KILLED proof of input_pKQbeey36s.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). (0) CpxTRS (1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (2) CpxTRS (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms] (4) CpxTRS (5) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms] (6) CpxWeightedTrs (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CpxTypedWeightedTrs (9) CompletionProof [UPPER BOUND(ID), 0 ms] (10) CpxTypedWeightedCompleteTrs (11) NarrowingProof [BOTH BOUNDS(ID, ID), 159 ms] (12) CpxTypedWeightedCompleteTrs (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (14) CpxRNTS (15) SimplificationProof [BOTH BOUNDS(ID, ID), 4 ms] (16) CpxRNTS (17) CompletionProof [UPPER BOUND(ID), 0 ms] (18) CpxTypedWeightedCompleteTrs (19) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (20) CpxRNTS (21) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (22) CdtProblem (23) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (24) CdtProblem (25) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (26) CdtProblem (27) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (28) CdtProblem (29) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (30) CdtProblem (31) CdtNarrowingProof [BOTH BOUNDS(ID, 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) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (38) CdtProblem (39) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (40) CdtProblem (41) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (42) CdtProblem (43) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (44) CdtProblem (45) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (46) CdtProblem (47) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (48) CdtProblem (49) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (50) CdtProblem (51) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (52) CdtProblem (53) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (54) CdtProblem (55) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (56) CdtProblem (57) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (58) CdtProblem (59) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (60) CdtProblem (61) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 13 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) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (70) CdtProblem (71) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (72) CdtProblem (73) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (74) CdtProblem (75) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (76) CdtProblem (77) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 12 ms] (78) CdtProblem (79) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (80) CdtProblem (81) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (82) CdtProblem (83) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (84) CdtProblem (85) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (86) CdtProblem (87) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (88) CdtProblem (89) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 22 ms] (90) CdtProblem (91) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (92) CdtProblem (93) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (94) CdtProblem (95) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (96) CdtProblem (97) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (98) CdtProblem (99) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (100) CdtProblem (101) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 35 ms] (102) CdtProblem (103) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (104) CdtProblem (105) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (106) CdtProblem (107) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (108) CdtProblem (109) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 2 ms] (110) CdtProblem (111) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (112) CdtProblem (113) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 24 ms] (114) CdtProblem (115) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (116) CdtProblem (117) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (118) CdtProblem (119) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (120) CdtProblem (121) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 80 ms] (122) CdtProblem (123) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (124) CdtProblem (125) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (126) CdtProblem (127) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (128) CdtProblem (129) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (130) CdtProblem (131) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (132) CdtProblem (133) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (134) CdtProblem (135) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (136) CdtProblem (137) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 49 ms] (138) CdtProblem (139) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (140) CdtProblem (141) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 39 ms] (142) CdtProblem (143) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (144) CdtProblem (145) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 38 ms] (146) CdtProblem (147) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (148) CdtProblem (149) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 17 ms] (150) CdtProblem (151) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (152) CdtProblem (153) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 58 ms] (154) CdtProblem (155) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (156) CdtProblem (157) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 39 ms] (158) CdtProblem (159) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (160) CdtProblem (161) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 39 ms] (162) CdtProblem (163) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (164) CdtProblem (165) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 79 ms] (166) CdtProblem (167) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (168) CdtProblem (169) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 83 ms] (170) CdtProblem (171) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (172) CdtProblem (173) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 68 ms] (174) CdtProblem (175) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 7 ms] (176) CdtProblem (177) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 96 ms] (178) CdtProblem (179) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 12 ms] (180) CdtProblem (181) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 108 ms] (182) CdtProblem (183) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 22 ms] (184) CdtProblem (185) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 13 ms] (186) CdtProblem (187) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 241 ms] (188) CdtProblem (189) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 24 ms] (190) CdtProblem (191) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (192) CdtProblem (193) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 263 ms] (194) CdtProblem (195) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 25 ms] (196) CdtProblem (197) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (198) CdtProblem (199) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 196 ms] (200) CdtProblem (201) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (202) CdtProblem (203) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 1 ms] (204) CdtProblem (205) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 295 ms] (206) CdtProblem (207) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 6 ms] (208) CdtProblem (209) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 14 ms] (210) CdtProblem (211) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 332 ms] (212) CdtProblem (213) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 30 ms] (214) CdtProblem (215) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 30 ms] (216) CdtProblem (217) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 473 ms] (218) CdtProblem (219) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 37 ms] (220) CdtProblem (221) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 47 ms] (222) 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) RelTrsToWeightedTrsProof (UPPER BOUND(ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (6) Obligation: The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: a__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 ---------------------------------------- (7) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (8) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: a__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 ---------------------------------------- (9) CompletionProof (UPPER BOUND(ID)) The transformation into a RNTS is sound, since: (a) The obligation is a constructor system where every type has a constant constructor, (b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols: none (c) The following functions are completely defined: mark_1 a__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 ---------------------------------------- (10) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: a__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 ---------------------------------------- (11) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (12) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: a__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 ---------------------------------------- (13) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: 0 => 0 true => 3 false => 1 primes => 2 ---------------------------------------- (14) 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 ---------------------------------------- (15) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (16) Obligation: Complexity RNTS consisting of the following rules: a__filter(z, z') -{ 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 ---------------------------------------- (17) CompletionProof (UPPER BOUND(ID)) The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added: none And the following fresh constants: none ---------------------------------------- (18) Obligation: Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: a__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) 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 ---------------------------------------- (20) 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. ---------------------------------------- (21) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (22) 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 ---------------------------------------- (23) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 10 trailing nodes: MARK(true) -> c27 MARK(false) -> c28 A__SIEVE(z0) -> c15 A__TAIL(z0) -> c7 A__PRIMES -> c1 A__FROM(z0) -> c3 MARK(0) -> c25 A__FILTER(z0, z1) -> c13 A__IF(z0, z1, z2) -> c10 A__HEAD(z0) -> c5 ---------------------------------------- (24) 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 ---------------------------------------- (25) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (26) 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 ---------------------------------------- (27) 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)))) ---------------------------------------- (28) 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 ---------------------------------------- (29) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (30) 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 ---------------------------------------- (31) 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))) ---------------------------------------- (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__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 ---------------------------------------- (33) 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)) ---------------------------------------- (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__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 ---------------------------------------- (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__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 ---------------------------------------- (37) 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))) ---------------------------------------- (38) 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 ---------------------------------------- (39) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing nodes: MARK(head(false)) -> c19(A__HEAD(false), MARK(false)) MARK(head(true)) -> c19(A__HEAD(true), MARK(true)) MARK(head(0)) -> c19(A__HEAD(0), MARK(0)) ---------------------------------------- (40) 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 ---------------------------------------- (41) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (42) 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 ---------------------------------------- (43) 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))) ---------------------------------------- (44) 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 ---------------------------------------- (45) 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)) ---------------------------------------- (46) 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 ---------------------------------------- (47) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (48) 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 ---------------------------------------- (49) 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))) ---------------------------------------- (50) 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 ---------------------------------------- (51) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: MARK(if(0, x1, x2)) -> c21(A__IF(0, x1, x2), MARK(0)) ---------------------------------------- (52) 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 ---------------------------------------- (53) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 5 trailing tuple parts ---------------------------------------- (54) 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 ---------------------------------------- (55) 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))) ---------------------------------------- (56) 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 ---------------------------------------- (57) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing nodes: MARK(filter(0, x1)) -> c22(A__FILTER(0, mark(x1)), MARK(0)) MARK(filter(true, x1)) -> c22(A__FILTER(true, mark(x1)), MARK(true)) MARK(filter(false, x1)) -> c22(A__FILTER(false, mark(x1)), MARK(false)) ---------------------------------------- (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(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 ---------------------------------------- (59) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 7 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(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 ---------------------------------------- (61) 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)) ---------------------------------------- (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(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 ---------------------------------------- (63) 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)) ---------------------------------------- (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(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 ---------------------------------------- (65) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 7 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(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 ---------------------------------------- (67) 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)))) ---------------------------------------- (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(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 ---------------------------------------- (69) 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)) ---------------------------------------- (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(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 ---------------------------------------- (71) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 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(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 ---------------------------------------- (73) 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)))) ---------------------------------------- (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(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 ---------------------------------------- (75) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing tuple parts ---------------------------------------- (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(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 ---------------------------------------- (77) 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))) ---------------------------------------- (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(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, 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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 ---------------------------------------- (79) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (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(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 ---------------------------------------- (81) 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)))) ---------------------------------------- (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(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 ---------------------------------------- (83) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 4 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(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))) -> 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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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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), 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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), 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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 ---------------------------------------- (85) 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)))) ---------------------------------------- (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(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)) 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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 ---------------------------------------- (87) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing tuple parts ---------------------------------------- (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(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))), 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))) -> 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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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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 ---------------------------------------- (89) 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))) ---------------------------------------- (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(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)))) -> 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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 ---------------------------------------- (91) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (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__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)))) 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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))) 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(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 ---------------------------------------- (93) 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))) ---------------------------------------- (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__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))) -> 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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))) 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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)) -> 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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, 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 ---------------------------------------- (95) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 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__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))) -> 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, 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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))) 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(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(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 ---------------------------------------- (97) 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)) ---------------------------------------- (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(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)) -> 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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, 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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 ---------------------------------------- (99) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (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(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), 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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 ---------------------------------------- (101) 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)))) ---------------------------------------- (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(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, 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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))), 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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, 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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 ---------------------------------------- (103) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing tuple parts ---------------------------------------- (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(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, 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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, 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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)) 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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 ---------------------------------------- (105) 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))) ---------------------------------------- (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(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))), 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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, 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, 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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 ---------------------------------------- (107) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (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(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)))) -> 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))) 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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))) 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))) 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) 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)))) ---------------------------------------- (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(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, 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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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))) 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(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 ---------------------------------------- (111) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing tuple parts ---------------------------------------- (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(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, 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), 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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)))) 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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, 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))) 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) 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)))) ---------------------------------------- (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(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))) -> 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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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))) -> 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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))) 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)))) 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(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)) -> 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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 ---------------------------------------- (115) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing tuple parts ---------------------------------------- (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(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))) -> 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, 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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 ---------------------------------------- (117) 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))) ---------------------------------------- (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(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)))) 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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 ---------------------------------------- (119) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (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(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, 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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 ---------------------------------------- (121) 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))) ---------------------------------------- (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(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))) 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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))) -> 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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, 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))) 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) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (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(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)))) 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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 ---------------------------------------- (125) 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)) ---------------------------------------- (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(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)), 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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(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 ---------------------------------------- (127) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (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(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, 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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 ---------------------------------------- (129) 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)))) ---------------------------------------- (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(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)) 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(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)))), 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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 ---------------------------------------- (131) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing tuple parts ---------------------------------------- (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(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)) 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)))), 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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), 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(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))) 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) 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))) ---------------------------------------- (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(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)))) 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))) 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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 ---------------------------------------- (135) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (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(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)))) 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)))) -> 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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 ---------------------------------------- (137) 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)))) ---------------------------------------- (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(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))) -> 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)))) -> 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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(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)))) 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) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing tuple parts ---------------------------------------- (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(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))) -> 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))) -> 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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)))) -> 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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 ---------------------------------------- (141) 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)))) ---------------------------------------- (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(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))) 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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))) -> 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(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)))) 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) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing tuple parts ---------------------------------------- (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))) -> 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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)))) -> 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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 ---------------------------------------- (145) 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))) ---------------------------------------- (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(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))), 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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, 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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))) -> 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(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))) 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) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (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(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)))) -> 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, 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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 ---------------------------------------- (149) 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))) ---------------------------------------- (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(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)) -> 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c20(A__TAIL(a__filter(divides(mark(z0), mark(z1)), mark(x1))), MARK(filter(divides(z0, z1), 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__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))) 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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), 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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, 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))) 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) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (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(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))) -> 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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)))) 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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))) 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) 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)) ---------------------------------------- (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(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)) -> 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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)))) -> 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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(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 ---------------------------------------- (155) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (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(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))) -> 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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)) 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) 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)))) ---------------------------------------- (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(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)))) 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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))), 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))) -> 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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(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 ---------------------------------------- (159) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing tuple parts ---------------------------------------- (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(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, 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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))) 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 ---------------------------------------- (161) 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))) ---------------------------------------- (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(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)))) -> 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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)) 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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 ---------------------------------------- (163) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (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(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))) -> 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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))) 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) 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)))) ---------------------------------------- (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(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, 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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, 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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), 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)))) 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(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))) 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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 ---------------------------------------- (167) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing tuple parts ---------------------------------------- (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(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), 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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)))) -> 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)))), 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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)) 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) 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)))) ---------------------------------------- (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(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)))) -> 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))) -> 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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, 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)) -> 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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(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 ---------------------------------------- (171) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing tuple parts ---------------------------------------- (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)) -> 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, 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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)))) 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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 ---------------------------------------- (173) 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))) ---------------------------------------- (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(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))) -> 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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(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))) 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) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (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(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))) 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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 ---------------------------------------- (177) 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))) ---------------------------------------- (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(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)))) 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))) 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)))), 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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))) 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))) -> 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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, 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 ---------------------------------------- (179) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (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(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)))), 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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)))) 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-> 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))) -> 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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)) -> 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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 ---------------------------------------- (181) 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))) ---------------------------------------- (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(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, 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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))) 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 ---------------------------------------- (183) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing nodes: MARK(filter(false, primes)) -> c22(A__FILTER(false, a__primes), MARK(false)) MARK(filter(0, primes)) -> c22(A__FILTER(0, a__primes), MARK(0)) MARK(filter(true, primes)) -> c22(A__FILTER(true, a__primes), MARK(true)) ---------------------------------------- (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(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, 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, 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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(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 ---------------------------------------- (185) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 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(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)))) -> 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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)) 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 ---------------------------------------- (187) 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))) ---------------------------------------- (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(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)), 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)))), 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-> 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 ---------------------------------------- (189) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing nodes: 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(0, sieve(x1))) -> c22(A__FILTER(0, a__sieve(mark(x1))), MARK(0)) ---------------------------------------- (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(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)), 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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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)), 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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))) 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 ---------------------------------------- (191) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (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(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)), 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)), 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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)) -> 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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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))) 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 ---------------------------------------- (193) 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))) ---------------------------------------- (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(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))) -> 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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, 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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, 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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 ---------------------------------------- (195) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing nodes: 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)) MARK(filter(true, from(x1))) -> c22(A__FILTER(true, a__from(mark(x1))), MARK(true)) ---------------------------------------- (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(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))) -> 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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, 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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)) -> 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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, 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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(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))) 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(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))), 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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(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(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 ---------------------------------------- (197) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 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(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))) -> 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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(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 ---------------------------------------- (199) 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))) ---------------------------------------- (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(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)))) 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)))), 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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, 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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(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 ---------------------------------------- (201) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing nodes: 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(0, head(x1))) -> c22(A__FILTER(0, a__head(mark(x1))), MARK(0)) ---------------------------------------- (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(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)))), 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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)))) 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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 ---------------------------------------- (203) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (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(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)))) 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))) -> 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-> 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))) 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(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)), 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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))), 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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))) 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 ---------------------------------------- (205) 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))) ---------------------------------------- (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(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)) 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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 ---------------------------------------- (207) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing nodes: MARK(filter(0, tail(x1))) -> c22(A__FILTER(0, a__tail(mark(x1))), MARK(0)) 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)) ---------------------------------------- (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, 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))) -> 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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 ---------------------------------------- (209) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 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, 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)))), 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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(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 ---------------------------------------- (211) 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))) ---------------------------------------- (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, 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)))) -> 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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(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 ---------------------------------------- (213) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing nodes: MARK(filter(0, if(x1, x2, x3))) -> c22(A__FILTER(0, a__if(mark(x1), x2, x3)), MARK(0)) 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)) ---------------------------------------- (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, 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)) -> 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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))) 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, 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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))) -> 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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))) 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 ---------------------------------------- (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))) -> 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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))) 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 ---------------------------------------- (217) 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))) ---------------------------------------- (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, 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, 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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))) 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 ---------------------------------------- (219) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing nodes: 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(0, filter(x1, x2))) -> c22(A__FILTER(0, a__filter(mark(x1), mark(x2))), MARK(0)) ---------------------------------------- (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, 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))) -> 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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(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 ---------------------------------------- (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, 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))) 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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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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