0, sieve(0))) -> c22(A__FILTER(mark(x0), a__sieve(0)), MARK(x0)) MARK(filter(x0, sieve(cons(z0, z1)))) -> c22(A__FILTER(mark(x0), a__sieve(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, sieve(true))) -> c22(A__FILTER(mark(x0), a__sieve(true)), MARK(x0)) MARK(filter(x0, sieve(false))) -> c22(A__FILTER(mark(x0), a__sieve(false)), MARK(x0)) MARK(filter(x0, sieve(divides(z0, z1)))) -> c22(A__FILTER(mark(x0), a__sieve(divides(mark(z0), mark(z1)))), MARK(x0)) MARK(filter(primes, sieve(x1))) -> c22(A__FILTER(a__primes, a__sieve(mark(x1))), MARK(primes)) MARK(filter(sieve(z0), sieve(x1))) -> c22(A__FILTER(a__sieve(mark(z0)), a__sieve(mark(x1))), MARK(sieve(z0))) MARK(filter(from(z0), sieve(x1))) -> c22(A__FILTER(a__from(mark(z0)), a__sieve(mark(x1))), MARK(from(z0))) MARK(filter(head(z0), sieve(x1))) -> c22(A__FILTER(a__head(mark(z0)), a__sieve(mark(x1))), MARK(head(z0))) MARK(filter(tail(z0), sieve(x1))) -> c22(A__FILTER(a__tail(mark(z0)), a__sieve(mark(x1))), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), sieve(x1))) -> c22(A__FILTER(a__if(mark(z0), z1, z2), a__sieve(mark(x1))), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), sieve(x1))) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), a__sieve(mark(x1))), MARK(filter(z0, z1))) MARK(filter(s(z0), sieve(x1))) -> c22(A__FILTER(s(mark(z0)), a__sieve(mark(x1))), MARK(s(z0))) MARK(filter(cons(z0, z1), sieve(x1))) -> c22(A__FILTER(cons(mark(z0), z1), a__sieve(mark(x1))), MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), sieve(x1))) -> c22(A__FILTER(divides(mark(z0), mark(z1)), a__sieve(mark(x1))), MARK(divides(z0, z1))) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c18_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_1, c17_2, c17_1, c19_2, c19_1, c20_2, c20_1, c21_1, c22_2, c22_1, c23_2, c23_1, c_2, c21_2 ---------------------------------------- (221) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (222) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, from(z0))) -> c22(A__FILTER(mark(x0), a__from(mark(z0))), MARK(x0)) MARK(filter(x0, head(z0))) -> c22(A__FILTER(mark(x0), a__head(mark(z0))), MARK(x0)) MARK(filter(x0, tail(z0))) -> c22(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(x0)) MARK(filter(x0, if(z0, z1, z2))) -> c22(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0)) MARK(filter(x0, filter(z0, z1))) -> c22(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(x0)) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(x0, s(z0))) -> c22(MARK(x0)) MARK(filter(x0, 0)) -> c22(MARK(x0)) MARK(filter(x0, true)) -> c22(MARK(x0)) MARK(filter(x0, false)) -> c22(MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), x1)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c23(A__FILTER(mark(x0), a__primes), MARK(primes)) MARK(filter(x0, sieve(z0))) -> c23(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, from(z0))) -> c23(A__FILTER(mark(x0), a__from(mark(z0))), MARK(from(z0))) MARK(filter(x0, head(z0))) -> c23(A__FILTER(mark(x0), a__head(mark(z0))), MARK(head(z0))) MARK(filter(x0, tail(z0))) -> c23(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(tail(z0))) MARK(filter(x0, if(z0, z1, z2))) -> c23(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(filter(x0, filter(z0, z1))) -> c23(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(filter(x0, cons(z0, z1))) -> c23(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(primes, x1)) -> c23(A__FILTER(a__primes, mark(x1)), MARK(x1)) MARK(filter(sieve(z0), x1)) -> c23(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(from(z0), x1)) -> c23(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(head(z0), x1)) -> c23(A__FILTER(a__head(mark(z0)), 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x2, x3), MARK(filter(x0, from(z0)))) MARK(if(filter(x0, head(z0)), x2, x3)) -> c21(A__IF(a__filter(mark(x0), a__head(mark(z0))), x2, x3), MARK(filter(x0, head(z0)))) MARK(if(filter(x0, tail(z0)), x2, x3)) -> c21(A__IF(a__filter(mark(x0), a__tail(mark(z0))), x2, x3), MARK(filter(x0, tail(z0)))) MARK(if(filter(x0, if(z0, z1, z2)), x2, x3)) -> c21(A__IF(a__filter(mark(x0), a__if(mark(z0), z1, z2)), x2, x3), MARK(filter(x0, if(z0, z1, z2)))) MARK(if(filter(x0, filter(z0, z1)), x2, x3)) -> c21(A__IF(a__filter(mark(x0), a__filter(mark(z0), mark(z1))), x2, x3), MARK(filter(x0, filter(z0, z1)))) MARK(if(filter(x0, s(z0)), x2, x3)) -> c21(A__IF(a__filter(mark(x0), s(mark(z0))), x2, x3), MARK(filter(x0, s(z0)))) MARK(if(filter(x0, 0), x2, x3)) -> c21(A__IF(a__filter(mark(x0), 0), x2, x3), MARK(filter(x0, 0))) MARK(if(filter(x0, cons(z0, z1)), x2, x3)) -> c21(A__IF(a__filter(mark(x0), cons(mark(z0), z1)), x2, x3), MARK(filter(x0, cons(z0, z1)))) MARK(if(filter(x0, true), x2, x3)) -> c21(A__IF(a__filter(mark(x0), true), x2, x3), MARK(filter(x0, true))) MARK(if(filter(x0, false), x2, x3)) -> c21(A__IF(a__filter(mark(x0), false), x2, x3), MARK(filter(x0, false))) MARK(if(filter(x0, divides(z0, z1)), x2, x3)) -> c21(A__IF(a__filter(mark(x0), divides(mark(z0), mark(z1))), x2, x3), MARK(filter(x0, divides(z0, z1)))) MARK(if(filter(primes, x1), x2, x3)) -> c21(A__IF(a__filter(a__primes, mark(x1)), x2, x3), MARK(filter(primes, x1))) MARK(if(filter(sieve(z0), x1), x2, x3)) -> c21(A__IF(a__filter(a__sieve(mark(z0)), mark(x1)), x2, x3), MARK(filter(sieve(z0), x1))) MARK(if(filter(from(z0), x1), x2, x3)) -> c21(A__IF(a__filter(a__from(mark(z0)), mark(x1)), x2, x3), MARK(filter(from(z0), x1))) MARK(if(filter(head(z0), x1), x2, x3)) -> c21(A__IF(a__filter(a__head(mark(z0)), mark(x1)), x2, x3), MARK(filter(head(z0), x1))) MARK(if(filter(tail(z0), x1), x2, x3)) -> c21(A__IF(a__filter(a__tail(mark(z0)), mark(x1)), x2, x3), MARK(filter(tail(z0), x1))) MARK(if(filter(if(z0, z1, z2), x1), x2, x3)) -> c21(A__IF(a__filter(a__if(mark(z0), z1, z2), mark(x1)), x2, x3), MARK(filter(if(z0, z1, z2), x1))) MARK(if(filter(filter(z0, z1), x1), x2, x3)) -> c21(A__IF(a__filter(a__filter(mark(z0), mark(z1)), mark(x1)), x2, x3), MARK(filter(filter(z0, z1), x1))) MARK(if(filter(s(z0), x1), x2, x3)) -> c21(A__IF(a__filter(s(mark(z0)), mark(x1)), x2, x3), MARK(filter(s(z0), x1))) MARK(if(filter(0, x1), x2, x3)) -> c21(A__IF(a__filter(0, mark(x1)), x2, x3), MARK(filter(0, x1))) MARK(if(filter(cons(z0, z1), x1), x2, x3)) -> c21(A__IF(a__filter(cons(mark(z0), z1), mark(x1)), x2, x3), MARK(filter(cons(z0, z1), x1))) MARK(if(filter(true, x1), x2, x3)) -> c21(A__IF(a__filter(true, mark(x1)), x2, x3), MARK(filter(true, x1))) MARK(if(filter(false, x1), x2, x3)) -> c21(A__IF(a__filter(false, mark(x1)), x2, x3), MARK(filter(false, x1))) MARK(if(filter(divides(z0, z1), x1), x2, x3)) -> c21(A__IF(a__filter(divides(mark(z0), mark(z1)), mark(x1)), x2, x3), MARK(filter(divides(z0, z1), x1))) MARK(if(filter(x0, x1), x2, x3)) -> c21(MARK(filter(x0, x1))) MARK(filter(x0, primes)) -> c22(A__FILTER(mark(x0), a__sieve(a__from(s(s(0))))), MARK(x0)) MARK(filter(primes, primes)) -> c22(A__FILTER(a__primes, a__primes), MARK(primes)) MARK(filter(sieve(z0), primes)) -> c22(A__FILTER(a__sieve(mark(z0)), a__primes), MARK(sieve(z0))) MARK(filter(from(z0), primes)) -> c22(A__FILTER(a__from(mark(z0)), a__primes), MARK(from(z0))) MARK(filter(head(z0), primes)) -> c22(A__FILTER(a__head(mark(z0)), a__primes), MARK(head(z0))) MARK(filter(tail(z0), primes)) -> c22(A__FILTER(a__tail(mark(z0)), a__primes), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), primes)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), a__primes), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), primes)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), a__primes), MARK(filter(z0, z1))) MARK(filter(s(z0), primes)) -> c22(A__FILTER(s(mark(z0)), a__primes), MARK(s(z0))) MARK(filter(x0, primes)) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), primes)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), primes)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, sieve(primes))) -> c22(A__FILTER(mark(x0), a__sieve(a__primes)), MARK(x0)) MARK(filter(x0, sieve(sieve(z0)))) -> c22(A__FILTER(mark(x0), a__sieve(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, sieve(from(z0)))) -> c22(A__FILTER(mark(x0), a__sieve(a__from(mark(z0)))), MARK(x0)) MARK(filter(x0, sieve(head(z0)))) -> c22(A__FILTER(mark(x0), a__sieve(a__head(mark(z0)))), MARK(x0)) MARK(filter(x0, sieve(tail(z0)))) -> c22(A__FILTER(mark(x0), a__sieve(a__tail(mark(z0)))), MARK(x0)) MARK(filter(x0, sieve(if(z0, z1, z2)))) -> c22(A__FILTER(mark(x0), a__sieve(a__if(mark(z0), z1, z2))), MARK(x0)) MARK(filter(x0, sieve(filter(z0, z1)))) -> c22(A__FILTER(mark(x0), a__sieve(a__filter(mark(z0), mark(z1)))), MARK(x0)) MARK(filter(x0, sieve(s(z0)))) -> c22(A__FILTER(mark(x0), a__sieve(s(mark(z0)))), MARK(x0)) MARK(filter(x0, sieve(0))) -> c22(A__FILTER(mark(x0), a__sieve(0)), MARK(x0)) MARK(filter(x0, sieve(cons(z0, z1)))) -> c22(A__FILTER(mark(x0), a__sieve(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, sieve(true))) -> c22(A__FILTER(mark(x0), a__sieve(true)), MARK(x0)) MARK(filter(x0, sieve(false))) -> c22(A__FILTER(mark(x0), a__sieve(false)), MARK(x0)) MARK(filter(x0, sieve(divides(z0, z1)))) -> c22(A__FILTER(mark(x0), a__sieve(divides(mark(z0), mark(z1)))), MARK(x0)) MARK(filter(primes, sieve(x1))) -> c22(A__FILTER(a__primes, a__sieve(mark(x1))), MARK(primes)) MARK(filter(sieve(z0), sieve(x1))) -> c22(A__FILTER(a__sieve(mark(z0)), a__sieve(mark(x1))), MARK(sieve(z0))) MARK(filter(from(z0), sieve(x1))) -> c22(A__FILTER(a__from(mark(z0)), a__sieve(mark(x1))), MARK(from(z0))) MARK(filter(head(z0), sieve(x1))) -> c22(A__FILTER(a__head(mark(z0)), a__sieve(mark(x1))), MARK(head(z0))) MARK(filter(tail(z0), sieve(x1))) -> c22(A__FILTER(a__tail(mark(z0)), a__sieve(mark(x1))), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), sieve(x1))) -> c22(A__FILTER(a__if(mark(z0), z1, z2), a__sieve(mark(x1))), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), sieve(x1))) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), a__sieve(mark(x1))), MARK(filter(z0, z1))) MARK(filter(s(z0), sieve(x1))) -> c22(A__FILTER(s(mark(z0)), a__sieve(mark(x1))), MARK(s(z0))) MARK(filter(x0, sieve(x1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), sieve(x1))) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), sieve(x1))) -> c22(MARK(divides(z0, z1))) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c18_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_1, c17_2, c17_1, c19_2, c19_1, c20_2, c20_1, c21_1, c22_2, c22_1, c23_2, c23_1, c_2, c21_2 ---------------------------------------- (223) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(filter(x0, from(z0))) -> c22(A__FILTER(mark(x0), a__from(mark(z0))), MARK(x0)) by MARK(filter(x0, from(x1))) -> c22(A__FILTER(mark(x0), cons(mark(mark(x1)), from(s(mark(x1))))), MARK(x0)) MARK(filter(x0, from(x1))) -> c22(A__FILTER(mark(x0), from(mark(x1))), MARK(x0)) MARK(filter(x0, from(primes))) -> c22(A__FILTER(mark(x0), a__from(a__primes)), MARK(x0)) MARK(filter(x0, from(sieve(z0)))) -> c22(A__FILTER(mark(x0), a__from(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, from(from(z0)))) -> c22(A__FILTER(mark(x0), a__from(a__from(mark(z0)))), MARK(x0)) MARK(filter(x0, from(head(z0)))) -> c22(A__FILTER(mark(x0), a__from(a__head(mark(z0)))), MARK(x0)) MARK(filter(x0, from(tail(z0)))) -> c22(A__FILTER(mark(x0), a__from(a__tail(mark(z0)))), MARK(x0)) MARK(filter(x0, from(if(z0, z1, z2)))) -> c22(A__FILTER(mark(x0), a__from(a__if(mark(z0), z1, z2))), MARK(x0)) MARK(filter(x0, from(filter(z0, z1)))) -> c22(A__FILTER(mark(x0), a__from(a__filter(mark(z0), mark(z1)))), MARK(x0)) MARK(filter(x0, from(s(z0)))) -> c22(A__FILTER(mark(x0), a__from(s(mark(z0)))), MARK(x0)) MARK(filter(x0, from(0))) -> c22(A__FILTER(mark(x0), a__from(0)), MARK(x0)) MARK(filter(x0, from(cons(z0, z1)))) -> c22(A__FILTER(mark(x0), a__from(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, from(true))) -> c22(A__FILTER(mark(x0), a__from(true)), MARK(x0)) MARK(filter(x0, from(false))) -> c22(A__FILTER(mark(x0), a__from(false)), MARK(x0)) MARK(filter(x0, from(divides(z0, z1)))) -> c22(A__FILTER(mark(x0), a__from(divides(mark(z0), mark(z1)))), MARK(x0)) MARK(filter(primes, from(x1))) -> c22(A__FILTER(a__primes, a__from(mark(x1))), MARK(primes)) MARK(filter(sieve(z0), from(x1))) -> c22(A__FILTER(a__sieve(mark(z0)), a__from(mark(x1))), MARK(sieve(z0))) MARK(filter(from(z0), from(x1))) -> c22(A__FILTER(a__from(mark(z0)), a__from(mark(x1))), MARK(from(z0))) MARK(filter(head(z0), from(x1))) -> c22(A__FILTER(a__head(mark(z0)), a__from(mark(x1))), MARK(head(z0))) MARK(filter(tail(z0), from(x1))) -> c22(A__FILTER(a__tail(mark(z0)), a__from(mark(x1))), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), from(x1))) -> c22(A__FILTER(a__if(mark(z0), z1, z2), a__from(mark(x1))), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), from(x1))) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), a__from(mark(x1))), MARK(filter(z0, z1))) MARK(filter(s(z0), from(x1))) -> c22(A__FILTER(s(mark(z0)), a__from(mark(x1))), MARK(s(z0))) MARK(filter(0, from(x1))) -> c22(A__FILTER(0, a__from(mark(x1))), MARK(0)) MARK(filter(cons(z0, z1), from(x1))) -> c22(A__FILTER(cons(mark(z0), z1), a__from(mark(x1))), MARK(cons(z0, z1))) MARK(filter(true, from(x1))) -> c22(A__FILTER(true, a__from(mark(x1))), MARK(true)) MARK(filter(false, from(x1))) -> c22(A__FILTER(false, a__from(mark(x1))), MARK(false)) MARK(filter(divides(z0, z1), from(x1))) -> c22(A__FILTER(divides(mark(z0), mark(z1)), a__from(mark(x1))), MARK(divides(z0, z1))) ---------------------------------------- (224) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, head(z0))) -> c22(A__FILTER(mark(x0), a__head(mark(z0))), MARK(x0)) MARK(filter(x0, tail(z0))) -> c22(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(x0)) MARK(filter(x0, if(z0, z1, z2))) -> c22(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0)) MARK(filter(x0, filter(z0, z1))) -> c22(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(x0)) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(x0, s(z0))) -> c22(MARK(x0)) MARK(filter(x0, 0)) -> c22(MARK(x0)) MARK(filter(x0, true)) -> c22(MARK(x0)) MARK(filter(x0, false)) -> c22(MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), x1)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c23(A__FILTER(mark(x0), a__primes), MARK(primes)) MARK(filter(x0, sieve(z0))) -> c23(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, from(z0))) -> c23(A__FILTER(mark(x0), a__from(mark(z0))), MARK(from(z0))) MARK(filter(x0, head(z0))) -> c23(A__FILTER(mark(x0), a__head(mark(z0))), MARK(head(z0))) MARK(filter(x0, tail(z0))) -> c23(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(tail(z0))) MARK(filter(x0, if(z0, z1, z2))) -> c23(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(filter(x0, filter(z0, z1))) -> c23(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(filter(x0, cons(z0, z1))) -> c23(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(primes, x1)) -> c23(A__FILTER(a__primes, mark(x1)), MARK(x1)) MARK(filter(sieve(z0), x1)) -> c23(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(from(z0), x1)) -> c23(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(head(z0), x1)) -> c23(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(tail(z0), x1)) -> c23(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(if(z0, z1, z2), x1)) -> c23(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1)) MARK(filter(filter(z0, z1), x1)) -> c23(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(x1)) MARK(filter(s(z0), x1)) -> c23(A__FILTER(s(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(x0, s(z0))) -> c23(MARK(s(z0))) MARK(filter(x0, divides(z0, z1))) -> c23(MARK(divides(z0, z1))) MARK(filter(0, x1)) -> c23(MARK(x1)) MARK(filter(cons(z0, z1), x1)) -> c23(MARK(x1)) MARK(filter(true, x1)) -> c23(MARK(x1)) MARK(filter(false, x1)) -> c23(MARK(x1)) MARK(filter(divides(z0, z1), x1)) -> c23(MARK(x1)) A__PRIMES -> c(A__SIEVE(cons(s(mark(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__sieve(a__from(s(s(0))))), 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x2), x3, x4), MARK(if(true, x1, x2))) MARK(if(if(false, x1, x2), x3, x4)) -> c21(A__IF(a__if(false, x1, x2), x3, x4), MARK(if(false, x1, x2))) MARK(if(if(divides(z0, z1), x1, x2), x3, x4)) -> c21(A__IF(a__if(divides(mark(z0), mark(z1)), x1, x2), x3, x4), MARK(if(divides(z0, z1), x1, x2))) MARK(if(if(x0, z1, z2), x3, x4)) -> c21(MARK(if(x0, z1, z2))) MARK(if(if(0, x1, x2), x3, x4)) -> c21(A__IF(a__if(0, x1, x2), x3, x4)) MARK(if(filter(x0, primes), x2, x3)) -> c21(A__IF(a__filter(mark(x0), a__primes), x2, x3), MARK(filter(x0, primes))) MARK(if(filter(x0, sieve(z0)), x2, x3)) -> c21(A__IF(a__filter(mark(x0), a__sieve(mark(z0))), x2, x3), MARK(filter(x0, sieve(z0)))) MARK(if(filter(x0, from(z0)), x2, x3)) -> c21(A__IF(a__filter(mark(x0), a__from(mark(z0))), x2, x3), MARK(filter(x0, from(z0)))) MARK(if(filter(x0, head(z0)), x2, x3)) -> c21(A__IF(a__filter(mark(x0), a__head(mark(z0))), x2, x3), MARK(filter(x0, head(z0)))) MARK(if(filter(x0, tail(z0)), x2, x3)) -> c21(A__IF(a__filter(mark(x0), a__tail(mark(z0))), x2, x3), MARK(filter(x0, tail(z0)))) MARK(if(filter(x0, if(z0, z1, z2)), x2, x3)) -> c21(A__IF(a__filter(mark(x0), a__if(mark(z0), z1, z2)), x2, x3), MARK(filter(x0, if(z0, z1, z2)))) MARK(if(filter(x0, filter(z0, z1)), x2, x3)) -> c21(A__IF(a__filter(mark(x0), a__filter(mark(z0), mark(z1))), x2, x3), MARK(filter(x0, filter(z0, z1)))) MARK(if(filter(x0, s(z0)), x2, x3)) -> c21(A__IF(a__filter(mark(x0), s(mark(z0))), x2, x3), MARK(filter(x0, s(z0)))) MARK(if(filter(x0, 0), x2, x3)) -> c21(A__IF(a__filter(mark(x0), 0), x2, x3), MARK(filter(x0, 0))) MARK(if(filter(x0, cons(z0, z1)), x2, x3)) -> c21(A__IF(a__filter(mark(x0), cons(mark(z0), z1)), x2, x3), MARK(filter(x0, cons(z0, z1)))) MARK(if(filter(x0, true), x2, x3)) -> c21(A__IF(a__filter(mark(x0), true), x2, x3), MARK(filter(x0, true))) MARK(if(filter(x0, false), x2, x3)) -> c21(A__IF(a__filter(mark(x0), false), x2, x3), MARK(filter(x0, false))) MARK(if(filter(x0, divides(z0, z1)), x2, x3)) -> c21(A__IF(a__filter(mark(x0), divides(mark(z0), mark(z1))), x2, x3), MARK(filter(x0, divides(z0, z1)))) MARK(if(filter(primes, x1), x2, x3)) -> c21(A__IF(a__filter(a__primes, mark(x1)), x2, x3), MARK(filter(primes, x1))) MARK(if(filter(sieve(z0), x1), x2, x3)) -> c21(A__IF(a__filter(a__sieve(mark(z0)), mark(x1)), x2, x3), MARK(filter(sieve(z0), x1))) MARK(if(filter(from(z0), x1), x2, x3)) -> c21(A__IF(a__filter(a__from(mark(z0)), mark(x1)), x2, x3), MARK(filter(from(z0), x1))) MARK(if(filter(head(z0), x1), x2, x3)) -> c21(A__IF(a__filter(a__head(mark(z0)), mark(x1)), x2, x3), MARK(filter(head(z0), x1))) MARK(if(filter(tail(z0), x1), x2, x3)) -> c21(A__IF(a__filter(a__tail(mark(z0)), mark(x1)), x2, x3), MARK(filter(tail(z0), x1))) MARK(if(filter(if(z0, z1, z2), x1), x2, x3)) -> c21(A__IF(a__filter(a__if(mark(z0), z1, z2), mark(x1)), x2, x3), MARK(filter(if(z0, z1, z2), x1))) MARK(if(filter(filter(z0, z1), x1), x2, x3)) -> c21(A__IF(a__filter(a__filter(mark(z0), mark(z1)), mark(x1)), x2, x3), MARK(filter(filter(z0, z1), x1))) MARK(if(filter(s(z0), x1), x2, x3)) -> c21(A__IF(a__filter(s(mark(z0)), mark(x1)), x2, x3), MARK(filter(s(z0), x1))) MARK(if(filter(0, x1), x2, x3)) -> c21(A__IF(a__filter(0, mark(x1)), x2, x3), MARK(filter(0, x1))) MARK(if(filter(cons(z0, z1), x1), x2, x3)) -> c21(A__IF(a__filter(cons(mark(z0), z1), mark(x1)), x2, x3), MARK(filter(cons(z0, z1), x1))) MARK(if(filter(true, x1), x2, x3)) -> c21(A__IF(a__filter(true, mark(x1)), x2, x3), MARK(filter(true, x1))) MARK(if(filter(false, x1), x2, x3)) -> c21(A__IF(a__filter(false, mark(x1)), x2, x3), MARK(filter(false, x1))) MARK(if(filter(divides(z0, z1), x1), x2, x3)) -> c21(A__IF(a__filter(divides(mark(z0), mark(z1)), mark(x1)), x2, x3), MARK(filter(divides(z0, z1), x1))) MARK(if(filter(x0, x1), x2, x3)) -> c21(MARK(filter(x0, x1))) MARK(filter(x0, primes)) -> c22(A__FILTER(mark(x0), a__sieve(a__from(s(s(0))))), MARK(x0)) MARK(filter(primes, primes)) -> c22(A__FILTER(a__primes, a__primes), MARK(primes)) MARK(filter(sieve(z0), primes)) -> c22(A__FILTER(a__sieve(mark(z0)), a__primes), MARK(sieve(z0))) MARK(filter(from(z0), primes)) -> c22(A__FILTER(a__from(mark(z0)), a__primes), MARK(from(z0))) MARK(filter(head(z0), primes)) -> c22(A__FILTER(a__head(mark(z0)), a__primes), MARK(head(z0))) MARK(filter(tail(z0), primes)) -> c22(A__FILTER(a__tail(mark(z0)), a__primes), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), primes)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), a__primes), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), primes)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), a__primes), MARK(filter(z0, z1))) MARK(filter(s(z0), primes)) -> c22(A__FILTER(s(mark(z0)), a__primes), MARK(s(z0))) MARK(filter(x0, primes)) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), primes)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), primes)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, sieve(primes))) -> c22(A__FILTER(mark(x0), a__sieve(a__primes)), MARK(x0)) MARK(filter(x0, sieve(sieve(z0)))) -> c22(A__FILTER(mark(x0), a__sieve(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, sieve(from(z0)))) -> c22(A__FILTER(mark(x0), a__sieve(a__from(mark(z0)))), MARK(x0)) MARK(filter(x0, sieve(head(z0)))) -> c22(A__FILTER(mark(x0), a__sieve(a__head(mark(z0)))), MARK(x0)) MARK(filter(x0, sieve(tail(z0)))) -> c22(A__FILTER(mark(x0), a__sieve(a__tail(mark(z0)))), MARK(x0)) MARK(filter(x0, sieve(if(z0, z1, z2)))) -> c22(A__FILTER(mark(x0), a__sieve(a__if(mark(z0), z1, z2))), MARK(x0)) MARK(filter(x0, sieve(filter(z0, z1)))) -> c22(A__FILTER(mark(x0), a__sieve(a__filter(mark(z0), mark(z1)))), MARK(x0)) MARK(filter(x0, sieve(s(z0)))) -> c22(A__FILTER(mark(x0), a__sieve(s(mark(z0)))), MARK(x0)) MARK(filter(x0, sieve(0))) -> c22(A__FILTER(mark(x0), a__sieve(0)), MARK(x0)) MARK(filter(x0, sieve(cons(z0, z1)))) -> c22(A__FILTER(mark(x0), a__sieve(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, sieve(true))) -> c22(A__FILTER(mark(x0), a__sieve(true)), MARK(x0)) MARK(filter(x0, sieve(false))) -> c22(A__FILTER(mark(x0), a__sieve(false)), MARK(x0)) MARK(filter(x0, sieve(divides(z0, z1)))) -> c22(A__FILTER(mark(x0), a__sieve(divides(mark(z0), mark(z1)))), MARK(x0)) MARK(filter(primes, sieve(x1))) -> c22(A__FILTER(a__primes, a__sieve(mark(x1))), MARK(primes)) MARK(filter(sieve(z0), sieve(x1))) -> c22(A__FILTER(a__sieve(mark(z0)), a__sieve(mark(x1))), MARK(sieve(z0))) MARK(filter(from(z0), sieve(x1))) -> c22(A__FILTER(a__from(mark(z0)), a__sieve(mark(x1))), MARK(from(z0))) MARK(filter(head(z0), sieve(x1))) -> c22(A__FILTER(a__head(mark(z0)), a__sieve(mark(x1))), MARK(head(z0))) MARK(filter(tail(z0), sieve(x1))) -> c22(A__FILTER(a__tail(mark(z0)), a__sieve(mark(x1))), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), sieve(x1))) -> c22(A__FILTER(a__if(mark(z0), z1, z2), a__sieve(mark(x1))), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), sieve(x1))) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), a__sieve(mark(x1))), MARK(filter(z0, z1))) MARK(filter(s(z0), sieve(x1))) -> c22(A__FILTER(s(mark(z0)), a__sieve(mark(x1))), MARK(s(z0))) MARK(filter(x0, sieve(x1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), sieve(x1))) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), sieve(x1))) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, from(x1))) -> c22(A__FILTER(mark(x0), cons(mark(mark(x1)), from(s(mark(x1))))), MARK(x0)) MARK(filter(x0, from(x1))) -> c22(A__FILTER(mark(x0), from(mark(x1))), MARK(x0)) MARK(filter(x0, from(primes))) -> c22(A__FILTER(mark(x0), a__from(a__primes)), MARK(x0)) MARK(filter(x0, from(sieve(z0)))) -> c22(A__FILTER(mark(x0), a__from(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, from(from(z0)))) -> c22(A__FILTER(mark(x0), a__from(a__from(mark(z0)))), MARK(x0)) MARK(filter(x0, from(head(z0)))) -> c22(A__FILTER(mark(x0), a__from(a__head(mark(z0)))), MARK(x0)) MARK(filter(x0, from(tail(z0)))) -> c22(A__FILTER(mark(x0), a__from(a__tail(mark(z0)))), MARK(x0)) MARK(filter(x0, from(if(z0, z1, z2)))) -> c22(A__FILTER(mark(x0), a__from(a__if(mark(z0), z1, z2))), MARK(x0)) MARK(filter(x0, from(filter(z0, z1)))) -> c22(A__FILTER(mark(x0), a__from(a__filter(mark(z0), mark(z1)))), MARK(x0)) MARK(filter(x0, from(s(z0)))) -> c22(A__FILTER(mark(x0), a__from(s(mark(z0)))), MARK(x0)) MARK(filter(x0, from(0))) -> c22(A__FILTER(mark(x0), a__from(0)), MARK(x0)) MARK(filter(x0, from(cons(z0, z1)))) -> c22(A__FILTER(mark(x0), a__from(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, from(true))) -> c22(A__FILTER(mark(x0), a__from(true)), MARK(x0)) MARK(filter(x0, from(false))) -> c22(A__FILTER(mark(x0), a__from(false)), MARK(x0)) MARK(filter(x0, from(divides(z0, z1)))) -> c22(A__FILTER(mark(x0), a__from(divides(mark(z0), mark(z1)))), MARK(x0)) MARK(filter(primes, from(x1))) -> c22(A__FILTER(a__primes, a__from(mark(x1))), MARK(primes)) MARK(filter(sieve(z0), from(x1))) -> c22(A__FILTER(a__sieve(mark(z0)), a__from(mark(x1))), MARK(sieve(z0))) MARK(filter(from(z0), from(x1))) -> c22(A__FILTER(a__from(mark(z0)), a__from(mark(x1))), MARK(from(z0))) MARK(filter(head(z0), from(x1))) -> c22(A__FILTER(a__head(mark(z0)), a__from(mark(x1))), MARK(head(z0))) MARK(filter(tail(z0), from(x1))) -> c22(A__FILTER(a__tail(mark(z0)), a__from(mark(x1))), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), from(x1))) -> c22(A__FILTER(a__if(mark(z0), z1, z2), a__from(mark(x1))), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), from(x1))) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), a__from(mark(x1))), MARK(filter(z0, z1))) MARK(filter(s(z0), from(x1))) -> c22(A__FILTER(s(mark(z0)), a__from(mark(x1))), MARK(s(z0))) MARK(filter(0, from(x1))) -> c22(A__FILTER(0, a__from(mark(x1))), MARK(0)) MARK(filter(cons(z0, z1), from(x1))) -> c22(A__FILTER(cons(mark(z0), z1), a__from(mark(x1))), MARK(cons(z0, z1))) MARK(filter(true, from(x1))) -> c22(A__FILTER(true, a__from(mark(x1))), MARK(true)) MARK(filter(false, from(x1))) -> c22(A__FILTER(false, a__from(mark(x1))), MARK(false)) MARK(filter(divides(z0, z1), from(x1))) -> c22(A__FILTER(divides(mark(z0), mark(z1)), a__from(mark(x1))), MARK(divides(z0, z1))) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c18_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_1, c17_2, c17_1, c19_2, c19_1, c20_2, c20_1, c21_1, c22_2, c22_1, c23_2, c23_1, c_2, c21_2 ---------------------------------------- (225) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing nodes: MARK(filter(true, from(x1))) -> c22(A__FILTER(true, a__from(mark(x1))), MARK(true)) MARK(filter(0, from(x1))) -> c22(A__FILTER(0, a__from(mark(x1))), MARK(0)) MARK(filter(false, from(x1))) -> c22(A__FILTER(false, a__from(mark(x1))), MARK(false)) ---------------------------------------- (226) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, head(z0))) -> c22(A__FILTER(mark(x0), a__head(mark(z0))), MARK(x0)) MARK(filter(x0, tail(z0))) -> c22(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(x0)) MARK(filter(x0, if(z0, z1, z2))) -> c22(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0)) MARK(filter(x0, filter(z0, z1))) -> c22(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(x0)) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(x0, s(z0))) -> c22(MARK(x0)) MARK(filter(x0, 0)) -> c22(MARK(x0)) MARK(filter(x0, true)) -> c22(MARK(x0)) MARK(filter(x0, false)) -> c22(MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), x1)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c23(A__FILTER(mark(x0), a__primes), MARK(primes)) MARK(filter(x0, sieve(z0))) -> c23(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, from(z0))) -> c23(A__FILTER(mark(x0), a__from(mark(z0))), MARK(from(z0))) MARK(filter(x0, head(z0))) -> c23(A__FILTER(mark(x0), a__head(mark(z0))), MARK(head(z0))) MARK(filter(x0, tail(z0))) -> c23(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(tail(z0))) MARK(filter(x0, if(z0, z1, z2))) -> c23(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(filter(x0, filter(z0, z1))) -> c23(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(filter(x0, cons(z0, z1))) -> c23(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(primes, x1)) -> c23(A__FILTER(a__primes, mark(x1)), MARK(x1)) MARK(filter(sieve(z0), x1)) -> c23(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(from(z0), x1)) -> c23(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(head(z0), x1)) -> c23(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(tail(z0), x1)) -> c23(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(if(z0, z1, z2), x1)) -> c23(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1)) MARK(filter(filter(z0, z1), x1)) -> c23(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(x1)) MARK(filter(s(z0), x1)) -> c23(A__FILTER(s(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(x0, s(z0))) -> c23(MARK(s(z0))) MARK(filter(x0, divides(z0, z1))) -> c23(MARK(divides(z0, z1))) MARK(filter(0, x1)) -> c23(MARK(x1)) MARK(filter(cons(z0, z1), x1)) -> c23(MARK(x1)) MARK(filter(true, x1)) -> c23(MARK(x1)) MARK(filter(false, x1)) -> c23(MARK(x1)) MARK(filter(divides(z0, z1), x1)) -> c23(MARK(x1)) A__PRIMES -> c(A__SIEVE(cons(s(mark(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(sieve(primes)) -> c17(MARK(primes)) MARK(sieve(sieve(primes))) -> c17(A__SIEVE(a__sieve(a__primes)), MARK(sieve(primes))) MARK(sieve(sieve(sieve(z0)))) -> c17(A__SIEVE(a__sieve(a__sieve(mark(z0)))), MARK(sieve(sieve(z0)))) MARK(sieve(sieve(from(z0)))) -> c17(A__SIEVE(a__sieve(a__from(mark(z0)))), MARK(sieve(from(z0)))) MARK(sieve(sieve(head(z0)))) -> c17(A__SIEVE(a__sieve(a__head(mark(z0)))), MARK(sieve(head(z0)))) MARK(sieve(sieve(tail(z0)))) -> c17(A__SIEVE(a__sieve(a__tail(mark(z0)))), MARK(sieve(tail(z0)))) MARK(sieve(sieve(if(z0, z1, z2)))) -> c17(A__SIEVE(a__sieve(a__if(mark(z0), z1, z2))), MARK(sieve(if(z0, z1, z2)))) MARK(sieve(sieve(filter(z0, z1)))) -> c17(A__SIEVE(a__sieve(a__filter(mark(z0), mark(z1)))), MARK(sieve(filter(z0, z1)))) MARK(sieve(sieve(s(z0)))) -> c17(A__SIEVE(a__sieve(s(mark(z0)))), MARK(sieve(s(z0)))) MARK(sieve(sieve(cons(z0, z1)))) -> c17(A__SIEVE(a__sieve(cons(mark(z0), z1))), MARK(sieve(cons(z0, z1)))) MARK(sieve(sieve(divides(z0, z1)))) -> c17(A__SIEVE(a__sieve(divides(mark(z0), mark(z1)))), MARK(sieve(divides(z0, z1)))) MARK(sieve(sieve(x0))) -> c17(MARK(sieve(x0))) MARK(sieve(sieve(0))) -> c17(A__SIEVE(a__sieve(0))) MARK(sieve(sieve(true))) -> c17(A__SIEVE(a__sieve(true))) MARK(sieve(sieve(false))) -> c17(A__SIEVE(a__sieve(false))) MARK(sieve(from(x0))) -> c17(A__SIEVE(cons(mark(mark(x0)), from(s(mark(x0))))), MARK(from(x0))) MARK(sieve(from(primes))) -> c17(A__SIEVE(a__from(a__primes)), MARK(from(primes))) MARK(sieve(from(sieve(z0)))) -> c17(A__SIEVE(a__from(a__sieve(mark(z0)))), MARK(from(sieve(z0)))) MARK(sieve(from(from(z0)))) -> c17(A__SIEVE(a__from(a__from(mark(z0)))), MARK(from(from(z0)))) MARK(sieve(from(head(z0)))) -> c17(A__SIEVE(a__from(a__head(mark(z0)))), MARK(from(head(z0)))) MARK(sieve(from(tail(z0)))) -> c17(A__SIEVE(a__from(a__tail(mark(z0)))), MARK(from(tail(z0)))) MARK(sieve(from(if(z0, z1, z2)))) -> c17(A__SIEVE(a__from(a__if(mark(z0), z1, z2))), MARK(from(if(z0, z1, z2)))) MARK(sieve(from(filter(z0, z1)))) -> c17(A__SIEVE(a__from(a__filter(mark(z0), mark(z1)))), MARK(from(filter(z0, z1)))) MARK(sieve(from(s(z0)))) -> c17(A__SIEVE(a__from(s(mark(z0)))), MARK(from(s(z0)))) MARK(sieve(from(0))) -> c17(A__SIEVE(a__from(0)), MARK(from(0))) MARK(sieve(from(cons(z0, z1)))) -> c17(A__SIEVE(a__from(cons(mark(z0), z1))), MARK(from(cons(z0, z1)))) MARK(sieve(from(true))) -> c17(A__SIEVE(a__from(true)), MARK(from(true))) MARK(sieve(from(false))) -> c17(A__SIEVE(a__from(false)), MARK(from(false))) MARK(sieve(from(divides(z0, z1)))) -> c17(A__SIEVE(a__from(divides(mark(z0), mark(z1)))), MARK(from(divides(z0, z1)))) MARK(sieve(from(x0))) -> c17(MARK(from(x0))) MARK(sieve(head(primes))) -> c17(A__SIEVE(a__head(a__primes)), MARK(head(primes))) MARK(sieve(head(sieve(z0)))) -> c17(A__SIEVE(a__head(a__sieve(mark(z0)))), MARK(head(sieve(z0)))) MARK(sieve(head(from(z0)))) -> c17(A__SIEVE(a__head(a__from(mark(z0)))), MARK(head(from(z0)))) MARK(sieve(head(head(z0)))) -> c17(A__SIEVE(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(sieve(head(tail(z0)))) -> c17(A__SIEVE(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(sieve(head(if(z0, z1, z2)))) -> c17(A__SIEVE(a__head(a__if(mark(z0), z1, z2))), MARK(head(if(z0, z1, z2)))) MARK(sieve(head(filter(z0, z1)))) -> c17(A__SIEVE(a__head(a__filter(mark(z0), mark(z1)))), MARK(head(filter(z0, z1)))) MARK(sieve(head(s(z0)))) -> c17(A__SIEVE(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(sieve(head(cons(z0, z1)))) -> c17(A__SIEVE(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(sieve(head(divides(z0, z1)))) -> c17(A__SIEVE(a__head(divides(mark(z0), mark(z1)))), MARK(head(divides(z0, z1)))) MARK(sieve(head(x0))) -> c17(MARK(head(x0))) MARK(sieve(head(0))) -> c17(A__SIEVE(a__head(0))) MARK(sieve(head(true))) -> c17(A__SIEVE(a__head(true))) MARK(sieve(head(false))) -> c17(A__SIEVE(a__head(false))) MARK(sieve(tail(primes))) -> c17(A__SIEVE(a__tail(a__primes)), MARK(tail(primes))) MARK(sieve(tail(sieve(z0)))) -> c17(A__SIEVE(a__tail(a__sieve(mark(z0)))), MARK(tail(sieve(z0)))) MARK(sieve(tail(from(z0)))) -> c17(A__SIEVE(a__tail(a__from(mark(z0)))), MARK(tail(from(z0)))) MARK(sieve(tail(head(z0)))) -> c17(A__SIEVE(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(sieve(tail(tail(z0)))) -> c17(A__SIEVE(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(sieve(tail(if(z0, z1, z2)))) -> c17(A__SIEVE(a__tail(a__if(mark(z0), z1, z2))), MARK(tail(if(z0, z1, z2)))) MARK(sieve(tail(filter(z0, z1)))) -> c17(A__SIEVE(a__tail(a__filter(mark(z0), mark(z1)))), MARK(tail(filter(z0, z1)))) MARK(sieve(tail(s(z0)))) -> c17(A__SIEVE(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(sieve(tail(cons(z0, z1)))) -> c17(A__SIEVE(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(sieve(tail(divides(z0, z1)))) -> c17(A__SIEVE(a__tail(divides(mark(z0), mark(z1)))), MARK(tail(divides(z0, z1)))) MARK(sieve(tail(x0))) -> c17(MARK(tail(x0))) MARK(sieve(tail(0))) -> c17(A__SIEVE(a__tail(0))) MARK(sieve(tail(true))) -> c17(A__SIEVE(a__tail(true))) MARK(sieve(tail(false))) -> c17(A__SIEVE(a__tail(false))) MARK(sieve(if(primes, x1, x2))) -> c17(A__SIEVE(a__if(a__primes, x1, x2)), MARK(if(primes, x1, x2))) MARK(sieve(if(sieve(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__sieve(mark(z0)), x1, x2)), MARK(if(sieve(z0), x1, x2))) MARK(sieve(if(from(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__from(mark(z0)), x1, x2)), MARK(if(from(z0), x1, x2))) MARK(sieve(if(head(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__head(mark(z0)), x1, x2)), MARK(if(head(z0), x1, x2))) MARK(sieve(if(tail(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__tail(mark(z0)), x1, x2)), MARK(if(tail(z0), x1, x2))) MARK(sieve(if(if(z0, z1, z2), x1, x2))) -> c17(A__SIEVE(a__if(a__if(mark(z0), z1, z2), x1, x2)), MARK(if(if(z0, z1, z2), x1, x2))) MARK(sieve(if(filter(z0, z1), x1, x2))) -> c17(A__SIEVE(a__if(a__filter(mark(z0), mark(z1)), x1, x2)), MARK(if(filter(z0, z1), x1, x2))) MARK(sieve(if(s(z0), x1, x2))) -> c17(A__SIEVE(a__if(s(mark(z0)), x1, x2)), MARK(if(s(z0), x1, x2))) MARK(sieve(if(cons(z0, z1), x1, x2))) -> c17(A__SIEVE(a__if(cons(mark(z0), z1), x1, x2)), MARK(if(cons(z0, z1), x1, x2))) MARK(sieve(if(true, x1, x2))) -> c17(A__SIEVE(a__if(true, x1, x2)), MARK(if(true, x1, x2))) MARK(sieve(if(false, x1, x2))) -> c17(A__SIEVE(a__if(false, x1, x2)), MARK(if(false, x1, x2))) MARK(sieve(if(divides(z0, z1), x1, x2))) -> c17(A__SIEVE(a__if(divides(mark(z0), mark(z1)), x1, x2)), MARK(if(divides(z0, z1), x1, x2))) MARK(sieve(if(x0, z1, z2))) -> c17(MARK(if(x0, z1, z2))) MARK(sieve(if(0, x1, x2))) -> c17(A__SIEVE(a__if(0, x1, x2))) MARK(sieve(filter(x0, primes))) -> c17(A__SIEVE(a__filter(mark(x0), a__primes)), MARK(filter(x0, primes))) MARK(sieve(filter(x0, sieve(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), a__sieve(mark(z0)))), MARK(filter(x0, sieve(z0)))) MARK(sieve(filter(x0, from(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), a__from(mark(z0)))), MARK(filter(x0, from(z0)))) MARK(sieve(filter(x0, head(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), a__head(mark(z0)))), MARK(filter(x0, head(z0)))) MARK(sieve(filter(x0, tail(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), a__tail(mark(z0)))), MARK(filter(x0, tail(z0)))) MARK(sieve(filter(x0, if(z0, z1, z2)))) -> c17(A__SIEVE(a__filter(mark(x0), a__if(mark(z0), z1, z2))), MARK(filter(x0, if(z0, z1, z2)))) MARK(sieve(filter(x0, filter(z0, z1)))) -> c17(A__SIEVE(a__filter(mark(x0), a__filter(mark(z0), mark(z1)))), MARK(filter(x0, filter(z0, z1)))) MARK(sieve(filter(x0, s(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), s(mark(z0)))), MARK(filter(x0, s(z0)))) MARK(sieve(filter(x0, 0))) -> c17(A__SIEVE(a__filter(mark(x0), 0)), MARK(filter(x0, 0))) MARK(sieve(filter(x0, cons(z0, z1)))) -> c17(A__SIEVE(a__filter(mark(x0), cons(mark(z0), z1))), MARK(filter(x0, cons(z0, z1)))) MARK(sieve(filter(x0, true))) -> c17(A__SIEVE(a__filter(mark(x0), true)), MARK(filter(x0, true))) MARK(sieve(filter(x0, false))) -> c17(A__SIEVE(a__filter(mark(x0), false)), MARK(filter(x0, false))) MARK(sieve(filter(x0, divides(z0, z1)))) -> c17(A__SIEVE(a__filter(mark(x0), divides(mark(z0), mark(z1)))), MARK(filter(x0, divides(z0, z1)))) MARK(sieve(filter(primes, x1))) -> c17(A__SIEVE(a__filter(a__primes, mark(x1))), MARK(filter(primes, x1))) MARK(sieve(filter(sieve(z0), x1))) -> c17(A__SIEVE(a__filter(a__sieve(mark(z0)), mark(x1))), MARK(filter(sieve(z0), x1))) MARK(sieve(filter(from(z0), x1))) -> c17(A__SIEVE(a__filter(a__from(mark(z0)), mark(x1))), MARK(filter(from(z0), x1))) MARK(sieve(filter(head(z0), x1))) -> c17(A__SIEVE(a__filter(a__head(mark(z0)), mark(x1))), MARK(filter(head(z0), x1))) MARK(sieve(filter(tail(z0), x1))) -> c17(A__SIEVE(a__filter(a__tail(mark(z0)), mark(x1))), MARK(filter(tail(z0), x1))) MARK(sieve(filter(if(z0, z1, z2), x1))) -> c17(A__SIEVE(a__filter(a__if(mark(z0), z1, z2), mark(x1))), MARK(filter(if(z0, z1, z2), x1))) MARK(sieve(filter(filter(z0, z1), x1))) -> 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MARK(filter(x0, from(0))) -> c22(A__FILTER(mark(x0), a__from(0)), MARK(x0)) MARK(filter(x0, from(cons(z0, z1)))) -> c22(A__FILTER(mark(x0), a__from(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, from(true))) -> c22(A__FILTER(mark(x0), a__from(true)), MARK(x0)) MARK(filter(x0, from(false))) -> c22(A__FILTER(mark(x0), a__from(false)), MARK(x0)) MARK(filter(x0, from(divides(z0, z1)))) -> c22(A__FILTER(mark(x0), a__from(divides(mark(z0), mark(z1)))), MARK(x0)) MARK(filter(primes, from(x1))) -> c22(A__FILTER(a__primes, a__from(mark(x1))), MARK(primes)) MARK(filter(sieve(z0), from(x1))) -> c22(A__FILTER(a__sieve(mark(z0)), a__from(mark(x1))), MARK(sieve(z0))) MARK(filter(from(z0), from(x1))) -> c22(A__FILTER(a__from(mark(z0)), a__from(mark(x1))), MARK(from(z0))) MARK(filter(head(z0), from(x1))) -> c22(A__FILTER(a__head(mark(z0)), a__from(mark(x1))), MARK(head(z0))) MARK(filter(tail(z0), from(x1))) -> c22(A__FILTER(a__tail(mark(z0)), a__from(mark(x1))), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), from(x1))) -> c22(A__FILTER(a__if(mark(z0), z1, z2), a__from(mark(x1))), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), from(x1))) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), a__from(mark(x1))), MARK(filter(z0, z1))) MARK(filter(s(z0), from(x1))) -> c22(A__FILTER(s(mark(z0)), a__from(mark(x1))), MARK(s(z0))) MARK(filter(cons(z0, z1), from(x1))) -> c22(A__FILTER(cons(mark(z0), z1), a__from(mark(x1))), MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), from(x1))) -> c22(A__FILTER(divides(mark(z0), mark(z1)), a__from(mark(x1))), MARK(divides(z0, z1))) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c18_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_1, c17_2, c17_1, c19_2, c19_1, c20_2, c20_1, c21_1, c22_2, c22_1, c23_2, c23_1, c_2, c21_2 ---------------------------------------- (227) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (228) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, head(z0))) -> c22(A__FILTER(mark(x0), a__head(mark(z0))), MARK(x0)) MARK(filter(x0, tail(z0))) -> c22(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(x0)) MARK(filter(x0, if(z0, z1, z2))) -> c22(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0)) MARK(filter(x0, filter(z0, z1))) -> c22(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(x0)) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(x0, s(z0))) -> c22(MARK(x0)) MARK(filter(x0, 0)) -> c22(MARK(x0)) MARK(filter(x0, true)) -> c22(MARK(x0)) MARK(filter(x0, false)) -> c22(MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), x1)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c23(A__FILTER(mark(x0), a__primes), MARK(primes)) MARK(filter(x0, sieve(z0))) -> c23(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, from(z0))) -> c23(A__FILTER(mark(x0), a__from(mark(z0))), MARK(from(z0))) MARK(filter(x0, head(z0))) -> c23(A__FILTER(mark(x0), a__head(mark(z0))), MARK(head(z0))) MARK(filter(x0, tail(z0))) -> c23(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(tail(z0))) MARK(filter(x0, if(z0, z1, z2))) -> c23(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(filter(x0, filter(z0, z1))) -> c23(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(filter(x0, cons(z0, z1))) -> c23(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(primes, x1)) -> c23(A__FILTER(a__primes, mark(x1)), MARK(x1)) MARK(filter(sieve(z0), x1)) -> c23(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(from(z0), x1)) -> c23(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(head(z0), x1)) -> c23(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(tail(z0), x1)) -> c23(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(if(z0, z1, z2), x1)) -> c23(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1)) MARK(filter(filter(z0, z1), x1)) -> c23(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(x1)) MARK(filter(s(z0), x1)) -> c23(A__FILTER(s(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(x0, s(z0))) -> c23(MARK(s(z0))) MARK(filter(x0, divides(z0, z1))) -> c23(MARK(divides(z0, z1))) MARK(filter(0, x1)) -> c23(MARK(x1)) MARK(filter(cons(z0, z1), x1)) -> c23(MARK(x1)) MARK(filter(true, x1)) -> c23(MARK(x1)) MARK(filter(false, x1)) -> c23(MARK(x1)) MARK(filter(divides(z0, z1), x1)) -> c23(MARK(x1)) A__PRIMES -> c(A__SIEVE(cons(s(mark(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(sieve(primes)) -> c17(MARK(primes)) MARK(sieve(sieve(primes))) -> c17(A__SIEVE(a__sieve(a__primes)), MARK(sieve(primes))) MARK(sieve(sieve(sieve(z0)))) -> c17(A__SIEVE(a__sieve(a__sieve(mark(z0)))), MARK(sieve(sieve(z0)))) MARK(sieve(sieve(from(z0)))) -> c17(A__SIEVE(a__sieve(a__from(mark(z0)))), MARK(sieve(from(z0)))) MARK(sieve(sieve(head(z0)))) -> c17(A__SIEVE(a__sieve(a__head(mark(z0)))), MARK(sieve(head(z0)))) MARK(sieve(sieve(tail(z0)))) -> c17(A__SIEVE(a__sieve(a__tail(mark(z0)))), MARK(sieve(tail(z0)))) MARK(sieve(sieve(if(z0, z1, z2)))) -> c17(A__SIEVE(a__sieve(a__if(mark(z0), z1, z2))), MARK(sieve(if(z0, z1, z2)))) MARK(sieve(sieve(filter(z0, z1)))) -> c17(A__SIEVE(a__sieve(a__filter(mark(z0), mark(z1)))), MARK(sieve(filter(z0, z1)))) MARK(sieve(sieve(s(z0)))) -> c17(A__SIEVE(a__sieve(s(mark(z0)))), MARK(sieve(s(z0)))) MARK(sieve(sieve(cons(z0, z1)))) -> c17(A__SIEVE(a__sieve(cons(mark(z0), z1))), MARK(sieve(cons(z0, z1)))) MARK(sieve(sieve(divides(z0, z1)))) -> c17(A__SIEVE(a__sieve(divides(mark(z0), mark(z1)))), MARK(sieve(divides(z0, z1)))) MARK(sieve(sieve(x0))) -> c17(MARK(sieve(x0))) MARK(sieve(sieve(0))) -> c17(A__SIEVE(a__sieve(0))) MARK(sieve(sieve(true))) -> c17(A__SIEVE(a__sieve(true))) MARK(sieve(sieve(false))) -> c17(A__SIEVE(a__sieve(false))) MARK(sieve(from(x0))) -> c17(A__SIEVE(cons(mark(mark(x0)), from(s(mark(x0))))), MARK(from(x0))) MARK(sieve(from(primes))) -> c17(A__SIEVE(a__from(a__primes)), MARK(from(primes))) MARK(sieve(from(sieve(z0)))) -> c17(A__SIEVE(a__from(a__sieve(mark(z0)))), MARK(from(sieve(z0)))) MARK(sieve(from(from(z0)))) -> c17(A__SIEVE(a__from(a__from(mark(z0)))), MARK(from(from(z0)))) MARK(sieve(from(head(z0)))) -> c17(A__SIEVE(a__from(a__head(mark(z0)))), MARK(from(head(z0)))) MARK(sieve(from(tail(z0)))) -> c17(A__SIEVE(a__from(a__tail(mark(z0)))), MARK(from(tail(z0)))) MARK(sieve(from(if(z0, z1, z2)))) -> c17(A__SIEVE(a__from(a__if(mark(z0), z1, z2))), MARK(from(if(z0, z1, z2)))) MARK(sieve(from(filter(z0, z1)))) -> c17(A__SIEVE(a__from(a__filter(mark(z0), mark(z1)))), MARK(from(filter(z0, z1)))) MARK(sieve(from(s(z0)))) -> c17(A__SIEVE(a__from(s(mark(z0)))), MARK(from(s(z0)))) MARK(sieve(from(0))) -> c17(A__SIEVE(a__from(0)), MARK(from(0))) MARK(sieve(from(cons(z0, z1)))) -> c17(A__SIEVE(a__from(cons(mark(z0), z1))), MARK(from(cons(z0, z1)))) MARK(sieve(from(true))) -> c17(A__SIEVE(a__from(true)), MARK(from(true))) MARK(sieve(from(false))) -> c17(A__SIEVE(a__from(false)), MARK(from(false))) MARK(sieve(from(divides(z0, z1)))) -> c17(A__SIEVE(a__from(divides(mark(z0), mark(z1)))), MARK(from(divides(z0, z1)))) MARK(sieve(from(x0))) -> c17(MARK(from(x0))) MARK(sieve(head(primes))) -> c17(A__SIEVE(a__head(a__primes)), MARK(head(primes))) MARK(sieve(head(sieve(z0)))) -> c17(A__SIEVE(a__head(a__sieve(mark(z0)))), MARK(head(sieve(z0)))) MARK(sieve(head(from(z0)))) -> c17(A__SIEVE(a__head(a__from(mark(z0)))), MARK(head(from(z0)))) MARK(sieve(head(head(z0)))) -> c17(A__SIEVE(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(sieve(head(tail(z0)))) -> c17(A__SIEVE(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(sieve(head(if(z0, z1, z2)))) -> c17(A__SIEVE(a__head(a__if(mark(z0), z1, z2))), MARK(head(if(z0, z1, z2)))) MARK(sieve(head(filter(z0, z1)))) -> c17(A__SIEVE(a__head(a__filter(mark(z0), mark(z1)))), MARK(head(filter(z0, z1)))) MARK(sieve(head(s(z0)))) -> c17(A__SIEVE(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(sieve(head(cons(z0, z1)))) -> c17(A__SIEVE(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(sieve(head(divides(z0, z1)))) -> c17(A__SIEVE(a__head(divides(mark(z0), mark(z1)))), MARK(head(divides(z0, z1)))) MARK(sieve(head(x0))) -> c17(MARK(head(x0))) MARK(sieve(head(0))) -> c17(A__SIEVE(a__head(0))) 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MARK(x0)) MARK(filter(x0, sieve(cons(z0, z1)))) -> c22(A__FILTER(mark(x0), a__sieve(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, sieve(true))) -> c22(A__FILTER(mark(x0), a__sieve(true)), MARK(x0)) MARK(filter(x0, sieve(false))) -> c22(A__FILTER(mark(x0), a__sieve(false)), MARK(x0)) MARK(filter(x0, sieve(divides(z0, z1)))) -> c22(A__FILTER(mark(x0), a__sieve(divides(mark(z0), mark(z1)))), MARK(x0)) MARK(filter(primes, sieve(x1))) -> c22(A__FILTER(a__primes, a__sieve(mark(x1))), MARK(primes)) MARK(filter(sieve(z0), sieve(x1))) -> c22(A__FILTER(a__sieve(mark(z0)), a__sieve(mark(x1))), MARK(sieve(z0))) MARK(filter(from(z0), sieve(x1))) -> c22(A__FILTER(a__from(mark(z0)), a__sieve(mark(x1))), MARK(from(z0))) MARK(filter(head(z0), sieve(x1))) -> c22(A__FILTER(a__head(mark(z0)), a__sieve(mark(x1))), MARK(head(z0))) MARK(filter(tail(z0), sieve(x1))) -> c22(A__FILTER(a__tail(mark(z0)), a__sieve(mark(x1))), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), sieve(x1))) -> c22(A__FILTER(a__if(mark(z0), z1, z2), a__sieve(mark(x1))), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), sieve(x1))) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), a__sieve(mark(x1))), MARK(filter(z0, z1))) MARK(filter(s(z0), sieve(x1))) -> c22(A__FILTER(s(mark(z0)), a__sieve(mark(x1))), MARK(s(z0))) MARK(filter(x0, sieve(x1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), sieve(x1))) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), sieve(x1))) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, from(x1))) -> c22(A__FILTER(mark(x0), cons(mark(mark(x1)), from(s(mark(x1))))), MARK(x0)) MARK(filter(x0, from(primes))) -> c22(A__FILTER(mark(x0), a__from(a__primes)), MARK(x0)) MARK(filter(x0, from(sieve(z0)))) -> c22(A__FILTER(mark(x0), a__from(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, from(from(z0)))) -> c22(A__FILTER(mark(x0), a__from(a__from(mark(z0)))), MARK(x0)) MARK(filter(x0, from(head(z0)))) -> c22(A__FILTER(mark(x0), a__from(a__head(mark(z0)))), MARK(x0)) MARK(filter(x0, 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c22(A__FILTER(a__sieve(mark(z0)), a__from(mark(x1))), MARK(sieve(z0))) MARK(filter(from(z0), from(x1))) -> c22(A__FILTER(a__from(mark(z0)), a__from(mark(x1))), MARK(from(z0))) MARK(filter(head(z0), from(x1))) -> c22(A__FILTER(a__head(mark(z0)), a__from(mark(x1))), MARK(head(z0))) MARK(filter(tail(z0), from(x1))) -> c22(A__FILTER(a__tail(mark(z0)), a__from(mark(x1))), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), from(x1))) -> c22(A__FILTER(a__if(mark(z0), z1, z2), a__from(mark(x1))), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), from(x1))) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), a__from(mark(x1))), MARK(filter(z0, z1))) MARK(filter(s(z0), from(x1))) -> c22(A__FILTER(s(mark(z0)), a__from(mark(x1))), MARK(s(z0))) MARK(filter(x0, from(x1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), from(x1))) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), from(x1))) -> c22(MARK(divides(z0, z1))) S tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) 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c22(A__FILTER(a__sieve(mark(z0)), a__primes), MARK(sieve(z0))) MARK(filter(from(z0), primes)) -> c22(A__FILTER(a__from(mark(z0)), a__primes), MARK(from(z0))) MARK(filter(head(z0), primes)) -> c22(A__FILTER(a__head(mark(z0)), a__primes), MARK(head(z0))) MARK(filter(tail(z0), primes)) -> c22(A__FILTER(a__tail(mark(z0)), a__primes), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), primes)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), a__primes), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), primes)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), a__primes), MARK(filter(z0, z1))) MARK(filter(s(z0), primes)) -> c22(A__FILTER(s(mark(z0)), a__primes), MARK(s(z0))) MARK(filter(x0, primes)) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), primes)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), primes)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, sieve(primes))) -> c22(A__FILTER(mark(x0), a__sieve(a__primes)), MARK(x0)) MARK(filter(x0, sieve(sieve(z0)))) -> c22(A__FILTER(mark(x0), a__sieve(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, sieve(from(z0)))) -> c22(A__FILTER(mark(x0), a__sieve(a__from(mark(z0)))), MARK(x0)) MARK(filter(x0, sieve(head(z0)))) -> c22(A__FILTER(mark(x0), a__sieve(a__head(mark(z0)))), MARK(x0)) MARK(filter(x0, sieve(tail(z0)))) -> c22(A__FILTER(mark(x0), a__sieve(a__tail(mark(z0)))), MARK(x0)) MARK(filter(x0, sieve(if(z0, z1, z2)))) -> c22(A__FILTER(mark(x0), a__sieve(a__if(mark(z0), z1, z2))), MARK(x0)) MARK(filter(x0, sieve(filter(z0, z1)))) -> c22(A__FILTER(mark(x0), a__sieve(a__filter(mark(z0), mark(z1)))), MARK(x0)) MARK(filter(x0, sieve(s(z0)))) -> c22(A__FILTER(mark(x0), a__sieve(s(mark(z0)))), MARK(x0)) MARK(filter(x0, sieve(0))) -> c22(A__FILTER(mark(x0), a__sieve(0)), MARK(x0)) MARK(filter(x0, sieve(cons(z0, z1)))) -> c22(A__FILTER(mark(x0), a__sieve(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, sieve(true))) -> c22(A__FILTER(mark(x0), a__sieve(true)), MARK(x0)) MARK(filter(x0, sieve(false))) -> c22(A__FILTER(mark(x0), a__sieve(false)), MARK(x0)) MARK(filter(x0, sieve(divides(z0, z1)))) -> c22(A__FILTER(mark(x0), a__sieve(divides(mark(z0), mark(z1)))), MARK(x0)) MARK(filter(primes, sieve(x1))) -> c22(A__FILTER(a__primes, a__sieve(mark(x1))), MARK(primes)) MARK(filter(sieve(z0), sieve(x1))) -> c22(A__FILTER(a__sieve(mark(z0)), a__sieve(mark(x1))), MARK(sieve(z0))) MARK(filter(from(z0), sieve(x1))) -> c22(A__FILTER(a__from(mark(z0)), a__sieve(mark(x1))), MARK(from(z0))) MARK(filter(head(z0), sieve(x1))) -> c22(A__FILTER(a__head(mark(z0)), a__sieve(mark(x1))), MARK(head(z0))) MARK(filter(tail(z0), sieve(x1))) -> c22(A__FILTER(a__tail(mark(z0)), a__sieve(mark(x1))), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), sieve(x1))) -> c22(A__FILTER(a__if(mark(z0), z1, z2), a__sieve(mark(x1))), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), sieve(x1))) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), a__sieve(mark(x1))), MARK(filter(z0, z1))) MARK(filter(s(z0), sieve(x1))) -> c22(A__FILTER(s(mark(z0)), a__sieve(mark(x1))), MARK(s(z0))) MARK(filter(x0, sieve(x1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), sieve(x1))) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), sieve(x1))) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, from(x1))) -> c22(A__FILTER(mark(x0), cons(mark(mark(x1)), from(s(mark(x1))))), MARK(x0)) MARK(filter(x0, from(primes))) -> c22(A__FILTER(mark(x0), a__from(a__primes)), MARK(x0)) MARK(filter(x0, from(sieve(z0)))) -> c22(A__FILTER(mark(x0), a__from(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, from(from(z0)))) -> c22(A__FILTER(mark(x0), a__from(a__from(mark(z0)))), MARK(x0)) MARK(filter(x0, from(head(z0)))) -> c22(A__FILTER(mark(x0), a__from(a__head(mark(z0)))), MARK(x0)) MARK(filter(x0, from(tail(z0)))) -> c22(A__FILTER(mark(x0), a__from(a__tail(mark(z0)))), MARK(x0)) MARK(filter(x0, from(if(z0, z1, z2)))) -> c22(A__FILTER(mark(x0), a__from(a__if(mark(z0), z1, z2))), MARK(x0)) MARK(filter(x0, from(filter(z0, z1)))) -> c22(A__FILTER(mark(x0), a__from(a__filter(mark(z0), mark(z1)))), MARK(x0)) MARK(filter(x0, from(s(z0)))) -> c22(A__FILTER(mark(x0), a__from(s(mark(z0)))), MARK(x0)) MARK(filter(x0, from(0))) -> c22(A__FILTER(mark(x0), a__from(0)), MARK(x0)) MARK(filter(x0, from(cons(z0, z1)))) -> c22(A__FILTER(mark(x0), a__from(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, from(true))) -> c22(A__FILTER(mark(x0), a__from(true)), MARK(x0)) MARK(filter(x0, from(false))) -> c22(A__FILTER(mark(x0), a__from(false)), MARK(x0)) MARK(filter(x0, from(divides(z0, z1)))) -> c22(A__FILTER(mark(x0), a__from(divides(mark(z0), mark(z1)))), MARK(x0)) MARK(filter(primes, from(x1))) -> c22(A__FILTER(a__primes, a__from(mark(x1))), MARK(primes)) MARK(filter(sieve(z0), from(x1))) -> c22(A__FILTER(a__sieve(mark(z0)), a__from(mark(x1))), MARK(sieve(z0))) MARK(filter(from(z0), from(x1))) -> c22(A__FILTER(a__from(mark(z0)), a__from(mark(x1))), MARK(from(z0))) MARK(filter(head(z0), from(x1))) -> c22(A__FILTER(a__head(mark(z0)), a__from(mark(x1))), MARK(head(z0))) MARK(filter(tail(z0), from(x1))) -> c22(A__FILTER(a__tail(mark(z0)), a__from(mark(x1))), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), from(x1))) -> c22(A__FILTER(a__if(mark(z0), z1, z2), a__from(mark(x1))), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), from(x1))) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), a__from(mark(x1))), MARK(filter(z0, z1))) MARK(filter(s(z0), from(x1))) -> c22(A__FILTER(s(mark(z0)), a__from(mark(x1))), MARK(s(z0))) MARK(filter(x0, from(x1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), from(x1))) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), from(x1))) -> c22(MARK(divides(z0, z1))) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c18_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_1, c17_2, c17_1, c19_2, c19_1, c20_2, c20_1, c21_1, c22_2, c22_1, c23_2, c23_1, c_2, c21_2 ---------------------------------------- (229) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(filter(x0, head(z0))) -> c22(A__FILTER(mark(x0), a__head(mark(z0))), MARK(x0)) by MARK(filter(x0, head(x1))) -> c22(A__FILTER(mark(x0), head(mark(x1))), MARK(x0)) MARK(filter(x0, head(primes))) -> c22(A__FILTER(mark(x0), a__head(a__primes)), MARK(x0)) MARK(filter(x0, head(sieve(z0)))) -> c22(A__FILTER(mark(x0), a__head(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, head(from(z0)))) -> c22(A__FILTER(mark(x0), a__head(a__from(mark(z0)))), MARK(x0)) MARK(filter(x0, head(head(z0)))) -> c22(A__FILTER(mark(x0), a__head(a__head(mark(z0)))), MARK(x0)) MARK(filter(x0, head(tail(z0)))) -> c22(A__FILTER(mark(x0), a__head(a__tail(mark(z0)))), MARK(x0)) MARK(filter(x0, head(if(z0, z1, z2)))) -> c22(A__FILTER(mark(x0), a__head(a__if(mark(z0), z1, z2))), MARK(x0)) MARK(filter(x0, head(filter(z0, z1)))) -> c22(A__FILTER(mark(x0), a__head(a__filter(mark(z0), mark(z1)))), MARK(x0)) MARK(filter(x0, head(s(z0)))) -> c22(A__FILTER(mark(x0), a__head(s(mark(z0)))), MARK(x0)) MARK(filter(x0, head(0))) -> c22(A__FILTER(mark(x0), a__head(0)), MARK(x0)) MARK(filter(x0, head(cons(z0, z1)))) -> c22(A__FILTER(mark(x0), a__head(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, head(true))) -> c22(A__FILTER(mark(x0), a__head(true)), MARK(x0)) MARK(filter(x0, head(false))) -> c22(A__FILTER(mark(x0), a__head(false)), MARK(x0)) MARK(filter(x0, head(divides(z0, z1)))) -> c22(A__FILTER(mark(x0), a__head(divides(mark(z0), mark(z1)))), MARK(x0)) MARK(filter(primes, head(x1))) -> c22(A__FILTER(a__primes, a__head(mark(x1))), MARK(primes)) MARK(filter(sieve(z0), head(x1))) -> c22(A__FILTER(a__sieve(mark(z0)), a__head(mark(x1))), MARK(sieve(z0))) MARK(filter(from(z0), head(x1))) -> c22(A__FILTER(a__from(mark(z0)), a__head(mark(x1))), MARK(from(z0))) MARK(filter(head(z0), head(x1))) -> c22(A__FILTER(a__head(mark(z0)), a__head(mark(x1))), MARK(head(z0))) MARK(filter(tail(z0), head(x1))) -> c22(A__FILTER(a__tail(mark(z0)), a__head(mark(x1))), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), head(x1))) -> c22(A__FILTER(a__if(mark(z0), z1, z2), a__head(mark(x1))), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), head(x1))) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), a__head(mark(x1))), MARK(filter(z0, z1))) MARK(filter(s(z0), head(x1))) -> c22(A__FILTER(s(mark(z0)), a__head(mark(x1))), MARK(s(z0))) MARK(filter(0, head(x1))) -> c22(A__FILTER(0, a__head(mark(x1))), MARK(0)) MARK(filter(cons(z0, z1), head(x1))) -> c22(A__FILTER(cons(mark(z0), z1), a__head(mark(x1))), MARK(cons(z0, z1))) MARK(filter(true, head(x1))) -> c22(A__FILTER(true, a__head(mark(x1))), MARK(true)) MARK(filter(false, head(x1))) -> c22(A__FILTER(false, a__head(mark(x1))), MARK(false)) MARK(filter(divides(z0, z1), head(x1))) -> c22(A__FILTER(divides(mark(z0), mark(z1)), a__head(mark(x1))), MARK(divides(z0, z1))) ---------------------------------------- (230) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, tail(z0))) -> c22(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(x0)) MARK(filter(x0, if(z0, z1, z2))) -> c22(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0)) MARK(filter(x0, filter(z0, z1))) -> c22(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(x0)) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(x0, s(z0))) -> c22(MARK(x0)) MARK(filter(x0, 0)) -> c22(MARK(x0)) MARK(filter(x0, true)) -> c22(MARK(x0)) MARK(filter(x0, false)) -> c22(MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), x1)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c23(A__FILTER(mark(x0), a__primes), MARK(primes)) MARK(filter(x0, sieve(z0))) -> c23(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, from(z0))) -> c23(A__FILTER(mark(x0), a__from(mark(z0))), MARK(from(z0))) MARK(filter(x0, head(z0))) -> c23(A__FILTER(mark(x0), a__head(mark(z0))), MARK(head(z0))) MARK(filter(x0, tail(z0))) -> c23(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(tail(z0))) MARK(filter(x0, if(z0, z1, z2))) -> c23(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(filter(x0, filter(z0, z1))) -> c23(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(filter(x0, cons(z0, z1))) -> c23(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(primes, x1)) -> c23(A__FILTER(a__primes, mark(x1)), MARK(x1)) MARK(filter(sieve(z0), x1)) -> c23(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(from(z0), x1)) -> c23(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(head(z0), x1)) -> c23(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(tail(z0), x1)) -> c23(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(if(z0, z1, z2), x1)) -> c23(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1)) MARK(filter(filter(z0, z1), x1)) -> c23(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(x1)) MARK(filter(s(z0), x1)) -> c23(A__FILTER(s(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(x0, s(z0))) -> c23(MARK(s(z0))) MARK(filter(x0, divides(z0, z1))) -> c23(MARK(divides(z0, z1))) MARK(filter(0, x1)) -> c23(MARK(x1)) MARK(filter(cons(z0, z1), x1)) -> c23(MARK(x1)) MARK(filter(true, x1)) -> c23(MARK(x1)) MARK(filter(false, x1)) -> c23(MARK(x1)) MARK(filter(divides(z0, z1), x1)) -> c23(MARK(x1)) A__PRIMES -> c(A__SIEVE(cons(s(mark(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(sieve(primes)) -> c17(MARK(primes)) MARK(sieve(sieve(primes))) -> c17(A__SIEVE(a__sieve(a__primes)), MARK(sieve(primes))) MARK(sieve(sieve(sieve(z0)))) -> c17(A__SIEVE(a__sieve(a__sieve(mark(z0)))), MARK(sieve(sieve(z0)))) MARK(sieve(sieve(from(z0)))) -> c17(A__SIEVE(a__sieve(a__from(mark(z0)))), MARK(sieve(from(z0)))) MARK(sieve(sieve(head(z0)))) -> c17(A__SIEVE(a__sieve(a__head(mark(z0)))), MARK(sieve(head(z0)))) MARK(sieve(sieve(tail(z0)))) -> c17(A__SIEVE(a__sieve(a__tail(mark(z0)))), MARK(sieve(tail(z0)))) MARK(sieve(sieve(if(z0, z1, z2)))) -> c17(A__SIEVE(a__sieve(a__if(mark(z0), z1, z2))), MARK(sieve(if(z0, z1, z2)))) MARK(sieve(sieve(filter(z0, z1)))) -> c17(A__SIEVE(a__sieve(a__filter(mark(z0), mark(z1)))), MARK(sieve(filter(z0, z1)))) MARK(sieve(sieve(s(z0)))) -> c17(A__SIEVE(a__sieve(s(mark(z0)))), MARK(sieve(s(z0)))) MARK(sieve(sieve(cons(z0, z1)))) -> c17(A__SIEVE(a__sieve(cons(mark(z0), z1))), MARK(sieve(cons(z0, z1)))) MARK(sieve(sieve(divides(z0, 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a__head(mark(x1))), MARK(filter(z0, z1))) MARK(filter(s(z0), head(x1))) -> c22(A__FILTER(s(mark(z0)), a__head(mark(x1))), MARK(s(z0))) MARK(filter(0, head(x1))) -> c22(A__FILTER(0, a__head(mark(x1))), MARK(0)) MARK(filter(cons(z0, z1), head(x1))) -> c22(A__FILTER(cons(mark(z0), z1), a__head(mark(x1))), MARK(cons(z0, z1))) MARK(filter(true, head(x1))) -> c22(A__FILTER(true, a__head(mark(x1))), MARK(true)) MARK(filter(false, head(x1))) -> c22(A__FILTER(false, a__head(mark(x1))), MARK(false)) MARK(filter(divides(z0, z1), head(x1))) -> c22(A__FILTER(divides(mark(z0), mark(z1)), a__head(mark(x1))), MARK(divides(z0, z1))) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c18_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_1, c17_2, c17_1, c19_2, c19_1, c20_2, c20_1, c21_1, c22_2, c22_1, c23_2, c23_1, c_2, c21_2 ---------------------------------------- (231) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing nodes: MARK(filter(0, head(x1))) -> c22(A__FILTER(0, a__head(mark(x1))), MARK(0)) MARK(filter(true, head(x1))) -> c22(A__FILTER(true, a__head(mark(x1))), MARK(true)) MARK(filter(false, head(x1))) -> c22(A__FILTER(false, a__head(mark(x1))), MARK(false)) ---------------------------------------- (232) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, tail(z0))) -> c22(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(x0)) MARK(filter(x0, if(z0, z1, z2))) -> c22(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0)) MARK(filter(x0, filter(z0, z1))) -> c22(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(x0)) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(x0, s(z0))) -> c22(MARK(x0)) MARK(filter(x0, 0)) -> c22(MARK(x0)) MARK(filter(x0, true)) -> c22(MARK(x0)) MARK(filter(x0, false)) -> c22(MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), x1)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c23(A__FILTER(mark(x0), a__primes), MARK(primes)) MARK(filter(x0, sieve(z0))) -> c23(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, from(z0))) -> c23(A__FILTER(mark(x0), a__from(mark(z0))), MARK(from(z0))) MARK(filter(x0, head(z0))) -> c23(A__FILTER(mark(x0), a__head(mark(z0))), MARK(head(z0))) MARK(filter(x0, tail(z0))) -> c23(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(tail(z0))) MARK(filter(x0, if(z0, z1, z2))) -> c23(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(filter(x0, filter(z0, z1))) -> c23(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(filter(x0, cons(z0, z1))) -> c23(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(primes, x1)) -> c23(A__FILTER(a__primes, mark(x1)), MARK(x1)) MARK(filter(sieve(z0), x1)) -> c23(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(from(z0), x1)) -> c23(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(head(z0), x1)) -> c23(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(tail(z0), x1)) -> c23(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(if(z0, z1, z2), x1)) -> c23(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1)) MARK(filter(filter(z0, z1), x1)) -> c23(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(x1)) MARK(filter(s(z0), x1)) -> c23(A__FILTER(s(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(x0, s(z0))) -> c23(MARK(s(z0))) MARK(filter(x0, divides(z0, z1))) -> c23(MARK(divides(z0, z1))) MARK(filter(0, x1)) -> c23(MARK(x1)) MARK(filter(cons(z0, z1), x1)) -> c23(MARK(x1)) MARK(filter(true, x1)) -> c23(MARK(x1)) MARK(filter(false, x1)) -> c23(MARK(x1)) MARK(filter(divides(z0, z1), x1)) -> c23(MARK(x1)) A__PRIMES -> c(A__SIEVE(cons(s(mark(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(sieve(primes)) -> c17(MARK(primes)) MARK(sieve(sieve(primes))) -> c17(A__SIEVE(a__sieve(a__primes)), MARK(sieve(primes))) MARK(sieve(sieve(sieve(z0)))) -> c17(A__SIEVE(a__sieve(a__sieve(mark(z0)))), MARK(sieve(sieve(z0)))) MARK(sieve(sieve(from(z0)))) -> c17(A__SIEVE(a__sieve(a__from(mark(z0)))), MARK(sieve(from(z0)))) MARK(sieve(sieve(head(z0)))) -> c17(A__SIEVE(a__sieve(a__head(mark(z0)))), MARK(sieve(head(z0)))) MARK(sieve(sieve(tail(z0)))) -> c17(A__SIEVE(a__sieve(a__tail(mark(z0)))), MARK(sieve(tail(z0)))) MARK(sieve(sieve(if(z0, z1, z2)))) -> c17(A__SIEVE(a__sieve(a__if(mark(z0), z1, z2))), MARK(sieve(if(z0, z1, z2)))) MARK(sieve(sieve(filter(z0, z1)))) -> c17(A__SIEVE(a__sieve(a__filter(mark(z0), mark(z1)))), MARK(sieve(filter(z0, z1)))) MARK(sieve(sieve(s(z0)))) -> c17(A__SIEVE(a__sieve(s(mark(z0)))), MARK(sieve(s(z0)))) MARK(sieve(sieve(cons(z0, z1)))) -> c17(A__SIEVE(a__sieve(cons(mark(z0), z1))), MARK(sieve(cons(z0, z1)))) MARK(sieve(sieve(divides(z0, z1)))) -> c17(A__SIEVE(a__sieve(divides(mark(z0), mark(z1)))), MARK(sieve(divides(z0, z1)))) MARK(sieve(sieve(x0))) -> c17(MARK(sieve(x0))) MARK(sieve(sieve(0))) -> c17(A__SIEVE(a__sieve(0))) MARK(sieve(sieve(true))) -> c17(A__SIEVE(a__sieve(true))) MARK(sieve(sieve(false))) -> c17(A__SIEVE(a__sieve(false))) MARK(sieve(from(x0))) -> c17(A__SIEVE(cons(mark(mark(x0)), from(s(mark(x0))))), MARK(from(x0))) MARK(sieve(from(primes))) -> c17(A__SIEVE(a__from(a__primes)), MARK(from(primes))) MARK(sieve(from(sieve(z0)))) -> c17(A__SIEVE(a__from(a__sieve(mark(z0)))), MARK(from(sieve(z0)))) MARK(sieve(from(from(z0)))) -> c17(A__SIEVE(a__from(a__from(mark(z0)))), MARK(from(from(z0)))) MARK(sieve(from(head(z0)))) -> c17(A__SIEVE(a__from(a__head(mark(z0)))), MARK(from(head(z0)))) MARK(sieve(from(tail(z0)))) -> c17(A__SIEVE(a__from(a__tail(mark(z0)))), MARK(from(tail(z0)))) MARK(sieve(from(if(z0, z1, z2)))) -> c17(A__SIEVE(a__from(a__if(mark(z0), z1, z2))), MARK(from(if(z0, z1, z2)))) MARK(sieve(from(filter(z0, z1)))) -> c17(A__SIEVE(a__from(a__filter(mark(z0), mark(z1)))), MARK(from(filter(z0, z1)))) MARK(sieve(from(s(z0)))) -> c17(A__SIEVE(a__from(s(mark(z0)))), MARK(from(s(z0)))) MARK(sieve(from(0))) -> c17(A__SIEVE(a__from(0)), MARK(from(0))) MARK(sieve(from(cons(z0, z1)))) -> c17(A__SIEVE(a__from(cons(mark(z0), z1))), MARK(from(cons(z0, z1)))) MARK(sieve(from(true))) -> c17(A__SIEVE(a__from(true)), MARK(from(true))) MARK(sieve(from(false))) -> c17(A__SIEVE(a__from(false)), MARK(from(false))) MARK(sieve(from(divides(z0, z1)))) -> c17(A__SIEVE(a__from(divides(mark(z0), mark(z1)))), MARK(from(divides(z0, z1)))) MARK(sieve(from(x0))) -> c17(MARK(from(x0))) MARK(sieve(head(primes))) -> c17(A__SIEVE(a__head(a__primes)), MARK(head(primes))) MARK(sieve(head(sieve(z0)))) -> c17(A__SIEVE(a__head(a__sieve(mark(z0)))), MARK(head(sieve(z0)))) MARK(sieve(head(from(z0)))) -> c17(A__SIEVE(a__head(a__from(mark(z0)))), MARK(head(from(z0)))) MARK(sieve(head(head(z0)))) -> c17(A__SIEVE(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(sieve(head(tail(z0)))) -> c17(A__SIEVE(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(sieve(head(if(z0, z1, z2)))) -> c17(A__SIEVE(a__head(a__if(mark(z0), z1, z2))), MARK(head(if(z0, z1, z2)))) MARK(sieve(head(filter(z0, z1)))) -> c17(A__SIEVE(a__head(a__filter(mark(z0), mark(z1)))), MARK(head(filter(z0, z1)))) MARK(sieve(head(s(z0)))) -> c17(A__SIEVE(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(sieve(head(cons(z0, z1)))) -> c17(A__SIEVE(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(sieve(head(divides(z0, z1)))) -> c17(A__SIEVE(a__head(divides(mark(z0), mark(z1)))), MARK(head(divides(z0, z1)))) MARK(sieve(head(x0))) -> c17(MARK(head(x0))) MARK(sieve(head(0))) -> c17(A__SIEVE(a__head(0))) MARK(sieve(head(true))) -> c17(A__SIEVE(a__head(true))) MARK(sieve(head(false))) -> c17(A__SIEVE(a__head(false))) MARK(sieve(tail(primes))) -> c17(A__SIEVE(a__tail(a__primes)), MARK(tail(primes))) MARK(sieve(tail(sieve(z0)))) -> c17(A__SIEVE(a__tail(a__sieve(mark(z0)))), MARK(tail(sieve(z0)))) MARK(sieve(tail(from(z0)))) -> c17(A__SIEVE(a__tail(a__from(mark(z0)))), MARK(tail(from(z0)))) MARK(sieve(tail(head(z0)))) -> c17(A__SIEVE(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(sieve(tail(tail(z0)))) -> c17(A__SIEVE(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(sieve(tail(if(z0, z1, z2)))) -> c17(A__SIEVE(a__tail(a__if(mark(z0), z1, z2))), MARK(tail(if(z0, z1, z2)))) MARK(sieve(tail(filter(z0, z1)))) -> c17(A__SIEVE(a__tail(a__filter(mark(z0), mark(z1)))), MARK(tail(filter(z0, z1)))) 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MARK(if(filter(tail(z0), x1), x2, x3)) -> c21(A__IF(a__filter(a__tail(mark(z0)), mark(x1)), x2, x3), MARK(filter(tail(z0), x1))) MARK(if(filter(if(z0, z1, z2), x1), x2, x3)) -> c21(A__IF(a__filter(a__if(mark(z0), z1, z2), mark(x1)), x2, x3), MARK(filter(if(z0, z1, z2), x1))) MARK(if(filter(filter(z0, z1), x1), x2, x3)) -> c21(A__IF(a__filter(a__filter(mark(z0), mark(z1)), mark(x1)), x2, x3), MARK(filter(filter(z0, z1), x1))) MARK(if(filter(s(z0), x1), x2, x3)) -> c21(A__IF(a__filter(s(mark(z0)), mark(x1)), x2, x3), MARK(filter(s(z0), x1))) MARK(if(filter(0, x1), x2, x3)) -> c21(A__IF(a__filter(0, mark(x1)), x2, x3), MARK(filter(0, x1))) MARK(if(filter(cons(z0, z1), x1), x2, x3)) -> c21(A__IF(a__filter(cons(mark(z0), z1), mark(x1)), x2, x3), MARK(filter(cons(z0, z1), x1))) MARK(if(filter(true, x1), x2, x3)) -> c21(A__IF(a__filter(true, mark(x1)), x2, x3), MARK(filter(true, x1))) MARK(if(filter(false, x1), x2, x3)) -> c21(A__IF(a__filter(false, mark(x1)), x2, x3), MARK(filter(false, x1))) MARK(if(filter(divides(z0, z1), x1), x2, x3)) -> c21(A__IF(a__filter(divides(mark(z0), mark(z1)), mark(x1)), x2, x3), MARK(filter(divides(z0, z1), x1))) MARK(if(filter(x0, x1), x2, x3)) -> c21(MARK(filter(x0, x1))) MARK(filter(x0, primes)) -> c22(A__FILTER(mark(x0), a__sieve(a__from(s(s(0))))), MARK(x0)) MARK(filter(primes, primes)) -> c22(A__FILTER(a__primes, a__primes), MARK(primes)) MARK(filter(sieve(z0), primes)) -> c22(A__FILTER(a__sieve(mark(z0)), a__primes), MARK(sieve(z0))) MARK(filter(from(z0), primes)) -> c22(A__FILTER(a__from(mark(z0)), a__primes), MARK(from(z0))) MARK(filter(head(z0), primes)) -> c22(A__FILTER(a__head(mark(z0)), a__primes), MARK(head(z0))) MARK(filter(tail(z0), primes)) -> c22(A__FILTER(a__tail(mark(z0)), a__primes), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), primes)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), a__primes), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), primes)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), a__primes), MARK(filter(z0, z1))) MARK(filter(s(z0), primes)) -> c22(A__FILTER(s(mark(z0)), a__primes), MARK(s(z0))) MARK(filter(x0, primes)) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), primes)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), primes)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, sieve(primes))) -> c22(A__FILTER(mark(x0), a__sieve(a__primes)), MARK(x0)) MARK(filter(x0, sieve(sieve(z0)))) -> c22(A__FILTER(mark(x0), a__sieve(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, sieve(from(z0)))) -> c22(A__FILTER(mark(x0), a__sieve(a__from(mark(z0)))), MARK(x0)) MARK(filter(x0, sieve(head(z0)))) -> c22(A__FILTER(mark(x0), a__sieve(a__head(mark(z0)))), MARK(x0)) MARK(filter(x0, sieve(tail(z0)))) -> c22(A__FILTER(mark(x0), a__sieve(a__tail(mark(z0)))), MARK(x0)) MARK(filter(x0, sieve(if(z0, z1, z2)))) -> c22(A__FILTER(mark(x0), a__sieve(a__if(mark(z0), z1, z2))), MARK(x0)) MARK(filter(x0, sieve(filter(z0, z1)))) -> c22(A__FILTER(mark(x0), a__sieve(a__filter(mark(z0), mark(z1)))), MARK(x0)) MARK(filter(x0, sieve(s(z0)))) -> c22(A__FILTER(mark(x0), a__sieve(s(mark(z0)))), MARK(x0)) MARK(filter(x0, sieve(0))) -> c22(A__FILTER(mark(x0), a__sieve(0)), MARK(x0)) MARK(filter(x0, sieve(cons(z0, z1)))) -> c22(A__FILTER(mark(x0), a__sieve(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, sieve(true))) -> c22(A__FILTER(mark(x0), a__sieve(true)), MARK(x0)) MARK(filter(x0, sieve(false))) -> c22(A__FILTER(mark(x0), a__sieve(false)), MARK(x0)) MARK(filter(x0, sieve(divides(z0, z1)))) -> c22(A__FILTER(mark(x0), a__sieve(divides(mark(z0), mark(z1)))), MARK(x0)) MARK(filter(primes, sieve(x1))) -> c22(A__FILTER(a__primes, a__sieve(mark(x1))), MARK(primes)) MARK(filter(sieve(z0), sieve(x1))) -> c22(A__FILTER(a__sieve(mark(z0)), a__sieve(mark(x1))), MARK(sieve(z0))) MARK(filter(from(z0), sieve(x1))) -> c22(A__FILTER(a__from(mark(z0)), a__sieve(mark(x1))), MARK(from(z0))) MARK(filter(head(z0), sieve(x1))) -> c22(A__FILTER(a__head(mark(z0)), a__sieve(mark(x1))), MARK(head(z0))) MARK(filter(tail(z0), sieve(x1))) -> c22(A__FILTER(a__tail(mark(z0)), a__sieve(mark(x1))), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), sieve(x1))) -> c22(A__FILTER(a__if(mark(z0), z1, z2), a__sieve(mark(x1))), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), sieve(x1))) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), a__sieve(mark(x1))), MARK(filter(z0, z1))) MARK(filter(s(z0), sieve(x1))) -> c22(A__FILTER(s(mark(z0)), a__sieve(mark(x1))), MARK(s(z0))) MARK(filter(x0, sieve(x1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), sieve(x1))) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), sieve(x1))) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, from(x1))) -> c22(A__FILTER(mark(x0), cons(mark(mark(x1)), from(s(mark(x1))))), MARK(x0)) MARK(filter(x0, from(primes))) -> c22(A__FILTER(mark(x0), a__from(a__primes)), MARK(x0)) MARK(filter(x0, from(sieve(z0)))) -> c22(A__FILTER(mark(x0), a__from(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, from(from(z0)))) -> c22(A__FILTER(mark(x0), a__from(a__from(mark(z0)))), MARK(x0)) MARK(filter(x0, from(head(z0)))) -> c22(A__FILTER(mark(x0), a__from(a__head(mark(z0)))), MARK(x0)) MARK(filter(x0, from(tail(z0)))) -> c22(A__FILTER(mark(x0), a__from(a__tail(mark(z0)))), MARK(x0)) MARK(filter(x0, from(if(z0, z1, z2)))) -> c22(A__FILTER(mark(x0), a__from(a__if(mark(z0), z1, z2))), MARK(x0)) MARK(filter(x0, from(filter(z0, z1)))) -> c22(A__FILTER(mark(x0), a__from(a__filter(mark(z0), mark(z1)))), MARK(x0)) MARK(filter(x0, from(s(z0)))) -> c22(A__FILTER(mark(x0), a__from(s(mark(z0)))), MARK(x0)) MARK(filter(x0, from(0))) -> c22(A__FILTER(mark(x0), a__from(0)), MARK(x0)) MARK(filter(x0, from(cons(z0, z1)))) -> c22(A__FILTER(mark(x0), a__from(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, from(true))) -> c22(A__FILTER(mark(x0), a__from(true)), MARK(x0)) MARK(filter(x0, from(false))) -> c22(A__FILTER(mark(x0), a__from(false)), MARK(x0)) MARK(filter(x0, from(divides(z0, z1)))) -> c22(A__FILTER(mark(x0), a__from(divides(mark(z0), mark(z1)))), MARK(x0)) MARK(filter(primes, from(x1))) -> c22(A__FILTER(a__primes, a__from(mark(x1))), MARK(primes)) MARK(filter(sieve(z0), from(x1))) -> c22(A__FILTER(a__sieve(mark(z0)), a__from(mark(x1))), MARK(sieve(z0))) MARK(filter(from(z0), from(x1))) -> c22(A__FILTER(a__from(mark(z0)), a__from(mark(x1))), MARK(from(z0))) MARK(filter(head(z0), from(x1))) -> c22(A__FILTER(a__head(mark(z0)), a__from(mark(x1))), MARK(head(z0))) MARK(filter(tail(z0), from(x1))) -> c22(A__FILTER(a__tail(mark(z0)), a__from(mark(x1))), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), from(x1))) -> c22(A__FILTER(a__if(mark(z0), z1, z2), a__from(mark(x1))), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), from(x1))) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), a__from(mark(x1))), MARK(filter(z0, z1))) MARK(filter(s(z0), from(x1))) -> c22(A__FILTER(s(mark(z0)), a__from(mark(x1))), MARK(s(z0))) MARK(filter(x0, from(x1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), from(x1))) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), from(x1))) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, head(x1))) -> c22(A__FILTER(mark(x0), head(mark(x1))), MARK(x0)) MARK(filter(x0, head(primes))) -> c22(A__FILTER(mark(x0), a__head(a__primes)), MARK(x0)) MARK(filter(x0, head(sieve(z0)))) -> c22(A__FILTER(mark(x0), a__head(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, head(from(z0)))) -> c22(A__FILTER(mark(x0), a__head(a__from(mark(z0)))), MARK(x0)) MARK(filter(x0, head(head(z0)))) -> c22(A__FILTER(mark(x0), a__head(a__head(mark(z0)))), MARK(x0)) MARK(filter(x0, head(tail(z0)))) -> c22(A__FILTER(mark(x0), a__head(a__tail(mark(z0)))), MARK(x0)) MARK(filter(x0, head(if(z0, z1, z2)))) -> c22(A__FILTER(mark(x0), a__head(a__if(mark(z0), z1, z2))), MARK(x0)) MARK(filter(x0, head(filter(z0, z1)))) -> c22(A__FILTER(mark(x0), a__head(a__filter(mark(z0), mark(z1)))), MARK(x0)) MARK(filter(x0, head(s(z0)))) -> c22(A__FILTER(mark(x0), a__head(s(mark(z0)))), MARK(x0)) MARK(filter(x0, head(0))) -> c22(A__FILTER(mark(x0), a__head(0)), MARK(x0)) MARK(filter(x0, head(cons(z0, z1)))) -> c22(A__FILTER(mark(x0), a__head(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, head(true))) -> c22(A__FILTER(mark(x0), a__head(true)), MARK(x0)) MARK(filter(x0, head(false))) -> c22(A__FILTER(mark(x0), a__head(false)), MARK(x0)) MARK(filter(x0, head(divides(z0, z1)))) -> c22(A__FILTER(mark(x0), a__head(divides(mark(z0), mark(z1)))), MARK(x0)) MARK(filter(primes, head(x1))) -> c22(A__FILTER(a__primes, a__head(mark(x1))), MARK(primes)) MARK(filter(sieve(z0), head(x1))) -> c22(A__FILTER(a__sieve(mark(z0)), a__head(mark(x1))), MARK(sieve(z0))) MARK(filter(from(z0), head(x1))) -> c22(A__FILTER(a__from(mark(z0)), a__head(mark(x1))), MARK(from(z0))) MARK(filter(head(z0), head(x1))) -> c22(A__FILTER(a__head(mark(z0)), a__head(mark(x1))), MARK(head(z0))) MARK(filter(tail(z0), head(x1))) -> c22(A__FILTER(a__tail(mark(z0)), a__head(mark(x1))), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), head(x1))) -> c22(A__FILTER(a__if(mark(z0), z1, z2), a__head(mark(x1))), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), head(x1))) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), a__head(mark(x1))), MARK(filter(z0, z1))) MARK(filter(s(z0), head(x1))) -> c22(A__FILTER(s(mark(z0)), a__head(mark(x1))), MARK(s(z0))) MARK(filter(cons(z0, z1), head(x1))) -> c22(A__FILTER(cons(mark(z0), z1), a__head(mark(x1))), MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), head(x1))) -> c22(A__FILTER(divides(mark(z0), mark(z1)), a__head(mark(x1))), MARK(divides(z0, z1))) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c18_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_1, c17_2, c17_1, c19_2, c19_1, c20_2, c20_1, c21_1, c22_2, c22_1, c23_2, c23_1, c_2, c21_2 ---------------------------------------- (233) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (234) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, tail(z0))) -> c22(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(x0)) MARK(filter(x0, if(z0, z1, z2))) -> c22(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0)) MARK(filter(x0, filter(z0, z1))) -> c22(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(x0)) MARK(filter(x0, cons(z0, z1))) -> 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c22(MARK(divides(z0, z1))) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c18_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_1, c17_2, c17_1, c19_2, c19_1, c20_2, c20_1, c21_1, c22_2, c22_1, c23_2, c23_1, c_2, c21_2 ---------------------------------------- (235) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(filter(x0, tail(z0))) -> c22(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(x0)) by MARK(filter(x0, tail(x1))) -> c22(A__FILTER(mark(x0), tail(mark(x1))), MARK(x0)) MARK(filter(x0, tail(primes))) -> c22(A__FILTER(mark(x0), a__tail(a__primes)), MARK(x0)) MARK(filter(x0, tail(sieve(z0)))) -> c22(A__FILTER(mark(x0), a__tail(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, tail(from(z0)))) -> c22(A__FILTER(mark(x0), a__tail(a__from(mark(z0)))), MARK(x0)) MARK(filter(x0, tail(head(z0)))) -> c22(A__FILTER(mark(x0), a__tail(a__head(mark(z0)))), MARK(x0)) MARK(filter(x0, tail(tail(z0)))) -> c22(A__FILTER(mark(x0), a__tail(a__tail(mark(z0)))), MARK(x0)) MARK(filter(x0, tail(if(z0, z1, z2)))) -> c22(A__FILTER(mark(x0), a__tail(a__if(mark(z0), z1, z2))), MARK(x0)) MARK(filter(x0, tail(filter(z0, z1)))) -> c22(A__FILTER(mark(x0), a__tail(a__filter(mark(z0), mark(z1)))), MARK(x0)) MARK(filter(x0, tail(s(z0)))) -> c22(A__FILTER(mark(x0), a__tail(s(mark(z0)))), MARK(x0)) MARK(filter(x0, tail(0))) -> c22(A__FILTER(mark(x0), a__tail(0)), MARK(x0)) MARK(filter(x0, tail(cons(z0, z1)))) -> c22(A__FILTER(mark(x0), a__tail(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, tail(true))) -> c22(A__FILTER(mark(x0), a__tail(true)), MARK(x0)) MARK(filter(x0, tail(false))) -> c22(A__FILTER(mark(x0), a__tail(false)), MARK(x0)) MARK(filter(x0, tail(divides(z0, z1)))) -> c22(A__FILTER(mark(x0), a__tail(divides(mark(z0), mark(z1)))), MARK(x0)) MARK(filter(primes, tail(x1))) -> c22(A__FILTER(a__primes, a__tail(mark(x1))), MARK(primes)) MARK(filter(sieve(z0), tail(x1))) -> c22(A__FILTER(a__sieve(mark(z0)), a__tail(mark(x1))), MARK(sieve(z0))) MARK(filter(from(z0), tail(x1))) -> c22(A__FILTER(a__from(mark(z0)), a__tail(mark(x1))), MARK(from(z0))) MARK(filter(head(z0), tail(x1))) -> c22(A__FILTER(a__head(mark(z0)), a__tail(mark(x1))), MARK(head(z0))) MARK(filter(tail(z0), tail(x1))) -> c22(A__FILTER(a__tail(mark(z0)), a__tail(mark(x1))), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), tail(x1))) -> c22(A__FILTER(a__if(mark(z0), z1, z2), a__tail(mark(x1))), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), tail(x1))) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), a__tail(mark(x1))), MARK(filter(z0, z1))) MARK(filter(s(z0), tail(x1))) -> c22(A__FILTER(s(mark(z0)), a__tail(mark(x1))), MARK(s(z0))) MARK(filter(0, tail(x1))) -> c22(A__FILTER(0, a__tail(mark(x1))), MARK(0)) MARK(filter(cons(z0, z1), tail(x1))) -> c22(A__FILTER(cons(mark(z0), z1), a__tail(mark(x1))), MARK(cons(z0, z1))) MARK(filter(true, tail(x1))) -> c22(A__FILTER(true, a__tail(mark(x1))), MARK(true)) MARK(filter(false, tail(x1))) -> c22(A__FILTER(false, a__tail(mark(x1))), MARK(false)) MARK(filter(divides(z0, z1), tail(x1))) -> c22(A__FILTER(divides(mark(z0), mark(z1)), a__tail(mark(x1))), MARK(divides(z0, z1))) ---------------------------------------- (236) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, if(z0, z1, z2))) -> c22(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0)) MARK(filter(x0, filter(z0, z1))) -> c22(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(x0)) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(x0, s(z0))) -> c22(MARK(x0)) MARK(filter(x0, 0)) -> c22(MARK(x0)) MARK(filter(x0, true)) -> c22(MARK(x0)) MARK(filter(x0, false)) -> c22(MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), x1)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c23(A__FILTER(mark(x0), a__primes), MARK(primes)) MARK(filter(x0, sieve(z0))) -> c23(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, from(z0))) -> c23(A__FILTER(mark(x0), a__from(mark(z0))), MARK(from(z0))) MARK(filter(x0, head(z0))) -> c23(A__FILTER(mark(x0), a__head(mark(z0))), MARK(head(z0))) MARK(filter(x0, tail(z0))) -> c23(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(tail(z0))) MARK(filter(x0, if(z0, z1, z2))) -> c23(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(filter(x0, filter(z0, z1))) -> c23(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(filter(x0, cons(z0, z1))) -> c23(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(primes, x1)) -> c23(A__FILTER(a__primes, mark(x1)), MARK(x1)) MARK(filter(sieve(z0), x1)) -> c23(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(from(z0), x1)) -> c23(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(head(z0), x1)) -> c23(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(tail(z0), x1)) -> c23(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(if(z0, z1, z2), x1)) -> c23(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1)) MARK(filter(filter(z0, z1), x1)) -> c23(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(x1)) MARK(filter(s(z0), x1)) -> c23(A__FILTER(s(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(x0, s(z0))) -> c23(MARK(s(z0))) MARK(filter(x0, divides(z0, z1))) -> c23(MARK(divides(z0, z1))) MARK(filter(0, x1)) -> c23(MARK(x1)) MARK(filter(cons(z0, z1), x1)) -> c23(MARK(x1)) MARK(filter(true, x1)) -> c23(MARK(x1)) MARK(filter(false, x1)) -> c23(MARK(x1)) MARK(filter(divides(z0, z1), x1)) -> c23(MARK(x1)) A__PRIMES -> c(A__SIEVE(cons(s(mark(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(sieve(primes)) -> c17(MARK(primes)) MARK(sieve(sieve(primes))) -> c17(A__SIEVE(a__sieve(a__primes)), MARK(sieve(primes))) MARK(sieve(sieve(sieve(z0)))) -> c17(A__SIEVE(a__sieve(a__sieve(mark(z0)))), MARK(sieve(sieve(z0)))) MARK(sieve(sieve(from(z0)))) -> c17(A__SIEVE(a__sieve(a__from(mark(z0)))), MARK(sieve(from(z0)))) MARK(sieve(sieve(head(z0)))) -> c17(A__SIEVE(a__sieve(a__head(mark(z0)))), MARK(sieve(head(z0)))) MARK(sieve(sieve(tail(z0)))) -> c17(A__SIEVE(a__sieve(a__tail(mark(z0)))), MARK(sieve(tail(z0)))) MARK(sieve(sieve(if(z0, z1, z2)))) -> c17(A__SIEVE(a__sieve(a__if(mark(z0), z1, z2))), MARK(sieve(if(z0, z1, z2)))) MARK(sieve(sieve(filter(z0, z1)))) -> c17(A__SIEVE(a__sieve(a__filter(mark(z0), mark(z1)))), MARK(sieve(filter(z0, z1)))) MARK(sieve(sieve(s(z0)))) -> c17(A__SIEVE(a__sieve(s(mark(z0)))), MARK(sieve(s(z0)))) MARK(sieve(sieve(cons(z0, z1)))) -> c17(A__SIEVE(a__sieve(cons(mark(z0), z1))), MARK(sieve(cons(z0, z1)))) MARK(sieve(sieve(divides(z0, z1)))) -> c17(A__SIEVE(a__sieve(divides(mark(z0), mark(z1)))), MARK(sieve(divides(z0, z1)))) MARK(sieve(sieve(x0))) -> c17(MARK(sieve(x0))) MARK(sieve(sieve(0))) -> c17(A__SIEVE(a__sieve(0))) MARK(sieve(sieve(true))) -> c17(A__SIEVE(a__sieve(true))) MARK(sieve(sieve(false))) -> c17(A__SIEVE(a__sieve(false))) MARK(sieve(from(x0))) -> c17(A__SIEVE(cons(mark(mark(x0)), from(s(mark(x0))))), MARK(from(x0))) MARK(sieve(from(primes))) -> c17(A__SIEVE(a__from(a__primes)), MARK(from(primes))) MARK(sieve(from(sieve(z0)))) -> c17(A__SIEVE(a__from(a__sieve(mark(z0)))), MARK(from(sieve(z0)))) MARK(sieve(from(from(z0)))) -> c17(A__SIEVE(a__from(a__from(mark(z0)))), MARK(from(from(z0)))) MARK(sieve(from(head(z0)))) -> c17(A__SIEVE(a__from(a__head(mark(z0)))), MARK(from(head(z0)))) MARK(sieve(from(tail(z0)))) -> c17(A__SIEVE(a__from(a__tail(mark(z0)))), MARK(from(tail(z0)))) MARK(sieve(from(if(z0, z1, z2)))) -> c17(A__SIEVE(a__from(a__if(mark(z0), z1, z2))), MARK(from(if(z0, z1, z2)))) MARK(sieve(from(filter(z0, z1)))) -> c17(A__SIEVE(a__from(a__filter(mark(z0), mark(z1)))), MARK(from(filter(z0, z1)))) MARK(sieve(from(s(z0)))) -> c17(A__SIEVE(a__from(s(mark(z0)))), MARK(from(s(z0)))) MARK(sieve(from(0))) -> c17(A__SIEVE(a__from(0)), MARK(from(0))) MARK(sieve(from(cons(z0, z1)))) -> c17(A__SIEVE(a__from(cons(mark(z0), z1))), MARK(from(cons(z0, z1)))) MARK(sieve(from(true))) -> c17(A__SIEVE(a__from(true)), MARK(from(true))) MARK(sieve(from(false))) -> c17(A__SIEVE(a__from(false)), MARK(from(false))) MARK(sieve(from(divides(z0, z1)))) -> c17(A__SIEVE(a__from(divides(mark(z0), mark(z1)))), MARK(from(divides(z0, z1)))) MARK(sieve(from(x0))) -> c17(MARK(from(x0))) MARK(sieve(head(primes))) -> 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c21(A__IF(a__filter(cons(mark(z0), z1), mark(x1)), x2, x3), MARK(filter(cons(z0, z1), x1))) MARK(if(filter(true, x1), x2, x3)) -> c21(A__IF(a__filter(true, mark(x1)), x2, x3), MARK(filter(true, x1))) MARK(if(filter(false, x1), x2, x3)) -> c21(A__IF(a__filter(false, mark(x1)), x2, x3), MARK(filter(false, x1))) MARK(if(filter(divides(z0, z1), x1), x2, x3)) -> c21(A__IF(a__filter(divides(mark(z0), mark(z1)), mark(x1)), x2, x3), MARK(filter(divides(z0, z1), x1))) MARK(if(filter(x0, x1), x2, x3)) -> c21(MARK(filter(x0, x1))) MARK(filter(x0, primes)) -> c22(A__FILTER(mark(x0), a__sieve(a__from(s(s(0))))), MARK(x0)) MARK(filter(primes, primes)) -> c22(A__FILTER(a__primes, a__primes), MARK(primes)) MARK(filter(sieve(z0), primes)) -> c22(A__FILTER(a__sieve(mark(z0)), a__primes), MARK(sieve(z0))) MARK(filter(from(z0), primes)) -> c22(A__FILTER(a__from(mark(z0)), a__primes), MARK(from(z0))) MARK(filter(head(z0), primes)) -> c22(A__FILTER(a__head(mark(z0)), a__primes), MARK(head(z0))) 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sieve(x1))) -> c22(A__FILTER(a__sieve(mark(z0)), a__sieve(mark(x1))), MARK(sieve(z0))) MARK(filter(from(z0), sieve(x1))) -> c22(A__FILTER(a__from(mark(z0)), a__sieve(mark(x1))), MARK(from(z0))) MARK(filter(head(z0), sieve(x1))) -> c22(A__FILTER(a__head(mark(z0)), a__sieve(mark(x1))), MARK(head(z0))) MARK(filter(tail(z0), sieve(x1))) -> c22(A__FILTER(a__tail(mark(z0)), a__sieve(mark(x1))), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), sieve(x1))) -> c22(A__FILTER(a__if(mark(z0), z1, z2), a__sieve(mark(x1))), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), sieve(x1))) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), a__sieve(mark(x1))), MARK(filter(z0, z1))) MARK(filter(s(z0), sieve(x1))) -> c22(A__FILTER(s(mark(z0)), a__sieve(mark(x1))), MARK(s(z0))) MARK(filter(x0, sieve(x1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), sieve(x1))) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), sieve(x1))) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, from(x1))) -> c22(A__FILTER(mark(x0), 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a__from(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, from(true))) -> c22(A__FILTER(mark(x0), a__from(true)), MARK(x0)) MARK(filter(x0, from(false))) -> c22(A__FILTER(mark(x0), a__from(false)), MARK(x0)) MARK(filter(x0, from(divides(z0, z1)))) -> c22(A__FILTER(mark(x0), a__from(divides(mark(z0), mark(z1)))), MARK(x0)) MARK(filter(primes, from(x1))) -> c22(A__FILTER(a__primes, a__from(mark(x1))), MARK(primes)) MARK(filter(sieve(z0), from(x1))) -> c22(A__FILTER(a__sieve(mark(z0)), a__from(mark(x1))), MARK(sieve(z0))) MARK(filter(from(z0), from(x1))) -> c22(A__FILTER(a__from(mark(z0)), a__from(mark(x1))), MARK(from(z0))) MARK(filter(head(z0), from(x1))) -> c22(A__FILTER(a__head(mark(z0)), a__from(mark(x1))), MARK(head(z0))) MARK(filter(tail(z0), from(x1))) -> c22(A__FILTER(a__tail(mark(z0)), a__from(mark(x1))), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), from(x1))) -> c22(A__FILTER(a__if(mark(z0), z1, z2), a__from(mark(x1))), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), from(x1))) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), a__from(mark(x1))), MARK(filter(z0, z1))) MARK(filter(s(z0), from(x1))) -> c22(A__FILTER(s(mark(z0)), a__from(mark(x1))), MARK(s(z0))) MARK(filter(x0, from(x1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), from(x1))) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), from(x1))) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, head(primes))) -> c22(A__FILTER(mark(x0), a__head(a__primes)), MARK(x0)) MARK(filter(x0, head(sieve(z0)))) -> c22(A__FILTER(mark(x0), a__head(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, head(from(z0)))) -> c22(A__FILTER(mark(x0), a__head(a__from(mark(z0)))), MARK(x0)) MARK(filter(x0, head(head(z0)))) -> c22(A__FILTER(mark(x0), a__head(a__head(mark(z0)))), MARK(x0)) MARK(filter(x0, head(tail(z0)))) -> c22(A__FILTER(mark(x0), a__head(a__tail(mark(z0)))), MARK(x0)) MARK(filter(x0, head(if(z0, z1, z2)))) -> c22(A__FILTER(mark(x0), a__head(a__if(mark(z0), z1, z2))), MARK(x0)) MARK(filter(x0, head(filter(z0, z1)))) -> c22(A__FILTER(mark(x0), a__head(a__filter(mark(z0), mark(z1)))), MARK(x0)) MARK(filter(x0, head(s(z0)))) -> c22(A__FILTER(mark(x0), a__head(s(mark(z0)))), MARK(x0)) MARK(filter(x0, head(0))) -> c22(A__FILTER(mark(x0), a__head(0)), MARK(x0)) MARK(filter(x0, head(cons(z0, z1)))) -> c22(A__FILTER(mark(x0), a__head(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, head(true))) -> c22(A__FILTER(mark(x0), a__head(true)), MARK(x0)) MARK(filter(x0, head(false))) -> c22(A__FILTER(mark(x0), a__head(false)), MARK(x0)) MARK(filter(x0, head(divides(z0, z1)))) -> c22(A__FILTER(mark(x0), a__head(divides(mark(z0), mark(z1)))), MARK(x0)) MARK(filter(primes, head(x1))) -> c22(A__FILTER(a__primes, a__head(mark(x1))), MARK(primes)) MARK(filter(sieve(z0), head(x1))) -> c22(A__FILTER(a__sieve(mark(z0)), a__head(mark(x1))), MARK(sieve(z0))) MARK(filter(from(z0), head(x1))) -> c22(A__FILTER(a__from(mark(z0)), a__head(mark(x1))), MARK(from(z0))) MARK(filter(head(z0), head(x1))) -> c22(A__FILTER(a__head(mark(z0)), a__head(mark(x1))), MARK(head(z0))) MARK(filter(tail(z0), head(x1))) -> c22(A__FILTER(a__tail(mark(z0)), a__head(mark(x1))), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), head(x1))) -> c22(A__FILTER(a__if(mark(z0), z1, z2), a__head(mark(x1))), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), head(x1))) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), a__head(mark(x1))), MARK(filter(z0, z1))) MARK(filter(s(z0), head(x1))) -> c22(A__FILTER(s(mark(z0)), a__head(mark(x1))), MARK(s(z0))) MARK(filter(x0, head(x1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), head(x1))) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), head(x1))) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, tail(x1))) -> c22(A__FILTER(mark(x0), tail(mark(x1))), MARK(x0)) MARK(filter(x0, tail(primes))) -> c22(A__FILTER(mark(x0), a__tail(a__primes)), MARK(x0)) MARK(filter(x0, tail(sieve(z0)))) -> c22(A__FILTER(mark(x0), a__tail(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, tail(from(z0)))) -> c22(A__FILTER(mark(x0), a__tail(a__from(mark(z0)))), MARK(x0)) MARK(filter(x0, tail(head(z0)))) -> c22(A__FILTER(mark(x0), a__tail(a__head(mark(z0)))), MARK(x0)) MARK(filter(x0, tail(tail(z0)))) -> c22(A__FILTER(mark(x0), a__tail(a__tail(mark(z0)))), MARK(x0)) MARK(filter(x0, tail(if(z0, z1, z2)))) -> c22(A__FILTER(mark(x0), a__tail(a__if(mark(z0), z1, z2))), MARK(x0)) MARK(filter(x0, tail(filter(z0, z1)))) -> c22(A__FILTER(mark(x0), a__tail(a__filter(mark(z0), mark(z1)))), MARK(x0)) MARK(filter(x0, tail(s(z0)))) -> c22(A__FILTER(mark(x0), a__tail(s(mark(z0)))), MARK(x0)) MARK(filter(x0, tail(0))) -> c22(A__FILTER(mark(x0), a__tail(0)), MARK(x0)) MARK(filter(x0, tail(cons(z0, z1)))) -> c22(A__FILTER(mark(x0), a__tail(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, tail(true))) -> c22(A__FILTER(mark(x0), a__tail(true)), MARK(x0)) MARK(filter(x0, tail(false))) -> c22(A__FILTER(mark(x0), a__tail(false)), MARK(x0)) MARK(filter(x0, tail(divides(z0, z1)))) -> c22(A__FILTER(mark(x0), a__tail(divides(mark(z0), mark(z1)))), MARK(x0)) MARK(filter(primes, tail(x1))) -> c22(A__FILTER(a__primes, a__tail(mark(x1))), MARK(primes)) MARK(filter(sieve(z0), tail(x1))) -> c22(A__FILTER(a__sieve(mark(z0)), a__tail(mark(x1))), MARK(sieve(z0))) MARK(filter(from(z0), tail(x1))) -> c22(A__FILTER(a__from(mark(z0)), a__tail(mark(x1))), MARK(from(z0))) MARK(filter(head(z0), tail(x1))) -> c22(A__FILTER(a__head(mark(z0)), a__tail(mark(x1))), MARK(head(z0))) MARK(filter(tail(z0), tail(x1))) -> c22(A__FILTER(a__tail(mark(z0)), a__tail(mark(x1))), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), tail(x1))) -> c22(A__FILTER(a__if(mark(z0), z1, z2), a__tail(mark(x1))), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), tail(x1))) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), a__tail(mark(x1))), MARK(filter(z0, z1))) MARK(filter(s(z0), tail(x1))) -> c22(A__FILTER(s(mark(z0)), a__tail(mark(x1))), MARK(s(z0))) MARK(filter(0, tail(x1))) -> c22(A__FILTER(0, a__tail(mark(x1))), MARK(0)) MARK(filter(cons(z0, z1), tail(x1))) -> c22(A__FILTER(cons(mark(z0), z1), a__tail(mark(x1))), MARK(cons(z0, z1))) MARK(filter(true, tail(x1))) -> c22(A__FILTER(true, a__tail(mark(x1))), MARK(true)) MARK(filter(false, tail(x1))) -> c22(A__FILTER(false, a__tail(mark(x1))), MARK(false)) MARK(filter(divides(z0, z1), tail(x1))) -> c22(A__FILTER(divides(mark(z0), mark(z1)), a__tail(mark(x1))), MARK(divides(z0, z1))) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c18_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_1, c17_2, c17_1, c19_2, c19_1, c20_2, c20_1, c21_1, c22_2, c22_1, c23_2, c23_1, c_2, c21_2 ---------------------------------------- (237) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing nodes: MARK(filter(true, tail(x1))) -> c22(A__FILTER(true, a__tail(mark(x1))), MARK(true)) MARK(filter(false, tail(x1))) -> c22(A__FILTER(false, a__tail(mark(x1))), MARK(false)) MARK(filter(0, tail(x1))) -> c22(A__FILTER(0, a__tail(mark(x1))), MARK(0)) ---------------------------------------- (238) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, if(z0, z1, z2))) -> c22(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0)) MARK(filter(x0, filter(z0, z1))) -> c22(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(x0)) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(x0, s(z0))) -> c22(MARK(x0)) MARK(filter(x0, 0)) -> c22(MARK(x0)) MARK(filter(x0, true)) -> c22(MARK(x0)) MARK(filter(x0, false)) -> c22(MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), x1)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c23(A__FILTER(mark(x0), a__primes), MARK(primes)) MARK(filter(x0, sieve(z0))) -> c23(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, from(z0))) -> c23(A__FILTER(mark(x0), a__from(mark(z0))), MARK(from(z0))) MARK(filter(x0, head(z0))) -> c23(A__FILTER(mark(x0), a__head(mark(z0))), MARK(head(z0))) MARK(filter(x0, tail(z0))) -> c23(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(tail(z0))) MARK(filter(x0, if(z0, z1, z2))) -> c23(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(filter(x0, filter(z0, z1))) -> c23(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(filter(x0, cons(z0, z1))) -> c23(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(primes, x1)) -> c23(A__FILTER(a__primes, mark(x1)), MARK(x1)) MARK(filter(sieve(z0), x1)) -> c23(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(from(z0), x1)) -> c23(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(head(z0), x1)) -> c23(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(tail(z0), x1)) -> c23(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(if(z0, z1, z2), x1)) -> c23(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1)) MARK(filter(filter(z0, z1), x1)) -> c23(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(x1)) MARK(filter(s(z0), x1)) -> c23(A__FILTER(s(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(x0, s(z0))) -> c23(MARK(s(z0))) MARK(filter(x0, divides(z0, z1))) -> c23(MARK(divides(z0, z1))) MARK(filter(0, x1)) -> c23(MARK(x1)) MARK(filter(cons(z0, z1), x1)) -> c23(MARK(x1)) MARK(filter(true, x1)) -> c23(MARK(x1)) MARK(filter(false, x1)) -> c23(MARK(x1)) MARK(filter(divides(z0, z1), x1)) -> c23(MARK(x1)) A__PRIMES -> c(A__SIEVE(cons(s(mark(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(sieve(primes)) -> c17(MARK(primes)) MARK(sieve(sieve(primes))) -> c17(A__SIEVE(a__sieve(a__primes)), MARK(sieve(primes))) MARK(sieve(sieve(sieve(z0)))) -> c17(A__SIEVE(a__sieve(a__sieve(mark(z0)))), MARK(sieve(sieve(z0)))) MARK(sieve(sieve(from(z0)))) -> c17(A__SIEVE(a__sieve(a__from(mark(z0)))), MARK(sieve(from(z0)))) MARK(sieve(sieve(head(z0)))) -> c17(A__SIEVE(a__sieve(a__head(mark(z0)))), MARK(sieve(head(z0)))) MARK(sieve(sieve(tail(z0)))) -> c17(A__SIEVE(a__sieve(a__tail(mark(z0)))), MARK(sieve(tail(z0)))) MARK(sieve(sieve(if(z0, z1, z2)))) -> c17(A__SIEVE(a__sieve(a__if(mark(z0), z1, z2))), MARK(sieve(if(z0, z1, z2)))) MARK(sieve(sieve(filter(z0, z1)))) -> c17(A__SIEVE(a__sieve(a__filter(mark(z0), mark(z1)))), MARK(sieve(filter(z0, z1)))) MARK(sieve(sieve(s(z0)))) -> c17(A__SIEVE(a__sieve(s(mark(z0)))), MARK(sieve(s(z0)))) MARK(sieve(sieve(cons(z0, z1)))) -> c17(A__SIEVE(a__sieve(cons(mark(z0), z1))), MARK(sieve(cons(z0, z1)))) MARK(sieve(sieve(divides(z0, z1)))) -> c17(A__SIEVE(a__sieve(divides(mark(z0), mark(z1)))), MARK(sieve(divides(z0, z1)))) MARK(sieve(sieve(x0))) -> c17(MARK(sieve(x0))) MARK(sieve(sieve(0))) -> c17(A__SIEVE(a__sieve(0))) MARK(sieve(sieve(true))) -> c17(A__SIEVE(a__sieve(true))) MARK(sieve(sieve(false))) -> c17(A__SIEVE(a__sieve(false))) MARK(sieve(from(x0))) -> c17(A__SIEVE(cons(mark(mark(x0)), from(s(mark(x0))))), MARK(from(x0))) MARK(sieve(from(primes))) -> c17(A__SIEVE(a__from(a__primes)), MARK(from(primes))) MARK(sieve(from(sieve(z0)))) -> c17(A__SIEVE(a__from(a__sieve(mark(z0)))), MARK(from(sieve(z0)))) MARK(sieve(from(from(z0)))) -> c17(A__SIEVE(a__from(a__from(mark(z0)))), MARK(from(from(z0)))) MARK(sieve(from(head(z0)))) -> c17(A__SIEVE(a__from(a__head(mark(z0)))), MARK(from(head(z0)))) MARK(sieve(from(tail(z0)))) -> c17(A__SIEVE(a__from(a__tail(mark(z0)))), MARK(from(tail(z0)))) MARK(sieve(from(if(z0, z1, z2)))) -> c17(A__SIEVE(a__from(a__if(mark(z0), z1, z2))), MARK(from(if(z0, z1, z2)))) MARK(sieve(from(filter(z0, z1)))) -> c17(A__SIEVE(a__from(a__filter(mark(z0), mark(z1)))), MARK(from(filter(z0, z1)))) MARK(sieve(from(s(z0)))) -> c17(A__SIEVE(a__from(s(mark(z0)))), MARK(from(s(z0)))) MARK(sieve(from(0))) -> c17(A__SIEVE(a__from(0)), MARK(from(0))) MARK(sieve(from(cons(z0, z1)))) -> c17(A__SIEVE(a__from(cons(mark(z0), z1))), MARK(from(cons(z0, z1)))) MARK(sieve(from(true))) -> c17(A__SIEVE(a__from(true)), MARK(from(true))) MARK(sieve(from(false))) -> c17(A__SIEVE(a__from(false)), MARK(from(false))) MARK(sieve(from(divides(z0, z1)))) -> c17(A__SIEVE(a__from(divides(mark(z0), mark(z1)))), MARK(from(divides(z0, z1)))) MARK(sieve(from(x0))) -> c17(MARK(from(x0))) MARK(sieve(head(primes))) -> c17(A__SIEVE(a__head(a__primes)), MARK(head(primes))) MARK(sieve(head(sieve(z0)))) -> c17(A__SIEVE(a__head(a__sieve(mark(z0)))), MARK(head(sieve(z0)))) MARK(sieve(head(from(z0)))) -> c17(A__SIEVE(a__head(a__from(mark(z0)))), MARK(head(from(z0)))) MARK(sieve(head(head(z0)))) -> c17(A__SIEVE(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(sieve(head(tail(z0)))) -> c17(A__SIEVE(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(sieve(head(if(z0, z1, z2)))) -> c17(A__SIEVE(a__head(a__if(mark(z0), z1, z2))), MARK(head(if(z0, z1, z2)))) MARK(sieve(head(filter(z0, z1)))) -> c17(A__SIEVE(a__head(a__filter(mark(z0), mark(z1)))), MARK(head(filter(z0, z1)))) MARK(sieve(head(s(z0)))) -> c17(A__SIEVE(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(sieve(head(cons(z0, z1)))) -> c17(A__SIEVE(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(sieve(head(divides(z0, z1)))) -> c17(A__SIEVE(a__head(divides(mark(z0), mark(z1)))), MARK(head(divides(z0, z1)))) MARK(sieve(head(x0))) -> c17(MARK(head(x0))) MARK(sieve(head(0))) -> c17(A__SIEVE(a__head(0))) MARK(sieve(head(true))) -> c17(A__SIEVE(a__head(true))) MARK(sieve(head(false))) -> c17(A__SIEVE(a__head(false))) 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a__sieve(mark(z0))), x2, x3), MARK(filter(x0, sieve(z0)))) MARK(if(filter(x0, from(z0)), x2, x3)) -> c21(A__IF(a__filter(mark(x0), a__from(mark(z0))), x2, x3), MARK(filter(x0, from(z0)))) MARK(if(filter(x0, head(z0)), x2, x3)) -> c21(A__IF(a__filter(mark(x0), a__head(mark(z0))), x2, x3), MARK(filter(x0, head(z0)))) MARK(if(filter(x0, tail(z0)), x2, x3)) -> c21(A__IF(a__filter(mark(x0), a__tail(mark(z0))), x2, x3), MARK(filter(x0, tail(z0)))) MARK(if(filter(x0, if(z0, z1, z2)), x2, x3)) -> c21(A__IF(a__filter(mark(x0), a__if(mark(z0), z1, z2)), x2, x3), MARK(filter(x0, if(z0, z1, z2)))) MARK(if(filter(x0, filter(z0, z1)), x2, x3)) -> c21(A__IF(a__filter(mark(x0), a__filter(mark(z0), mark(z1))), x2, x3), MARK(filter(x0, filter(z0, z1)))) MARK(if(filter(x0, s(z0)), x2, x3)) -> c21(A__IF(a__filter(mark(x0), s(mark(z0))), x2, x3), MARK(filter(x0, s(z0)))) MARK(if(filter(x0, 0), x2, x3)) -> c21(A__IF(a__filter(mark(x0), 0), x2, x3), MARK(filter(x0, 0))) MARK(if(filter(x0, cons(z0, z1)), x2, x3)) -> c21(A__IF(a__filter(mark(x0), cons(mark(z0), z1)), x2, x3), MARK(filter(x0, cons(z0, z1)))) MARK(if(filter(x0, true), x2, x3)) -> c21(A__IF(a__filter(mark(x0), true), x2, x3), MARK(filter(x0, true))) MARK(if(filter(x0, false), x2, x3)) -> c21(A__IF(a__filter(mark(x0), false), x2, x3), MARK(filter(x0, false))) MARK(if(filter(x0, divides(z0, z1)), x2, x3)) -> c21(A__IF(a__filter(mark(x0), divides(mark(z0), mark(z1))), x2, x3), MARK(filter(x0, divides(z0, z1)))) MARK(if(filter(primes, x1), x2, x3)) -> c21(A__IF(a__filter(a__primes, mark(x1)), x2, x3), MARK(filter(primes, x1))) MARK(if(filter(sieve(z0), x1), x2, x3)) -> c21(A__IF(a__filter(a__sieve(mark(z0)), mark(x1)), x2, x3), MARK(filter(sieve(z0), x1))) MARK(if(filter(from(z0), x1), x2, x3)) -> c21(A__IF(a__filter(a__from(mark(z0)), mark(x1)), x2, x3), MARK(filter(from(z0), x1))) MARK(if(filter(head(z0), x1), x2, x3)) -> c21(A__IF(a__filter(a__head(mark(z0)), mark(x1)), x2, x3), MARK(filter(head(z0), x1))) MARK(if(filter(tail(z0), x1), x2, x3)) -> c21(A__IF(a__filter(a__tail(mark(z0)), mark(x1)), x2, x3), MARK(filter(tail(z0), x1))) MARK(if(filter(if(z0, z1, z2), x1), x2, x3)) -> c21(A__IF(a__filter(a__if(mark(z0), z1, z2), mark(x1)), x2, x3), MARK(filter(if(z0, z1, z2), x1))) MARK(if(filter(filter(z0, z1), x1), x2, x3)) -> c21(A__IF(a__filter(a__filter(mark(z0), mark(z1)), mark(x1)), x2, x3), MARK(filter(filter(z0, z1), x1))) MARK(if(filter(s(z0), x1), x2, x3)) -> c21(A__IF(a__filter(s(mark(z0)), mark(x1)), x2, x3), MARK(filter(s(z0), x1))) MARK(if(filter(0, x1), x2, x3)) -> c21(A__IF(a__filter(0, mark(x1)), x2, x3), MARK(filter(0, x1))) MARK(if(filter(cons(z0, z1), x1), x2, x3)) -> c21(A__IF(a__filter(cons(mark(z0), z1), mark(x1)), x2, x3), MARK(filter(cons(z0, z1), x1))) MARK(if(filter(true, x1), x2, x3)) -> c21(A__IF(a__filter(true, mark(x1)), x2, x3), MARK(filter(true, x1))) MARK(if(filter(false, x1), x2, x3)) -> c21(A__IF(a__filter(false, mark(x1)), x2, x3), MARK(filter(false, 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MARK(filter(z0, z1))) MARK(filter(s(z0), primes)) -> c22(A__FILTER(s(mark(z0)), a__primes), MARK(s(z0))) MARK(filter(x0, primes)) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), primes)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), primes)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, sieve(primes))) -> c22(A__FILTER(mark(x0), a__sieve(a__primes)), MARK(x0)) MARK(filter(x0, sieve(sieve(z0)))) -> c22(A__FILTER(mark(x0), a__sieve(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, sieve(from(z0)))) -> c22(A__FILTER(mark(x0), a__sieve(a__from(mark(z0)))), MARK(x0)) MARK(filter(x0, sieve(head(z0)))) -> c22(A__FILTER(mark(x0), a__sieve(a__head(mark(z0)))), MARK(x0)) MARK(filter(x0, sieve(tail(z0)))) -> c22(A__FILTER(mark(x0), a__sieve(a__tail(mark(z0)))), MARK(x0)) MARK(filter(x0, sieve(if(z0, z1, z2)))) -> c22(A__FILTER(mark(x0), a__sieve(a__if(mark(z0), z1, z2))), MARK(x0)) MARK(filter(x0, sieve(filter(z0, z1)))) -> c22(A__FILTER(mark(x0), a__sieve(a__filter(mark(z0), mark(z1)))), MARK(x0)) MARK(filter(x0, sieve(s(z0)))) -> c22(A__FILTER(mark(x0), a__sieve(s(mark(z0)))), MARK(x0)) MARK(filter(x0, sieve(0))) -> c22(A__FILTER(mark(x0), a__sieve(0)), MARK(x0)) MARK(filter(x0, sieve(cons(z0, z1)))) -> c22(A__FILTER(mark(x0), a__sieve(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, sieve(true))) -> c22(A__FILTER(mark(x0), a__sieve(true)), MARK(x0)) MARK(filter(x0, sieve(false))) -> c22(A__FILTER(mark(x0), a__sieve(false)), MARK(x0)) MARK(filter(x0, sieve(divides(z0, z1)))) -> c22(A__FILTER(mark(x0), a__sieve(divides(mark(z0), mark(z1)))), MARK(x0)) MARK(filter(primes, sieve(x1))) -> c22(A__FILTER(a__primes, a__sieve(mark(x1))), MARK(primes)) MARK(filter(sieve(z0), sieve(x1))) -> c22(A__FILTER(a__sieve(mark(z0)), a__sieve(mark(x1))), MARK(sieve(z0))) MARK(filter(from(z0), sieve(x1))) -> c22(A__FILTER(a__from(mark(z0)), a__sieve(mark(x1))), MARK(from(z0))) MARK(filter(head(z0), sieve(x1))) -> c22(A__FILTER(a__head(mark(z0)), a__sieve(mark(x1))), MARK(head(z0))) MARK(filter(tail(z0), sieve(x1))) -> c22(A__FILTER(a__tail(mark(z0)), a__sieve(mark(x1))), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), sieve(x1))) -> c22(A__FILTER(a__if(mark(z0), z1, z2), a__sieve(mark(x1))), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), sieve(x1))) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), a__sieve(mark(x1))), MARK(filter(z0, z1))) MARK(filter(s(z0), sieve(x1))) -> c22(A__FILTER(s(mark(z0)), a__sieve(mark(x1))), MARK(s(z0))) MARK(filter(x0, sieve(x1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), sieve(x1))) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), sieve(x1))) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, from(x1))) -> c22(A__FILTER(mark(x0), cons(mark(mark(x1)), from(s(mark(x1))))), MARK(x0)) MARK(filter(x0, from(primes))) -> c22(A__FILTER(mark(x0), a__from(a__primes)), MARK(x0)) MARK(filter(x0, from(sieve(z0)))) -> c22(A__FILTER(mark(x0), a__from(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, from(from(z0)))) -> c22(A__FILTER(mark(x0), a__from(a__from(mark(z0)))), MARK(x0)) MARK(filter(x0, from(head(z0)))) -> c22(A__FILTER(mark(x0), a__from(a__head(mark(z0)))), MARK(x0)) MARK(filter(x0, from(tail(z0)))) -> c22(A__FILTER(mark(x0), a__from(a__tail(mark(z0)))), MARK(x0)) MARK(filter(x0, from(if(z0, z1, z2)))) -> c22(A__FILTER(mark(x0), a__from(a__if(mark(z0), z1, z2))), MARK(x0)) MARK(filter(x0, from(filter(z0, z1)))) -> c22(A__FILTER(mark(x0), a__from(a__filter(mark(z0), mark(z1)))), MARK(x0)) MARK(filter(x0, from(s(z0)))) -> c22(A__FILTER(mark(x0), a__from(s(mark(z0)))), MARK(x0)) MARK(filter(x0, from(0))) -> c22(A__FILTER(mark(x0), a__from(0)), MARK(x0)) MARK(filter(x0, from(cons(z0, z1)))) -> c22(A__FILTER(mark(x0), a__from(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, from(true))) -> c22(A__FILTER(mark(x0), a__from(true)), MARK(x0)) MARK(filter(x0, from(false))) -> c22(A__FILTER(mark(x0), a__from(false)), MARK(x0)) MARK(filter(x0, from(divides(z0, z1)))) -> c22(A__FILTER(mark(x0), a__from(divides(mark(z0), mark(z1)))), MARK(x0)) MARK(filter(primes, from(x1))) -> c22(A__FILTER(a__primes, a__from(mark(x1))), MARK(primes)) MARK(filter(sieve(z0), from(x1))) -> c22(A__FILTER(a__sieve(mark(z0)), a__from(mark(x1))), MARK(sieve(z0))) MARK(filter(from(z0), from(x1))) -> c22(A__FILTER(a__from(mark(z0)), a__from(mark(x1))), MARK(from(z0))) MARK(filter(head(z0), from(x1))) -> c22(A__FILTER(a__head(mark(z0)), a__from(mark(x1))), MARK(head(z0))) MARK(filter(tail(z0), from(x1))) -> c22(A__FILTER(a__tail(mark(z0)), a__from(mark(x1))), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), from(x1))) -> c22(A__FILTER(a__if(mark(z0), z1, z2), a__from(mark(x1))), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), from(x1))) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), a__from(mark(x1))), MARK(filter(z0, z1))) MARK(filter(s(z0), from(x1))) -> c22(A__FILTER(s(mark(z0)), a__from(mark(x1))), MARK(s(z0))) MARK(filter(x0, from(x1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), from(x1))) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), from(x1))) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, head(primes))) -> c22(A__FILTER(mark(x0), a__head(a__primes)), MARK(x0)) MARK(filter(x0, head(sieve(z0)))) -> c22(A__FILTER(mark(x0), a__head(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, head(from(z0)))) -> c22(A__FILTER(mark(x0), a__head(a__from(mark(z0)))), MARK(x0)) MARK(filter(x0, head(head(z0)))) -> c22(A__FILTER(mark(x0), a__head(a__head(mark(z0)))), MARK(x0)) MARK(filter(x0, head(tail(z0)))) -> c22(A__FILTER(mark(x0), a__head(a__tail(mark(z0)))), MARK(x0)) MARK(filter(x0, head(if(z0, z1, z2)))) -> c22(A__FILTER(mark(x0), a__head(a__if(mark(z0), z1, z2))), MARK(x0)) MARK(filter(x0, head(filter(z0, z1)))) -> c22(A__FILTER(mark(x0), a__head(a__filter(mark(z0), mark(z1)))), MARK(x0)) MARK(filter(x0, head(s(z0)))) -> c22(A__FILTER(mark(x0), a__head(s(mark(z0)))), MARK(x0)) MARK(filter(x0, head(0))) -> c22(A__FILTER(mark(x0), a__head(0)), MARK(x0)) MARK(filter(x0, head(cons(z0, z1)))) -> c22(A__FILTER(mark(x0), a__head(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, head(true))) -> c22(A__FILTER(mark(x0), a__head(true)), MARK(x0)) MARK(filter(x0, head(false))) -> c22(A__FILTER(mark(x0), a__head(false)), MARK(x0)) MARK(filter(x0, head(divides(z0, z1)))) -> c22(A__FILTER(mark(x0), a__head(divides(mark(z0), mark(z1)))), MARK(x0)) MARK(filter(primes, head(x1))) -> c22(A__FILTER(a__primes, a__head(mark(x1))), MARK(primes)) MARK(filter(sieve(z0), head(x1))) -> c22(A__FILTER(a__sieve(mark(z0)), a__head(mark(x1))), MARK(sieve(z0))) MARK(filter(from(z0), head(x1))) -> c22(A__FILTER(a__from(mark(z0)), a__head(mark(x1))), MARK(from(z0))) MARK(filter(head(z0), head(x1))) -> c22(A__FILTER(a__head(mark(z0)), a__head(mark(x1))), MARK(head(z0))) MARK(filter(tail(z0), head(x1))) -> c22(A__FILTER(a__tail(mark(z0)), a__head(mark(x1))), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), head(x1))) -> c22(A__FILTER(a__if(mark(z0), z1, z2), a__head(mark(x1))), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), head(x1))) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), a__head(mark(x1))), MARK(filter(z0, z1))) MARK(filter(s(z0), head(x1))) -> c22(A__FILTER(s(mark(z0)), a__head(mark(x1))), MARK(s(z0))) MARK(filter(x0, head(x1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), head(x1))) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), head(x1))) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, tail(x1))) -> c22(A__FILTER(mark(x0), tail(mark(x1))), MARK(x0)) MARK(filter(x0, tail(primes))) -> c22(A__FILTER(mark(x0), a__tail(a__primes)), MARK(x0)) MARK(filter(x0, tail(sieve(z0)))) -> c22(A__FILTER(mark(x0), a__tail(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, tail(from(z0)))) -> c22(A__FILTER(mark(x0), a__tail(a__from(mark(z0)))), MARK(x0)) MARK(filter(x0, tail(head(z0)))) -> c22(A__FILTER(mark(x0), a__tail(a__head(mark(z0)))), MARK(x0)) MARK(filter(x0, tail(tail(z0)))) -> c22(A__FILTER(mark(x0), a__tail(a__tail(mark(z0)))), MARK(x0)) MARK(filter(x0, tail(if(z0, z1, z2)))) -> c22(A__FILTER(mark(x0), a__tail(a__if(mark(z0), z1, z2))), MARK(x0)) MARK(filter(x0, tail(filter(z0, z1)))) -> c22(A__FILTER(mark(x0), a__tail(a__filter(mark(z0), mark(z1)))), MARK(x0)) MARK(filter(x0, tail(s(z0)))) -> c22(A__FILTER(mark(x0), a__tail(s(mark(z0)))), MARK(x0)) MARK(filter(x0, tail(0))) -> c22(A__FILTER(mark(x0), a__tail(0)), MARK(x0)) MARK(filter(x0, tail(cons(z0, z1)))) -> c22(A__FILTER(mark(x0), a__tail(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, tail(true))) -> c22(A__FILTER(mark(x0), a__tail(true)), MARK(x0)) MARK(filter(x0, tail(false))) -> c22(A__FILTER(mark(x0), a__tail(false)), MARK(x0)) MARK(filter(x0, tail(divides(z0, z1)))) -> c22(A__FILTER(mark(x0), a__tail(divides(mark(z0), mark(z1)))), MARK(x0)) MARK(filter(primes, tail(x1))) -> c22(A__FILTER(a__primes, a__tail(mark(x1))), MARK(primes)) MARK(filter(sieve(z0), tail(x1))) -> c22(A__FILTER(a__sieve(mark(z0)), a__tail(mark(x1))), MARK(sieve(z0))) MARK(filter(from(z0), tail(x1))) -> c22(A__FILTER(a__from(mark(z0)), a__tail(mark(x1))), MARK(from(z0))) MARK(filter(head(z0), tail(x1))) -> c22(A__FILTER(a__head(mark(z0)), a__tail(mark(x1))), MARK(head(z0))) MARK(filter(tail(z0), tail(x1))) -> c22(A__FILTER(a__tail(mark(z0)), a__tail(mark(x1))), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), tail(x1))) -> c22(A__FILTER(a__if(mark(z0), z1, z2), a__tail(mark(x1))), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), tail(x1))) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), a__tail(mark(x1))), MARK(filter(z0, z1))) MARK(filter(s(z0), tail(x1))) -> c22(A__FILTER(s(mark(z0)), a__tail(mark(x1))), MARK(s(z0))) MARK(filter(cons(z0, z1), tail(x1))) -> c22(A__FILTER(cons(mark(z0), z1), a__tail(mark(x1))), MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), tail(x1))) -> c22(A__FILTER(divides(mark(z0), mark(z1)), a__tail(mark(x1))), MARK(divides(z0, z1))) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c18_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_1, c17_2, c17_1, c19_2, c19_1, c20_2, c20_1, c21_1, c22_2, c22_1, c23_2, c23_1, c_2, c21_2 ---------------------------------------- (239) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (240) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, if(z0, z1, z2))) -> c22(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0)) MARK(filter(x0, filter(z0, z1))) -> c22(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(x0)) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(x0, s(z0))) -> c22(MARK(x0)) MARK(filter(x0, 0)) -> c22(MARK(x0)) MARK(filter(x0, true)) -> c22(MARK(x0)) MARK(filter(x0, false)) -> c22(MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), x1)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c23(A__FILTER(mark(x0), a__primes), MARK(primes)) MARK(filter(x0, sieve(z0))) -> c23(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, from(z0))) -> c23(A__FILTER(mark(x0), a__from(mark(z0))), MARK(from(z0))) MARK(filter(x0, head(z0))) -> c23(A__FILTER(mark(x0), a__head(mark(z0))), MARK(head(z0))) MARK(filter(x0, tail(z0))) -> c23(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(tail(z0))) MARK(filter(x0, if(z0, z1, z2))) -> c23(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(filter(x0, filter(z0, z1))) -> c23(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(filter(x0, cons(z0, z1))) -> c23(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(primes, x1)) -> c23(A__FILTER(a__primes, mark(x1)), MARK(x1)) MARK(filter(sieve(z0), x1)) -> c23(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(from(z0), x1)) -> c23(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(head(z0), x1)) -> c23(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(tail(z0), x1)) -> c23(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(if(z0, z1, z2), x1)) -> c23(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1)) MARK(filter(filter(z0, z1), x1)) -> c23(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(x1)) MARK(filter(s(z0), x1)) -> c23(A__FILTER(s(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(x0, s(z0))) -> c23(MARK(s(z0))) MARK(filter(x0, divides(z0, z1))) -> c23(MARK(divides(z0, z1))) MARK(filter(0, x1)) -> c23(MARK(x1)) MARK(filter(cons(z0, z1), x1)) -> c23(MARK(x1)) MARK(filter(true, x1)) -> c23(MARK(x1)) MARK(filter(false, x1)) -> c23(MARK(x1)) MARK(filter(divides(z0, z1), x1)) -> c23(MARK(x1)) A__PRIMES -> c(A__SIEVE(cons(s(mark(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(sieve(primes)) -> c17(MARK(primes)) MARK(sieve(sieve(primes))) -> c17(A__SIEVE(a__sieve(a__primes)), MARK(sieve(primes))) MARK(sieve(sieve(sieve(z0)))) -> c17(A__SIEVE(a__sieve(a__sieve(mark(z0)))), MARK(sieve(sieve(z0)))) MARK(sieve(sieve(from(z0)))) -> c17(A__SIEVE(a__sieve(a__from(mark(z0)))), MARK(sieve(from(z0)))) MARK(sieve(sieve(head(z0)))) -> c17(A__SIEVE(a__sieve(a__head(mark(z0)))), MARK(sieve(head(z0)))) MARK(sieve(sieve(tail(z0)))) -> c17(A__SIEVE(a__sieve(a__tail(mark(z0)))), MARK(sieve(tail(z0)))) MARK(sieve(sieve(if(z0, z1, z2)))) -> c17(A__SIEVE(a__sieve(a__if(mark(z0), z1, z2))), MARK(sieve(if(z0, z1, z2)))) MARK(sieve(sieve(filter(z0, z1)))) -> c17(A__SIEVE(a__sieve(a__filter(mark(z0), mark(z1)))), MARK(sieve(filter(z0, z1)))) MARK(sieve(sieve(s(z0)))) -> c17(A__SIEVE(a__sieve(s(mark(z0)))), MARK(sieve(s(z0)))) MARK(sieve(sieve(cons(z0, z1)))) -> c17(A__SIEVE(a__sieve(cons(mark(z0), z1))), MARK(sieve(cons(z0, z1)))) MARK(sieve(sieve(divides(z0, z1)))) -> c17(A__SIEVE(a__sieve(divides(mark(z0), mark(z1)))), MARK(sieve(divides(z0, z1)))) MARK(sieve(sieve(x0))) -> c17(MARK(sieve(x0))) MARK(sieve(sieve(0))) -> c17(A__SIEVE(a__sieve(0))) MARK(sieve(sieve(true))) -> c17(A__SIEVE(a__sieve(true))) MARK(sieve(sieve(false))) -> c17(A__SIEVE(a__sieve(false))) MARK(sieve(from(x0))) -> c17(A__SIEVE(cons(mark(mark(x0)), from(s(mark(x0))))), MARK(from(x0))) MARK(sieve(from(primes))) -> c17(A__SIEVE(a__from(a__primes)), MARK(from(primes))) MARK(sieve(from(sieve(z0)))) -> c17(A__SIEVE(a__from(a__sieve(mark(z0)))), MARK(from(sieve(z0)))) MARK(sieve(from(from(z0)))) -> c17(A__SIEVE(a__from(a__from(mark(z0)))), MARK(from(from(z0)))) MARK(sieve(from(head(z0)))) -> c17(A__SIEVE(a__from(a__head(mark(z0)))), MARK(from(head(z0)))) MARK(sieve(from(tail(z0)))) -> c17(A__SIEVE(a__from(a__tail(mark(z0)))), MARK(from(tail(z0)))) MARK(sieve(from(if(z0, z1, z2)))) -> c17(A__SIEVE(a__from(a__if(mark(z0), z1, z2))), MARK(from(if(z0, z1, z2)))) MARK(sieve(from(filter(z0, z1)))) -> c17(A__SIEVE(a__from(a__filter(mark(z0), mark(z1)))), MARK(from(filter(z0, z1)))) MARK(sieve(from(s(z0)))) -> c17(A__SIEVE(a__from(s(mark(z0)))), MARK(from(s(z0)))) MARK(sieve(from(0))) -> c17(A__SIEVE(a__from(0)), MARK(from(0))) MARK(sieve(from(cons(z0, z1)))) -> c17(A__SIEVE(a__from(cons(mark(z0), z1))), MARK(from(cons(z0, z1)))) MARK(sieve(from(true))) -> c17(A__SIEVE(a__from(true)), MARK(from(true))) MARK(sieve(from(false))) -> c17(A__SIEVE(a__from(false)), MARK(from(false))) MARK(sieve(from(divides(z0, z1)))) -> c17(A__SIEVE(a__from(divides(mark(z0), mark(z1)))), MARK(from(divides(z0, z1)))) MARK(sieve(from(x0))) -> c17(MARK(from(x0))) MARK(sieve(head(primes))) -> c17(A__SIEVE(a__head(a__primes)), MARK(head(primes))) MARK(sieve(head(sieve(z0)))) -> c17(A__SIEVE(a__head(a__sieve(mark(z0)))), MARK(head(sieve(z0)))) MARK(sieve(head(from(z0)))) -> c17(A__SIEVE(a__head(a__from(mark(z0)))), MARK(head(from(z0)))) MARK(sieve(head(head(z0)))) -> c17(A__SIEVE(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(sieve(head(tail(z0)))) -> c17(A__SIEVE(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(sieve(head(if(z0, z1, z2)))) -> c17(A__SIEVE(a__head(a__if(mark(z0), z1, z2))), MARK(head(if(z0, z1, z2)))) MARK(sieve(head(filter(z0, z1)))) -> c17(A__SIEVE(a__head(a__filter(mark(z0), mark(z1)))), MARK(head(filter(z0, z1)))) MARK(sieve(head(s(z0)))) -> c17(A__SIEVE(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(sieve(head(cons(z0, z1)))) -> c17(A__SIEVE(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(sieve(head(divides(z0, z1)))) -> c17(A__SIEVE(a__head(divides(mark(z0), mark(z1)))), MARK(head(divides(z0, z1)))) MARK(sieve(head(x0))) -> c17(MARK(head(x0))) MARK(sieve(head(0))) -> c17(A__SIEVE(a__head(0))) MARK(sieve(head(true))) -> c17(A__SIEVE(a__head(true))) MARK(sieve(head(false))) -> c17(A__SIEVE(a__head(false))) MARK(sieve(tail(primes))) -> c17(A__SIEVE(a__tail(a__primes)), MARK(tail(primes))) MARK(sieve(tail(sieve(z0)))) -> c17(A__SIEVE(a__tail(a__sieve(mark(z0)))), MARK(tail(sieve(z0)))) MARK(sieve(tail(from(z0)))) -> c17(A__SIEVE(a__tail(a__from(mark(z0)))), MARK(tail(from(z0)))) MARK(sieve(tail(head(z0)))) -> c17(A__SIEVE(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(sieve(tail(tail(z0)))) -> c17(A__SIEVE(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(sieve(tail(if(z0, z1, z2)))) -> c17(A__SIEVE(a__tail(a__if(mark(z0), z1, z2))), MARK(tail(if(z0, z1, z2)))) MARK(sieve(tail(filter(z0, z1)))) -> c17(A__SIEVE(a__tail(a__filter(mark(z0), mark(z1)))), MARK(tail(filter(z0, z1)))) MARK(sieve(tail(s(z0)))) -> c17(A__SIEVE(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(sieve(tail(cons(z0, z1)))) -> c17(A__SIEVE(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(sieve(tail(divides(z0, z1)))) -> c17(A__SIEVE(a__tail(divides(mark(z0), mark(z1)))), MARK(tail(divides(z0, z1)))) MARK(sieve(tail(x0))) -> c17(MARK(tail(x0))) MARK(sieve(tail(0))) -> c17(A__SIEVE(a__tail(0))) MARK(sieve(tail(true))) -> c17(A__SIEVE(a__tail(true))) MARK(sieve(tail(false))) -> c17(A__SIEVE(a__tail(false))) MARK(sieve(if(primes, x1, x2))) -> c17(A__SIEVE(a__if(a__primes, x1, x2)), MARK(if(primes, x1, x2))) MARK(sieve(if(sieve(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__sieve(mark(z0)), x1, x2)), MARK(if(sieve(z0), x1, x2))) MARK(sieve(if(from(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__from(mark(z0)), x1, x2)), MARK(if(from(z0), x1, x2))) MARK(sieve(if(head(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__head(mark(z0)), x1, x2)), MARK(if(head(z0), x1, x2))) MARK(sieve(if(tail(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__tail(mark(z0)), x1, x2)), MARK(if(tail(z0), x1, x2))) 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MARK(filter(head(z0), x1))) MARK(if(filter(tail(z0), x1), x2, x3)) -> c21(A__IF(a__filter(a__tail(mark(z0)), mark(x1)), x2, x3), MARK(filter(tail(z0), x1))) MARK(if(filter(if(z0, z1, z2), x1), x2, x3)) -> c21(A__IF(a__filter(a__if(mark(z0), z1, z2), mark(x1)), x2, x3), MARK(filter(if(z0, z1, z2), x1))) MARK(if(filter(filter(z0, z1), x1), x2, x3)) -> c21(A__IF(a__filter(a__filter(mark(z0), mark(z1)), mark(x1)), x2, x3), MARK(filter(filter(z0, z1), x1))) MARK(if(filter(s(z0), x1), x2, x3)) -> c21(A__IF(a__filter(s(mark(z0)), mark(x1)), x2, x3), MARK(filter(s(z0), x1))) MARK(if(filter(0, x1), x2, x3)) -> c21(A__IF(a__filter(0, mark(x1)), x2, x3), MARK(filter(0, x1))) MARK(if(filter(cons(z0, z1), x1), x2, x3)) -> c21(A__IF(a__filter(cons(mark(z0), z1), mark(x1)), x2, x3), MARK(filter(cons(z0, z1), x1))) MARK(if(filter(true, x1), x2, x3)) -> c21(A__IF(a__filter(true, mark(x1)), x2, x3), MARK(filter(true, x1))) MARK(if(filter(false, x1), x2, x3)) -> c21(A__IF(a__filter(false, mark(x1)), x2, x3), MARK(filter(false, x1))) MARK(if(filter(divides(z0, z1), x1), x2, x3)) -> c21(A__IF(a__filter(divides(mark(z0), mark(z1)), mark(x1)), x2, x3), MARK(filter(divides(z0, z1), x1))) MARK(if(filter(x0, x1), x2, x3)) -> c21(MARK(filter(x0, x1))) MARK(filter(x0, primes)) -> c22(A__FILTER(mark(x0), a__sieve(a__from(s(s(0))))), MARK(x0)) MARK(filter(primes, primes)) -> c22(A__FILTER(a__primes, a__primes), MARK(primes)) MARK(filter(sieve(z0), primes)) -> c22(A__FILTER(a__sieve(mark(z0)), a__primes), MARK(sieve(z0))) MARK(filter(from(z0), primes)) -> c22(A__FILTER(a__from(mark(z0)), a__primes), MARK(from(z0))) MARK(filter(head(z0), primes)) -> c22(A__FILTER(a__head(mark(z0)), a__primes), MARK(head(z0))) MARK(filter(tail(z0), primes)) -> c22(A__FILTER(a__tail(mark(z0)), a__primes), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), primes)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), a__primes), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), primes)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), a__primes), MARK(filter(z0, z1))) MARK(filter(s(z0), primes)) -> c22(A__FILTER(s(mark(z0)), a__primes), MARK(s(z0))) MARK(filter(x0, primes)) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), primes)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), primes)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, sieve(primes))) -> c22(A__FILTER(mark(x0), a__sieve(a__primes)), MARK(x0)) MARK(filter(x0, sieve(sieve(z0)))) -> c22(A__FILTER(mark(x0), a__sieve(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, sieve(from(z0)))) -> c22(A__FILTER(mark(x0), a__sieve(a__from(mark(z0)))), MARK(x0)) MARK(filter(x0, sieve(head(z0)))) -> c22(A__FILTER(mark(x0), a__sieve(a__head(mark(z0)))), MARK(x0)) MARK(filter(x0, sieve(tail(z0)))) -> c22(A__FILTER(mark(x0), a__sieve(a__tail(mark(z0)))), MARK(x0)) MARK(filter(x0, sieve(if(z0, z1, z2)))) -> c22(A__FILTER(mark(x0), a__sieve(a__if(mark(z0), z1, z2))), MARK(x0)) MARK(filter(x0, sieve(filter(z0, z1)))) -> c22(A__FILTER(mark(x0), a__sieve(a__filter(mark(z0), mark(z1)))), MARK(x0)) MARK(filter(x0, sieve(s(z0)))) -> c22(A__FILTER(mark(x0), a__sieve(s(mark(z0)))), MARK(x0)) MARK(filter(x0, sieve(0))) -> c22(A__FILTER(mark(x0), a__sieve(0)), MARK(x0)) MARK(filter(x0, sieve(cons(z0, z1)))) -> c22(A__FILTER(mark(x0), a__sieve(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, sieve(true))) -> c22(A__FILTER(mark(x0), a__sieve(true)), MARK(x0)) MARK(filter(x0, sieve(false))) -> c22(A__FILTER(mark(x0), a__sieve(false)), MARK(x0)) MARK(filter(x0, sieve(divides(z0, z1)))) -> c22(A__FILTER(mark(x0), a__sieve(divides(mark(z0), mark(z1)))), MARK(x0)) MARK(filter(primes, sieve(x1))) -> c22(A__FILTER(a__primes, a__sieve(mark(x1))), MARK(primes)) MARK(filter(sieve(z0), sieve(x1))) -> c22(A__FILTER(a__sieve(mark(z0)), a__sieve(mark(x1))), MARK(sieve(z0))) MARK(filter(from(z0), sieve(x1))) -> c22(A__FILTER(a__from(mark(z0)), a__sieve(mark(x1))), MARK(from(z0))) MARK(filter(head(z0), sieve(x1))) -> c22(A__FILTER(a__head(mark(z0)), a__sieve(mark(x1))), MARK(head(z0))) MARK(filter(tail(z0), sieve(x1))) -> c22(A__FILTER(a__tail(mark(z0)), a__sieve(mark(x1))), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), sieve(x1))) -> c22(A__FILTER(a__if(mark(z0), z1, z2), a__sieve(mark(x1))), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), sieve(x1))) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), a__sieve(mark(x1))), MARK(filter(z0, z1))) MARK(filter(s(z0), sieve(x1))) -> c22(A__FILTER(s(mark(z0)), a__sieve(mark(x1))), MARK(s(z0))) MARK(filter(x0, sieve(x1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), sieve(x1))) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), sieve(x1))) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, from(x1))) -> c22(A__FILTER(mark(x0), cons(mark(mark(x1)), from(s(mark(x1))))), MARK(x0)) MARK(filter(x0, from(primes))) -> c22(A__FILTER(mark(x0), a__from(a__primes)), MARK(x0)) MARK(filter(x0, from(sieve(z0)))) -> c22(A__FILTER(mark(x0), a__from(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, from(from(z0)))) -> c22(A__FILTER(mark(x0), a__from(a__from(mark(z0)))), MARK(x0)) MARK(filter(x0, from(head(z0)))) -> c22(A__FILTER(mark(x0), a__from(a__head(mark(z0)))), MARK(x0)) MARK(filter(x0, from(tail(z0)))) -> c22(A__FILTER(mark(x0), a__from(a__tail(mark(z0)))), MARK(x0)) MARK(filter(x0, from(if(z0, z1, z2)))) -> c22(A__FILTER(mark(x0), a__from(a__if(mark(z0), z1, z2))), MARK(x0)) MARK(filter(x0, from(filter(z0, z1)))) -> c22(A__FILTER(mark(x0), a__from(a__filter(mark(z0), mark(z1)))), MARK(x0)) MARK(filter(x0, from(s(z0)))) -> c22(A__FILTER(mark(x0), a__from(s(mark(z0)))), MARK(x0)) MARK(filter(x0, from(0))) -> c22(A__FILTER(mark(x0), a__from(0)), MARK(x0)) MARK(filter(x0, from(cons(z0, z1)))) -> c22(A__FILTER(mark(x0), a__from(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, from(true))) -> c22(A__FILTER(mark(x0), a__from(true)), MARK(x0)) MARK(filter(x0, from(false))) -> c22(A__FILTER(mark(x0), a__from(false)), MARK(x0)) MARK(filter(x0, from(divides(z0, z1)))) -> c22(A__FILTER(mark(x0), a__from(divides(mark(z0), mark(z1)))), MARK(x0)) MARK(filter(primes, from(x1))) -> c22(A__FILTER(a__primes, a__from(mark(x1))), MARK(primes)) MARK(filter(sieve(z0), from(x1))) -> c22(A__FILTER(a__sieve(mark(z0)), a__from(mark(x1))), MARK(sieve(z0))) MARK(filter(from(z0), from(x1))) -> c22(A__FILTER(a__from(mark(z0)), a__from(mark(x1))), MARK(from(z0))) MARK(filter(head(z0), from(x1))) -> c22(A__FILTER(a__head(mark(z0)), a__from(mark(x1))), MARK(head(z0))) MARK(filter(tail(z0), from(x1))) -> c22(A__FILTER(a__tail(mark(z0)), a__from(mark(x1))), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), from(x1))) -> c22(A__FILTER(a__if(mark(z0), z1, z2), a__from(mark(x1))), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), from(x1))) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), a__from(mark(x1))), MARK(filter(z0, z1))) MARK(filter(s(z0), from(x1))) -> c22(A__FILTER(s(mark(z0)), a__from(mark(x1))), MARK(s(z0))) MARK(filter(x0, from(x1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), from(x1))) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), from(x1))) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, head(primes))) -> c22(A__FILTER(mark(x0), a__head(a__primes)), MARK(x0)) MARK(filter(x0, head(sieve(z0)))) -> c22(A__FILTER(mark(x0), a__head(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, head(from(z0)))) -> c22(A__FILTER(mark(x0), a__head(a__from(mark(z0)))), MARK(x0)) MARK(filter(x0, head(head(z0)))) -> c22(A__FILTER(mark(x0), a__head(a__head(mark(z0)))), MARK(x0)) MARK(filter(x0, head(tail(z0)))) -> c22(A__FILTER(mark(x0), a__head(a__tail(mark(z0)))), MARK(x0)) MARK(filter(x0, head(if(z0, z1, z2)))) -> c22(A__FILTER(mark(x0), a__head(a__if(mark(z0), z1, z2))), MARK(x0)) MARK(filter(x0, head(filter(z0, z1)))) -> c22(A__FILTER(mark(x0), a__head(a__filter(mark(z0), mark(z1)))), MARK(x0)) MARK(filter(x0, head(s(z0)))) -> c22(A__FILTER(mark(x0), a__head(s(mark(z0)))), MARK(x0)) MARK(filter(x0, head(0))) -> c22(A__FILTER(mark(x0), a__head(0)), MARK(x0)) MARK(filter(x0, head(cons(z0, z1)))) -> c22(A__FILTER(mark(x0), a__head(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, head(true))) -> c22(A__FILTER(mark(x0), a__head(true)), MARK(x0)) MARK(filter(x0, head(false))) -> c22(A__FILTER(mark(x0), a__head(false)), MARK(x0)) MARK(filter(x0, head(divides(z0, z1)))) -> c22(A__FILTER(mark(x0), a__head(divides(mark(z0), mark(z1)))), MARK(x0)) MARK(filter(primes, head(x1))) -> c22(A__FILTER(a__primes, a__head(mark(x1))), MARK(primes)) MARK(filter(sieve(z0), head(x1))) -> c22(A__FILTER(a__sieve(mark(z0)), a__head(mark(x1))), MARK(sieve(z0))) MARK(filter(from(z0), head(x1))) -> c22(A__FILTER(a__from(mark(z0)), a__head(mark(x1))), MARK(from(z0))) MARK(filter(head(z0), head(x1))) -> c22(A__FILTER(a__head(mark(z0)), a__head(mark(x1))), MARK(head(z0))) MARK(filter(tail(z0), head(x1))) -> c22(A__FILTER(a__tail(mark(z0)), a__head(mark(x1))), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), head(x1))) -> c22(A__FILTER(a__if(mark(z0), z1, z2), a__head(mark(x1))), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), head(x1))) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), a__head(mark(x1))), MARK(filter(z0, z1))) MARK(filter(s(z0), head(x1))) -> c22(A__FILTER(s(mark(z0)), a__head(mark(x1))), MARK(s(z0))) MARK(filter(x0, head(x1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), head(x1))) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), head(x1))) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, tail(primes))) -> c22(A__FILTER(mark(x0), a__tail(a__primes)), MARK(x0)) MARK(filter(x0, tail(sieve(z0)))) -> c22(A__FILTER(mark(x0), a__tail(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, tail(from(z0)))) -> c22(A__FILTER(mark(x0), a__tail(a__from(mark(z0)))), MARK(x0)) MARK(filter(x0, tail(head(z0)))) -> c22(A__FILTER(mark(x0), a__tail(a__head(mark(z0)))), MARK(x0)) MARK(filter(x0, tail(tail(z0)))) -> c22(A__FILTER(mark(x0), a__tail(a__tail(mark(z0)))), MARK(x0)) MARK(filter(x0, tail(if(z0, z1, z2)))) -> c22(A__FILTER(mark(x0), a__tail(a__if(mark(z0), z1, z2))), MARK(x0)) MARK(filter(x0, tail(filter(z0, z1)))) -> c22(A__FILTER(mark(x0), a__tail(a__filter(mark(z0), mark(z1)))), MARK(x0)) MARK(filter(x0, tail(s(z0)))) -> c22(A__FILTER(mark(x0), a__tail(s(mark(z0)))), MARK(x0)) MARK(filter(x0, tail(0))) -> c22(A__FILTER(mark(x0), a__tail(0)), MARK(x0)) MARK(filter(x0, tail(cons(z0, z1)))) -> c22(A__FILTER(mark(x0), a__tail(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, tail(true))) -> c22(A__FILTER(mark(x0), a__tail(true)), MARK(x0)) MARK(filter(x0, tail(false))) -> c22(A__FILTER(mark(x0), a__tail(false)), MARK(x0)) MARK(filter(x0, tail(divides(z0, z1)))) -> c22(A__FILTER(mark(x0), a__tail(divides(mark(z0), mark(z1)))), MARK(x0)) MARK(filter(primes, tail(x1))) -> c22(A__FILTER(a__primes, a__tail(mark(x1))), MARK(primes)) MARK(filter(sieve(z0), tail(x1))) -> c22(A__FILTER(a__sieve(mark(z0)), a__tail(mark(x1))), MARK(sieve(z0))) MARK(filter(from(z0), tail(x1))) -> c22(A__FILTER(a__from(mark(z0)), a__tail(mark(x1))), MARK(from(z0))) MARK(filter(head(z0), tail(x1))) -> c22(A__FILTER(a__head(mark(z0)), a__tail(mark(x1))), MARK(head(z0))) MARK(filter(tail(z0), tail(x1))) -> c22(A__FILTER(a__tail(mark(z0)), a__tail(mark(x1))), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), tail(x1))) -> c22(A__FILTER(a__if(mark(z0), z1, z2), a__tail(mark(x1))), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), tail(x1))) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), a__tail(mark(x1))), MARK(filter(z0, z1))) MARK(filter(s(z0), tail(x1))) -> c22(A__FILTER(s(mark(z0)), a__tail(mark(x1))), MARK(s(z0))) MARK(filter(x0, tail(x1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), tail(x1))) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), tail(x1))) -> c22(MARK(divides(z0, z1))) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c18_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_1, c17_2, c17_1, c19_2, c19_1, c20_2, c20_1, c21_1, c22_2, c22_1, c23_2, c23_1, c_2, c21_2 ---------------------------------------- (241) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(filter(x0, if(z0, z1, z2))) -> c22(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(x0)) by MARK(filter(x0, if(x1, z1, z2))) -> c22(A__FILTER(mark(x0), if(mark(x1), z1, z2)), MARK(x0)) MARK(filter(x0, if(primes, x2, x3))) -> c22(A__FILTER(mark(x0), a__if(a__primes, x2, x3)), MARK(x0)) MARK(filter(x0, if(sieve(z0), x2, x3))) -> c22(A__FILTER(mark(x0), a__if(a__sieve(mark(z0)), x2, x3)), MARK(x0)) MARK(filter(x0, if(from(z0), x2, x3))) -> c22(A__FILTER(mark(x0), a__if(a__from(mark(z0)), x2, x3)), MARK(x0)) MARK(filter(x0, if(head(z0), x2, x3))) -> c22(A__FILTER(mark(x0), a__if(a__head(mark(z0)), x2, x3)), MARK(x0)) MARK(filter(x0, if(tail(z0), x2, x3))) -> c22(A__FILTER(mark(x0), a__if(a__tail(mark(z0)), x2, x3)), MARK(x0)) MARK(filter(x0, if(if(z0, z1, z2), x2, x3))) -> c22(A__FILTER(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0)) MARK(filter(x0, if(filter(z0, z1), x2, x3))) -> c22(A__FILTER(mark(x0), a__if(a__filter(mark(z0), mark(z1)), x2, x3)), MARK(x0)) MARK(filter(x0, if(s(z0), x2, x3))) -> c22(A__FILTER(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0)) MARK(filter(x0, if(0, x2, x3))) -> c22(A__FILTER(mark(x0), a__if(0, x2, x3)), MARK(x0)) MARK(filter(x0, if(cons(z0, z1), x2, x3))) -> c22(A__FILTER(mark(x0), a__if(cons(mark(z0), z1), x2, x3)), MARK(x0)) MARK(filter(x0, if(true, x2, x3))) -> c22(A__FILTER(mark(x0), a__if(true, x2, x3)), MARK(x0)) MARK(filter(x0, if(false, x2, x3))) -> c22(A__FILTER(mark(x0), a__if(false, x2, x3)), MARK(x0)) MARK(filter(x0, if(divides(z0, z1), x2, x3))) -> c22(A__FILTER(mark(x0), a__if(divides(mark(z0), mark(z1)), x2, x3)), MARK(x0)) MARK(filter(primes, if(x1, x2, x3))) -> c22(A__FILTER(a__primes, a__if(mark(x1), x2, x3)), MARK(primes)) MARK(filter(sieve(z0), if(x1, x2, x3))) -> c22(A__FILTER(a__sieve(mark(z0)), a__if(mark(x1), x2, x3)), MARK(sieve(z0))) MARK(filter(from(z0), if(x1, x2, x3))) -> c22(A__FILTER(a__from(mark(z0)), a__if(mark(x1), x2, x3)), MARK(from(z0))) MARK(filter(head(z0), if(x1, x2, x3))) -> c22(A__FILTER(a__head(mark(z0)), a__if(mark(x1), x2, x3)), MARK(head(z0))) MARK(filter(tail(z0), if(x1, x2, x3))) -> c22(A__FILTER(a__tail(mark(z0)), a__if(mark(x1), x2, x3)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), if(x1, x2, x3))) -> c22(A__FILTER(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), if(x1, x2, x3))) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(filter(z0, z1))) MARK(filter(s(z0), if(x1, x2, x3))) -> c22(A__FILTER(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0))) MARK(filter(0, if(x1, x2, x3))) -> c22(A__FILTER(0, a__if(mark(x1), x2, x3)), MARK(0)) MARK(filter(cons(z0, z1), if(x1, x2, x3))) -> c22(A__FILTER(cons(mark(z0), z1), a__if(mark(x1), x2, x3)), MARK(cons(z0, z1))) MARK(filter(true, if(x1, x2, x3))) -> c22(A__FILTER(true, a__if(mark(x1), x2, x3)), MARK(true)) MARK(filter(false, if(x1, x2, x3))) -> c22(A__FILTER(false, a__if(mark(x1), x2, x3)), MARK(false)) MARK(filter(divides(z0, z1), if(x1, x2, x3))) -> c22(A__FILTER(divides(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(divides(z0, z1))) ---------------------------------------- (242) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, filter(z0, z1))) -> c22(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(x0)) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(x0, s(z0))) -> c22(MARK(x0)) MARK(filter(x0, 0)) -> c22(MARK(x0)) MARK(filter(x0, true)) -> c22(MARK(x0)) MARK(filter(x0, false)) -> c22(MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), x1)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c23(A__FILTER(mark(x0), a__primes), MARK(primes)) MARK(filter(x0, sieve(z0))) -> c23(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, from(z0))) -> c23(A__FILTER(mark(x0), a__from(mark(z0))), MARK(from(z0))) MARK(filter(x0, head(z0))) -> c23(A__FILTER(mark(x0), a__head(mark(z0))), MARK(head(z0))) MARK(filter(x0, tail(z0))) -> c23(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(tail(z0))) MARK(filter(x0, if(z0, z1, z2))) -> c23(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(filter(x0, filter(z0, z1))) -> c23(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(filter(x0, cons(z0, z1))) -> c23(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(primes, x1)) -> c23(A__FILTER(a__primes, mark(x1)), MARK(x1)) MARK(filter(sieve(z0), x1)) -> c23(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(from(z0), x1)) -> c23(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(head(z0), x1)) -> c23(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(tail(z0), x1)) -> c23(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(if(z0, z1, z2), x1)) -> c23(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1)) MARK(filter(filter(z0, z1), x1)) -> c23(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(x1)) MARK(filter(s(z0), x1)) -> c23(A__FILTER(s(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(x0, s(z0))) -> c23(MARK(s(z0))) MARK(filter(x0, divides(z0, z1))) -> c23(MARK(divides(z0, z1))) MARK(filter(0, x1)) -> c23(MARK(x1)) MARK(filter(cons(z0, z1), x1)) -> c23(MARK(x1)) MARK(filter(true, x1)) -> c23(MARK(x1)) MARK(filter(false, x1)) -> c23(MARK(x1)) MARK(filter(divides(z0, z1), x1)) -> c23(MARK(x1)) A__PRIMES -> c(A__SIEVE(cons(s(mark(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(sieve(primes)) -> c17(MARK(primes)) MARK(sieve(sieve(primes))) -> c17(A__SIEVE(a__sieve(a__primes)), MARK(sieve(primes))) MARK(sieve(sieve(sieve(z0)))) -> c17(A__SIEVE(a__sieve(a__sieve(mark(z0)))), MARK(sieve(sieve(z0)))) MARK(sieve(sieve(from(z0)))) -> c17(A__SIEVE(a__sieve(a__from(mark(z0)))), MARK(sieve(from(z0)))) MARK(sieve(sieve(head(z0)))) -> c17(A__SIEVE(a__sieve(a__head(mark(z0)))), MARK(sieve(head(z0)))) MARK(sieve(sieve(tail(z0)))) -> c17(A__SIEVE(a__sieve(a__tail(mark(z0)))), MARK(sieve(tail(z0)))) MARK(sieve(sieve(if(z0, z1, z2)))) -> c17(A__SIEVE(a__sieve(a__if(mark(z0), z1, z2))), MARK(sieve(if(z0, z1, z2)))) MARK(sieve(sieve(filter(z0, z1)))) -> c17(A__SIEVE(a__sieve(a__filter(mark(z0), mark(z1)))), MARK(sieve(filter(z0, z1)))) MARK(sieve(sieve(s(z0)))) -> c17(A__SIEVE(a__sieve(s(mark(z0)))), MARK(sieve(s(z0)))) MARK(sieve(sieve(cons(z0, z1)))) -> c17(A__SIEVE(a__sieve(cons(mark(z0), z1))), MARK(sieve(cons(z0, z1)))) MARK(sieve(sieve(divides(z0, z1)))) -> c17(A__SIEVE(a__sieve(divides(mark(z0), mark(z1)))), MARK(sieve(divides(z0, z1)))) MARK(sieve(sieve(x0))) -> c17(MARK(sieve(x0))) MARK(sieve(sieve(0))) -> c17(A__SIEVE(a__sieve(0))) MARK(sieve(sieve(true))) -> c17(A__SIEVE(a__sieve(true))) MARK(sieve(sieve(false))) -> c17(A__SIEVE(a__sieve(false))) MARK(sieve(from(x0))) -> c17(A__SIEVE(cons(mark(mark(x0)), from(s(mark(x0))))), MARK(from(x0))) MARK(sieve(from(primes))) -> c17(A__SIEVE(a__from(a__primes)), MARK(from(primes))) MARK(sieve(from(sieve(z0)))) -> c17(A__SIEVE(a__from(a__sieve(mark(z0)))), MARK(from(sieve(z0)))) MARK(sieve(from(from(z0)))) -> c17(A__SIEVE(a__from(a__from(mark(z0)))), MARK(from(from(z0)))) MARK(sieve(from(head(z0)))) -> c17(A__SIEVE(a__from(a__head(mark(z0)))), MARK(from(head(z0)))) MARK(sieve(from(tail(z0)))) -> c17(A__SIEVE(a__from(a__tail(mark(z0)))), MARK(from(tail(z0)))) MARK(sieve(from(if(z0, z1, z2)))) -> c17(A__SIEVE(a__from(a__if(mark(z0), z1, z2))), MARK(from(if(z0, z1, z2)))) MARK(sieve(from(filter(z0, z1)))) -> c17(A__SIEVE(a__from(a__filter(mark(z0), mark(z1)))), MARK(from(filter(z0, z1)))) MARK(sieve(from(s(z0)))) -> c17(A__SIEVE(a__from(s(mark(z0)))), MARK(from(s(z0)))) MARK(sieve(from(0))) -> c17(A__SIEVE(a__from(0)), MARK(from(0))) MARK(sieve(from(cons(z0, z1)))) -> c17(A__SIEVE(a__from(cons(mark(z0), z1))), MARK(from(cons(z0, z1)))) MARK(sieve(from(true))) -> c17(A__SIEVE(a__from(true)), MARK(from(true))) MARK(sieve(from(false))) -> c17(A__SIEVE(a__from(false)), MARK(from(false))) MARK(sieve(from(divides(z0, z1)))) -> c17(A__SIEVE(a__from(divides(mark(z0), mark(z1)))), MARK(from(divides(z0, z1)))) MARK(sieve(from(x0))) -> c17(MARK(from(x0))) MARK(sieve(head(primes))) -> c17(A__SIEVE(a__head(a__primes)), MARK(head(primes))) MARK(sieve(head(sieve(z0)))) -> c17(A__SIEVE(a__head(a__sieve(mark(z0)))), MARK(head(sieve(z0)))) MARK(sieve(head(from(z0)))) -> c17(A__SIEVE(a__head(a__from(mark(z0)))), MARK(head(from(z0)))) MARK(sieve(head(head(z0)))) -> c17(A__SIEVE(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(sieve(head(tail(z0)))) -> c17(A__SIEVE(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(sieve(head(if(z0, z1, z2)))) -> c17(A__SIEVE(a__head(a__if(mark(z0), z1, z2))), MARK(head(if(z0, z1, z2)))) MARK(sieve(head(filter(z0, z1)))) -> c17(A__SIEVE(a__head(a__filter(mark(z0), mark(z1)))), MARK(head(filter(z0, z1)))) MARK(sieve(head(s(z0)))) -> c17(A__SIEVE(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(sieve(head(cons(z0, z1)))) -> c17(A__SIEVE(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(sieve(head(divides(z0, z1)))) -> c17(A__SIEVE(a__head(divides(mark(z0), mark(z1)))), MARK(head(divides(z0, z1)))) MARK(sieve(head(x0))) -> c17(MARK(head(x0))) MARK(sieve(head(0))) -> c17(A__SIEVE(a__head(0))) MARK(sieve(head(true))) -> c17(A__SIEVE(a__head(true))) MARK(sieve(head(false))) -> c17(A__SIEVE(a__head(false))) MARK(sieve(tail(primes))) -> c17(A__SIEVE(a__tail(a__primes)), MARK(tail(primes))) MARK(sieve(tail(sieve(z0)))) -> c17(A__SIEVE(a__tail(a__sieve(mark(z0)))), MARK(tail(sieve(z0)))) MARK(sieve(tail(from(z0)))) -> c17(A__SIEVE(a__tail(a__from(mark(z0)))), MARK(tail(from(z0)))) MARK(sieve(tail(head(z0)))) -> c17(A__SIEVE(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(sieve(tail(tail(z0)))) -> c17(A__SIEVE(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(sieve(tail(if(z0, z1, z2)))) -> c17(A__SIEVE(a__tail(a__if(mark(z0), z1, z2))), MARK(tail(if(z0, z1, z2)))) MARK(sieve(tail(filter(z0, z1)))) -> c17(A__SIEVE(a__tail(a__filter(mark(z0), mark(z1)))), MARK(tail(filter(z0, z1)))) MARK(sieve(tail(s(z0)))) -> c17(A__SIEVE(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(sieve(tail(cons(z0, z1)))) -> c17(A__SIEVE(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) 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c22(A__FILTER(mark(x0), a__tail(s(mark(z0)))), MARK(x0)) MARK(filter(x0, tail(0))) -> c22(A__FILTER(mark(x0), a__tail(0)), MARK(x0)) MARK(filter(x0, tail(cons(z0, z1)))) -> c22(A__FILTER(mark(x0), a__tail(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, tail(true))) -> c22(A__FILTER(mark(x0), a__tail(true)), MARK(x0)) MARK(filter(x0, tail(false))) -> c22(A__FILTER(mark(x0), a__tail(false)), MARK(x0)) MARK(filter(x0, tail(divides(z0, z1)))) -> c22(A__FILTER(mark(x0), a__tail(divides(mark(z0), mark(z1)))), MARK(x0)) MARK(filter(primes, tail(x1))) -> c22(A__FILTER(a__primes, a__tail(mark(x1))), MARK(primes)) MARK(filter(sieve(z0), tail(x1))) -> c22(A__FILTER(a__sieve(mark(z0)), a__tail(mark(x1))), MARK(sieve(z0))) MARK(filter(from(z0), tail(x1))) -> c22(A__FILTER(a__from(mark(z0)), a__tail(mark(x1))), MARK(from(z0))) MARK(filter(head(z0), tail(x1))) -> c22(A__FILTER(a__head(mark(z0)), a__tail(mark(x1))), MARK(head(z0))) MARK(filter(tail(z0), tail(x1))) -> c22(A__FILTER(a__tail(mark(z0)), a__tail(mark(x1))), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), tail(x1))) -> c22(A__FILTER(a__if(mark(z0), z1, z2), a__tail(mark(x1))), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), tail(x1))) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), a__tail(mark(x1))), MARK(filter(z0, z1))) MARK(filter(s(z0), tail(x1))) -> c22(A__FILTER(s(mark(z0)), a__tail(mark(x1))), MARK(s(z0))) MARK(filter(x0, tail(x1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), tail(x1))) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), tail(x1))) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, if(x1, z1, z2))) -> c22(A__FILTER(mark(x0), if(mark(x1), z1, z2)), MARK(x0)) MARK(filter(x0, if(primes, x2, x3))) -> c22(A__FILTER(mark(x0), a__if(a__primes, x2, x3)), MARK(x0)) MARK(filter(x0, if(sieve(z0), x2, x3))) -> c22(A__FILTER(mark(x0), a__if(a__sieve(mark(z0)), x2, x3)), MARK(x0)) MARK(filter(x0, if(from(z0), x2, x3))) -> c22(A__FILTER(mark(x0), a__if(a__from(mark(z0)), x2, x3)), MARK(x0)) MARK(filter(x0, if(head(z0), x2, x3))) -> c22(A__FILTER(mark(x0), a__if(a__head(mark(z0)), x2, x3)), MARK(x0)) MARK(filter(x0, if(tail(z0), x2, x3))) -> c22(A__FILTER(mark(x0), a__if(a__tail(mark(z0)), x2, x3)), MARK(x0)) MARK(filter(x0, if(if(z0, z1, z2), x2, x3))) -> c22(A__FILTER(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0)) MARK(filter(x0, if(filter(z0, z1), x2, x3))) -> c22(A__FILTER(mark(x0), a__if(a__filter(mark(z0), mark(z1)), x2, x3)), MARK(x0)) MARK(filter(x0, if(s(z0), x2, x3))) -> c22(A__FILTER(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0)) MARK(filter(x0, if(0, x2, x3))) -> c22(A__FILTER(mark(x0), a__if(0, x2, x3)), MARK(x0)) MARK(filter(x0, if(cons(z0, z1), x2, x3))) -> c22(A__FILTER(mark(x0), a__if(cons(mark(z0), z1), x2, x3)), MARK(x0)) MARK(filter(x0, if(true, x2, x3))) -> c22(A__FILTER(mark(x0), a__if(true, x2, x3)), MARK(x0)) MARK(filter(x0, if(false, x2, x3))) -> c22(A__FILTER(mark(x0), a__if(false, x2, x3)), MARK(x0)) MARK(filter(x0, if(divides(z0, z1), x2, x3))) -> c22(A__FILTER(mark(x0), a__if(divides(mark(z0), mark(z1)), x2, x3)), MARK(x0)) MARK(filter(primes, if(x1, x2, x3))) -> c22(A__FILTER(a__primes, a__if(mark(x1), x2, x3)), MARK(primes)) MARK(filter(sieve(z0), if(x1, x2, x3))) -> c22(A__FILTER(a__sieve(mark(z0)), a__if(mark(x1), x2, x3)), MARK(sieve(z0))) MARK(filter(from(z0), if(x1, x2, x3))) -> c22(A__FILTER(a__from(mark(z0)), a__if(mark(x1), x2, x3)), MARK(from(z0))) MARK(filter(head(z0), if(x1, x2, x3))) -> c22(A__FILTER(a__head(mark(z0)), a__if(mark(x1), x2, x3)), MARK(head(z0))) MARK(filter(tail(z0), if(x1, x2, x3))) -> c22(A__FILTER(a__tail(mark(z0)), a__if(mark(x1), x2, x3)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), if(x1, x2, x3))) -> c22(A__FILTER(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), if(x1, x2, x3))) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(filter(z0, z1))) MARK(filter(s(z0), if(x1, x2, x3))) -> c22(A__FILTER(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0))) MARK(filter(0, if(x1, x2, x3))) -> c22(A__FILTER(0, a__if(mark(x1), x2, x3)), MARK(0)) MARK(filter(cons(z0, z1), if(x1, x2, x3))) -> c22(A__FILTER(cons(mark(z0), z1), a__if(mark(x1), x2, x3)), MARK(cons(z0, z1))) MARK(filter(true, if(x1, x2, x3))) -> c22(A__FILTER(true, a__if(mark(x1), x2, x3)), MARK(true)) MARK(filter(false, if(x1, x2, x3))) -> c22(A__FILTER(false, a__if(mark(x1), x2, x3)), MARK(false)) MARK(filter(divides(z0, z1), if(x1, x2, x3))) -> c22(A__FILTER(divides(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(divides(z0, z1))) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c18_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_1, c17_2, c17_1, c19_2, c19_1, c20_2, c20_1, c21_1, c22_2, c22_1, c23_2, c23_1, c_2, c21_2 ---------------------------------------- (243) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing nodes: MARK(filter(true, if(x1, x2, x3))) -> c22(A__FILTER(true, a__if(mark(x1), x2, x3)), MARK(true)) MARK(filter(0, if(x1, x2, x3))) -> c22(A__FILTER(0, a__if(mark(x1), x2, x3)), MARK(0)) MARK(filter(false, if(x1, x2, x3))) -> c22(A__FILTER(false, a__if(mark(x1), x2, x3)), MARK(false)) ---------------------------------------- (244) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, filter(z0, z1))) -> c22(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(x0)) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(x0, s(z0))) -> c22(MARK(x0)) MARK(filter(x0, 0)) -> c22(MARK(x0)) MARK(filter(x0, true)) -> c22(MARK(x0)) MARK(filter(x0, false)) -> c22(MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), x1)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c23(A__FILTER(mark(x0), a__primes), MARK(primes)) MARK(filter(x0, sieve(z0))) -> c23(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, from(z0))) -> c23(A__FILTER(mark(x0), a__from(mark(z0))), MARK(from(z0))) MARK(filter(x0, head(z0))) -> c23(A__FILTER(mark(x0), a__head(mark(z0))), MARK(head(z0))) MARK(filter(x0, tail(z0))) -> c23(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(tail(z0))) MARK(filter(x0, if(z0, z1, z2))) -> c23(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(filter(x0, filter(z0, z1))) -> c23(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(filter(x0, cons(z0, z1))) -> c23(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(primes, x1)) -> c23(A__FILTER(a__primes, mark(x1)), MARK(x1)) MARK(filter(sieve(z0), x1)) -> c23(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(from(z0), x1)) -> c23(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(head(z0), x1)) -> c23(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(tail(z0), x1)) -> c23(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(if(z0, z1, z2), x1)) -> c23(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1)) MARK(filter(filter(z0, z1), x1)) -> c23(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(x1)) MARK(filter(s(z0), x1)) -> c23(A__FILTER(s(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(x0, s(z0))) -> c23(MARK(s(z0))) MARK(filter(x0, divides(z0, z1))) -> c23(MARK(divides(z0, z1))) MARK(filter(0, x1)) -> c23(MARK(x1)) MARK(filter(cons(z0, z1), x1)) -> c23(MARK(x1)) MARK(filter(true, x1)) -> c23(MARK(x1)) MARK(filter(false, x1)) -> c23(MARK(x1)) MARK(filter(divides(z0, z1), x1)) -> c23(MARK(x1)) A__PRIMES -> c(A__SIEVE(cons(s(mark(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(sieve(primes)) -> c17(MARK(primes)) MARK(sieve(sieve(primes))) -> c17(A__SIEVE(a__sieve(a__primes)), MARK(sieve(primes))) MARK(sieve(sieve(sieve(z0)))) -> c17(A__SIEVE(a__sieve(a__sieve(mark(z0)))), MARK(sieve(sieve(z0)))) MARK(sieve(sieve(from(z0)))) -> c17(A__SIEVE(a__sieve(a__from(mark(z0)))), MARK(sieve(from(z0)))) MARK(sieve(sieve(head(z0)))) -> c17(A__SIEVE(a__sieve(a__head(mark(z0)))), MARK(sieve(head(z0)))) MARK(sieve(sieve(tail(z0)))) -> c17(A__SIEVE(a__sieve(a__tail(mark(z0)))), MARK(sieve(tail(z0)))) MARK(sieve(sieve(if(z0, z1, z2)))) -> c17(A__SIEVE(a__sieve(a__if(mark(z0), z1, z2))), MARK(sieve(if(z0, z1, z2)))) MARK(sieve(sieve(filter(z0, z1)))) -> c17(A__SIEVE(a__sieve(a__filter(mark(z0), mark(z1)))), MARK(sieve(filter(z0, z1)))) MARK(sieve(sieve(s(z0)))) -> 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a__from(a__filter(mark(z0), mark(z1)))), MARK(x0)) MARK(filter(x0, from(s(z0)))) -> c22(A__FILTER(mark(x0), a__from(s(mark(z0)))), MARK(x0)) MARK(filter(x0, from(0))) -> c22(A__FILTER(mark(x0), a__from(0)), MARK(x0)) MARK(filter(x0, from(cons(z0, z1)))) -> c22(A__FILTER(mark(x0), a__from(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, from(true))) -> c22(A__FILTER(mark(x0), a__from(true)), MARK(x0)) MARK(filter(x0, from(false))) -> c22(A__FILTER(mark(x0), a__from(false)), MARK(x0)) MARK(filter(x0, from(divides(z0, z1)))) -> c22(A__FILTER(mark(x0), a__from(divides(mark(z0), mark(z1)))), MARK(x0)) MARK(filter(primes, from(x1))) -> c22(A__FILTER(a__primes, a__from(mark(x1))), MARK(primes)) MARK(filter(sieve(z0), from(x1))) -> c22(A__FILTER(a__sieve(mark(z0)), a__from(mark(x1))), MARK(sieve(z0))) MARK(filter(from(z0), from(x1))) -> c22(A__FILTER(a__from(mark(z0)), a__from(mark(x1))), MARK(from(z0))) MARK(filter(head(z0), from(x1))) -> c22(A__FILTER(a__head(mark(z0)), a__from(mark(x1))), MARK(head(z0))) MARK(filter(tail(z0), from(x1))) -> c22(A__FILTER(a__tail(mark(z0)), a__from(mark(x1))), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), from(x1))) -> c22(A__FILTER(a__if(mark(z0), z1, z2), a__from(mark(x1))), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), from(x1))) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), a__from(mark(x1))), MARK(filter(z0, z1))) MARK(filter(s(z0), from(x1))) -> c22(A__FILTER(s(mark(z0)), a__from(mark(x1))), MARK(s(z0))) MARK(filter(x0, from(x1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), from(x1))) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), from(x1))) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, head(primes))) -> c22(A__FILTER(mark(x0), a__head(a__primes)), MARK(x0)) MARK(filter(x0, head(sieve(z0)))) -> c22(A__FILTER(mark(x0), a__head(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, head(from(z0)))) -> c22(A__FILTER(mark(x0), a__head(a__from(mark(z0)))), MARK(x0)) MARK(filter(x0, head(head(z0)))) -> c22(A__FILTER(mark(x0), a__head(a__head(mark(z0)))), MARK(x0)) MARK(filter(x0, head(tail(z0)))) -> c22(A__FILTER(mark(x0), a__head(a__tail(mark(z0)))), MARK(x0)) MARK(filter(x0, head(if(z0, z1, z2)))) -> c22(A__FILTER(mark(x0), a__head(a__if(mark(z0), z1, z2))), MARK(x0)) MARK(filter(x0, head(filter(z0, z1)))) -> c22(A__FILTER(mark(x0), a__head(a__filter(mark(z0), mark(z1)))), MARK(x0)) MARK(filter(x0, head(s(z0)))) -> c22(A__FILTER(mark(x0), a__head(s(mark(z0)))), MARK(x0)) MARK(filter(x0, head(0))) -> c22(A__FILTER(mark(x0), a__head(0)), MARK(x0)) MARK(filter(x0, head(cons(z0, z1)))) -> c22(A__FILTER(mark(x0), a__head(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, head(true))) -> c22(A__FILTER(mark(x0), a__head(true)), MARK(x0)) MARK(filter(x0, head(false))) -> c22(A__FILTER(mark(x0), a__head(false)), MARK(x0)) MARK(filter(x0, head(divides(z0, z1)))) -> c22(A__FILTER(mark(x0), a__head(divides(mark(z0), mark(z1)))), MARK(x0)) MARK(filter(primes, head(x1))) -> c22(A__FILTER(a__primes, a__head(mark(x1))), MARK(primes)) MARK(filter(sieve(z0), head(x1))) -> c22(A__FILTER(a__sieve(mark(z0)), a__head(mark(x1))), MARK(sieve(z0))) MARK(filter(from(z0), head(x1))) -> c22(A__FILTER(a__from(mark(z0)), a__head(mark(x1))), MARK(from(z0))) MARK(filter(head(z0), head(x1))) -> c22(A__FILTER(a__head(mark(z0)), a__head(mark(x1))), MARK(head(z0))) MARK(filter(tail(z0), head(x1))) -> c22(A__FILTER(a__tail(mark(z0)), a__head(mark(x1))), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), head(x1))) -> c22(A__FILTER(a__if(mark(z0), z1, z2), a__head(mark(x1))), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), head(x1))) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), a__head(mark(x1))), MARK(filter(z0, z1))) MARK(filter(s(z0), head(x1))) -> c22(A__FILTER(s(mark(z0)), a__head(mark(x1))), MARK(s(z0))) MARK(filter(x0, head(x1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), head(x1))) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), head(x1))) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, tail(primes))) -> c22(A__FILTER(mark(x0), a__tail(a__primes)), MARK(x0)) MARK(filter(x0, tail(sieve(z0)))) -> c22(A__FILTER(mark(x0), a__tail(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, tail(from(z0)))) -> c22(A__FILTER(mark(x0), a__tail(a__from(mark(z0)))), MARK(x0)) MARK(filter(x0, tail(head(z0)))) -> c22(A__FILTER(mark(x0), a__tail(a__head(mark(z0)))), MARK(x0)) MARK(filter(x0, tail(tail(z0)))) -> c22(A__FILTER(mark(x0), a__tail(a__tail(mark(z0)))), MARK(x0)) MARK(filter(x0, tail(if(z0, z1, z2)))) -> c22(A__FILTER(mark(x0), a__tail(a__if(mark(z0), z1, z2))), MARK(x0)) MARK(filter(x0, tail(filter(z0, z1)))) -> c22(A__FILTER(mark(x0), a__tail(a__filter(mark(z0), mark(z1)))), MARK(x0)) MARK(filter(x0, tail(s(z0)))) -> c22(A__FILTER(mark(x0), a__tail(s(mark(z0)))), MARK(x0)) MARK(filter(x0, tail(0))) -> c22(A__FILTER(mark(x0), a__tail(0)), MARK(x0)) MARK(filter(x0, tail(cons(z0, z1)))) -> c22(A__FILTER(mark(x0), a__tail(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, tail(true))) -> c22(A__FILTER(mark(x0), a__tail(true)), MARK(x0)) MARK(filter(x0, tail(false))) -> c22(A__FILTER(mark(x0), a__tail(false)), MARK(x0)) MARK(filter(x0, tail(divides(z0, z1)))) -> c22(A__FILTER(mark(x0), a__tail(divides(mark(z0), mark(z1)))), MARK(x0)) MARK(filter(primes, tail(x1))) -> c22(A__FILTER(a__primes, a__tail(mark(x1))), MARK(primes)) MARK(filter(sieve(z0), tail(x1))) -> c22(A__FILTER(a__sieve(mark(z0)), a__tail(mark(x1))), MARK(sieve(z0))) MARK(filter(from(z0), tail(x1))) -> c22(A__FILTER(a__from(mark(z0)), a__tail(mark(x1))), MARK(from(z0))) MARK(filter(head(z0), tail(x1))) -> c22(A__FILTER(a__head(mark(z0)), a__tail(mark(x1))), MARK(head(z0))) MARK(filter(tail(z0), tail(x1))) -> c22(A__FILTER(a__tail(mark(z0)), a__tail(mark(x1))), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), tail(x1))) -> c22(A__FILTER(a__if(mark(z0), z1, z2), a__tail(mark(x1))), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), tail(x1))) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), a__tail(mark(x1))), MARK(filter(z0, z1))) MARK(filter(s(z0), tail(x1))) -> c22(A__FILTER(s(mark(z0)), a__tail(mark(x1))), MARK(s(z0))) MARK(filter(x0, tail(x1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), tail(x1))) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), tail(x1))) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, if(x1, z1, z2))) -> c22(A__FILTER(mark(x0), if(mark(x1), z1, z2)), MARK(x0)) MARK(filter(x0, if(primes, x2, x3))) -> c22(A__FILTER(mark(x0), a__if(a__primes, x2, x3)), MARK(x0)) MARK(filter(x0, if(sieve(z0), x2, x3))) -> c22(A__FILTER(mark(x0), a__if(a__sieve(mark(z0)), x2, x3)), MARK(x0)) MARK(filter(x0, if(from(z0), x2, x3))) -> c22(A__FILTER(mark(x0), a__if(a__from(mark(z0)), x2, x3)), MARK(x0)) MARK(filter(x0, if(head(z0), x2, x3))) -> c22(A__FILTER(mark(x0), a__if(a__head(mark(z0)), x2, x3)), MARK(x0)) MARK(filter(x0, if(tail(z0), x2, x3))) -> c22(A__FILTER(mark(x0), a__if(a__tail(mark(z0)), x2, x3)), MARK(x0)) MARK(filter(x0, if(if(z0, z1, z2), x2, x3))) -> c22(A__FILTER(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0)) MARK(filter(x0, if(filter(z0, z1), x2, x3))) -> c22(A__FILTER(mark(x0), a__if(a__filter(mark(z0), mark(z1)), x2, x3)), MARK(x0)) MARK(filter(x0, if(s(z0), x2, x3))) -> c22(A__FILTER(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0)) MARK(filter(x0, if(0, x2, x3))) -> c22(A__FILTER(mark(x0), a__if(0, x2, x3)), MARK(x0)) MARK(filter(x0, if(cons(z0, z1), x2, x3))) -> c22(A__FILTER(mark(x0), a__if(cons(mark(z0), z1), x2, x3)), MARK(x0)) MARK(filter(x0, if(true, x2, x3))) -> c22(A__FILTER(mark(x0), a__if(true, x2, x3)), MARK(x0)) MARK(filter(x0, if(false, x2, x3))) -> c22(A__FILTER(mark(x0), a__if(false, x2, x3)), MARK(x0)) MARK(filter(x0, if(divides(z0, z1), x2, x3))) -> c22(A__FILTER(mark(x0), a__if(divides(mark(z0), mark(z1)), x2, x3)), MARK(x0)) MARK(filter(primes, if(x1, x2, x3))) -> c22(A__FILTER(a__primes, a__if(mark(x1), x2, x3)), MARK(primes)) MARK(filter(sieve(z0), if(x1, x2, x3))) -> c22(A__FILTER(a__sieve(mark(z0)), a__if(mark(x1), x2, x3)), MARK(sieve(z0))) MARK(filter(from(z0), if(x1, x2, x3))) -> c22(A__FILTER(a__from(mark(z0)), a__if(mark(x1), x2, x3)), MARK(from(z0))) MARK(filter(head(z0), if(x1, x2, x3))) -> c22(A__FILTER(a__head(mark(z0)), a__if(mark(x1), x2, x3)), MARK(head(z0))) MARK(filter(tail(z0), if(x1, x2, x3))) -> c22(A__FILTER(a__tail(mark(z0)), a__if(mark(x1), x2, x3)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), if(x1, x2, x3))) -> c22(A__FILTER(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), if(x1, x2, x3))) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(filter(z0, z1))) MARK(filter(s(z0), if(x1, x2, x3))) -> c22(A__FILTER(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0))) MARK(filter(cons(z0, z1), if(x1, x2, x3))) -> c22(A__FILTER(cons(mark(z0), z1), a__if(mark(x1), x2, x3)), MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), if(x1, x2, x3))) -> c22(A__FILTER(divides(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(divides(z0, z1))) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c18_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_1, c17_2, c17_1, c19_2, c19_1, c20_2, c20_1, c21_1, c22_2, c22_1, c23_2, c23_1, c_2, c21_2 ---------------------------------------- (245) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (246) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, filter(z0, z1))) -> c22(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(x0)) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(x0, s(z0))) -> c22(MARK(x0)) MARK(filter(x0, 0)) -> c22(MARK(x0)) MARK(filter(x0, true)) -> c22(MARK(x0)) MARK(filter(x0, false)) -> c22(MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), x1)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c23(A__FILTER(mark(x0), a__primes), MARK(primes)) MARK(filter(x0, sieve(z0))) -> c23(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, from(z0))) -> c23(A__FILTER(mark(x0), a__from(mark(z0))), MARK(from(z0))) MARK(filter(x0, head(z0))) -> c23(A__FILTER(mark(x0), a__head(mark(z0))), MARK(head(z0))) MARK(filter(x0, tail(z0))) -> c23(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(tail(z0))) MARK(filter(x0, if(z0, z1, z2))) -> c23(A__FILTER(mark(x0), 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MARK(sieve(filter(x0, s(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), s(mark(z0)))), MARK(filter(x0, s(z0)))) MARK(sieve(filter(x0, 0))) -> c17(A__SIEVE(a__filter(mark(x0), 0)), MARK(filter(x0, 0))) MARK(sieve(filter(x0, cons(z0, z1)))) -> c17(A__SIEVE(a__filter(mark(x0), cons(mark(z0), z1))), MARK(filter(x0, cons(z0, z1)))) MARK(sieve(filter(x0, true))) -> c17(A__SIEVE(a__filter(mark(x0), true)), MARK(filter(x0, true))) MARK(sieve(filter(x0, false))) -> c17(A__SIEVE(a__filter(mark(x0), false)), MARK(filter(x0, false))) MARK(sieve(filter(x0, divides(z0, z1)))) -> c17(A__SIEVE(a__filter(mark(x0), divides(mark(z0), mark(z1)))), MARK(filter(x0, divides(z0, z1)))) MARK(sieve(filter(primes, x1))) -> c17(A__SIEVE(a__filter(a__primes, mark(x1))), MARK(filter(primes, x1))) MARK(sieve(filter(sieve(z0), x1))) -> c17(A__SIEVE(a__filter(a__sieve(mark(z0)), mark(x1))), MARK(filter(sieve(z0), x1))) MARK(sieve(filter(from(z0), x1))) -> c17(A__SIEVE(a__filter(a__from(mark(z0)), mark(x1))), MARK(filter(from(z0), x1))) MARK(sieve(filter(head(z0), x1))) -> c17(A__SIEVE(a__filter(a__head(mark(z0)), mark(x1))), MARK(filter(head(z0), x1))) MARK(sieve(filter(tail(z0), x1))) -> c17(A__SIEVE(a__filter(a__tail(mark(z0)), mark(x1))), MARK(filter(tail(z0), x1))) MARK(sieve(filter(if(z0, z1, z2), x1))) -> c17(A__SIEVE(a__filter(a__if(mark(z0), z1, z2), mark(x1))), MARK(filter(if(z0, z1, z2), x1))) MARK(sieve(filter(filter(z0, z1), x1))) -> c17(A__SIEVE(a__filter(a__filter(mark(z0), mark(z1)), mark(x1))), MARK(filter(filter(z0, z1), x1))) MARK(sieve(filter(s(z0), x1))) -> c17(A__SIEVE(a__filter(s(mark(z0)), mark(x1))), MARK(filter(s(z0), x1))) MARK(sieve(filter(0, x1))) -> c17(A__SIEVE(a__filter(0, mark(x1))), MARK(filter(0, x1))) MARK(sieve(filter(cons(z0, z1), x1))) -> c17(A__SIEVE(a__filter(cons(mark(z0), z1), mark(x1))), MARK(filter(cons(z0, z1), x1))) MARK(sieve(filter(true, x1))) -> c17(A__SIEVE(a__filter(true, mark(x1))), MARK(filter(true, x1))) MARK(sieve(filter(false, x1))) 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MARK(filter(x0, sieve(false))) -> c22(A__FILTER(mark(x0), a__sieve(false)), MARK(x0)) MARK(filter(x0, sieve(divides(z0, z1)))) -> c22(A__FILTER(mark(x0), a__sieve(divides(mark(z0), mark(z1)))), MARK(x0)) MARK(filter(primes, sieve(x1))) -> c22(A__FILTER(a__primes, a__sieve(mark(x1))), MARK(primes)) MARK(filter(sieve(z0), sieve(x1))) -> c22(A__FILTER(a__sieve(mark(z0)), a__sieve(mark(x1))), MARK(sieve(z0))) MARK(filter(from(z0), sieve(x1))) -> c22(A__FILTER(a__from(mark(z0)), a__sieve(mark(x1))), MARK(from(z0))) MARK(filter(head(z0), sieve(x1))) -> c22(A__FILTER(a__head(mark(z0)), a__sieve(mark(x1))), MARK(head(z0))) MARK(filter(tail(z0), sieve(x1))) -> c22(A__FILTER(a__tail(mark(z0)), a__sieve(mark(x1))), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), sieve(x1))) -> c22(A__FILTER(a__if(mark(z0), z1, z2), a__sieve(mark(x1))), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), sieve(x1))) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), a__sieve(mark(x1))), MARK(filter(z0, z1))) 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from(filter(z0, z1)))) -> c22(A__FILTER(mark(x0), a__from(a__filter(mark(z0), mark(z1)))), MARK(x0)) MARK(filter(x0, from(s(z0)))) -> c22(A__FILTER(mark(x0), a__from(s(mark(z0)))), MARK(x0)) MARK(filter(x0, from(0))) -> c22(A__FILTER(mark(x0), a__from(0)), MARK(x0)) MARK(filter(x0, from(cons(z0, z1)))) -> c22(A__FILTER(mark(x0), a__from(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, from(true))) -> c22(A__FILTER(mark(x0), a__from(true)), MARK(x0)) MARK(filter(x0, from(false))) -> c22(A__FILTER(mark(x0), a__from(false)), MARK(x0)) MARK(filter(x0, from(divides(z0, z1)))) -> c22(A__FILTER(mark(x0), a__from(divides(mark(z0), mark(z1)))), MARK(x0)) MARK(filter(primes, from(x1))) -> c22(A__FILTER(a__primes, a__from(mark(x1))), MARK(primes)) MARK(filter(sieve(z0), from(x1))) -> c22(A__FILTER(a__sieve(mark(z0)), a__from(mark(x1))), MARK(sieve(z0))) MARK(filter(from(z0), from(x1))) -> c22(A__FILTER(a__from(mark(z0)), a__from(mark(x1))), MARK(from(z0))) MARK(filter(head(z0), from(x1))) -> c22(A__FILTER(a__head(mark(z0)), a__from(mark(x1))), MARK(head(z0))) MARK(filter(tail(z0), from(x1))) -> c22(A__FILTER(a__tail(mark(z0)), a__from(mark(x1))), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), from(x1))) -> c22(A__FILTER(a__if(mark(z0), z1, z2), a__from(mark(x1))), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), from(x1))) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), a__from(mark(x1))), MARK(filter(z0, z1))) MARK(filter(s(z0), from(x1))) -> c22(A__FILTER(s(mark(z0)), a__from(mark(x1))), MARK(s(z0))) MARK(filter(x0, from(x1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), from(x1))) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), from(x1))) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, head(primes))) -> c22(A__FILTER(mark(x0), a__head(a__primes)), MARK(x0)) MARK(filter(x0, head(sieve(z0)))) -> c22(A__FILTER(mark(x0), a__head(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, head(from(z0)))) -> c22(A__FILTER(mark(x0), a__head(a__from(mark(z0)))), MARK(x0)) MARK(filter(x0, head(head(z0)))) -> c22(A__FILTER(mark(x0), a__head(a__head(mark(z0)))), MARK(x0)) MARK(filter(x0, head(tail(z0)))) -> c22(A__FILTER(mark(x0), a__head(a__tail(mark(z0)))), MARK(x0)) MARK(filter(x0, head(if(z0, z1, z2)))) -> c22(A__FILTER(mark(x0), a__head(a__if(mark(z0), z1, z2))), MARK(x0)) MARK(filter(x0, head(filter(z0, z1)))) -> c22(A__FILTER(mark(x0), a__head(a__filter(mark(z0), mark(z1)))), MARK(x0)) MARK(filter(x0, head(s(z0)))) -> c22(A__FILTER(mark(x0), a__head(s(mark(z0)))), MARK(x0)) MARK(filter(x0, head(0))) -> c22(A__FILTER(mark(x0), a__head(0)), MARK(x0)) MARK(filter(x0, head(cons(z0, z1)))) -> c22(A__FILTER(mark(x0), a__head(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, head(true))) -> c22(A__FILTER(mark(x0), a__head(true)), MARK(x0)) MARK(filter(x0, head(false))) -> c22(A__FILTER(mark(x0), a__head(false)), MARK(x0)) MARK(filter(x0, head(divides(z0, z1)))) -> c22(A__FILTER(mark(x0), a__head(divides(mark(z0), mark(z1)))), MARK(x0)) MARK(filter(primes, head(x1))) -> c22(A__FILTER(a__primes, a__head(mark(x1))), MARK(primes)) MARK(filter(sieve(z0), head(x1))) -> c22(A__FILTER(a__sieve(mark(z0)), a__head(mark(x1))), MARK(sieve(z0))) MARK(filter(from(z0), head(x1))) -> c22(A__FILTER(a__from(mark(z0)), a__head(mark(x1))), MARK(from(z0))) MARK(filter(head(z0), head(x1))) -> c22(A__FILTER(a__head(mark(z0)), a__head(mark(x1))), MARK(head(z0))) MARK(filter(tail(z0), head(x1))) -> c22(A__FILTER(a__tail(mark(z0)), a__head(mark(x1))), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), head(x1))) -> c22(A__FILTER(a__if(mark(z0), z1, z2), a__head(mark(x1))), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), head(x1))) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), a__head(mark(x1))), MARK(filter(z0, z1))) MARK(filter(s(z0), head(x1))) -> c22(A__FILTER(s(mark(z0)), a__head(mark(x1))), MARK(s(z0))) MARK(filter(x0, head(x1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), head(x1))) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), head(x1))) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, tail(primes))) -> c22(A__FILTER(mark(x0), a__tail(a__primes)), MARK(x0)) MARK(filter(x0, tail(sieve(z0)))) -> c22(A__FILTER(mark(x0), a__tail(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, tail(from(z0)))) -> c22(A__FILTER(mark(x0), a__tail(a__from(mark(z0)))), MARK(x0)) MARK(filter(x0, tail(head(z0)))) -> c22(A__FILTER(mark(x0), a__tail(a__head(mark(z0)))), MARK(x0)) MARK(filter(x0, tail(tail(z0)))) -> c22(A__FILTER(mark(x0), a__tail(a__tail(mark(z0)))), MARK(x0)) MARK(filter(x0, tail(if(z0, z1, z2)))) -> c22(A__FILTER(mark(x0), a__tail(a__if(mark(z0), z1, z2))), MARK(x0)) MARK(filter(x0, tail(filter(z0, z1)))) -> c22(A__FILTER(mark(x0), a__tail(a__filter(mark(z0), mark(z1)))), MARK(x0)) MARK(filter(x0, tail(s(z0)))) -> c22(A__FILTER(mark(x0), a__tail(s(mark(z0)))), MARK(x0)) MARK(filter(x0, tail(0))) -> c22(A__FILTER(mark(x0), a__tail(0)), MARK(x0)) MARK(filter(x0, tail(cons(z0, z1)))) -> c22(A__FILTER(mark(x0), a__tail(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, tail(true))) -> c22(A__FILTER(mark(x0), a__tail(true)), MARK(x0)) MARK(filter(x0, tail(false))) -> c22(A__FILTER(mark(x0), a__tail(false)), MARK(x0)) MARK(filter(x0, tail(divides(z0, z1)))) -> c22(A__FILTER(mark(x0), a__tail(divides(mark(z0), mark(z1)))), MARK(x0)) MARK(filter(primes, tail(x1))) -> c22(A__FILTER(a__primes, a__tail(mark(x1))), MARK(primes)) MARK(filter(sieve(z0), tail(x1))) -> c22(A__FILTER(a__sieve(mark(z0)), a__tail(mark(x1))), MARK(sieve(z0))) MARK(filter(from(z0), tail(x1))) -> c22(A__FILTER(a__from(mark(z0)), a__tail(mark(x1))), MARK(from(z0))) MARK(filter(head(z0), tail(x1))) -> c22(A__FILTER(a__head(mark(z0)), a__tail(mark(x1))), MARK(head(z0))) MARK(filter(tail(z0), tail(x1))) -> c22(A__FILTER(a__tail(mark(z0)), a__tail(mark(x1))), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), tail(x1))) -> c22(A__FILTER(a__if(mark(z0), z1, z2), a__tail(mark(x1))), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), tail(x1))) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), a__tail(mark(x1))), MARK(filter(z0, z1))) MARK(filter(s(z0), tail(x1))) -> c22(A__FILTER(s(mark(z0)), a__tail(mark(x1))), MARK(s(z0))) MARK(filter(x0, tail(x1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), tail(x1))) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), tail(x1))) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, if(primes, x2, x3))) -> c22(A__FILTER(mark(x0), a__if(a__primes, x2, x3)), MARK(x0)) MARK(filter(x0, if(sieve(z0), x2, x3))) -> c22(A__FILTER(mark(x0), a__if(a__sieve(mark(z0)), x2, x3)), MARK(x0)) MARK(filter(x0, if(from(z0), x2, x3))) -> c22(A__FILTER(mark(x0), a__if(a__from(mark(z0)), x2, x3)), MARK(x0)) MARK(filter(x0, if(head(z0), x2, x3))) -> c22(A__FILTER(mark(x0), a__if(a__head(mark(z0)), x2, x3)), MARK(x0)) MARK(filter(x0, if(tail(z0), x2, x3))) -> c22(A__FILTER(mark(x0), a__if(a__tail(mark(z0)), x2, x3)), MARK(x0)) MARK(filter(x0, if(if(z0, z1, z2), x2, x3))) -> c22(A__FILTER(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0)) MARK(filter(x0, if(filter(z0, z1), x2, x3))) -> c22(A__FILTER(mark(x0), a__if(a__filter(mark(z0), mark(z1)), x2, x3)), MARK(x0)) MARK(filter(x0, if(s(z0), x2, x3))) -> c22(A__FILTER(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0)) MARK(filter(x0, if(0, x2, x3))) -> c22(A__FILTER(mark(x0), a__if(0, x2, x3)), MARK(x0)) MARK(filter(x0, if(cons(z0, z1), x2, x3))) -> c22(A__FILTER(mark(x0), a__if(cons(mark(z0), z1), x2, x3)), MARK(x0)) MARK(filter(x0, if(true, x2, x3))) -> c22(A__FILTER(mark(x0), a__if(true, x2, x3)), MARK(x0)) MARK(filter(x0, if(false, x2, x3))) -> c22(A__FILTER(mark(x0), a__if(false, x2, x3)), MARK(x0)) MARK(filter(x0, if(divides(z0, z1), x2, x3))) -> c22(A__FILTER(mark(x0), a__if(divides(mark(z0), mark(z1)), x2, x3)), MARK(x0)) MARK(filter(primes, if(x1, x2, x3))) -> c22(A__FILTER(a__primes, a__if(mark(x1), x2, x3)), MARK(primes)) MARK(filter(sieve(z0), if(x1, x2, x3))) -> c22(A__FILTER(a__sieve(mark(z0)), a__if(mark(x1), x2, x3)), MARK(sieve(z0))) MARK(filter(from(z0), if(x1, x2, x3))) -> c22(A__FILTER(a__from(mark(z0)), a__if(mark(x1), x2, x3)), MARK(from(z0))) MARK(filter(head(z0), if(x1, x2, x3))) -> c22(A__FILTER(a__head(mark(z0)), a__if(mark(x1), x2, x3)), MARK(head(z0))) MARK(filter(tail(z0), if(x1, x2, x3))) -> c22(A__FILTER(a__tail(mark(z0)), a__if(mark(x1), x2, x3)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), if(x1, x2, x3))) -> c22(A__FILTER(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), if(x1, x2, x3))) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(filter(z0, z1))) MARK(filter(s(z0), if(x1, x2, x3))) -> c22(A__FILTER(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0))) MARK(filter(x0, if(x1, z1, z2))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), if(x1, x2, x3))) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), if(x1, x2, x3))) -> c22(MARK(divides(z0, z1))) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c18_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_1, c17_2, c17_1, c19_2, c19_1, c20_2, c20_1, c21_1, c22_2, c22_1, c23_2, c23_1, c_2, c21_2 ---------------------------------------- (247) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace MARK(filter(x0, filter(z0, z1))) -> c22(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(x0)) by MARK(filter(x0, filter(x1, x2))) -> c22(A__FILTER(mark(x0), filter(mark(x1), mark(x2))), MARK(x0)) MARK(filter(x0, filter(x1, primes))) -> c22(A__FILTER(mark(x0), a__filter(mark(x1), a__primes)), MARK(x0)) MARK(filter(x0, filter(x1, sieve(z0)))) -> c22(A__FILTER(mark(x0), a__filter(mark(x1), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(x1, from(z0)))) -> c22(A__FILTER(mark(x0), a__filter(mark(x1), a__from(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(x1, head(z0)))) -> c22(A__FILTER(mark(x0), a__filter(mark(x1), a__head(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(x1, tail(z0)))) -> c22(A__FILTER(mark(x0), a__filter(mark(x1), a__tail(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(x1, if(z0, z1, z2)))) -> c22(A__FILTER(mark(x0), a__filter(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0)) MARK(filter(x0, filter(x1, filter(z0, z1)))) -> c22(A__FILTER(mark(x0), a__filter(mark(x1), a__filter(mark(z0), mark(z1)))), MARK(x0)) MARK(filter(x0, filter(x1, s(z0)))) -> c22(A__FILTER(mark(x0), a__filter(mark(x1), s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(x1, 0))) -> c22(A__FILTER(mark(x0), a__filter(mark(x1), 0)), MARK(x0)) MARK(filter(x0, filter(x1, cons(z0, z1)))) -> c22(A__FILTER(mark(x0), a__filter(mark(x1), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, filter(x1, true))) -> c22(A__FILTER(mark(x0), a__filter(mark(x1), true)), MARK(x0)) MARK(filter(x0, filter(x1, false))) -> c22(A__FILTER(mark(x0), a__filter(mark(x1), false)), MARK(x0)) MARK(filter(x0, filter(x1, divides(z0, z1)))) -> c22(A__FILTER(mark(x0), a__filter(mark(x1), divides(mark(z0), mark(z1)))), MARK(x0)) MARK(filter(x0, filter(primes, x2))) -> c22(A__FILTER(mark(x0), a__filter(a__primes, mark(x2))), MARK(x0)) MARK(filter(x0, filter(sieve(z0), x2))) -> c22(A__FILTER(mark(x0), a__filter(a__sieve(mark(z0)), mark(x2))), MARK(x0)) MARK(filter(x0, filter(from(z0), x2))) -> c22(A__FILTER(mark(x0), a__filter(a__from(mark(z0)), mark(x2))), MARK(x0)) MARK(filter(x0, filter(head(z0), x2))) -> c22(A__FILTER(mark(x0), a__filter(a__head(mark(z0)), mark(x2))), MARK(x0)) MARK(filter(x0, filter(tail(z0), x2))) -> c22(A__FILTER(mark(x0), a__filter(a__tail(mark(z0)), mark(x2))), MARK(x0)) MARK(filter(x0, filter(if(z0, z1, z2), x2))) -> c22(A__FILTER(mark(x0), a__filter(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0)) MARK(filter(x0, filter(filter(z0, z1), x2))) -> c22(A__FILTER(mark(x0), a__filter(a__filter(mark(z0), mark(z1)), mark(x2))), MARK(x0)) MARK(filter(x0, filter(s(z0), x2))) -> c22(A__FILTER(mark(x0), a__filter(s(mark(z0)), mark(x2))), MARK(x0)) MARK(filter(x0, filter(0, x2))) -> c22(A__FILTER(mark(x0), a__filter(0, mark(x2))), MARK(x0)) MARK(filter(x0, filter(cons(z0, z1), x2))) -> c22(A__FILTER(mark(x0), a__filter(cons(mark(z0), z1), mark(x2))), MARK(x0)) MARK(filter(x0, filter(true, x2))) -> c22(A__FILTER(mark(x0), a__filter(true, mark(x2))), MARK(x0)) MARK(filter(x0, filter(false, x2))) -> c22(A__FILTER(mark(x0), a__filter(false, mark(x2))), MARK(x0)) MARK(filter(x0, filter(divides(z0, z1), x2))) -> c22(A__FILTER(mark(x0), a__filter(divides(mark(z0), mark(z1)), mark(x2))), MARK(x0)) MARK(filter(primes, filter(x1, x2))) -> c22(A__FILTER(a__primes, a__filter(mark(x1), mark(x2))), MARK(primes)) MARK(filter(sieve(z0), filter(x1, x2))) -> c22(A__FILTER(a__sieve(mark(z0)), a__filter(mark(x1), mark(x2))), MARK(sieve(z0))) MARK(filter(from(z0), filter(x1, x2))) -> c22(A__FILTER(a__from(mark(z0)), a__filter(mark(x1), mark(x2))), MARK(from(z0))) MARK(filter(head(z0), filter(x1, x2))) -> c22(A__FILTER(a__head(mark(z0)), a__filter(mark(x1), mark(x2))), MARK(head(z0))) MARK(filter(tail(z0), filter(x1, x2))) -> c22(A__FILTER(a__tail(mark(z0)), a__filter(mark(x1), mark(x2))), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), filter(x1, x2))) -> c22(A__FILTER(a__if(mark(z0), z1, z2), a__filter(mark(x1), mark(x2))), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), filter(x1, x2))) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), a__filter(mark(x1), mark(x2))), MARK(filter(z0, z1))) MARK(filter(s(z0), filter(x1, x2))) -> c22(A__FILTER(s(mark(z0)), a__filter(mark(x1), mark(x2))), MARK(s(z0))) MARK(filter(0, filter(x1, x2))) -> c22(A__FILTER(0, a__filter(mark(x1), mark(x2))), MARK(0)) MARK(filter(cons(z0, z1), filter(x1, x2))) -> c22(A__FILTER(cons(mark(z0), z1), a__filter(mark(x1), mark(x2))), MARK(cons(z0, z1))) MARK(filter(true, filter(x1, x2))) -> c22(A__FILTER(true, a__filter(mark(x1), mark(x2))), MARK(true)) MARK(filter(false, filter(x1, x2))) -> c22(A__FILTER(false, a__filter(mark(x1), mark(x2))), MARK(false)) MARK(filter(divides(z0, z1), filter(x1, x2))) -> c22(A__FILTER(divides(mark(z0), mark(z1)), a__filter(mark(x1), mark(x2))), MARK(divides(z0, z1))) ---------------------------------------- (248) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(x0, s(z0))) -> c22(MARK(x0)) MARK(filter(x0, 0)) -> c22(MARK(x0)) MARK(filter(x0, true)) -> c22(MARK(x0)) MARK(filter(x0, false)) -> c22(MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), x1)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c23(A__FILTER(mark(x0), a__primes), MARK(primes)) MARK(filter(x0, sieve(z0))) -> c23(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, from(z0))) -> c23(A__FILTER(mark(x0), a__from(mark(z0))), MARK(from(z0))) MARK(filter(x0, head(z0))) -> c23(A__FILTER(mark(x0), a__head(mark(z0))), MARK(head(z0))) MARK(filter(x0, tail(z0))) -> c23(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(tail(z0))) MARK(filter(x0, if(z0, z1, z2))) -> c23(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(filter(x0, filter(z0, z1))) -> c23(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(filter(x0, cons(z0, z1))) -> c23(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(primes, x1)) -> c23(A__FILTER(a__primes, mark(x1)), MARK(x1)) MARK(filter(sieve(z0), x1)) -> c23(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(from(z0), x1)) -> c23(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(head(z0), x1)) -> c23(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(tail(z0), x1)) -> c23(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(if(z0, z1, z2), x1)) -> c23(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(x1)) MARK(filter(filter(z0, z1), x1)) -> c23(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(x1)) MARK(filter(s(z0), x1)) -> c23(A__FILTER(s(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(x0, s(z0))) -> c23(MARK(s(z0))) MARK(filter(x0, divides(z0, z1))) -> c23(MARK(divides(z0, z1))) MARK(filter(0, x1)) -> c23(MARK(x1)) MARK(filter(cons(z0, z1), x1)) -> c23(MARK(x1)) MARK(filter(true, x1)) -> c23(MARK(x1)) MARK(filter(false, x1)) -> c23(MARK(x1)) MARK(filter(divides(z0, z1), x1)) -> c23(MARK(x1)) A__PRIMES -> c(A__SIEVE(cons(s(mark(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(sieve(primes)) -> c17(MARK(primes)) MARK(sieve(sieve(primes))) -> c17(A__SIEVE(a__sieve(a__primes)), MARK(sieve(primes))) MARK(sieve(sieve(sieve(z0)))) -> c17(A__SIEVE(a__sieve(a__sieve(mark(z0)))), MARK(sieve(sieve(z0)))) MARK(sieve(sieve(from(z0)))) -> c17(A__SIEVE(a__sieve(a__from(mark(z0)))), MARK(sieve(from(z0)))) MARK(sieve(sieve(head(z0)))) -> c17(A__SIEVE(a__sieve(a__head(mark(z0)))), MARK(sieve(head(z0)))) MARK(sieve(sieve(tail(z0)))) -> c17(A__SIEVE(a__sieve(a__tail(mark(z0)))), MARK(sieve(tail(z0)))) MARK(sieve(sieve(if(z0, z1, z2)))) -> c17(A__SIEVE(a__sieve(a__if(mark(z0), z1, z2))), MARK(sieve(if(z0, z1, z2)))) MARK(sieve(sieve(filter(z0, z1)))) -> c17(A__SIEVE(a__sieve(a__filter(mark(z0), mark(z1)))), MARK(sieve(filter(z0, z1)))) MARK(sieve(sieve(s(z0)))) -> c17(A__SIEVE(a__sieve(s(mark(z0)))), MARK(sieve(s(z0)))) MARK(sieve(sieve(cons(z0, z1)))) -> c17(A__SIEVE(a__sieve(cons(mark(z0), z1))), MARK(sieve(cons(z0, z1)))) MARK(sieve(sieve(divides(z0, z1)))) -> c17(A__SIEVE(a__sieve(divides(mark(z0), mark(z1)))), MARK(sieve(divides(z0, z1)))) MARK(sieve(sieve(x0))) -> c17(MARK(sieve(x0))) MARK(sieve(sieve(0))) -> c17(A__SIEVE(a__sieve(0))) MARK(sieve(sieve(true))) -> c17(A__SIEVE(a__sieve(true))) MARK(sieve(sieve(false))) -> c17(A__SIEVE(a__sieve(false))) MARK(sieve(from(x0))) -> c17(A__SIEVE(cons(mark(mark(x0)), from(s(mark(x0))))), MARK(from(x0))) MARK(sieve(from(primes))) -> c17(A__SIEVE(a__from(a__primes)), MARK(from(primes))) MARK(sieve(from(sieve(z0)))) -> c17(A__SIEVE(a__from(a__sieve(mark(z0)))), MARK(from(sieve(z0)))) MARK(sieve(from(from(z0)))) -> c17(A__SIEVE(a__from(a__from(mark(z0)))), MARK(from(from(z0)))) MARK(sieve(from(head(z0)))) -> c17(A__SIEVE(a__from(a__head(mark(z0)))), MARK(from(head(z0)))) MARK(sieve(from(tail(z0)))) -> c17(A__SIEVE(a__from(a__tail(mark(z0)))), MARK(from(tail(z0)))) MARK(sieve(from(if(z0, z1, z2)))) -> c17(A__SIEVE(a__from(a__if(mark(z0), z1, z2))), MARK(from(if(z0, z1, z2)))) MARK(sieve(from(filter(z0, z1)))) -> c17(A__SIEVE(a__from(a__filter(mark(z0), mark(z1)))), MARK(from(filter(z0, z1)))) MARK(sieve(from(s(z0)))) -> c17(A__SIEVE(a__from(s(mark(z0)))), MARK(from(s(z0)))) MARK(sieve(from(0))) -> c17(A__SIEVE(a__from(0)), MARK(from(0))) MARK(sieve(from(cons(z0, z1)))) -> c17(A__SIEVE(a__from(cons(mark(z0), z1))), MARK(from(cons(z0, z1)))) MARK(sieve(from(true))) -> c17(A__SIEVE(a__from(true)), MARK(from(true))) MARK(sieve(from(false))) -> c17(A__SIEVE(a__from(false)), MARK(from(false))) MARK(sieve(from(divides(z0, z1)))) -> c17(A__SIEVE(a__from(divides(mark(z0), mark(z1)))), MARK(from(divides(z0, z1)))) MARK(sieve(from(x0))) -> c17(MARK(from(x0))) MARK(sieve(head(primes))) -> c17(A__SIEVE(a__head(a__primes)), MARK(head(primes))) MARK(sieve(head(sieve(z0)))) -> c17(A__SIEVE(a__head(a__sieve(mark(z0)))), MARK(head(sieve(z0)))) MARK(sieve(head(from(z0)))) -> c17(A__SIEVE(a__head(a__from(mark(z0)))), MARK(head(from(z0)))) MARK(sieve(head(head(z0)))) -> c17(A__SIEVE(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(sieve(head(tail(z0)))) -> c17(A__SIEVE(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(sieve(head(if(z0, z1, z2)))) -> c17(A__SIEVE(a__head(a__if(mark(z0), z1, z2))), MARK(head(if(z0, z1, z2)))) MARK(sieve(head(filter(z0, z1)))) -> c17(A__SIEVE(a__head(a__filter(mark(z0), mark(z1)))), MARK(head(filter(z0, z1)))) MARK(sieve(head(s(z0)))) -> c17(A__SIEVE(a__head(s(mark(z0)))), MARK(head(s(z0)))) MARK(sieve(head(cons(z0, z1)))) -> c17(A__SIEVE(a__head(cons(mark(z0), z1))), MARK(head(cons(z0, z1)))) MARK(sieve(head(divides(z0, z1)))) -> c17(A__SIEVE(a__head(divides(mark(z0), mark(z1)))), MARK(head(divides(z0, z1)))) MARK(sieve(head(x0))) -> c17(MARK(head(x0))) MARK(sieve(head(0))) -> c17(A__SIEVE(a__head(0))) MARK(sieve(head(true))) -> c17(A__SIEVE(a__head(true))) MARK(sieve(head(false))) -> c17(A__SIEVE(a__head(false))) MARK(sieve(tail(primes))) -> c17(A__SIEVE(a__tail(a__primes)), MARK(tail(primes))) MARK(sieve(tail(sieve(z0)))) -> c17(A__SIEVE(a__tail(a__sieve(mark(z0)))), MARK(tail(sieve(z0)))) MARK(sieve(tail(from(z0)))) -> c17(A__SIEVE(a__tail(a__from(mark(z0)))), MARK(tail(from(z0)))) MARK(sieve(tail(head(z0)))) -> c17(A__SIEVE(a__tail(a__head(mark(z0)))), MARK(tail(head(z0)))) MARK(sieve(tail(tail(z0)))) -> c17(A__SIEVE(a__tail(a__tail(mark(z0)))), MARK(tail(tail(z0)))) MARK(sieve(tail(if(z0, z1, z2)))) -> c17(A__SIEVE(a__tail(a__if(mark(z0), z1, z2))), MARK(tail(if(z0, z1, z2)))) MARK(sieve(tail(filter(z0, z1)))) -> c17(A__SIEVE(a__tail(a__filter(mark(z0), mark(z1)))), MARK(tail(filter(z0, z1)))) MARK(sieve(tail(s(z0)))) -> c17(A__SIEVE(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(sieve(tail(cons(z0, z1)))) -> c17(A__SIEVE(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(sieve(tail(divides(z0, z1)))) -> c17(A__SIEVE(a__tail(divides(mark(z0), mark(z1)))), MARK(tail(divides(z0, z1)))) MARK(sieve(tail(x0))) -> c17(MARK(tail(x0))) MARK(sieve(tail(0))) -> c17(A__SIEVE(a__tail(0))) MARK(sieve(tail(true))) -> c17(A__SIEVE(a__tail(true))) MARK(sieve(tail(false))) -> c17(A__SIEVE(a__tail(false))) MARK(sieve(if(primes, x1, x2))) -> c17(A__SIEVE(a__if(a__primes, x1, x2)), MARK(if(primes, x1, x2))) MARK(sieve(if(sieve(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__sieve(mark(z0)), x1, x2)), MARK(if(sieve(z0), x1, x2))) MARK(sieve(if(from(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__from(mark(z0)), x1, x2)), MARK(if(from(z0), x1, x2))) MARK(sieve(if(head(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__head(mark(z0)), x1, x2)), MARK(if(head(z0), x1, x2))) MARK(sieve(if(tail(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__tail(mark(z0)), x1, x2)), MARK(if(tail(z0), x1, x2))) MARK(sieve(if(if(z0, z1, z2), x1, x2))) -> c17(A__SIEVE(a__if(a__if(mark(z0), z1, z2), x1, x2)), MARK(if(if(z0, z1, z2), x1, x2))) MARK(sieve(if(filter(z0, z1), x1, x2))) -> c17(A__SIEVE(a__if(a__filter(mark(z0), mark(z1)), x1, x2)), MARK(if(filter(z0, z1), x1, x2))) MARK(sieve(if(s(z0), x1, x2))) -> c17(A__SIEVE(a__if(s(mark(z0)), x1, x2)), MARK(if(s(z0), x1, x2))) MARK(sieve(if(cons(z0, z1), x1, x2))) -> c17(A__SIEVE(a__if(cons(mark(z0), z1), x1, x2)), MARK(if(cons(z0, z1), x1, x2))) MARK(sieve(if(true, x1, x2))) -> c17(A__SIEVE(a__if(true, x1, x2)), MARK(if(true, x1, x2))) MARK(sieve(if(false, x1, x2))) -> c17(A__SIEVE(a__if(false, x1, x2)), MARK(if(false, x1, x2))) MARK(sieve(if(divides(z0, z1), x1, x2))) -> c17(A__SIEVE(a__if(divides(mark(z0), mark(z1)), x1, x2)), MARK(if(divides(z0, z1), x1, x2))) MARK(sieve(if(x0, z1, z2))) -> c17(MARK(if(x0, z1, z2))) MARK(sieve(if(0, x1, x2))) -> c17(A__SIEVE(a__if(0, x1, x2))) MARK(sieve(filter(x0, primes))) -> c17(A__SIEVE(a__filter(mark(x0), a__primes)), MARK(filter(x0, primes))) MARK(sieve(filter(x0, sieve(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), a__sieve(mark(z0)))), MARK(filter(x0, sieve(z0)))) MARK(sieve(filter(x0, from(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), a__from(mark(z0)))), MARK(filter(x0, from(z0)))) 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head(false))) -> c22(A__FILTER(mark(x0), a__head(false)), MARK(x0)) MARK(filter(x0, head(divides(z0, z1)))) -> c22(A__FILTER(mark(x0), a__head(divides(mark(z0), mark(z1)))), MARK(x0)) MARK(filter(primes, head(x1))) -> c22(A__FILTER(a__primes, a__head(mark(x1))), MARK(primes)) MARK(filter(sieve(z0), head(x1))) -> c22(A__FILTER(a__sieve(mark(z0)), a__head(mark(x1))), MARK(sieve(z0))) MARK(filter(from(z0), head(x1))) -> c22(A__FILTER(a__from(mark(z0)), a__head(mark(x1))), MARK(from(z0))) MARK(filter(head(z0), head(x1))) -> c22(A__FILTER(a__head(mark(z0)), a__head(mark(x1))), MARK(head(z0))) MARK(filter(tail(z0), head(x1))) -> c22(A__FILTER(a__tail(mark(z0)), a__head(mark(x1))), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), head(x1))) -> c22(A__FILTER(a__if(mark(z0), z1, z2), a__head(mark(x1))), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), head(x1))) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), a__head(mark(x1))), MARK(filter(z0, z1))) MARK(filter(s(z0), head(x1))) -> c22(A__FILTER(s(mark(z0)), a__head(mark(x1))), MARK(s(z0))) MARK(filter(x0, head(x1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), head(x1))) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), head(x1))) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, tail(primes))) -> c22(A__FILTER(mark(x0), a__tail(a__primes)), MARK(x0)) MARK(filter(x0, tail(sieve(z0)))) -> c22(A__FILTER(mark(x0), a__tail(a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, tail(from(z0)))) -> c22(A__FILTER(mark(x0), a__tail(a__from(mark(z0)))), MARK(x0)) MARK(filter(x0, tail(head(z0)))) -> c22(A__FILTER(mark(x0), a__tail(a__head(mark(z0)))), MARK(x0)) MARK(filter(x0, tail(tail(z0)))) -> c22(A__FILTER(mark(x0), a__tail(a__tail(mark(z0)))), MARK(x0)) MARK(filter(x0, tail(if(z0, z1, z2)))) -> c22(A__FILTER(mark(x0), a__tail(a__if(mark(z0), z1, z2))), MARK(x0)) MARK(filter(x0, tail(filter(z0, z1)))) -> c22(A__FILTER(mark(x0), a__tail(a__filter(mark(z0), mark(z1)))), MARK(x0)) MARK(filter(x0, tail(s(z0)))) -> c22(A__FILTER(mark(x0), a__tail(s(mark(z0)))), MARK(x0)) MARK(filter(x0, tail(0))) -> c22(A__FILTER(mark(x0), a__tail(0)), MARK(x0)) MARK(filter(x0, tail(cons(z0, z1)))) -> c22(A__FILTER(mark(x0), a__tail(cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, tail(true))) -> c22(A__FILTER(mark(x0), a__tail(true)), MARK(x0)) MARK(filter(x0, tail(false))) -> c22(A__FILTER(mark(x0), a__tail(false)), MARK(x0)) MARK(filter(x0, tail(divides(z0, z1)))) -> c22(A__FILTER(mark(x0), a__tail(divides(mark(z0), mark(z1)))), MARK(x0)) MARK(filter(primes, tail(x1))) -> c22(A__FILTER(a__primes, a__tail(mark(x1))), MARK(primes)) MARK(filter(sieve(z0), tail(x1))) -> c22(A__FILTER(a__sieve(mark(z0)), a__tail(mark(x1))), MARK(sieve(z0))) MARK(filter(from(z0), tail(x1))) -> c22(A__FILTER(a__from(mark(z0)), a__tail(mark(x1))), MARK(from(z0))) MARK(filter(head(z0), tail(x1))) -> c22(A__FILTER(a__head(mark(z0)), a__tail(mark(x1))), MARK(head(z0))) MARK(filter(tail(z0), tail(x1))) -> c22(A__FILTER(a__tail(mark(z0)), a__tail(mark(x1))), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), tail(x1))) -> c22(A__FILTER(a__if(mark(z0), z1, z2), a__tail(mark(x1))), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), tail(x1))) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), a__tail(mark(x1))), MARK(filter(z0, z1))) MARK(filter(s(z0), tail(x1))) -> c22(A__FILTER(s(mark(z0)), a__tail(mark(x1))), MARK(s(z0))) MARK(filter(x0, tail(x1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), tail(x1))) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), tail(x1))) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, if(primes, x2, x3))) -> c22(A__FILTER(mark(x0), a__if(a__primes, x2, x3)), MARK(x0)) MARK(filter(x0, if(sieve(z0), x2, x3))) -> c22(A__FILTER(mark(x0), a__if(a__sieve(mark(z0)), x2, x3)), MARK(x0)) MARK(filter(x0, if(from(z0), x2, x3))) -> c22(A__FILTER(mark(x0), a__if(a__from(mark(z0)), x2, x3)), MARK(x0)) MARK(filter(x0, if(head(z0), x2, x3))) -> c22(A__FILTER(mark(x0), a__if(a__head(mark(z0)), x2, x3)), MARK(x0)) MARK(filter(x0, if(tail(z0), x2, x3))) -> c22(A__FILTER(mark(x0), a__if(a__tail(mark(z0)), x2, x3)), MARK(x0)) MARK(filter(x0, if(if(z0, z1, z2), x2, x3))) -> c22(A__FILTER(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0)) MARK(filter(x0, if(filter(z0, z1), x2, x3))) -> c22(A__FILTER(mark(x0), a__if(a__filter(mark(z0), mark(z1)), x2, x3)), MARK(x0)) MARK(filter(x0, if(s(z0), x2, x3))) -> c22(A__FILTER(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0)) MARK(filter(x0, if(0, x2, x3))) -> c22(A__FILTER(mark(x0), a__if(0, x2, x3)), MARK(x0)) MARK(filter(x0, if(cons(z0, z1), x2, x3))) -> c22(A__FILTER(mark(x0), a__if(cons(mark(z0), z1), x2, x3)), MARK(x0)) MARK(filter(x0, if(true, x2, x3))) -> c22(A__FILTER(mark(x0), a__if(true, x2, x3)), MARK(x0)) MARK(filter(x0, if(false, x2, x3))) -> c22(A__FILTER(mark(x0), a__if(false, x2, x3)), MARK(x0)) MARK(filter(x0, if(divides(z0, z1), x2, x3))) -> c22(A__FILTER(mark(x0), a__if(divides(mark(z0), mark(z1)), x2, x3)), MARK(x0)) MARK(filter(primes, if(x1, x2, x3))) -> c22(A__FILTER(a__primes, a__if(mark(x1), x2, x3)), MARK(primes)) MARK(filter(sieve(z0), if(x1, x2, x3))) -> c22(A__FILTER(a__sieve(mark(z0)), a__if(mark(x1), x2, x3)), MARK(sieve(z0))) MARK(filter(from(z0), if(x1, x2, x3))) -> c22(A__FILTER(a__from(mark(z0)), a__if(mark(x1), x2, x3)), MARK(from(z0))) MARK(filter(head(z0), if(x1, x2, x3))) -> c22(A__FILTER(a__head(mark(z0)), a__if(mark(x1), x2, x3)), MARK(head(z0))) MARK(filter(tail(z0), if(x1, x2, x3))) -> c22(A__FILTER(a__tail(mark(z0)), a__if(mark(x1), x2, x3)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), if(x1, x2, x3))) -> c22(A__FILTER(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), if(x1, x2, x3))) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(filter(z0, z1))) MARK(filter(s(z0), if(x1, x2, x3))) -> c22(A__FILTER(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0))) MARK(filter(x0, if(x1, z1, z2))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), if(x1, x2, x3))) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), if(x1, x2, x3))) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, filter(x1, x2))) -> c22(A__FILTER(mark(x0), filter(mark(x1), mark(x2))), MARK(x0)) MARK(filter(x0, filter(x1, primes))) -> c22(A__FILTER(mark(x0), a__filter(mark(x1), a__primes)), MARK(x0)) MARK(filter(x0, filter(x1, sieve(z0)))) -> c22(A__FILTER(mark(x0), a__filter(mark(x1), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(x1, from(z0)))) -> c22(A__FILTER(mark(x0), a__filter(mark(x1), a__from(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(x1, head(z0)))) -> c22(A__FILTER(mark(x0), a__filter(mark(x1), a__head(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(x1, tail(z0)))) -> c22(A__FILTER(mark(x0), a__filter(mark(x1), a__tail(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(x1, if(z0, z1, z2)))) -> c22(A__FILTER(mark(x0), a__filter(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0)) MARK(filter(x0, filter(x1, filter(z0, z1)))) -> c22(A__FILTER(mark(x0), a__filter(mark(x1), a__filter(mark(z0), mark(z1)))), MARK(x0)) MARK(filter(x0, filter(x1, s(z0)))) -> c22(A__FILTER(mark(x0), a__filter(mark(x1), s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(x1, 0))) -> c22(A__FILTER(mark(x0), a__filter(mark(x1), 0)), MARK(x0)) MARK(filter(x0, filter(x1, cons(z0, z1)))) -> c22(A__FILTER(mark(x0), a__filter(mark(x1), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, filter(x1, true))) -> c22(A__FILTER(mark(x0), a__filter(mark(x1), true)), MARK(x0)) MARK(filter(x0, filter(x1, false))) -> c22(A__FILTER(mark(x0), a__filter(mark(x1), false)), MARK(x0)) MARK(filter(x0, filter(x1, divides(z0, z1)))) -> c22(A__FILTER(mark(x0), a__filter(mark(x1), divides(mark(z0), mark(z1)))), MARK(x0)) MARK(filter(x0, filter(primes, x2))) -> c22(A__FILTER(mark(x0), a__filter(a__primes, mark(x2))), MARK(x0)) MARK(filter(x0, filter(sieve(z0), x2))) -> c22(A__FILTER(mark(x0), a__filter(a__sieve(mark(z0)), mark(x2))), MARK(x0)) MARK(filter(x0, filter(from(z0), x2))) -> c22(A__FILTER(mark(x0), a__filter(a__from(mark(z0)), mark(x2))), MARK(x0)) MARK(filter(x0, filter(head(z0), x2))) -> c22(A__FILTER(mark(x0), a__filter(a__head(mark(z0)), mark(x2))), MARK(x0)) MARK(filter(x0, filter(tail(z0), x2))) -> c22(A__FILTER(mark(x0), a__filter(a__tail(mark(z0)), mark(x2))), MARK(x0)) MARK(filter(x0, filter(if(z0, z1, z2), x2))) -> c22(A__FILTER(mark(x0), a__filter(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0)) MARK(filter(x0, filter(filter(z0, z1), x2))) -> c22(A__FILTER(mark(x0), a__filter(a__filter(mark(z0), mark(z1)), mark(x2))), MARK(x0)) MARK(filter(x0, filter(s(z0), x2))) -> c22(A__FILTER(mark(x0), a__filter(s(mark(z0)), mark(x2))), MARK(x0)) MARK(filter(x0, filter(0, x2))) -> c22(A__FILTER(mark(x0), a__filter(0, mark(x2))), MARK(x0)) MARK(filter(x0, filter(cons(z0, z1), x2))) -> c22(A__FILTER(mark(x0), a__filter(cons(mark(z0), z1), mark(x2))), MARK(x0)) MARK(filter(x0, filter(true, x2))) -> c22(A__FILTER(mark(x0), a__filter(true, mark(x2))), MARK(x0)) MARK(filter(x0, filter(false, x2))) -> c22(A__FILTER(mark(x0), a__filter(false, mark(x2))), MARK(x0)) MARK(filter(x0, filter(divides(z0, z1), x2))) -> c22(A__FILTER(mark(x0), a__filter(divides(mark(z0), mark(z1)), mark(x2))), MARK(x0)) MARK(filter(primes, filter(x1, x2))) -> c22(A__FILTER(a__primes, a__filter(mark(x1), mark(x2))), MARK(primes)) MARK(filter(sieve(z0), filter(x1, x2))) -> c22(A__FILTER(a__sieve(mark(z0)), a__filter(mark(x1), mark(x2))), MARK(sieve(z0))) MARK(filter(from(z0), filter(x1, x2))) -> c22(A__FILTER(a__from(mark(z0)), a__filter(mark(x1), mark(x2))), MARK(from(z0))) MARK(filter(head(z0), filter(x1, x2))) -> c22(A__FILTER(a__head(mark(z0)), a__filter(mark(x1), mark(x2))), MARK(head(z0))) MARK(filter(tail(z0), filter(x1, x2))) -> c22(A__FILTER(a__tail(mark(z0)), a__filter(mark(x1), mark(x2))), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), filter(x1, x2))) -> c22(A__FILTER(a__if(mark(z0), z1, z2), a__filter(mark(x1), mark(x2))), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), filter(x1, x2))) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), a__filter(mark(x1), mark(x2))), MARK(filter(z0, z1))) MARK(filter(s(z0), filter(x1, x2))) -> c22(A__FILTER(s(mark(z0)), a__filter(mark(x1), mark(x2))), MARK(s(z0))) MARK(filter(0, filter(x1, x2))) -> c22(A__FILTER(0, a__filter(mark(x1), mark(x2))), MARK(0)) MARK(filter(cons(z0, z1), filter(x1, x2))) -> c22(A__FILTER(cons(mark(z0), z1), a__filter(mark(x1), mark(x2))), MARK(cons(z0, z1))) MARK(filter(true, filter(x1, x2))) -> c22(A__FILTER(true, a__filter(mark(x1), mark(x2))), MARK(true)) MARK(filter(false, filter(x1, x2))) -> c22(A__FILTER(false, a__filter(mark(x1), mark(x2))), MARK(false)) MARK(filter(divides(z0, z1), filter(x1, x2))) -> c22(A__FILTER(divides(mark(z0), mark(z1)), a__filter(mark(x1), mark(x2))), MARK(divides(z0, z1))) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c18_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_1, c17_2, c17_1, c19_2, c19_1, c20_2, c20_1, c21_1, c22_2, c22_1, c23_2, c23_1, c_2, c21_2 ---------------------------------------- (249) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing nodes: MARK(filter(0, filter(x1, x2))) -> c22(A__FILTER(0, a__filter(mark(x1), mark(x2))), MARK(0)) MARK(filter(true, filter(x1, x2))) -> c22(A__FILTER(true, a__filter(mark(x1), mark(x2))), MARK(true)) MARK(filter(false, filter(x1, x2))) -> c22(A__FILTER(false, a__filter(mark(x1), mark(x2))), MARK(false)) ---------------------------------------- (250) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> 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a__if(a__head(mark(z0)), x2, x3)), MARK(x0)) MARK(filter(x0, if(tail(z0), x2, x3))) -> c22(A__FILTER(mark(x0), a__if(a__tail(mark(z0)), x2, x3)), MARK(x0)) MARK(filter(x0, if(if(z0, z1, z2), x2, x3))) -> c22(A__FILTER(mark(x0), a__if(a__if(mark(z0), z1, z2), x2, x3)), MARK(x0)) MARK(filter(x0, if(filter(z0, z1), x2, x3))) -> c22(A__FILTER(mark(x0), a__if(a__filter(mark(z0), mark(z1)), x2, x3)), MARK(x0)) MARK(filter(x0, if(s(z0), x2, x3))) -> c22(A__FILTER(mark(x0), a__if(s(mark(z0)), x2, x3)), MARK(x0)) MARK(filter(x0, if(0, x2, x3))) -> c22(A__FILTER(mark(x0), a__if(0, x2, x3)), MARK(x0)) MARK(filter(x0, if(cons(z0, z1), x2, x3))) -> c22(A__FILTER(mark(x0), a__if(cons(mark(z0), z1), x2, x3)), MARK(x0)) MARK(filter(x0, if(true, x2, x3))) -> c22(A__FILTER(mark(x0), a__if(true, x2, x3)), MARK(x0)) MARK(filter(x0, if(false, x2, x3))) -> c22(A__FILTER(mark(x0), a__if(false, x2, x3)), MARK(x0)) MARK(filter(x0, if(divides(z0, z1), x2, x3))) -> c22(A__FILTER(mark(x0), a__if(divides(mark(z0), mark(z1)), x2, x3)), MARK(x0)) MARK(filter(primes, if(x1, x2, x3))) -> c22(A__FILTER(a__primes, a__if(mark(x1), x2, x3)), MARK(primes)) MARK(filter(sieve(z0), if(x1, x2, x3))) -> c22(A__FILTER(a__sieve(mark(z0)), a__if(mark(x1), x2, x3)), MARK(sieve(z0))) MARK(filter(from(z0), if(x1, x2, x3))) -> c22(A__FILTER(a__from(mark(z0)), a__if(mark(x1), x2, x3)), MARK(from(z0))) MARK(filter(head(z0), if(x1, x2, x3))) -> c22(A__FILTER(a__head(mark(z0)), a__if(mark(x1), x2, x3)), MARK(head(z0))) MARK(filter(tail(z0), if(x1, x2, x3))) -> c22(A__FILTER(a__tail(mark(z0)), a__if(mark(x1), x2, x3)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), if(x1, x2, x3))) -> c22(A__FILTER(a__if(mark(z0), z1, z2), a__if(mark(x1), x2, x3)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), if(x1, x2, x3))) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), a__if(mark(x1), x2, x3)), MARK(filter(z0, z1))) MARK(filter(s(z0), if(x1, x2, x3))) -> c22(A__FILTER(s(mark(z0)), a__if(mark(x1), x2, x3)), MARK(s(z0))) MARK(filter(x0, if(x1, z1, z2))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), if(x1, x2, x3))) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), if(x1, x2, x3))) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, filter(x1, x2))) -> c22(A__FILTER(mark(x0), filter(mark(x1), mark(x2))), MARK(x0)) MARK(filter(x0, filter(x1, primes))) -> c22(A__FILTER(mark(x0), a__filter(mark(x1), a__primes)), MARK(x0)) MARK(filter(x0, filter(x1, sieve(z0)))) -> c22(A__FILTER(mark(x0), a__filter(mark(x1), a__sieve(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(x1, from(z0)))) -> c22(A__FILTER(mark(x0), a__filter(mark(x1), a__from(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(x1, head(z0)))) -> c22(A__FILTER(mark(x0), a__filter(mark(x1), a__head(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(x1, tail(z0)))) -> c22(A__FILTER(mark(x0), a__filter(mark(x1), a__tail(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(x1, if(z0, z1, z2)))) -> c22(A__FILTER(mark(x0), a__filter(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0)) MARK(filter(x0, filter(x1, filter(z0, z1)))) -> c22(A__FILTER(mark(x0), a__filter(mark(x1), a__filter(mark(z0), mark(z1)))), MARK(x0)) MARK(filter(x0, filter(x1, s(z0)))) -> c22(A__FILTER(mark(x0), a__filter(mark(x1), s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(x1, 0))) -> c22(A__FILTER(mark(x0), a__filter(mark(x1), 0)), MARK(x0)) MARK(filter(x0, filter(x1, cons(z0, z1)))) -> c22(A__FILTER(mark(x0), a__filter(mark(x1), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, filter(x1, true))) -> c22(A__FILTER(mark(x0), a__filter(mark(x1), true)), MARK(x0)) MARK(filter(x0, filter(x1, false))) -> c22(A__FILTER(mark(x0), a__filter(mark(x1), false)), MARK(x0)) MARK(filter(x0, filter(x1, divides(z0, z1)))) -> c22(A__FILTER(mark(x0), a__filter(mark(x1), divides(mark(z0), mark(z1)))), MARK(x0)) MARK(filter(x0, filter(primes, x2))) -> c22(A__FILTER(mark(x0), a__filter(a__primes, mark(x2))), MARK(x0)) MARK(filter(x0, filter(sieve(z0), x2))) -> c22(A__FILTER(mark(x0), a__filter(a__sieve(mark(z0)), mark(x2))), MARK(x0)) MARK(filter(x0, filter(from(z0), x2))) -> c22(A__FILTER(mark(x0), a__filter(a__from(mark(z0)), mark(x2))), MARK(x0)) MARK(filter(x0, filter(head(z0), x2))) -> c22(A__FILTER(mark(x0), a__filter(a__head(mark(z0)), mark(x2))), MARK(x0)) MARK(filter(x0, filter(tail(z0), x2))) -> c22(A__FILTER(mark(x0), a__filter(a__tail(mark(z0)), mark(x2))), MARK(x0)) MARK(filter(x0, filter(if(z0, z1, z2), x2))) -> c22(A__FILTER(mark(x0), a__filter(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0)) MARK(filter(x0, filter(filter(z0, z1), x2))) -> c22(A__FILTER(mark(x0), a__filter(a__filter(mark(z0), mark(z1)), mark(x2))), MARK(x0)) MARK(filter(x0, filter(s(z0), x2))) -> c22(A__FILTER(mark(x0), a__filter(s(mark(z0)), mark(x2))), MARK(x0)) MARK(filter(x0, filter(0, x2))) -> c22(A__FILTER(mark(x0), a__filter(0, mark(x2))), MARK(x0)) MARK(filter(x0, filter(cons(z0, z1), x2))) -> c22(A__FILTER(mark(x0), a__filter(cons(mark(z0), z1), mark(x2))), MARK(x0)) MARK(filter(x0, filter(true, x2))) -> c22(A__FILTER(mark(x0), a__filter(true, mark(x2))), MARK(x0)) MARK(filter(x0, filter(false, x2))) -> c22(A__FILTER(mark(x0), a__filter(false, mark(x2))), MARK(x0)) MARK(filter(x0, filter(divides(z0, z1), x2))) -> c22(A__FILTER(mark(x0), a__filter(divides(mark(z0), mark(z1)), mark(x2))), MARK(x0)) MARK(filter(primes, filter(x1, x2))) -> c22(A__FILTER(a__primes, a__filter(mark(x1), mark(x2))), MARK(primes)) MARK(filter(sieve(z0), filter(x1, x2))) -> c22(A__FILTER(a__sieve(mark(z0)), a__filter(mark(x1), mark(x2))), MARK(sieve(z0))) MARK(filter(from(z0), filter(x1, x2))) -> c22(A__FILTER(a__from(mark(z0)), a__filter(mark(x1), mark(x2))), MARK(from(z0))) MARK(filter(head(z0), filter(x1, x2))) -> c22(A__FILTER(a__head(mark(z0)), a__filter(mark(x1), mark(x2))), MARK(head(z0))) MARK(filter(tail(z0), filter(x1, x2))) -> c22(A__FILTER(a__tail(mark(z0)), a__filter(mark(x1), mark(x2))), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), filter(x1, x2))) -> c22(A__FILTER(a__if(mark(z0), z1, z2), a__filter(mark(x1), mark(x2))), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), filter(x1, x2))) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), a__filter(mark(x1), mark(x2))), MARK(filter(z0, z1))) MARK(filter(s(z0), filter(x1, x2))) -> c22(A__FILTER(s(mark(z0)), a__filter(mark(x1), mark(x2))), MARK(s(z0))) MARK(filter(cons(z0, z1), filter(x1, x2))) -> c22(A__FILTER(cons(mark(z0), z1), a__filter(mark(x1), mark(x2))), MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), filter(x1, x2))) -> c22(A__FILTER(divides(mark(z0), mark(z1)), a__filter(mark(x1), mark(x2))), MARK(divides(z0, z1))) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c18_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_1, c17_2, c17_1, c19_2, c19_1, c20_2, c20_1, c21_1, c22_2, c22_1, c23_2, c23_1, c_2, c21_2 ---------------------------------------- (251) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (252) Obligation: Complexity Dependency Tuples Problem Rules: a__primes -> a__sieve(a__from(s(s(0)))) a__primes -> primes a__from(z0) -> cons(mark(z0), from(s(z0))) a__from(z0) -> from(z0) a__head(cons(z0, z1)) -> mark(z0) a__head(z0) -> head(z0) a__tail(cons(z0, z1)) -> mark(z1) a__tail(z0) -> tail(z0) a__if(true, z0, z1) -> mark(z0) a__if(false, z0, z1) -> mark(z1) a__if(z0, z1, z2) -> if(z0, z1, z2) a__filter(s(s(z0)), cons(z1, z2)) -> a__if(divides(s(s(mark(z0))), mark(z1)), filter(s(s(z0)), z2), cons(z1, filter(z0, sieve(z1)))) a__filter(z0, z1) -> filter(z0, z1) a__sieve(cons(z0, z1)) -> cons(mark(z0), filter(z0, sieve(z1))) a__sieve(z0) -> sieve(z0) mark(primes) -> a__primes mark(sieve(z0)) -> a__sieve(mark(z0)) mark(from(z0)) -> a__from(mark(z0)) mark(head(z0)) -> a__head(mark(z0)) mark(tail(z0)) -> a__tail(mark(z0)) mark(if(z0, z1, z2)) -> a__if(mark(z0), z1, z2) mark(filter(z0, z1)) -> a__filter(mark(z0), mark(z1)) mark(s(z0)) -> s(mark(z0)) mark(0) -> 0 mark(cons(z0, z1)) -> cons(mark(z0), z1) mark(true) -> true mark(false) -> false mark(divides(z0, z1)) -> divides(mark(z0), mark(z1)) Tuples: A__FROM(z0) -> c2(MARK(z0)) A__HEAD(cons(z0, z1)) -> c4(MARK(z0)) A__TAIL(cons(z0, z1)) -> c6(MARK(z1)) A__IF(true, z0, z1) -> c8(MARK(z0)) A__IF(false, z0, z1) -> c9(MARK(z1)) A__SIEVE(cons(z0, z1)) -> c14(MARK(z0)) MARK(primes) -> c16(A__PRIMES) MARK(from(z0)) -> c18(A__FROM(mark(z0)), MARK(z0)) MARK(s(z0)) -> c24(MARK(z0)) MARK(cons(z0, z1)) -> c26(MARK(z0)) MARK(divides(z0, z1)) -> c29(MARK(z0)) MARK(divides(z0, z1)) -> c30(MARK(z1)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c11(MARK(z0)) A__FILTER(s(s(z0)), cons(z1, z2)) -> c12(MARK(z1)) A__PRIMES -> c(A__FROM(s(s(0)))) MARK(sieve(cons(z0, z1))) -> c17(A__SIEVE(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(sieve(s(z0))) -> c17(MARK(s(z0))) MARK(sieve(divides(z0, z1))) -> c17(MARK(divides(z0, z1))) MARK(head(cons(z0, z1))) -> c19(A__HEAD(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(head(s(z0))) -> c19(MARK(s(z0))) MARK(head(divides(z0, z1))) -> c19(MARK(divides(z0, z1))) MARK(tail(cons(z0, z1))) -> c20(A__TAIL(cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(tail(s(z0))) -> c20(MARK(s(z0))) MARK(tail(divides(z0, z1))) -> c20(MARK(divides(z0, z1))) MARK(if(s(z0), x1, x2)) -> c21(MARK(s(z0))) MARK(if(cons(z0, z1), x1, x2)) -> c21(MARK(cons(z0, z1))) MARK(if(true, x1, x2)) -> c21(A__IF(true, x1, x2)) MARK(if(false, x1, x2)) -> c21(A__IF(false, x1, x2)) MARK(if(divides(z0, z1), x1, x2)) -> c21(MARK(divides(z0, z1))) MARK(filter(x0, cons(z0, z1))) -> c22(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(x0)) MARK(filter(primes, x1)) -> c22(A__FILTER(a__primes, mark(x1)), MARK(primes)) MARK(filter(sieve(z0), x1)) -> c22(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(sieve(z0))) MARK(filter(from(z0), x1)) -> c22(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(from(z0))) MARK(filter(head(z0), x1)) -> c22(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(head(z0))) MARK(filter(tail(z0), x1)) -> c22(A__FILTER(a__tail(mark(z0)), mark(x1)), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), x1)) -> c22(A__FILTER(a__if(mark(z0), z1, z2), mark(x1)), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), x1)) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(filter(z0, z1))) MARK(filter(s(z0), x1)) -> c22(A__FILTER(s(mark(z0)), mark(x1)), MARK(s(z0))) MARK(filter(x0, s(z0))) -> c22(MARK(x0)) MARK(filter(x0, 0)) -> c22(MARK(x0)) MARK(filter(x0, true)) -> c22(MARK(x0)) MARK(filter(x0, false)) -> c22(MARK(x0)) MARK(filter(x0, divides(z0, z1))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), x1)) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), x1)) -> c22(MARK(divides(z0, z1))) MARK(filter(x0, primes)) -> c23(A__FILTER(mark(x0), a__primes), MARK(primes)) MARK(filter(x0, sieve(z0))) -> c23(A__FILTER(mark(x0), a__sieve(mark(z0))), MARK(sieve(z0))) MARK(filter(x0, from(z0))) -> c23(A__FILTER(mark(x0), a__from(mark(z0))), MARK(from(z0))) MARK(filter(x0, head(z0))) -> c23(A__FILTER(mark(x0), a__head(mark(z0))), MARK(head(z0))) MARK(filter(x0, tail(z0))) -> c23(A__FILTER(mark(x0), a__tail(mark(z0))), MARK(tail(z0))) MARK(filter(x0, if(z0, z1, z2))) -> c23(A__FILTER(mark(x0), a__if(mark(z0), z1, z2)), MARK(if(z0, z1, z2))) MARK(filter(x0, filter(z0, z1))) -> c23(A__FILTER(mark(x0), a__filter(mark(z0), mark(z1))), MARK(filter(z0, z1))) MARK(filter(x0, cons(z0, z1))) -> c23(A__FILTER(mark(x0), cons(mark(z0), z1)), MARK(cons(z0, z1))) MARK(filter(primes, x1)) -> c23(A__FILTER(a__primes, mark(x1)), MARK(x1)) MARK(filter(sieve(z0), x1)) -> c23(A__FILTER(a__sieve(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(from(z0), x1)) -> c23(A__FILTER(a__from(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(head(z0), x1)) -> c23(A__FILTER(a__head(mark(z0)), mark(x1)), MARK(x1)) 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MARK(filter(filter(z0, z1), x1)) -> c23(A__FILTER(a__filter(mark(z0), mark(z1)), mark(x1)), MARK(x1)) MARK(filter(s(z0), x1)) -> c23(A__FILTER(s(mark(z0)), mark(x1)), MARK(x1)) MARK(filter(x0, s(z0))) -> c23(MARK(s(z0))) MARK(filter(x0, divides(z0, z1))) -> c23(MARK(divides(z0, z1))) MARK(filter(0, x1)) -> c23(MARK(x1)) MARK(filter(cons(z0, z1), x1)) -> c23(MARK(x1)) MARK(filter(true, x1)) -> c23(MARK(x1)) MARK(filter(false, x1)) -> c23(MARK(x1)) MARK(filter(divides(z0, z1), x1)) -> c23(MARK(x1)) A__PRIMES -> c(A__SIEVE(cons(s(mark(s(0))), from(s(s(s(0)))))), A__FROM(s(s(0)))) MARK(sieve(primes)) -> c17(A__SIEVE(a__sieve(a__from(s(s(0))))), MARK(primes)) MARK(sieve(primes)) -> c17(MARK(primes)) MARK(sieve(sieve(primes))) -> c17(A__SIEVE(a__sieve(a__primes)), MARK(sieve(primes))) MARK(sieve(sieve(sieve(z0)))) -> c17(A__SIEVE(a__sieve(a__sieve(mark(z0)))), MARK(sieve(sieve(z0)))) MARK(sieve(sieve(from(z0)))) -> c17(A__SIEVE(a__sieve(a__from(mark(z0)))), MARK(sieve(from(z0)))) 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c17(A__SIEVE(cons(mark(mark(x0)), from(s(mark(x0))))), MARK(from(x0))) MARK(sieve(from(primes))) -> c17(A__SIEVE(a__from(a__primes)), MARK(from(primes))) MARK(sieve(from(sieve(z0)))) -> c17(A__SIEVE(a__from(a__sieve(mark(z0)))), MARK(from(sieve(z0)))) MARK(sieve(from(from(z0)))) -> c17(A__SIEVE(a__from(a__from(mark(z0)))), MARK(from(from(z0)))) MARK(sieve(from(head(z0)))) -> c17(A__SIEVE(a__from(a__head(mark(z0)))), MARK(from(head(z0)))) MARK(sieve(from(tail(z0)))) -> c17(A__SIEVE(a__from(a__tail(mark(z0)))), MARK(from(tail(z0)))) MARK(sieve(from(if(z0, z1, z2)))) -> c17(A__SIEVE(a__from(a__if(mark(z0), z1, z2))), MARK(from(if(z0, z1, z2)))) MARK(sieve(from(filter(z0, z1)))) -> c17(A__SIEVE(a__from(a__filter(mark(z0), mark(z1)))), MARK(from(filter(z0, z1)))) MARK(sieve(from(s(z0)))) -> c17(A__SIEVE(a__from(s(mark(z0)))), MARK(from(s(z0)))) MARK(sieve(from(0))) -> c17(A__SIEVE(a__from(0)), MARK(from(0))) MARK(sieve(from(cons(z0, z1)))) -> c17(A__SIEVE(a__from(cons(mark(z0), z1))), MARK(from(cons(z0, z1)))) MARK(sieve(from(true))) -> c17(A__SIEVE(a__from(true)), MARK(from(true))) MARK(sieve(from(false))) -> c17(A__SIEVE(a__from(false)), MARK(from(false))) MARK(sieve(from(divides(z0, z1)))) -> c17(A__SIEVE(a__from(divides(mark(z0), mark(z1)))), MARK(from(divides(z0, z1)))) MARK(sieve(from(x0))) -> c17(MARK(from(x0))) MARK(sieve(head(primes))) -> c17(A__SIEVE(a__head(a__primes)), MARK(head(primes))) MARK(sieve(head(sieve(z0)))) -> c17(A__SIEVE(a__head(a__sieve(mark(z0)))), MARK(head(sieve(z0)))) MARK(sieve(head(from(z0)))) -> c17(A__SIEVE(a__head(a__from(mark(z0)))), MARK(head(from(z0)))) MARK(sieve(head(head(z0)))) -> c17(A__SIEVE(a__head(a__head(mark(z0)))), MARK(head(head(z0)))) MARK(sieve(head(tail(z0)))) -> c17(A__SIEVE(a__head(a__tail(mark(z0)))), MARK(head(tail(z0)))) MARK(sieve(head(if(z0, z1, z2)))) -> c17(A__SIEVE(a__head(a__if(mark(z0), z1, z2))), MARK(head(if(z0, z1, z2)))) MARK(sieve(head(filter(z0, z1)))) -> c17(A__SIEVE(a__head(a__filter(mark(z0), 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MARK(tail(tail(z0)))) MARK(sieve(tail(if(z0, z1, z2)))) -> c17(A__SIEVE(a__tail(a__if(mark(z0), z1, z2))), MARK(tail(if(z0, z1, z2)))) MARK(sieve(tail(filter(z0, z1)))) -> c17(A__SIEVE(a__tail(a__filter(mark(z0), mark(z1)))), MARK(tail(filter(z0, z1)))) MARK(sieve(tail(s(z0)))) -> c17(A__SIEVE(a__tail(s(mark(z0)))), MARK(tail(s(z0)))) MARK(sieve(tail(cons(z0, z1)))) -> c17(A__SIEVE(a__tail(cons(mark(z0), z1))), MARK(tail(cons(z0, z1)))) MARK(sieve(tail(divides(z0, z1)))) -> c17(A__SIEVE(a__tail(divides(mark(z0), mark(z1)))), MARK(tail(divides(z0, z1)))) MARK(sieve(tail(x0))) -> c17(MARK(tail(x0))) MARK(sieve(tail(0))) -> c17(A__SIEVE(a__tail(0))) MARK(sieve(tail(true))) -> c17(A__SIEVE(a__tail(true))) MARK(sieve(tail(false))) -> c17(A__SIEVE(a__tail(false))) MARK(sieve(if(primes, x1, x2))) -> c17(A__SIEVE(a__if(a__primes, x1, x2)), MARK(if(primes, x1, x2))) MARK(sieve(if(sieve(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__sieve(mark(z0)), x1, x2)), MARK(if(sieve(z0), x1, x2))) MARK(sieve(if(from(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__from(mark(z0)), x1, x2)), MARK(if(from(z0), x1, x2))) MARK(sieve(if(head(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__head(mark(z0)), x1, x2)), MARK(if(head(z0), x1, x2))) MARK(sieve(if(tail(z0), x1, x2))) -> c17(A__SIEVE(a__if(a__tail(mark(z0)), x1, x2)), MARK(if(tail(z0), x1, x2))) MARK(sieve(if(if(z0, z1, z2), x1, x2))) -> c17(A__SIEVE(a__if(a__if(mark(z0), z1, z2), x1, x2)), MARK(if(if(z0, z1, z2), x1, x2))) MARK(sieve(if(filter(z0, z1), x1, x2))) -> c17(A__SIEVE(a__if(a__filter(mark(z0), mark(z1)), x1, x2)), MARK(if(filter(z0, z1), x1, x2))) MARK(sieve(if(s(z0), x1, x2))) -> c17(A__SIEVE(a__if(s(mark(z0)), x1, x2)), MARK(if(s(z0), x1, x2))) MARK(sieve(if(cons(z0, z1), x1, x2))) -> c17(A__SIEVE(a__if(cons(mark(z0), z1), x1, x2)), MARK(if(cons(z0, z1), x1, x2))) MARK(sieve(if(true, x1, x2))) -> c17(A__SIEVE(a__if(true, x1, x2)), MARK(if(true, x1, x2))) MARK(sieve(if(false, x1, x2))) -> c17(A__SIEVE(a__if(false, x1, x2)), MARK(if(false, x1, x2))) MARK(sieve(if(divides(z0, z1), x1, x2))) -> c17(A__SIEVE(a__if(divides(mark(z0), mark(z1)), x1, x2)), MARK(if(divides(z0, z1), x1, x2))) MARK(sieve(if(x0, z1, z2))) -> c17(MARK(if(x0, z1, z2))) MARK(sieve(if(0, x1, x2))) -> c17(A__SIEVE(a__if(0, x1, x2))) MARK(sieve(filter(x0, primes))) -> c17(A__SIEVE(a__filter(mark(x0), a__primes)), MARK(filter(x0, primes))) MARK(sieve(filter(x0, sieve(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), a__sieve(mark(z0)))), MARK(filter(x0, sieve(z0)))) MARK(sieve(filter(x0, from(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), a__from(mark(z0)))), MARK(filter(x0, from(z0)))) MARK(sieve(filter(x0, head(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), a__head(mark(z0)))), MARK(filter(x0, head(z0)))) MARK(sieve(filter(x0, tail(z0)))) -> c17(A__SIEVE(a__filter(mark(x0), a__tail(mark(z0)))), MARK(filter(x0, tail(z0)))) MARK(sieve(filter(x0, if(z0, z1, z2)))) -> c17(A__SIEVE(a__filter(mark(x0), a__if(mark(z0), z1, z2))), MARK(filter(x0, if(z0, z1, 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c17(A__SIEVE(a__filter(a__sieve(mark(z0)), mark(x1))), MARK(filter(sieve(z0), x1))) MARK(sieve(filter(from(z0), x1))) -> c17(A__SIEVE(a__filter(a__from(mark(z0)), mark(x1))), MARK(filter(from(z0), x1))) MARK(sieve(filter(head(z0), x1))) -> c17(A__SIEVE(a__filter(a__head(mark(z0)), mark(x1))), MARK(filter(head(z0), x1))) MARK(sieve(filter(tail(z0), x1))) -> c17(A__SIEVE(a__filter(a__tail(mark(z0)), mark(x1))), MARK(filter(tail(z0), x1))) MARK(sieve(filter(if(z0, z1, z2), x1))) -> c17(A__SIEVE(a__filter(a__if(mark(z0), z1, z2), mark(x1))), MARK(filter(if(z0, z1, z2), x1))) MARK(sieve(filter(filter(z0, z1), x1))) -> c17(A__SIEVE(a__filter(a__filter(mark(z0), mark(z1)), mark(x1))), MARK(filter(filter(z0, z1), x1))) MARK(sieve(filter(s(z0), x1))) -> c17(A__SIEVE(a__filter(s(mark(z0)), mark(x1))), MARK(filter(s(z0), x1))) MARK(sieve(filter(0, x1))) -> c17(A__SIEVE(a__filter(0, mark(x1))), MARK(filter(0, x1))) MARK(sieve(filter(cons(z0, z1), x1))) -> c17(A__SIEVE(a__filter(cons(mark(z0), z1), 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c22(A__FILTER(mark(x0), a__filter(mark(x1), a__head(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(x1, tail(z0)))) -> c22(A__FILTER(mark(x0), a__filter(mark(x1), a__tail(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(x1, if(z0, z1, z2)))) -> c22(A__FILTER(mark(x0), a__filter(mark(x1), a__if(mark(z0), z1, z2))), MARK(x0)) MARK(filter(x0, filter(x1, filter(z0, z1)))) -> c22(A__FILTER(mark(x0), a__filter(mark(x1), a__filter(mark(z0), mark(z1)))), MARK(x0)) MARK(filter(x0, filter(x1, s(z0)))) -> c22(A__FILTER(mark(x0), a__filter(mark(x1), s(mark(z0)))), MARK(x0)) MARK(filter(x0, filter(x1, 0))) -> c22(A__FILTER(mark(x0), a__filter(mark(x1), 0)), MARK(x0)) MARK(filter(x0, filter(x1, cons(z0, z1)))) -> c22(A__FILTER(mark(x0), a__filter(mark(x1), cons(mark(z0), z1))), MARK(x0)) MARK(filter(x0, filter(x1, true))) -> c22(A__FILTER(mark(x0), a__filter(mark(x1), true)), MARK(x0)) MARK(filter(x0, filter(x1, false))) -> c22(A__FILTER(mark(x0), a__filter(mark(x1), false)), MARK(x0)) MARK(filter(x0, filter(x1, divides(z0, z1)))) -> c22(A__FILTER(mark(x0), a__filter(mark(x1), divides(mark(z0), mark(z1)))), MARK(x0)) MARK(filter(x0, filter(primes, x2))) -> c22(A__FILTER(mark(x0), a__filter(a__primes, mark(x2))), MARK(x0)) MARK(filter(x0, filter(sieve(z0), x2))) -> c22(A__FILTER(mark(x0), a__filter(a__sieve(mark(z0)), mark(x2))), MARK(x0)) MARK(filter(x0, filter(from(z0), x2))) -> c22(A__FILTER(mark(x0), a__filter(a__from(mark(z0)), mark(x2))), MARK(x0)) MARK(filter(x0, filter(head(z0), x2))) -> c22(A__FILTER(mark(x0), a__filter(a__head(mark(z0)), mark(x2))), MARK(x0)) MARK(filter(x0, filter(tail(z0), x2))) -> c22(A__FILTER(mark(x0), a__filter(a__tail(mark(z0)), mark(x2))), MARK(x0)) MARK(filter(x0, filter(if(z0, z1, z2), x2))) -> c22(A__FILTER(mark(x0), a__filter(a__if(mark(z0), z1, z2), mark(x2))), MARK(x0)) MARK(filter(x0, filter(filter(z0, z1), x2))) -> c22(A__FILTER(mark(x0), a__filter(a__filter(mark(z0), mark(z1)), mark(x2))), MARK(x0)) MARK(filter(x0, filter(s(z0), x2))) -> c22(A__FILTER(mark(x0), a__filter(s(mark(z0)), mark(x2))), MARK(x0)) MARK(filter(x0, filter(0, x2))) -> c22(A__FILTER(mark(x0), a__filter(0, mark(x2))), MARK(x0)) MARK(filter(x0, filter(cons(z0, z1), x2))) -> c22(A__FILTER(mark(x0), a__filter(cons(mark(z0), z1), mark(x2))), MARK(x0)) MARK(filter(x0, filter(true, x2))) -> c22(A__FILTER(mark(x0), a__filter(true, mark(x2))), MARK(x0)) MARK(filter(x0, filter(false, x2))) -> c22(A__FILTER(mark(x0), a__filter(false, mark(x2))), MARK(x0)) MARK(filter(x0, filter(divides(z0, z1), x2))) -> c22(A__FILTER(mark(x0), a__filter(divides(mark(z0), mark(z1)), mark(x2))), MARK(x0)) MARK(filter(primes, filter(x1, x2))) -> c22(A__FILTER(a__primes, a__filter(mark(x1), mark(x2))), MARK(primes)) MARK(filter(sieve(z0), filter(x1, x2))) -> c22(A__FILTER(a__sieve(mark(z0)), a__filter(mark(x1), mark(x2))), MARK(sieve(z0))) MARK(filter(from(z0), filter(x1, x2))) -> c22(A__FILTER(a__from(mark(z0)), a__filter(mark(x1), mark(x2))), MARK(from(z0))) MARK(filter(head(z0), filter(x1, x2))) -> c22(A__FILTER(a__head(mark(z0)), a__filter(mark(x1), mark(x2))), MARK(head(z0))) MARK(filter(tail(z0), filter(x1, x2))) -> c22(A__FILTER(a__tail(mark(z0)), a__filter(mark(x1), mark(x2))), MARK(tail(z0))) MARK(filter(if(z0, z1, z2), filter(x1, x2))) -> c22(A__FILTER(a__if(mark(z0), z1, z2), a__filter(mark(x1), mark(x2))), MARK(if(z0, z1, z2))) MARK(filter(filter(z0, z1), filter(x1, x2))) -> c22(A__FILTER(a__filter(mark(z0), mark(z1)), a__filter(mark(x1), mark(x2))), MARK(filter(z0, z1))) MARK(filter(s(z0), filter(x1, x2))) -> c22(A__FILTER(s(mark(z0)), a__filter(mark(x1), mark(x2))), MARK(s(z0))) MARK(filter(x0, filter(x1, x2))) -> c22(MARK(x0)) MARK(filter(cons(z0, z1), filter(x1, x2))) -> c22(MARK(cons(z0, z1))) MARK(filter(divides(z0, z1), filter(x1, x2))) -> c22(MARK(divides(z0, z1))) K tuples:none Defined Rule Symbols: a__primes, a__from_1, a__head_1, a__tail_1, a__if_3, a__filter_2, a__sieve_1, mark_1 Defined Pair Symbols: A__FROM_1, A__HEAD_1, A__TAIL_1, A__IF_3, A__SIEVE_1, MARK_1, A__FILTER_2, A__PRIMES Compound Symbols: c2_1, c4_1, c6_1, c8_1, c9_1, c14_1, c16_1, c18_2, c24_1, c26_1, c29_1, c30_1, c11_1, c12_1, c_1, c17_2, c17_1, c19_2, c19_1, c20_2, c20_1, c21_1, c22_2, c22_1, c23_2, c23_1, c_2, c21_2