/export/starexec/sandbox/solver/bin/starexec_run_standard /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- YES proof of /export/starexec/sandbox/benchmark/theBenchmark.xml # AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished Termination w.r.t. Q of the given QTRS could be proven: (0) QTRS (1) DependencyPairsProof [EQUIVALENT, 197 ms] (2) QDP (3) DependencyGraphProof [EQUIVALENT, 0 ms] (4) AND (5) QDP (6) UsableRulesProof [EQUIVALENT, 0 ms] (7) QDP (8) QDPSizeChangeProof [EQUIVALENT, 0 ms] (9) YES (10) QDP (11) UsableRulesProof [EQUIVALENT, 0 ms] (12) QDP (13) QDPSizeChangeProof [EQUIVALENT, 0 ms] (14) YES (15) QDP (16) UsableRulesProof [EQUIVALENT, 0 ms] (17) QDP (18) QDPSizeChangeProof [EQUIVALENT, 0 ms] (19) YES (20) QDP (21) UsableRulesProof [EQUIVALENT, 0 ms] (22) QDP (23) QDPSizeChangeProof [EQUIVALENT, 0 ms] (24) YES (25) QDP (26) UsableRulesProof [EQUIVALENT, 0 ms] (27) QDP (28) QDPSizeChangeProof [EQUIVALENT, 0 ms] (29) YES (30) QDP (31) UsableRulesProof [EQUIVALENT, 0 ms] (32) QDP (33) QDPSizeChangeProof [EQUIVALENT, 0 ms] (34) YES (35) QDP (36) UsableRulesProof [EQUIVALENT, 0 ms] (37) QDP (38) QDPSizeChangeProof [EQUIVALENT, 0 ms] (39) YES (40) QDP (41) UsableRulesProof [EQUIVALENT, 0 ms] (42) QDP (43) QDPSizeChangeProof [EQUIVALENT, 0 ms] (44) YES (45) QDP (46) UsableRulesProof [EQUIVALENT, 0 ms] (47) QDP (48) QDPSizeChangeProof [EQUIVALENT, 0 ms] (49) YES (50) QDP (51) UsableRulesProof [EQUIVALENT, 0 ms] (52) QDP (53) QDPSizeChangeProof [EQUIVALENT, 0 ms] (54) YES (55) QDP (56) UsableRulesProof [EQUIVALENT, 0 ms] (57) QDP (58) QDPSizeChangeProof [EQUIVALENT, 0 ms] (59) YES (60) QDP (61) UsableRulesProof [EQUIVALENT, 0 ms] (62) QDP (63) QDPSizeChangeProof [EQUIVALENT, 0 ms] (64) YES (65) QDP (66) UsableRulesProof [EQUIVALENT, 0 ms] (67) QDP (68) QDPSizeChangeProof [EQUIVALENT, 0 ms] (69) YES (70) QDP (71) UsableRulesProof [EQUIVALENT, 0 ms] (72) QDP (73) QDPSizeChangeProof [EQUIVALENT, 0 ms] (74) YES (75) QDP (76) UsableRulesProof [EQUIVALENT, 0 ms] (77) QDP (78) QDPSizeChangeProof [EQUIVALENT, 0 ms] (79) YES (80) QDP (81) UsableRulesProof [EQUIVALENT, 0 ms] (82) QDP (83) QDPSizeChangeProof [EQUIVALENT, 0 ms] (84) YES (85) QDP (86) UsableRulesProof [EQUIVALENT, 0 ms] (87) QDP (88) QDPSizeChangeProof [EQUIVALENT, 0 ms] (89) YES (90) QDP (91) UsableRulesProof [EQUIVALENT, 0 ms] (92) QDP (93) QDPSizeChangeProof [EQUIVALENT, 0 ms] (94) YES (95) QDP (96) UsableRulesProof [EQUIVALENT, 0 ms] (97) QDP (98) QDPSizeChangeProof [EQUIVALENT, 0 ms] (99) YES (100) QDP (101) UsableRulesProof [EQUIVALENT, 0 ms] (102) QDP (103) QDPSizeChangeProof [EQUIVALENT, 0 ms] (104) YES (105) QDP (106) UsableRulesProof [EQUIVALENT, 0 ms] (107) QDP (108) QDPSizeChangeProof [EQUIVALENT, 0 ms] (109) YES (110) QDP (111) UsableRulesProof [EQUIVALENT, 0 ms] (112) QDP (113) QDPSizeChangeProof [EQUIVALENT, 0 ms] (114) YES (115) QDP (116) QDPOrderProof [EQUIVALENT, 806 ms] (117) QDP (118) QDPOrderProof [EQUIVALENT, 90 ms] (119) QDP (120) QDPOrderProof [EQUIVALENT, 1010 ms] (121) QDP (122) DependencyGraphProof [EQUIVALENT, 0 ms] (123) QDP (124) QDPOrderProof [EQUIVALENT, 552 ms] (125) QDP (126) QDPOrderProof [EQUIVALENT, 573 ms] (127) QDP (128) QDPOrderProof [EQUIVALENT, 612 ms] (129) QDP (130) QDPOrderProof [EQUIVALENT, 526 ms] (131) QDP (132) QDPOrderProof [EQUIVALENT, 541 ms] (133) QDP (134) QDPOrderProof [EQUIVALENT, 561 ms] (135) QDP (136) QDPOrderProof [EQUIVALENT, 539 ms] (137) QDP (138) QDPOrderProof [EQUIVALENT, 1137 ms] (139) QDP (140) QDPOrderProof [EQUIVALENT, 516 ms] (141) QDP (142) QDPOrderProof [EQUIVALENT, 934 ms] (143) QDP (144) QDPOrderProof [EQUIVALENT, 1099 ms] (145) QDP (146) QDPOrderProof [EQUIVALENT, 804 ms] (147) QDP (148) QDPOrderProof [EQUIVALENT, 695 ms] (149) QDP (150) QDPOrderProof [EQUIVALENT, 436 ms] (151) QDP (152) QDPOrderProof [EQUIVALENT, 503 ms] (153) QDP (154) QDPOrderProof [EQUIVALENT, 1244 ms] (155) QDP (156) QDPOrderProof [EQUIVALENT, 453 ms] (157) QDP (158) QDPOrderProof [EQUIVALENT, 543 ms] (159) QDP (160) QDPOrderProof [EQUIVALENT, 389 ms] (161) QDP (162) QDPOrderProof [EQUIVALENT, 577 ms] (163) QDP (164) QDPOrderProof [EQUIVALENT, 988 ms] (165) QDP (166) QDPOrderProof [EQUIVALENT, 390 ms] (167) QDP (168) QDPOrderProof [EQUIVALENT, 392 ms] (169) QDP (170) QDPOrderProof [EQUIVALENT, 432 ms] (171) QDP (172) QDPOrderProof [EQUIVALENT, 356 ms] (173) QDP (174) QDPOrderProof [EQUIVALENT, 1427 ms] (175) QDP (176) QDPOrderProof [EQUIVALENT, 405 ms] (177) QDP (178) QDPOrderProof [EQUIVALENT, 284 ms] (179) QDP (180) DependencyGraphProof [EQUIVALENT, 0 ms] (181) TRUE ---------------------------------------- (0) Obligation: Q restricted rewrite system: The TRS R consists of the following rules: active(zeros) -> mark(cons(0, zeros)) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatIList(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, L, N)) -> mark(U72(isNat(N), L)) active(U72(tt, L)) -> mark(s(length(L))) active(U81(tt)) -> mark(nil) active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N)) active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N)) active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(isNat(0)) -> mark(tt) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2)) active(length(nil)) -> mark(0) active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N)) active(take(0, IL)) -> mark(U81(isNatIList(IL))) active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N)) mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) mark(0) -> active(0) mark(U11(X)) -> active(U11(mark(X))) mark(tt) -> active(tt) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1, X2)) -> active(U51(mark(X1), X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil) -> active(nil) mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4)) mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4)) mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4)) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) cons(mark(X1), X2) -> cons(X1, X2) cons(X1, mark(X2)) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1), X2) -> U41(X1, X2) U41(X1, mark(X2)) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1), X2) -> U51(X1, X2) U51(X1, mark(X2)) -> U51(X1, X2) U51(active(X1), X2) -> U51(X1, X2) U51(X1, active(X2)) -> U51(X1, X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4) U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4) U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4) U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4) U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4) U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4) take(mark(X1), X2) -> take(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) Q is empty. ---------------------------------------- (1) DependencyPairsProof (EQUIVALENT) Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem. ---------------------------------------- (2) Obligation: Q DP problem: The TRS P consists of the following rules: ACTIVE(zeros) -> MARK(cons(0, zeros)) ACTIVE(zeros) -> CONS(0, zeros) ACTIVE(U11(tt)) -> MARK(tt) ACTIVE(U21(tt)) -> MARK(tt) ACTIVE(U31(tt)) -> MARK(tt) ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2))) ACTIVE(U41(tt, V2)) -> U42^1(isNatIList(V2)) ACTIVE(U41(tt, V2)) -> ISNATILIST(V2) ACTIVE(U42(tt)) -> MARK(tt) ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2))) ACTIVE(U51(tt, V2)) -> U52^1(isNatList(V2)) ACTIVE(U51(tt, V2)) -> ISNATLIST(V2) ACTIVE(U52(tt)) -> MARK(tt) ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIList(V2))) ACTIVE(U61(tt, V2)) -> U62^1(isNatIList(V2)) ACTIVE(U61(tt, V2)) -> ISNATILIST(V2) ACTIVE(U62(tt)) -> MARK(tt) ACTIVE(U71(tt, L, N)) -> MARK(U72(isNat(N), L)) ACTIVE(U71(tt, L, N)) -> U72^1(isNat(N), L) ACTIVE(U71(tt, L, N)) -> ISNAT(N) ACTIVE(U72(tt, L)) -> MARK(s(length(L))) ACTIVE(U72(tt, L)) -> S(length(L)) ACTIVE(U72(tt, L)) -> LENGTH(L) ACTIVE(U81(tt)) -> MARK(nil) ACTIVE(U91(tt, IL, M, N)) -> MARK(U92(isNat(M), IL, M, N)) ACTIVE(U91(tt, IL, M, N)) -> U92^1(isNat(M), IL, M, N) ACTIVE(U91(tt, IL, M, N)) -> ISNAT(M) ACTIVE(U92(tt, IL, M, N)) -> MARK(U93(isNat(N), IL, M, N)) ACTIVE(U92(tt, IL, M, N)) -> U93^1(isNat(N), IL, M, N) ACTIVE(U92(tt, IL, M, N)) -> ISNAT(N) ACTIVE(U93(tt, IL, M, N)) -> MARK(cons(N, take(M, IL))) ACTIVE(U93(tt, IL, M, N)) -> CONS(N, take(M, IL)) ACTIVE(U93(tt, IL, M, N)) -> TAKE(M, IL) ACTIVE(isNat(0)) -> MARK(tt) ACTIVE(isNat(length(V1))) -> MARK(U11(isNatList(V1))) ACTIVE(isNat(length(V1))) -> U11^1(isNatList(V1)) ACTIVE(isNat(length(V1))) -> ISNATLIST(V1) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) ACTIVE(isNat(s(V1))) -> U21^1(isNat(V1)) ACTIVE(isNat(s(V1))) -> ISNAT(V1) ACTIVE(isNatIList(V)) -> MARK(U31(isNatList(V))) ACTIVE(isNatIList(V)) -> U31^1(isNatList(V)) ACTIVE(isNatIList(V)) -> ISNATLIST(V) ACTIVE(isNatIList(zeros)) -> MARK(tt) ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNat(V1), V2)) ACTIVE(isNatIList(cons(V1, V2))) -> U41^1(isNat(V1), V2) ACTIVE(isNatIList(cons(V1, V2))) -> ISNAT(V1) ACTIVE(isNatList(nil)) -> MARK(tt) ACTIVE(isNatList(cons(V1, V2))) -> MARK(U51(isNat(V1), V2)) ACTIVE(isNatList(cons(V1, V2))) -> U51^1(isNat(V1), V2) ACTIVE(isNatList(cons(V1, V2))) -> ISNAT(V1) ACTIVE(isNatList(take(V1, V2))) -> MARK(U61(isNat(V1), V2)) ACTIVE(isNatList(take(V1, V2))) -> U61^1(isNat(V1), V2) ACTIVE(isNatList(take(V1, V2))) -> ISNAT(V1) ACTIVE(length(nil)) -> MARK(0) ACTIVE(length(cons(N, L))) -> MARK(U71(isNatList(L), L, N)) ACTIVE(length(cons(N, L))) -> U71^1(isNatList(L), L, N) ACTIVE(length(cons(N, L))) -> ISNATLIST(L) ACTIVE(take(0, IL)) -> MARK(U81(isNatIList(IL))) ACTIVE(take(0, IL)) -> U81^1(isNatIList(IL)) ACTIVE(take(0, IL)) -> ISNATILIST(IL) ACTIVE(take(s(M), cons(N, IL))) -> MARK(U91(isNatIList(IL), IL, M, N)) ACTIVE(take(s(M), cons(N, IL))) -> U91^1(isNatIList(IL), IL, M, N) ACTIVE(take(s(M), cons(N, IL))) -> ISNATILIST(IL) MARK(zeros) -> ACTIVE(zeros) MARK(cons(X1, X2)) -> ACTIVE(cons(mark(X1), X2)) MARK(cons(X1, X2)) -> CONS(mark(X1), X2) MARK(cons(X1, X2)) -> MARK(X1) MARK(0) -> ACTIVE(0) MARK(U11(X)) -> ACTIVE(U11(mark(X))) MARK(U11(X)) -> U11^1(mark(X)) MARK(U11(X)) -> MARK(X) MARK(tt) -> ACTIVE(tt) MARK(U21(X)) -> ACTIVE(U21(mark(X))) MARK(U21(X)) -> U21^1(mark(X)) MARK(U21(X)) -> MARK(X) MARK(U31(X)) -> ACTIVE(U31(mark(X))) MARK(U31(X)) -> U31^1(mark(X)) MARK(U31(X)) -> MARK(X) MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) MARK(U41(X1, X2)) -> U41^1(mark(X1), X2) MARK(U41(X1, X2)) -> MARK(X1) MARK(U42(X)) -> ACTIVE(U42(mark(X))) MARK(U42(X)) -> U42^1(mark(X)) MARK(U42(X)) -> MARK(X) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2)) MARK(U51(X1, X2)) -> U51^1(mark(X1), X2) MARK(U51(X1, X2)) -> MARK(X1) MARK(U52(X)) -> ACTIVE(U52(mark(X))) MARK(U52(X)) -> U52^1(mark(X)) MARK(U52(X)) -> MARK(X) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2)) MARK(U61(X1, X2)) -> U61^1(mark(X1), X2) MARK(U61(X1, X2)) -> MARK(X1) MARK(U62(X)) -> ACTIVE(U62(mark(X))) MARK(U62(X)) -> U62^1(mark(X)) MARK(U62(X)) -> MARK(X) MARK(U71(X1, X2, X3)) -> ACTIVE(U71(mark(X1), X2, X3)) MARK(U71(X1, X2, X3)) -> U71^1(mark(X1), X2, X3) MARK(U71(X1, X2, X3)) -> MARK(X1) MARK(U72(X1, X2)) -> ACTIVE(U72(mark(X1), X2)) MARK(U72(X1, X2)) -> U72^1(mark(X1), X2) MARK(U72(X1, X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(s(X)) -> S(mark(X)) MARK(s(X)) -> MARK(X) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(length(X)) -> LENGTH(mark(X)) MARK(length(X)) -> MARK(X) MARK(U81(X)) -> ACTIVE(U81(mark(X))) MARK(U81(X)) -> U81^1(mark(X)) MARK(U81(X)) -> MARK(X) MARK(nil) -> ACTIVE(nil) MARK(U91(X1, X2, X3, X4)) -> ACTIVE(U91(mark(X1), X2, X3, X4)) MARK(U91(X1, X2, X3, X4)) -> U91^1(mark(X1), X2, X3, X4) MARK(U91(X1, X2, X3, X4)) -> MARK(X1) MARK(U92(X1, X2, X3, X4)) -> ACTIVE(U92(mark(X1), X2, X3, X4)) MARK(U92(X1, X2, X3, X4)) -> U92^1(mark(X1), X2, X3, X4) MARK(U92(X1, X2, X3, X4)) -> MARK(X1) MARK(U93(X1, X2, X3, X4)) -> ACTIVE(U93(mark(X1), X2, X3, X4)) MARK(U93(X1, X2, X3, X4)) -> U93^1(mark(X1), X2, X3, X4) MARK(U93(X1, X2, X3, X4)) -> MARK(X1) MARK(take(X1, X2)) -> ACTIVE(take(mark(X1), mark(X2))) MARK(take(X1, X2)) -> TAKE(mark(X1), mark(X2)) MARK(take(X1, X2)) -> MARK(X1) MARK(take(X1, X2)) -> MARK(X2) CONS(mark(X1), X2) -> CONS(X1, X2) CONS(X1, mark(X2)) -> CONS(X1, X2) CONS(active(X1), X2) -> CONS(X1, X2) CONS(X1, active(X2)) -> CONS(X1, X2) U11^1(mark(X)) -> U11^1(X) U11^1(active(X)) -> U11^1(X) U21^1(mark(X)) -> U21^1(X) U21^1(active(X)) -> U21^1(X) U31^1(mark(X)) -> U31^1(X) U31^1(active(X)) -> U31^1(X) U41^1(mark(X1), X2) -> U41^1(X1, X2) U41^1(X1, mark(X2)) -> U41^1(X1, X2) U41^1(active(X1), X2) -> U41^1(X1, X2) U41^1(X1, active(X2)) -> U41^1(X1, X2) U42^1(mark(X)) -> U42^1(X) U42^1(active(X)) -> U42^1(X) ISNATILIST(mark(X)) -> ISNATILIST(X) ISNATILIST(active(X)) -> ISNATILIST(X) U51^1(mark(X1), X2) -> U51^1(X1, X2) U51^1(X1, mark(X2)) -> U51^1(X1, X2) U51^1(active(X1), X2) -> U51^1(X1, X2) U51^1(X1, active(X2)) -> U51^1(X1, X2) U52^1(mark(X)) -> U52^1(X) U52^1(active(X)) -> U52^1(X) ISNATLIST(mark(X)) -> ISNATLIST(X) ISNATLIST(active(X)) -> ISNATLIST(X) U61^1(mark(X1), X2) -> U61^1(X1, X2) U61^1(X1, mark(X2)) -> U61^1(X1, X2) U61^1(active(X1), X2) -> U61^1(X1, X2) U61^1(X1, active(X2)) -> U61^1(X1, X2) U62^1(mark(X)) -> U62^1(X) U62^1(active(X)) -> U62^1(X) U71^1(mark(X1), X2, X3) -> U71^1(X1, X2, X3) U71^1(X1, mark(X2), X3) -> U71^1(X1, X2, X3) U71^1(X1, X2, mark(X3)) -> U71^1(X1, X2, X3) U71^1(active(X1), X2, X3) -> U71^1(X1, X2, X3) U71^1(X1, active(X2), X3) -> U71^1(X1, X2, X3) U71^1(X1, X2, active(X3)) -> U71^1(X1, X2, X3) U72^1(mark(X1), X2) -> U72^1(X1, X2) U72^1(X1, mark(X2)) -> U72^1(X1, X2) U72^1(active(X1), X2) -> U72^1(X1, X2) U72^1(X1, active(X2)) -> U72^1(X1, X2) ISNAT(mark(X)) -> ISNAT(X) ISNAT(active(X)) -> ISNAT(X) S(mark(X)) -> S(X) S(active(X)) -> S(X) LENGTH(mark(X)) -> LENGTH(X) LENGTH(active(X)) -> LENGTH(X) U81^1(mark(X)) -> U81^1(X) U81^1(active(X)) -> U81^1(X) U91^1(mark(X1), X2, X3, X4) -> U91^1(X1, X2, X3, X4) U91^1(X1, mark(X2), X3, X4) -> U91^1(X1, X2, X3, X4) U91^1(X1, X2, mark(X3), X4) -> U91^1(X1, X2, X3, X4) U91^1(X1, X2, X3, mark(X4)) -> U91^1(X1, X2, X3, X4) U91^1(active(X1), X2, X3, X4) -> U91^1(X1, X2, X3, X4) U91^1(X1, active(X2), X3, X4) -> U91^1(X1, X2, X3, X4) U91^1(X1, X2, active(X3), X4) -> U91^1(X1, X2, X3, X4) U91^1(X1, X2, X3, active(X4)) -> U91^1(X1, X2, X3, X4) U92^1(mark(X1), X2, X3, X4) -> U92^1(X1, X2, X3, X4) U92^1(X1, mark(X2), X3, X4) -> U92^1(X1, X2, X3, X4) U92^1(X1, X2, mark(X3), X4) -> U92^1(X1, X2, X3, X4) U92^1(X1, X2, X3, mark(X4)) -> U92^1(X1, X2, X3, X4) U92^1(active(X1), X2, X3, X4) -> U92^1(X1, X2, X3, X4) U92^1(X1, active(X2), X3, X4) -> U92^1(X1, X2, X3, X4) U92^1(X1, X2, active(X3), X4) -> U92^1(X1, X2, X3, X4) U92^1(X1, X2, X3, active(X4)) -> U92^1(X1, X2, X3, X4) U93^1(mark(X1), X2, X3, X4) -> U93^1(X1, X2, X3, X4) U93^1(X1, mark(X2), X3, X4) -> U93^1(X1, X2, X3, X4) U93^1(X1, X2, mark(X3), X4) -> U93^1(X1, X2, X3, X4) U93^1(X1, X2, X3, mark(X4)) -> U93^1(X1, X2, X3, X4) U93^1(active(X1), X2, X3, X4) -> U93^1(X1, X2, X3, X4) U93^1(X1, active(X2), X3, X4) -> U93^1(X1, X2, X3, X4) U93^1(X1, X2, active(X3), X4) -> U93^1(X1, X2, X3, X4) U93^1(X1, X2, X3, active(X4)) -> U93^1(X1, X2, X3, X4) TAKE(mark(X1), X2) -> TAKE(X1, X2) TAKE(X1, mark(X2)) -> TAKE(X1, X2) TAKE(active(X1), X2) -> TAKE(X1, X2) TAKE(X1, active(X2)) -> TAKE(X1, X2) The TRS R consists of the following rules: active(zeros) -> mark(cons(0, zeros)) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatIList(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, L, N)) -> mark(U72(isNat(N), L)) active(U72(tt, L)) -> mark(s(length(L))) active(U81(tt)) -> mark(nil) active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N)) active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N)) active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(isNat(0)) -> mark(tt) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2)) active(length(nil)) -> mark(0) active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N)) active(take(0, IL)) -> mark(U81(isNatIList(IL))) active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N)) mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) mark(0) -> active(0) mark(U11(X)) -> active(U11(mark(X))) mark(tt) -> active(tt) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1, X2)) -> active(U51(mark(X1), X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil) -> active(nil) mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4)) mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4)) mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4)) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) cons(mark(X1), X2) -> cons(X1, X2) cons(X1, mark(X2)) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1), X2) -> U41(X1, X2) U41(X1, mark(X2)) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1), X2) -> U51(X1, X2) U51(X1, mark(X2)) -> U51(X1, X2) U51(active(X1), X2) -> U51(X1, X2) U51(X1, active(X2)) -> U51(X1, X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4) U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4) U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4) U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4) U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4) U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4) take(mark(X1), X2) -> take(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (3) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 23 SCCs with 68 less nodes. ---------------------------------------- (4) Complex Obligation (AND) ---------------------------------------- (5) Obligation: Q DP problem: The TRS P consists of the following rules: TAKE(X1, mark(X2)) -> TAKE(X1, X2) TAKE(mark(X1), X2) -> TAKE(X1, X2) TAKE(active(X1), X2) -> TAKE(X1, X2) TAKE(X1, active(X2)) -> TAKE(X1, X2) The TRS R consists of the following rules: active(zeros) -> mark(cons(0, zeros)) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatIList(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, L, N)) -> mark(U72(isNat(N), L)) active(U72(tt, L)) -> mark(s(length(L))) active(U81(tt)) -> mark(nil) active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N)) active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N)) active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(isNat(0)) -> mark(tt) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2)) active(length(nil)) -> mark(0) active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N)) active(take(0, IL)) -> mark(U81(isNatIList(IL))) active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N)) mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) mark(0) -> active(0) mark(U11(X)) -> active(U11(mark(X))) mark(tt) -> active(tt) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1, X2)) -> active(U51(mark(X1), X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil) -> active(nil) mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4)) mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4)) mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4)) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) cons(mark(X1), X2) -> cons(X1, X2) cons(X1, mark(X2)) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1), X2) -> U41(X1, X2) U41(X1, mark(X2)) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1), X2) -> U51(X1, X2) U51(X1, mark(X2)) -> U51(X1, X2) U51(active(X1), X2) -> U51(X1, X2) U51(X1, active(X2)) -> U51(X1, X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4) U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4) U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4) U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4) U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4) U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4) take(mark(X1), X2) -> take(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (6) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (7) Obligation: Q DP problem: The TRS P consists of the following rules: TAKE(X1, mark(X2)) -> TAKE(X1, X2) TAKE(mark(X1), X2) -> TAKE(X1, X2) TAKE(active(X1), X2) -> TAKE(X1, X2) TAKE(X1, active(X2)) -> TAKE(X1, X2) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (8) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *TAKE(X1, mark(X2)) -> TAKE(X1, X2) The graph contains the following edges 1 >= 1, 2 > 2 *TAKE(mark(X1), X2) -> TAKE(X1, X2) The graph contains the following edges 1 > 1, 2 >= 2 *TAKE(active(X1), X2) -> TAKE(X1, X2) The graph contains the following edges 1 > 1, 2 >= 2 *TAKE(X1, active(X2)) -> TAKE(X1, X2) The graph contains the following edges 1 >= 1, 2 > 2 ---------------------------------------- (9) YES ---------------------------------------- (10) Obligation: Q DP problem: The TRS P consists of the following rules: U93^1(X1, mark(X2), X3, X4) -> U93^1(X1, X2, X3, X4) U93^1(mark(X1), X2, X3, X4) -> U93^1(X1, X2, X3, X4) U93^1(X1, X2, mark(X3), X4) -> U93^1(X1, X2, X3, X4) U93^1(X1, X2, X3, mark(X4)) -> U93^1(X1, X2, X3, X4) U93^1(active(X1), X2, X3, X4) -> U93^1(X1, X2, X3, X4) U93^1(X1, active(X2), X3, X4) -> U93^1(X1, X2, X3, X4) U93^1(X1, X2, active(X3), X4) -> U93^1(X1, X2, X3, X4) U93^1(X1, X2, X3, active(X4)) -> U93^1(X1, X2, X3, X4) The TRS R consists of the following rules: active(zeros) -> mark(cons(0, zeros)) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatIList(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, L, N)) -> mark(U72(isNat(N), L)) active(U72(tt, L)) -> mark(s(length(L))) active(U81(tt)) -> mark(nil) active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N)) active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N)) active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(isNat(0)) -> mark(tt) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2)) active(length(nil)) -> mark(0) active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N)) active(take(0, IL)) -> mark(U81(isNatIList(IL))) active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N)) mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) mark(0) -> active(0) mark(U11(X)) -> active(U11(mark(X))) mark(tt) -> active(tt) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1, X2)) -> active(U51(mark(X1), X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil) -> active(nil) mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4)) mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4)) mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4)) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) cons(mark(X1), X2) -> cons(X1, X2) cons(X1, mark(X2)) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1), X2) -> U41(X1, X2) U41(X1, mark(X2)) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1), X2) -> U51(X1, X2) U51(X1, mark(X2)) -> U51(X1, X2) U51(active(X1), X2) -> U51(X1, X2) U51(X1, active(X2)) -> U51(X1, X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4) U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4) U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4) U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4) U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4) U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4) take(mark(X1), X2) -> take(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (11) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (12) Obligation: Q DP problem: The TRS P consists of the following rules: U93^1(X1, mark(X2), X3, X4) -> U93^1(X1, X2, X3, X4) U93^1(mark(X1), X2, X3, X4) -> U93^1(X1, X2, X3, X4) U93^1(X1, X2, mark(X3), X4) -> U93^1(X1, X2, X3, X4) U93^1(X1, X2, X3, mark(X4)) -> U93^1(X1, X2, X3, X4) U93^1(active(X1), X2, X3, X4) -> U93^1(X1, X2, X3, X4) U93^1(X1, active(X2), X3, X4) -> U93^1(X1, X2, X3, X4) U93^1(X1, X2, active(X3), X4) -> U93^1(X1, X2, X3, X4) U93^1(X1, X2, X3, active(X4)) -> U93^1(X1, X2, X3, X4) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (13) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *U93^1(X1, mark(X2), X3, X4) -> U93^1(X1, X2, X3, X4) The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4 *U93^1(mark(X1), X2, X3, X4) -> U93^1(X1, X2, X3, X4) The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4 *U93^1(X1, X2, mark(X3), X4) -> U93^1(X1, X2, X3, X4) The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 >= 4 *U93^1(X1, X2, X3, mark(X4)) -> U93^1(X1, X2, X3, X4) The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 > 4 *U93^1(active(X1), X2, X3, X4) -> U93^1(X1, X2, X3, X4) The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4 *U93^1(X1, active(X2), X3, X4) -> U93^1(X1, X2, X3, X4) The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4 *U93^1(X1, X2, active(X3), X4) -> U93^1(X1, X2, X3, X4) The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 >= 4 *U93^1(X1, X2, X3, active(X4)) -> U93^1(X1, X2, X3, X4) The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 > 4 ---------------------------------------- (14) YES ---------------------------------------- (15) Obligation: Q DP problem: The TRS P consists of the following rules: U92^1(X1, mark(X2), X3, X4) -> U92^1(X1, X2, X3, X4) U92^1(mark(X1), X2, X3, X4) -> U92^1(X1, X2, X3, X4) U92^1(X1, X2, mark(X3), X4) -> U92^1(X1, X2, X3, X4) U92^1(X1, X2, X3, mark(X4)) -> U92^1(X1, X2, X3, X4) U92^1(active(X1), X2, X3, X4) -> U92^1(X1, X2, X3, X4) U92^1(X1, active(X2), X3, X4) -> U92^1(X1, X2, X3, X4) U92^1(X1, X2, active(X3), X4) -> U92^1(X1, X2, X3, X4) U92^1(X1, X2, X3, active(X4)) -> U92^1(X1, X2, X3, X4) The TRS R consists of the following rules: active(zeros) -> mark(cons(0, zeros)) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatIList(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, L, N)) -> mark(U72(isNat(N), L)) active(U72(tt, L)) -> mark(s(length(L))) active(U81(tt)) -> mark(nil) active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N)) active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N)) active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(isNat(0)) -> mark(tt) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2)) active(length(nil)) -> mark(0) active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N)) active(take(0, IL)) -> mark(U81(isNatIList(IL))) active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N)) mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) mark(0) -> active(0) mark(U11(X)) -> active(U11(mark(X))) mark(tt) -> active(tt) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1, X2)) -> active(U51(mark(X1), X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil) -> active(nil) mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4)) mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4)) mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4)) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) cons(mark(X1), X2) -> cons(X1, X2) cons(X1, mark(X2)) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1), X2) -> U41(X1, X2) U41(X1, mark(X2)) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1), X2) -> U51(X1, X2) U51(X1, mark(X2)) -> U51(X1, X2) U51(active(X1), X2) -> U51(X1, X2) U51(X1, active(X2)) -> U51(X1, X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4) U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4) U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4) U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4) U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4) U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4) take(mark(X1), X2) -> take(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (16) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (17) Obligation: Q DP problem: The TRS P consists of the following rules: U92^1(X1, mark(X2), X3, X4) -> U92^1(X1, X2, X3, X4) U92^1(mark(X1), X2, X3, X4) -> U92^1(X1, X2, X3, X4) U92^1(X1, X2, mark(X3), X4) -> U92^1(X1, X2, X3, X4) U92^1(X1, X2, X3, mark(X4)) -> U92^1(X1, X2, X3, X4) U92^1(active(X1), X2, X3, X4) -> U92^1(X1, X2, X3, X4) U92^1(X1, active(X2), X3, X4) -> U92^1(X1, X2, X3, X4) U92^1(X1, X2, active(X3), X4) -> U92^1(X1, X2, X3, X4) U92^1(X1, X2, X3, active(X4)) -> U92^1(X1, X2, X3, X4) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (18) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *U92^1(X1, mark(X2), X3, X4) -> U92^1(X1, X2, X3, X4) The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4 *U92^1(mark(X1), X2, X3, X4) -> U92^1(X1, X2, X3, X4) The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4 *U92^1(X1, X2, mark(X3), X4) -> U92^1(X1, X2, X3, X4) The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 >= 4 *U92^1(X1, X2, X3, mark(X4)) -> U92^1(X1, X2, X3, X4) The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 > 4 *U92^1(active(X1), X2, X3, X4) -> U92^1(X1, X2, X3, X4) The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4 *U92^1(X1, active(X2), X3, X4) -> U92^1(X1, X2, X3, X4) The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4 *U92^1(X1, X2, active(X3), X4) -> U92^1(X1, X2, X3, X4) The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 >= 4 *U92^1(X1, X2, X3, active(X4)) -> U92^1(X1, X2, X3, X4) The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 > 4 ---------------------------------------- (19) YES ---------------------------------------- (20) Obligation: Q DP problem: The TRS P consists of the following rules: U91^1(X1, mark(X2), X3, X4) -> U91^1(X1, X2, X3, X4) U91^1(mark(X1), X2, X3, X4) -> U91^1(X1, X2, X3, X4) U91^1(X1, X2, mark(X3), X4) -> U91^1(X1, X2, X3, X4) U91^1(X1, X2, X3, mark(X4)) -> U91^1(X1, X2, X3, X4) U91^1(active(X1), X2, X3, X4) -> U91^1(X1, X2, X3, X4) U91^1(X1, active(X2), X3, X4) -> U91^1(X1, X2, X3, X4) U91^1(X1, X2, active(X3), X4) -> U91^1(X1, X2, X3, X4) U91^1(X1, X2, X3, active(X4)) -> U91^1(X1, X2, X3, X4) The TRS R consists of the following rules: active(zeros) -> mark(cons(0, zeros)) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatIList(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, L, N)) -> mark(U72(isNat(N), L)) active(U72(tt, L)) -> mark(s(length(L))) active(U81(tt)) -> mark(nil) active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N)) active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N)) active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(isNat(0)) -> mark(tt) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2)) active(length(nil)) -> mark(0) active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N)) active(take(0, IL)) -> mark(U81(isNatIList(IL))) active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N)) mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) mark(0) -> active(0) mark(U11(X)) -> active(U11(mark(X))) mark(tt) -> active(tt) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1, X2)) -> active(U51(mark(X1), X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil) -> active(nil) mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4)) mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4)) mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4)) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) cons(mark(X1), X2) -> cons(X1, X2) cons(X1, mark(X2)) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1), X2) -> U41(X1, X2) U41(X1, mark(X2)) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1), X2) -> U51(X1, X2) U51(X1, mark(X2)) -> U51(X1, X2) U51(active(X1), X2) -> U51(X1, X2) U51(X1, active(X2)) -> U51(X1, X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4) U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4) U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4) U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4) U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4) U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4) take(mark(X1), X2) -> take(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (21) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (22) Obligation: Q DP problem: The TRS P consists of the following rules: U91^1(X1, mark(X2), X3, X4) -> U91^1(X1, X2, X3, X4) U91^1(mark(X1), X2, X3, X4) -> U91^1(X1, X2, X3, X4) U91^1(X1, X2, mark(X3), X4) -> U91^1(X1, X2, X3, X4) U91^1(X1, X2, X3, mark(X4)) -> U91^1(X1, X2, X3, X4) U91^1(active(X1), X2, X3, X4) -> U91^1(X1, X2, X3, X4) U91^1(X1, active(X2), X3, X4) -> U91^1(X1, X2, X3, X4) U91^1(X1, X2, active(X3), X4) -> U91^1(X1, X2, X3, X4) U91^1(X1, X2, X3, active(X4)) -> U91^1(X1, X2, X3, X4) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (23) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *U91^1(X1, mark(X2), X3, X4) -> U91^1(X1, X2, X3, X4) The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4 *U91^1(mark(X1), X2, X3, X4) -> U91^1(X1, X2, X3, X4) The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4 *U91^1(X1, X2, mark(X3), X4) -> U91^1(X1, X2, X3, X4) The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 >= 4 *U91^1(X1, X2, X3, mark(X4)) -> U91^1(X1, X2, X3, X4) The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 > 4 *U91^1(active(X1), X2, X3, X4) -> U91^1(X1, X2, X3, X4) The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4 *U91^1(X1, active(X2), X3, X4) -> U91^1(X1, X2, X3, X4) The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4 *U91^1(X1, X2, active(X3), X4) -> U91^1(X1, X2, X3, X4) The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 >= 4 *U91^1(X1, X2, X3, active(X4)) -> U91^1(X1, X2, X3, X4) The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 > 4 ---------------------------------------- (24) YES ---------------------------------------- (25) Obligation: Q DP problem: The TRS P consists of the following rules: U81^1(active(X)) -> U81^1(X) U81^1(mark(X)) -> U81^1(X) The TRS R consists of the following rules: active(zeros) -> mark(cons(0, zeros)) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatIList(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, L, N)) -> mark(U72(isNat(N), L)) active(U72(tt, L)) -> mark(s(length(L))) active(U81(tt)) -> mark(nil) active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N)) active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N)) active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(isNat(0)) -> mark(tt) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2)) active(length(nil)) -> mark(0) active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N)) active(take(0, IL)) -> mark(U81(isNatIList(IL))) active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N)) mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) mark(0) -> active(0) mark(U11(X)) -> active(U11(mark(X))) mark(tt) -> active(tt) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1, X2)) -> active(U51(mark(X1), X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil) -> active(nil) mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4)) mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4)) mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4)) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) cons(mark(X1), X2) -> cons(X1, X2) cons(X1, mark(X2)) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1), X2) -> U41(X1, X2) U41(X1, mark(X2)) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1), X2) -> U51(X1, X2) U51(X1, mark(X2)) -> U51(X1, X2) U51(active(X1), X2) -> U51(X1, X2) U51(X1, active(X2)) -> U51(X1, X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4) U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4) U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4) U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4) U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4) U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4) take(mark(X1), X2) -> take(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (26) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (27) Obligation: Q DP problem: The TRS P consists of the following rules: U81^1(active(X)) -> U81^1(X) U81^1(mark(X)) -> U81^1(X) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (28) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *U81^1(active(X)) -> U81^1(X) The graph contains the following edges 1 > 1 *U81^1(mark(X)) -> U81^1(X) The graph contains the following edges 1 > 1 ---------------------------------------- (29) YES ---------------------------------------- (30) Obligation: Q DP problem: The TRS P consists of the following rules: LENGTH(active(X)) -> LENGTH(X) LENGTH(mark(X)) -> LENGTH(X) The TRS R consists of the following rules: active(zeros) -> mark(cons(0, zeros)) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatIList(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, L, N)) -> mark(U72(isNat(N), L)) active(U72(tt, L)) -> mark(s(length(L))) active(U81(tt)) -> mark(nil) active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N)) active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N)) active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(isNat(0)) -> mark(tt) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2)) active(length(nil)) -> mark(0) active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N)) active(take(0, IL)) -> mark(U81(isNatIList(IL))) active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N)) mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) mark(0) -> active(0) mark(U11(X)) -> active(U11(mark(X))) mark(tt) -> active(tt) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1, X2)) -> active(U51(mark(X1), X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil) -> active(nil) mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4)) mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4)) mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4)) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) cons(mark(X1), X2) -> cons(X1, X2) cons(X1, mark(X2)) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1), X2) -> U41(X1, X2) U41(X1, mark(X2)) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1), X2) -> U51(X1, X2) U51(X1, mark(X2)) -> U51(X1, X2) U51(active(X1), X2) -> U51(X1, X2) U51(X1, active(X2)) -> U51(X1, X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4) U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4) U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4) U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4) U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4) U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4) take(mark(X1), X2) -> take(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (31) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (32) Obligation: Q DP problem: The TRS P consists of the following rules: LENGTH(active(X)) -> LENGTH(X) LENGTH(mark(X)) -> LENGTH(X) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (33) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *LENGTH(active(X)) -> LENGTH(X) The graph contains the following edges 1 > 1 *LENGTH(mark(X)) -> LENGTH(X) The graph contains the following edges 1 > 1 ---------------------------------------- (34) YES ---------------------------------------- (35) Obligation: Q DP problem: The TRS P consists of the following rules: S(active(X)) -> S(X) S(mark(X)) -> S(X) The TRS R consists of the following rules: active(zeros) -> mark(cons(0, zeros)) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatIList(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, L, N)) -> mark(U72(isNat(N), L)) active(U72(tt, L)) -> mark(s(length(L))) active(U81(tt)) -> mark(nil) active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N)) active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N)) active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(isNat(0)) -> mark(tt) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2)) active(length(nil)) -> mark(0) active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N)) active(take(0, IL)) -> mark(U81(isNatIList(IL))) active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N)) mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) mark(0) -> active(0) mark(U11(X)) -> active(U11(mark(X))) mark(tt) -> active(tt) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1, X2)) -> active(U51(mark(X1), X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil) -> active(nil) mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4)) mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4)) mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4)) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) cons(mark(X1), X2) -> cons(X1, X2) cons(X1, mark(X2)) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1), X2) -> U41(X1, X2) U41(X1, mark(X2)) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1), X2) -> U51(X1, X2) U51(X1, mark(X2)) -> U51(X1, X2) U51(active(X1), X2) -> U51(X1, X2) U51(X1, active(X2)) -> U51(X1, X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4) U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4) U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4) U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4) U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4) U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4) take(mark(X1), X2) -> take(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (36) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (37) Obligation: Q DP problem: The TRS P consists of the following rules: S(active(X)) -> S(X) S(mark(X)) -> S(X) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (38) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *S(active(X)) -> S(X) The graph contains the following edges 1 > 1 *S(mark(X)) -> S(X) The graph contains the following edges 1 > 1 ---------------------------------------- (39) YES ---------------------------------------- (40) Obligation: Q DP problem: The TRS P consists of the following rules: ISNAT(active(X)) -> ISNAT(X) ISNAT(mark(X)) -> ISNAT(X) The TRS R consists of the following rules: active(zeros) -> mark(cons(0, zeros)) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatIList(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, L, N)) -> mark(U72(isNat(N), L)) active(U72(tt, L)) -> mark(s(length(L))) active(U81(tt)) -> mark(nil) active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N)) active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N)) active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(isNat(0)) -> mark(tt) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2)) active(length(nil)) -> mark(0) active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N)) active(take(0, IL)) -> mark(U81(isNatIList(IL))) active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N)) mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) mark(0) -> active(0) mark(U11(X)) -> active(U11(mark(X))) mark(tt) -> active(tt) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1, X2)) -> active(U51(mark(X1), X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil) -> active(nil) mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4)) mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4)) mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4)) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) cons(mark(X1), X2) -> cons(X1, X2) cons(X1, mark(X2)) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1), X2) -> U41(X1, X2) U41(X1, mark(X2)) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1), X2) -> U51(X1, X2) U51(X1, mark(X2)) -> U51(X1, X2) U51(active(X1), X2) -> U51(X1, X2) U51(X1, active(X2)) -> U51(X1, X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4) U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4) U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4) U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4) U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4) U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4) take(mark(X1), X2) -> take(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (41) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (42) Obligation: Q DP problem: The TRS P consists of the following rules: ISNAT(active(X)) -> ISNAT(X) ISNAT(mark(X)) -> ISNAT(X) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (43) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *ISNAT(active(X)) -> ISNAT(X) The graph contains the following edges 1 > 1 *ISNAT(mark(X)) -> ISNAT(X) The graph contains the following edges 1 > 1 ---------------------------------------- (44) YES ---------------------------------------- (45) Obligation: Q DP problem: The TRS P consists of the following rules: U72^1(X1, mark(X2)) -> U72^1(X1, X2) U72^1(mark(X1), X2) -> U72^1(X1, X2) U72^1(active(X1), X2) -> U72^1(X1, X2) U72^1(X1, active(X2)) -> U72^1(X1, X2) The TRS R consists of the following rules: active(zeros) -> mark(cons(0, zeros)) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatIList(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, L, N)) -> mark(U72(isNat(N), L)) active(U72(tt, L)) -> mark(s(length(L))) active(U81(tt)) -> mark(nil) active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N)) active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N)) active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(isNat(0)) -> mark(tt) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2)) active(length(nil)) -> mark(0) active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N)) active(take(0, IL)) -> mark(U81(isNatIList(IL))) active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N)) mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) mark(0) -> active(0) mark(U11(X)) -> active(U11(mark(X))) mark(tt) -> active(tt) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1, X2)) -> active(U51(mark(X1), X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil) -> active(nil) mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4)) mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4)) mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4)) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) cons(mark(X1), X2) -> cons(X1, X2) cons(X1, mark(X2)) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1), X2) -> U41(X1, X2) U41(X1, mark(X2)) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1), X2) -> U51(X1, X2) U51(X1, mark(X2)) -> U51(X1, X2) U51(active(X1), X2) -> U51(X1, X2) U51(X1, active(X2)) -> U51(X1, X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4) U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4) U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4) U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4) U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4) U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4) take(mark(X1), X2) -> take(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (46) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (47) Obligation: Q DP problem: The TRS P consists of the following rules: U72^1(X1, mark(X2)) -> U72^1(X1, X2) U72^1(mark(X1), X2) -> U72^1(X1, X2) U72^1(active(X1), X2) -> U72^1(X1, X2) U72^1(X1, active(X2)) -> U72^1(X1, X2) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (48) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *U72^1(X1, mark(X2)) -> U72^1(X1, X2) The graph contains the following edges 1 >= 1, 2 > 2 *U72^1(mark(X1), X2) -> U72^1(X1, X2) The graph contains the following edges 1 > 1, 2 >= 2 *U72^1(active(X1), X2) -> U72^1(X1, X2) The graph contains the following edges 1 > 1, 2 >= 2 *U72^1(X1, active(X2)) -> U72^1(X1, X2) The graph contains the following edges 1 >= 1, 2 > 2 ---------------------------------------- (49) YES ---------------------------------------- (50) Obligation: Q DP problem: The TRS P consists of the following rules: U71^1(X1, mark(X2), X3) -> U71^1(X1, X2, X3) U71^1(mark(X1), X2, X3) -> U71^1(X1, X2, X3) U71^1(X1, X2, mark(X3)) -> U71^1(X1, X2, X3) U71^1(active(X1), X2, X3) -> U71^1(X1, X2, X3) U71^1(X1, active(X2), X3) -> U71^1(X1, X2, X3) U71^1(X1, X2, active(X3)) -> U71^1(X1, X2, X3) The TRS R consists of the following rules: active(zeros) -> mark(cons(0, zeros)) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatIList(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, L, N)) -> mark(U72(isNat(N), L)) active(U72(tt, L)) -> mark(s(length(L))) active(U81(tt)) -> mark(nil) active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N)) active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N)) active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(isNat(0)) -> mark(tt) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2)) active(length(nil)) -> mark(0) active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N)) active(take(0, IL)) -> mark(U81(isNatIList(IL))) active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N)) mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) mark(0) -> active(0) mark(U11(X)) -> active(U11(mark(X))) mark(tt) -> active(tt) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1, X2)) -> active(U51(mark(X1), X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil) -> active(nil) mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4)) mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4)) mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4)) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) cons(mark(X1), X2) -> cons(X1, X2) cons(X1, mark(X2)) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1), X2) -> U41(X1, X2) U41(X1, mark(X2)) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1), X2) -> U51(X1, X2) U51(X1, mark(X2)) -> U51(X1, X2) U51(active(X1), X2) -> U51(X1, X2) U51(X1, active(X2)) -> U51(X1, X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4) U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4) U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4) U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4) U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4) U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4) take(mark(X1), X2) -> take(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (51) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (52) Obligation: Q DP problem: The TRS P consists of the following rules: U71^1(X1, mark(X2), X3) -> U71^1(X1, X2, X3) U71^1(mark(X1), X2, X3) -> U71^1(X1, X2, X3) U71^1(X1, X2, mark(X3)) -> U71^1(X1, X2, X3) U71^1(active(X1), X2, X3) -> U71^1(X1, X2, X3) U71^1(X1, active(X2), X3) -> U71^1(X1, X2, X3) U71^1(X1, X2, active(X3)) -> U71^1(X1, X2, X3) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (53) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *U71^1(X1, mark(X2), X3) -> U71^1(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3 *U71^1(mark(X1), X2, X3) -> U71^1(X1, X2, X3) The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3 *U71^1(X1, X2, mark(X3)) -> U71^1(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3 *U71^1(active(X1), X2, X3) -> U71^1(X1, X2, X3) The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3 *U71^1(X1, active(X2), X3) -> U71^1(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3 *U71^1(X1, X2, active(X3)) -> U71^1(X1, X2, X3) The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3 ---------------------------------------- (54) YES ---------------------------------------- (55) Obligation: Q DP problem: The TRS P consists of the following rules: U62^1(active(X)) -> U62^1(X) U62^1(mark(X)) -> U62^1(X) The TRS R consists of the following rules: active(zeros) -> mark(cons(0, zeros)) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatIList(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, L, N)) -> mark(U72(isNat(N), L)) active(U72(tt, L)) -> mark(s(length(L))) active(U81(tt)) -> mark(nil) active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N)) active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N)) active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(isNat(0)) -> mark(tt) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2)) active(length(nil)) -> mark(0) active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N)) active(take(0, IL)) -> mark(U81(isNatIList(IL))) active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N)) mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) mark(0) -> active(0) mark(U11(X)) -> active(U11(mark(X))) mark(tt) -> active(tt) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1, X2)) -> active(U51(mark(X1), X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil) -> active(nil) mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4)) mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4)) mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4)) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) cons(mark(X1), X2) -> cons(X1, X2) cons(X1, mark(X2)) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1), X2) -> U41(X1, X2) U41(X1, mark(X2)) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1), X2) -> U51(X1, X2) U51(X1, mark(X2)) -> U51(X1, X2) U51(active(X1), X2) -> U51(X1, X2) U51(X1, active(X2)) -> U51(X1, X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4) U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4) U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4) U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4) U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4) U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4) take(mark(X1), X2) -> take(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (56) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (57) Obligation: Q DP problem: The TRS P consists of the following rules: U62^1(active(X)) -> U62^1(X) U62^1(mark(X)) -> U62^1(X) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (58) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *U62^1(active(X)) -> U62^1(X) The graph contains the following edges 1 > 1 *U62^1(mark(X)) -> U62^1(X) The graph contains the following edges 1 > 1 ---------------------------------------- (59) YES ---------------------------------------- (60) Obligation: Q DP problem: The TRS P consists of the following rules: U61^1(X1, mark(X2)) -> U61^1(X1, X2) U61^1(mark(X1), X2) -> U61^1(X1, X2) U61^1(active(X1), X2) -> U61^1(X1, X2) U61^1(X1, active(X2)) -> U61^1(X1, X2) The TRS R consists of the following rules: active(zeros) -> mark(cons(0, zeros)) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatIList(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, L, N)) -> mark(U72(isNat(N), L)) active(U72(tt, L)) -> mark(s(length(L))) active(U81(tt)) -> mark(nil) active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N)) active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N)) active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(isNat(0)) -> mark(tt) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2)) active(length(nil)) -> mark(0) active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N)) active(take(0, IL)) -> mark(U81(isNatIList(IL))) active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N)) mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) mark(0) -> active(0) mark(U11(X)) -> active(U11(mark(X))) mark(tt) -> active(tt) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1, X2)) -> active(U51(mark(X1), X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil) -> active(nil) mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4)) mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4)) mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4)) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) cons(mark(X1), X2) -> cons(X1, X2) cons(X1, mark(X2)) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1), X2) -> U41(X1, X2) U41(X1, mark(X2)) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1), X2) -> U51(X1, X2) U51(X1, mark(X2)) -> U51(X1, X2) U51(active(X1), X2) -> U51(X1, X2) U51(X1, active(X2)) -> U51(X1, X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4) U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4) U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4) U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4) U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4) U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4) take(mark(X1), X2) -> take(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (61) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (62) Obligation: Q DP problem: The TRS P consists of the following rules: U61^1(X1, mark(X2)) -> U61^1(X1, X2) U61^1(mark(X1), X2) -> U61^1(X1, X2) U61^1(active(X1), X2) -> U61^1(X1, X2) U61^1(X1, active(X2)) -> U61^1(X1, X2) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (63) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *U61^1(X1, mark(X2)) -> U61^1(X1, X2) The graph contains the following edges 1 >= 1, 2 > 2 *U61^1(mark(X1), X2) -> U61^1(X1, X2) The graph contains the following edges 1 > 1, 2 >= 2 *U61^1(active(X1), X2) -> U61^1(X1, X2) The graph contains the following edges 1 > 1, 2 >= 2 *U61^1(X1, active(X2)) -> U61^1(X1, X2) The graph contains the following edges 1 >= 1, 2 > 2 ---------------------------------------- (64) YES ---------------------------------------- (65) Obligation: Q DP problem: The TRS P consists of the following rules: ISNATLIST(active(X)) -> ISNATLIST(X) ISNATLIST(mark(X)) -> ISNATLIST(X) The TRS R consists of the following rules: active(zeros) -> mark(cons(0, zeros)) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatIList(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, L, N)) -> mark(U72(isNat(N), L)) active(U72(tt, L)) -> mark(s(length(L))) active(U81(tt)) -> mark(nil) active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N)) active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N)) active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(isNat(0)) -> mark(tt) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2)) active(length(nil)) -> mark(0) active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N)) active(take(0, IL)) -> mark(U81(isNatIList(IL))) active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N)) mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) mark(0) -> active(0) mark(U11(X)) -> active(U11(mark(X))) mark(tt) -> active(tt) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1, X2)) -> active(U51(mark(X1), X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil) -> active(nil) mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4)) mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4)) mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4)) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) cons(mark(X1), X2) -> cons(X1, X2) cons(X1, mark(X2)) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1), X2) -> U41(X1, X2) U41(X1, mark(X2)) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1), X2) -> U51(X1, X2) U51(X1, mark(X2)) -> U51(X1, X2) U51(active(X1), X2) -> U51(X1, X2) U51(X1, active(X2)) -> U51(X1, X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4) U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4) U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4) U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4) U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4) U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4) take(mark(X1), X2) -> take(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (66) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (67) Obligation: Q DP problem: The TRS P consists of the following rules: ISNATLIST(active(X)) -> ISNATLIST(X) ISNATLIST(mark(X)) -> ISNATLIST(X) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (68) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *ISNATLIST(active(X)) -> ISNATLIST(X) The graph contains the following edges 1 > 1 *ISNATLIST(mark(X)) -> ISNATLIST(X) The graph contains the following edges 1 > 1 ---------------------------------------- (69) YES ---------------------------------------- (70) Obligation: Q DP problem: The TRS P consists of the following rules: U52^1(active(X)) -> U52^1(X) U52^1(mark(X)) -> U52^1(X) The TRS R consists of the following rules: active(zeros) -> mark(cons(0, zeros)) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatIList(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, L, N)) -> mark(U72(isNat(N), L)) active(U72(tt, L)) -> mark(s(length(L))) active(U81(tt)) -> mark(nil) active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N)) active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N)) active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(isNat(0)) -> mark(tt) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2)) active(length(nil)) -> mark(0) active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N)) active(take(0, IL)) -> mark(U81(isNatIList(IL))) active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N)) mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) mark(0) -> active(0) mark(U11(X)) -> active(U11(mark(X))) mark(tt) -> active(tt) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1, X2)) -> active(U51(mark(X1), X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil) -> active(nil) mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4)) mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4)) mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4)) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) cons(mark(X1), X2) -> cons(X1, X2) cons(X1, mark(X2)) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1), X2) -> U41(X1, X2) U41(X1, mark(X2)) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1), X2) -> U51(X1, X2) U51(X1, mark(X2)) -> U51(X1, X2) U51(active(X1), X2) -> U51(X1, X2) U51(X1, active(X2)) -> U51(X1, X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4) U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4) U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4) U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4) U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4) U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4) take(mark(X1), X2) -> take(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (71) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (72) Obligation: Q DP problem: The TRS P consists of the following rules: U52^1(active(X)) -> U52^1(X) U52^1(mark(X)) -> U52^1(X) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (73) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *U52^1(active(X)) -> U52^1(X) The graph contains the following edges 1 > 1 *U52^1(mark(X)) -> U52^1(X) The graph contains the following edges 1 > 1 ---------------------------------------- (74) YES ---------------------------------------- (75) Obligation: Q DP problem: The TRS P consists of the following rules: U51^1(X1, mark(X2)) -> U51^1(X1, X2) U51^1(mark(X1), X2) -> U51^1(X1, X2) U51^1(active(X1), X2) -> U51^1(X1, X2) U51^1(X1, active(X2)) -> U51^1(X1, X2) The TRS R consists of the following rules: active(zeros) -> mark(cons(0, zeros)) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatIList(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, L, N)) -> mark(U72(isNat(N), L)) active(U72(tt, L)) -> mark(s(length(L))) active(U81(tt)) -> mark(nil) active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N)) active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N)) active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(isNat(0)) -> mark(tt) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2)) active(length(nil)) -> mark(0) active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N)) active(take(0, IL)) -> mark(U81(isNatIList(IL))) active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N)) mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) mark(0) -> active(0) mark(U11(X)) -> active(U11(mark(X))) mark(tt) -> active(tt) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1, X2)) -> active(U51(mark(X1), X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil) -> active(nil) mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4)) mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4)) mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4)) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) cons(mark(X1), X2) -> cons(X1, X2) cons(X1, mark(X2)) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1), X2) -> U41(X1, X2) U41(X1, mark(X2)) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1), X2) -> U51(X1, X2) U51(X1, mark(X2)) -> U51(X1, X2) U51(active(X1), X2) -> U51(X1, X2) U51(X1, active(X2)) -> U51(X1, X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4) U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4) U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4) U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4) U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4) U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4) take(mark(X1), X2) -> take(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (76) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (77) Obligation: Q DP problem: The TRS P consists of the following rules: U51^1(X1, mark(X2)) -> U51^1(X1, X2) U51^1(mark(X1), X2) -> U51^1(X1, X2) U51^1(active(X1), X2) -> U51^1(X1, X2) U51^1(X1, active(X2)) -> U51^1(X1, X2) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (78) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *U51^1(X1, mark(X2)) -> U51^1(X1, X2) The graph contains the following edges 1 >= 1, 2 > 2 *U51^1(mark(X1), X2) -> U51^1(X1, X2) The graph contains the following edges 1 > 1, 2 >= 2 *U51^1(active(X1), X2) -> U51^1(X1, X2) The graph contains the following edges 1 > 1, 2 >= 2 *U51^1(X1, active(X2)) -> U51^1(X1, X2) The graph contains the following edges 1 >= 1, 2 > 2 ---------------------------------------- (79) YES ---------------------------------------- (80) Obligation: Q DP problem: The TRS P consists of the following rules: ISNATILIST(active(X)) -> ISNATILIST(X) ISNATILIST(mark(X)) -> ISNATILIST(X) The TRS R consists of the following rules: active(zeros) -> mark(cons(0, zeros)) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatIList(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, L, N)) -> mark(U72(isNat(N), L)) active(U72(tt, L)) -> mark(s(length(L))) active(U81(tt)) -> mark(nil) active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N)) active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N)) active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(isNat(0)) -> mark(tt) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2)) active(length(nil)) -> mark(0) active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N)) active(take(0, IL)) -> mark(U81(isNatIList(IL))) active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N)) mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) mark(0) -> active(0) mark(U11(X)) -> active(U11(mark(X))) mark(tt) -> active(tt) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1, X2)) -> active(U51(mark(X1), X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil) -> active(nil) mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4)) mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4)) mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4)) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) cons(mark(X1), X2) -> cons(X1, X2) cons(X1, mark(X2)) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1), X2) -> U41(X1, X2) U41(X1, mark(X2)) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1), X2) -> U51(X1, X2) U51(X1, mark(X2)) -> U51(X1, X2) U51(active(X1), X2) -> U51(X1, X2) U51(X1, active(X2)) -> U51(X1, X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4) U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4) U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4) U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4) U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4) U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4) take(mark(X1), X2) -> take(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (81) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (82) Obligation: Q DP problem: The TRS P consists of the following rules: ISNATILIST(active(X)) -> ISNATILIST(X) ISNATILIST(mark(X)) -> ISNATILIST(X) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (83) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *ISNATILIST(active(X)) -> ISNATILIST(X) The graph contains the following edges 1 > 1 *ISNATILIST(mark(X)) -> ISNATILIST(X) The graph contains the following edges 1 > 1 ---------------------------------------- (84) YES ---------------------------------------- (85) Obligation: Q DP problem: The TRS P consists of the following rules: U42^1(active(X)) -> U42^1(X) U42^1(mark(X)) -> U42^1(X) The TRS R consists of the following rules: active(zeros) -> mark(cons(0, zeros)) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatIList(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, L, N)) -> mark(U72(isNat(N), L)) active(U72(tt, L)) -> mark(s(length(L))) active(U81(tt)) -> mark(nil) active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N)) active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N)) active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(isNat(0)) -> mark(tt) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2)) active(length(nil)) -> mark(0) active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N)) active(take(0, IL)) -> mark(U81(isNatIList(IL))) active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N)) mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) mark(0) -> active(0) mark(U11(X)) -> active(U11(mark(X))) mark(tt) -> active(tt) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1, X2)) -> active(U51(mark(X1), X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil) -> active(nil) mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4)) mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4)) mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4)) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) cons(mark(X1), X2) -> cons(X1, X2) cons(X1, mark(X2)) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1), X2) -> U41(X1, X2) U41(X1, mark(X2)) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1), X2) -> U51(X1, X2) U51(X1, mark(X2)) -> U51(X1, X2) U51(active(X1), X2) -> U51(X1, X2) U51(X1, active(X2)) -> U51(X1, X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4) U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4) U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4) U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4) U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4) U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4) take(mark(X1), X2) -> take(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (86) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (87) Obligation: Q DP problem: The TRS P consists of the following rules: U42^1(active(X)) -> U42^1(X) U42^1(mark(X)) -> U42^1(X) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (88) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *U42^1(active(X)) -> U42^1(X) The graph contains the following edges 1 > 1 *U42^1(mark(X)) -> U42^1(X) The graph contains the following edges 1 > 1 ---------------------------------------- (89) YES ---------------------------------------- (90) Obligation: Q DP problem: The TRS P consists of the following rules: U41^1(X1, mark(X2)) -> U41^1(X1, X2) U41^1(mark(X1), X2) -> U41^1(X1, X2) U41^1(active(X1), X2) -> U41^1(X1, X2) U41^1(X1, active(X2)) -> U41^1(X1, X2) The TRS R consists of the following rules: active(zeros) -> mark(cons(0, zeros)) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatIList(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, L, N)) -> mark(U72(isNat(N), L)) active(U72(tt, L)) -> mark(s(length(L))) active(U81(tt)) -> mark(nil) active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N)) active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N)) active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(isNat(0)) -> mark(tt) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2)) active(length(nil)) -> mark(0) active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N)) active(take(0, IL)) -> mark(U81(isNatIList(IL))) active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N)) mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) mark(0) -> active(0) mark(U11(X)) -> active(U11(mark(X))) mark(tt) -> active(tt) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1, X2)) -> active(U51(mark(X1), X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil) -> active(nil) mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4)) mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4)) mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4)) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) cons(mark(X1), X2) -> cons(X1, X2) cons(X1, mark(X2)) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1), X2) -> U41(X1, X2) U41(X1, mark(X2)) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1), X2) -> U51(X1, X2) U51(X1, mark(X2)) -> U51(X1, X2) U51(active(X1), X2) -> U51(X1, X2) U51(X1, active(X2)) -> U51(X1, X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4) U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4) U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4) U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4) U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4) U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4) take(mark(X1), X2) -> take(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (91) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (92) Obligation: Q DP problem: The TRS P consists of the following rules: U41^1(X1, mark(X2)) -> U41^1(X1, X2) U41^1(mark(X1), X2) -> U41^1(X1, X2) U41^1(active(X1), X2) -> U41^1(X1, X2) U41^1(X1, active(X2)) -> U41^1(X1, X2) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (93) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *U41^1(X1, mark(X2)) -> U41^1(X1, X2) The graph contains the following edges 1 >= 1, 2 > 2 *U41^1(mark(X1), X2) -> U41^1(X1, X2) The graph contains the following edges 1 > 1, 2 >= 2 *U41^1(active(X1), X2) -> U41^1(X1, X2) The graph contains the following edges 1 > 1, 2 >= 2 *U41^1(X1, active(X2)) -> U41^1(X1, X2) The graph contains the following edges 1 >= 1, 2 > 2 ---------------------------------------- (94) YES ---------------------------------------- (95) Obligation: Q DP problem: The TRS P consists of the following rules: U31^1(active(X)) -> U31^1(X) U31^1(mark(X)) -> U31^1(X) The TRS R consists of the following rules: active(zeros) -> mark(cons(0, zeros)) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatIList(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, L, N)) -> mark(U72(isNat(N), L)) active(U72(tt, L)) -> mark(s(length(L))) active(U81(tt)) -> mark(nil) active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N)) active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N)) active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(isNat(0)) -> mark(tt) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2)) active(length(nil)) -> mark(0) active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N)) active(take(0, IL)) -> mark(U81(isNatIList(IL))) active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N)) mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) mark(0) -> active(0) mark(U11(X)) -> active(U11(mark(X))) mark(tt) -> active(tt) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1, X2)) -> active(U51(mark(X1), X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil) -> active(nil) mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4)) mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4)) mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4)) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) cons(mark(X1), X2) -> cons(X1, X2) cons(X1, mark(X2)) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1), X2) -> U41(X1, X2) U41(X1, mark(X2)) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1), X2) -> U51(X1, X2) U51(X1, mark(X2)) -> U51(X1, X2) U51(active(X1), X2) -> U51(X1, X2) U51(X1, active(X2)) -> U51(X1, X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4) U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4) U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4) U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4) U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4) U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4) take(mark(X1), X2) -> take(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (96) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (97) Obligation: Q DP problem: The TRS P consists of the following rules: U31^1(active(X)) -> U31^1(X) U31^1(mark(X)) -> U31^1(X) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (98) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *U31^1(active(X)) -> U31^1(X) The graph contains the following edges 1 > 1 *U31^1(mark(X)) -> U31^1(X) The graph contains the following edges 1 > 1 ---------------------------------------- (99) YES ---------------------------------------- (100) Obligation: Q DP problem: The TRS P consists of the following rules: U21^1(active(X)) -> U21^1(X) U21^1(mark(X)) -> U21^1(X) The TRS R consists of the following rules: active(zeros) -> mark(cons(0, zeros)) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatIList(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, L, N)) -> mark(U72(isNat(N), L)) active(U72(tt, L)) -> mark(s(length(L))) active(U81(tt)) -> mark(nil) active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N)) active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N)) active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(isNat(0)) -> mark(tt) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2)) active(length(nil)) -> mark(0) active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N)) active(take(0, IL)) -> mark(U81(isNatIList(IL))) active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N)) mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) mark(0) -> active(0) mark(U11(X)) -> active(U11(mark(X))) mark(tt) -> active(tt) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1, X2)) -> active(U51(mark(X1), X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil) -> active(nil) mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4)) mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4)) mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4)) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) cons(mark(X1), X2) -> cons(X1, X2) cons(X1, mark(X2)) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1), X2) -> U41(X1, X2) U41(X1, mark(X2)) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1), X2) -> U51(X1, X2) U51(X1, mark(X2)) -> U51(X1, X2) U51(active(X1), X2) -> U51(X1, X2) U51(X1, active(X2)) -> U51(X1, X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4) U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4) U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4) U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4) U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4) U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4) take(mark(X1), X2) -> take(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (101) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (102) Obligation: Q DP problem: The TRS P consists of the following rules: U21^1(active(X)) -> U21^1(X) U21^1(mark(X)) -> U21^1(X) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (103) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *U21^1(active(X)) -> U21^1(X) The graph contains the following edges 1 > 1 *U21^1(mark(X)) -> U21^1(X) The graph contains the following edges 1 > 1 ---------------------------------------- (104) YES ---------------------------------------- (105) Obligation: Q DP problem: The TRS P consists of the following rules: U11^1(active(X)) -> U11^1(X) U11^1(mark(X)) -> U11^1(X) The TRS R consists of the following rules: active(zeros) -> mark(cons(0, zeros)) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatIList(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, L, N)) -> mark(U72(isNat(N), L)) active(U72(tt, L)) -> mark(s(length(L))) active(U81(tt)) -> mark(nil) active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N)) active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N)) active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(isNat(0)) -> mark(tt) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2)) active(length(nil)) -> mark(0) active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N)) active(take(0, IL)) -> mark(U81(isNatIList(IL))) active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N)) mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) mark(0) -> active(0) mark(U11(X)) -> active(U11(mark(X))) mark(tt) -> active(tt) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1, X2)) -> active(U51(mark(X1), X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil) -> active(nil) mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4)) mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4)) mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4)) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) cons(mark(X1), X2) -> cons(X1, X2) cons(X1, mark(X2)) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1), X2) -> U41(X1, X2) U41(X1, mark(X2)) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1), X2) -> U51(X1, X2) U51(X1, mark(X2)) -> U51(X1, X2) U51(active(X1), X2) -> U51(X1, X2) U51(X1, active(X2)) -> U51(X1, X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4) U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4) U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4) U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4) U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4) U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4) take(mark(X1), X2) -> take(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (106) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (107) Obligation: Q DP problem: The TRS P consists of the following rules: U11^1(active(X)) -> U11^1(X) U11^1(mark(X)) -> U11^1(X) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (108) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *U11^1(active(X)) -> U11^1(X) The graph contains the following edges 1 > 1 *U11^1(mark(X)) -> U11^1(X) The graph contains the following edges 1 > 1 ---------------------------------------- (109) YES ---------------------------------------- (110) Obligation: Q DP problem: The TRS P consists of the following rules: CONS(X1, mark(X2)) -> CONS(X1, X2) CONS(mark(X1), X2) -> CONS(X1, X2) CONS(active(X1), X2) -> CONS(X1, X2) CONS(X1, active(X2)) -> CONS(X1, X2) The TRS R consists of the following rules: active(zeros) -> mark(cons(0, zeros)) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatIList(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, L, N)) -> mark(U72(isNat(N), L)) active(U72(tt, L)) -> mark(s(length(L))) active(U81(tt)) -> mark(nil) active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N)) active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N)) active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(isNat(0)) -> mark(tt) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2)) active(length(nil)) -> mark(0) active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N)) active(take(0, IL)) -> mark(U81(isNatIList(IL))) active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N)) mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) mark(0) -> active(0) mark(U11(X)) -> active(U11(mark(X))) mark(tt) -> active(tt) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1, X2)) -> active(U51(mark(X1), X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil) -> active(nil) mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4)) mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4)) mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4)) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) cons(mark(X1), X2) -> cons(X1, X2) cons(X1, mark(X2)) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1), X2) -> U41(X1, X2) U41(X1, mark(X2)) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1), X2) -> U51(X1, X2) U51(X1, mark(X2)) -> U51(X1, X2) U51(active(X1), X2) -> U51(X1, X2) U51(X1, active(X2)) -> U51(X1, X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4) U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4) U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4) U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4) U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4) U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4) take(mark(X1), X2) -> take(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (111) UsableRulesProof (EQUIVALENT) We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (112) Obligation: Q DP problem: The TRS P consists of the following rules: CONS(X1, mark(X2)) -> CONS(X1, X2) CONS(mark(X1), X2) -> CONS(X1, X2) CONS(active(X1), X2) -> CONS(X1, X2) CONS(X1, active(X2)) -> CONS(X1, X2) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (113) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *CONS(X1, mark(X2)) -> CONS(X1, X2) The graph contains the following edges 1 >= 1, 2 > 2 *CONS(mark(X1), X2) -> CONS(X1, X2) The graph contains the following edges 1 > 1, 2 >= 2 *CONS(active(X1), X2) -> CONS(X1, X2) The graph contains the following edges 1 > 1, 2 >= 2 *CONS(X1, active(X2)) -> CONS(X1, X2) The graph contains the following edges 1 >= 1, 2 > 2 ---------------------------------------- (114) YES ---------------------------------------- (115) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(cons(X1, X2)) -> ACTIVE(cons(mark(X1), X2)) ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2))) MARK(cons(X1, X2)) -> MARK(X1) MARK(zeros) -> ACTIVE(zeros) ACTIVE(zeros) -> MARK(cons(0, zeros)) MARK(U11(X)) -> ACTIVE(U11(mark(X))) ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2))) MARK(U11(X)) -> MARK(X) MARK(U21(X)) -> ACTIVE(U21(mark(X))) ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIList(V2))) MARK(U21(X)) -> MARK(X) MARK(U31(X)) -> ACTIVE(U31(mark(X))) ACTIVE(U71(tt, L, N)) -> MARK(U72(isNat(N), L)) MARK(U31(X)) -> MARK(X) MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) ACTIVE(U72(tt, L)) -> MARK(s(length(L))) MARK(U41(X1, X2)) -> MARK(X1) MARK(U42(X)) -> ACTIVE(U42(mark(X))) ACTIVE(U91(tt, IL, M, N)) -> MARK(U92(isNat(M), IL, M, N)) MARK(U42(X)) -> MARK(X) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) ACTIVE(U92(tt, IL, M, N)) -> MARK(U93(isNat(N), IL, M, N)) MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2)) ACTIVE(U93(tt, IL, M, N)) -> MARK(cons(N, take(M, IL))) MARK(U51(X1, X2)) -> MARK(X1) MARK(U52(X)) -> ACTIVE(U52(mark(X))) ACTIVE(isNat(length(V1))) -> MARK(U11(isNatList(V1))) MARK(U52(X)) -> MARK(X) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2)) ACTIVE(isNatIList(V)) -> MARK(U31(isNatList(V))) MARK(U61(X1, X2)) -> MARK(X1) MARK(U62(X)) -> ACTIVE(U62(mark(X))) ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNat(V1), V2)) MARK(U62(X)) -> MARK(X) MARK(U71(X1, X2, X3)) -> ACTIVE(U71(mark(X1), X2, X3)) ACTIVE(isNatList(cons(V1, V2))) -> MARK(U51(isNat(V1), V2)) MARK(U71(X1, X2, X3)) -> MARK(X1) MARK(U72(X1, X2)) -> ACTIVE(U72(mark(X1), X2)) ACTIVE(isNatList(take(V1, V2))) -> MARK(U61(isNat(V1), V2)) MARK(U72(X1, X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(length(cons(N, L))) -> MARK(U71(isNatList(L), L, N)) MARK(s(X)) -> ACTIVE(s(mark(X))) ACTIVE(take(0, IL)) -> MARK(U81(isNatIList(IL))) MARK(s(X)) -> MARK(X) MARK(length(X)) -> ACTIVE(length(mark(X))) ACTIVE(take(s(M), cons(N, IL))) -> MARK(U91(isNatIList(IL), IL, M, N)) MARK(length(X)) -> MARK(X) MARK(U81(X)) -> ACTIVE(U81(mark(X))) MARK(U81(X)) -> MARK(X) MARK(U91(X1, X2, X3, X4)) -> ACTIVE(U91(mark(X1), X2, X3, X4)) MARK(U91(X1, X2, X3, X4)) -> MARK(X1) MARK(U92(X1, X2, X3, X4)) -> ACTIVE(U92(mark(X1), X2, X3, X4)) MARK(U92(X1, X2, X3, X4)) -> MARK(X1) MARK(U93(X1, X2, X3, X4)) -> ACTIVE(U93(mark(X1), X2, X3, X4)) MARK(U93(X1, X2, X3, X4)) -> MARK(X1) MARK(take(X1, X2)) -> ACTIVE(take(mark(X1), mark(X2))) MARK(take(X1, X2)) -> MARK(X1) MARK(take(X1, X2)) -> MARK(X2) The TRS R consists of the following rules: active(zeros) -> mark(cons(0, zeros)) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatIList(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, L, N)) -> mark(U72(isNat(N), L)) active(U72(tt, L)) -> mark(s(length(L))) active(U81(tt)) -> mark(nil) active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N)) active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N)) active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(isNat(0)) -> mark(tt) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2)) active(length(nil)) -> mark(0) active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N)) active(take(0, IL)) -> mark(U81(isNatIList(IL))) active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N)) mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) mark(0) -> active(0) mark(U11(X)) -> active(U11(mark(X))) mark(tt) -> active(tt) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1, X2)) -> active(U51(mark(X1), X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil) -> active(nil) mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4)) mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4)) mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4)) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) cons(mark(X1), X2) -> cons(X1, X2) cons(X1, mark(X2)) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1), X2) -> U41(X1, X2) U41(X1, mark(X2)) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1), X2) -> U51(X1, X2) U51(X1, mark(X2)) -> U51(X1, X2) U51(active(X1), X2) -> U51(X1, X2) U51(X1, active(X2)) -> U51(X1, X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4) U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4) U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4) U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4) U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4) U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4) take(mark(X1), X2) -> take(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (116) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MARK(cons(X1, X2)) -> ACTIVE(cons(mark(X1), X2)) MARK(U11(X)) -> ACTIVE(U11(mark(X))) MARK(U21(X)) -> ACTIVE(U21(mark(X))) MARK(U31(X)) -> ACTIVE(U31(mark(X))) MARK(U42(X)) -> ACTIVE(U42(mark(X))) MARK(U52(X)) -> ACTIVE(U52(mark(X))) MARK(U62(X)) -> ACTIVE(U62(mark(X))) MARK(s(X)) -> ACTIVE(s(mark(X))) MARK(U81(X)) -> ACTIVE(U81(mark(X))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( ACTIVE_1(x_1) ) = max{0, x_1 - 1} POL( U11_1(x_1) ) = 0 POL( U21_1(x_1) ) = max{0, -2} POL( U31_1(x_1) ) = max{0, -2} POL( U41_2(x_1, x_2) ) = 2 POL( U42_1(x_1) ) = max{0, -2} POL( U51_2(x_1, x_2) ) = 2 POL( U52_1(x_1) ) = 0 POL( U61_2(x_1, x_2) ) = 2 POL( U62_1(x_1) ) = max{0, -2} POL( U71_3(x_1, ..., x_3) ) = 2 POL( U72_2(x_1, x_2) ) = 2 POL( U81_1(x_1) ) = max{0, -2} POL( U91_4(x_1, ..., x_4) ) = 2 POL( U92_4(x_1, ..., x_4) ) = 2 POL( U93_4(x_1, ..., x_4) ) = 2 POL( cons_2(x_1, x_2) ) = max{0, -2} POL( length_1(x_1) ) = 2 POL( s_1(x_1) ) = 0 POL( take_2(x_1, x_2) ) = 2 POL( mark_1(x_1) ) = max{0, -2} POL( active_1(x_1) ) = 2x_1 POL( zeros ) = 2 POL( 0 ) = 1 POL( tt ) = 1 POL( isNatIList_1(x_1) ) = 2 POL( isNatList_1(x_1) ) = 2 POL( isNat_1(x_1) ) = 2 POL( nil ) = 1 POL( MARK_1(x_1) ) = 1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: cons(X1, mark(X2)) -> cons(X1, X2) cons(mark(X1), X2) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U62(active(X)) -> U62(X) U62(mark(X)) -> U62(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) U72(X1, mark(X2)) -> U72(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U41(X1, mark(X2)) -> U41(X1, X2) U41(mark(X1), X2) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) length(active(X)) -> length(X) length(mark(X)) -> length(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4) U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4) U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4) U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4) U51(X1, mark(X2)) -> U51(X1, X2) U51(mark(X1), X2) -> U51(X1, X2) U51(active(X1), X2) -> U51(X1, X2) U51(X1, active(X2)) -> U51(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(mark(X1), X2) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(mark(X1), X2) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U81(active(X)) -> U81(X) U81(mark(X)) -> U81(X) U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4) U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4) ---------------------------------------- (117) Obligation: Q DP problem: The TRS P consists of the following rules: ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2))) MARK(cons(X1, X2)) -> MARK(X1) MARK(zeros) -> ACTIVE(zeros) ACTIVE(zeros) -> MARK(cons(0, zeros)) ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2))) MARK(U11(X)) -> MARK(X) ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIList(V2))) MARK(U21(X)) -> MARK(X) ACTIVE(U71(tt, L, N)) -> MARK(U72(isNat(N), L)) MARK(U31(X)) -> MARK(X) MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) ACTIVE(U72(tt, L)) -> MARK(s(length(L))) MARK(U41(X1, X2)) -> MARK(X1) ACTIVE(U91(tt, IL, M, N)) -> MARK(U92(isNat(M), IL, M, N)) MARK(U42(X)) -> MARK(X) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) ACTIVE(U92(tt, IL, M, N)) -> MARK(U93(isNat(N), IL, M, N)) MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2)) ACTIVE(U93(tt, IL, M, N)) -> MARK(cons(N, take(M, IL))) MARK(U51(X1, X2)) -> MARK(X1) ACTIVE(isNat(length(V1))) -> MARK(U11(isNatList(V1))) MARK(U52(X)) -> MARK(X) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2)) ACTIVE(isNatIList(V)) -> MARK(U31(isNatList(V))) MARK(U61(X1, X2)) -> MARK(X1) ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNat(V1), V2)) MARK(U62(X)) -> MARK(X) MARK(U71(X1, X2, X3)) -> ACTIVE(U71(mark(X1), X2, X3)) ACTIVE(isNatList(cons(V1, V2))) -> MARK(U51(isNat(V1), V2)) MARK(U71(X1, X2, X3)) -> MARK(X1) MARK(U72(X1, X2)) -> ACTIVE(U72(mark(X1), X2)) ACTIVE(isNatList(take(V1, V2))) -> MARK(U61(isNat(V1), V2)) MARK(U72(X1, X2)) -> MARK(X1) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(length(cons(N, L))) -> MARK(U71(isNatList(L), L, N)) ACTIVE(take(0, IL)) -> MARK(U81(isNatIList(IL))) MARK(s(X)) -> MARK(X) MARK(length(X)) -> ACTIVE(length(mark(X))) ACTIVE(take(s(M), cons(N, IL))) -> MARK(U91(isNatIList(IL), IL, M, N)) MARK(length(X)) -> MARK(X) MARK(U81(X)) -> MARK(X) MARK(U91(X1, X2, X3, X4)) -> ACTIVE(U91(mark(X1), X2, X3, X4)) MARK(U91(X1, X2, X3, X4)) -> MARK(X1) MARK(U92(X1, X2, X3, X4)) -> ACTIVE(U92(mark(X1), X2, X3, X4)) MARK(U92(X1, X2, X3, X4)) -> MARK(X1) MARK(U93(X1, X2, X3, X4)) -> ACTIVE(U93(mark(X1), X2, X3, X4)) MARK(U93(X1, X2, X3, X4)) -> MARK(X1) MARK(take(X1, X2)) -> ACTIVE(take(mark(X1), mark(X2))) MARK(take(X1, X2)) -> MARK(X1) MARK(take(X1, X2)) -> MARK(X2) The TRS R consists of the following rules: active(zeros) -> mark(cons(0, zeros)) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatIList(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, L, N)) -> mark(U72(isNat(N), L)) active(U72(tt, L)) -> mark(s(length(L))) active(U81(tt)) -> mark(nil) active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N)) active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N)) active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(isNat(0)) -> mark(tt) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2)) active(length(nil)) -> mark(0) active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N)) active(take(0, IL)) -> mark(U81(isNatIList(IL))) active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N)) mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) mark(0) -> active(0) mark(U11(X)) -> active(U11(mark(X))) mark(tt) -> active(tt) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1, X2)) -> active(U51(mark(X1), X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil) -> active(nil) mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4)) mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4)) mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4)) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) cons(mark(X1), X2) -> cons(X1, X2) cons(X1, mark(X2)) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1), X2) -> U41(X1, X2) U41(X1, mark(X2)) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1), X2) -> U51(X1, X2) U51(X1, mark(X2)) -> U51(X1, X2) U51(active(X1), X2) -> U51(X1, X2) U51(X1, active(X2)) -> U51(X1, X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4) U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4) U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4) U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4) U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4) U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4) take(mark(X1), X2) -> take(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (118) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MARK(U71(X1, X2, X3)) -> MARK(X1) MARK(U72(X1, X2)) -> MARK(X1) MARK(length(X)) -> MARK(X) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(0) = 0 POL(ACTIVE(x_1)) = x_1 POL(MARK(x_1)) = x_1 POL(U11(x_1)) = x_1 POL(U21(x_1)) = x_1 POL(U31(x_1)) = x_1 POL(U41(x_1, x_2)) = x_1 POL(U42(x_1)) = x_1 POL(U51(x_1, x_2)) = x_1 POL(U52(x_1)) = x_1 POL(U61(x_1, x_2)) = x_1 POL(U62(x_1)) = x_1 POL(U71(x_1, x_2, x_3)) = 1 + x_1 + x_2 POL(U72(x_1, x_2)) = 1 + x_1 + x_2 POL(U81(x_1)) = x_1 POL(U91(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4 POL(U92(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4 POL(U93(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4 POL(active(x_1)) = x_1 POL(cons(x_1, x_2)) = x_1 + x_2 POL(isNat(x_1)) = 0 POL(isNatIList(x_1)) = 0 POL(isNatList(x_1)) = 0 POL(length(x_1)) = 1 + x_1 POL(mark(x_1)) = x_1 POL(nil) = 0 POL(s(x_1)) = x_1 POL(take(x_1, x_2)) = x_1 + x_2 POL(tt) = 0 POL(zeros) = 0 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U62(active(X)) -> U62(X) U62(mark(X)) -> U62(X) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) U72(X1, mark(X2)) -> U72(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) active(zeros) -> mark(cons(0, zeros)) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) mark(zeros) -> active(zeros) mark(U11(X)) -> active(U11(mark(X))) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) mark(U21(X)) -> active(U21(mark(X))) active(U61(tt, V2)) -> mark(U62(isNatIList(V2))) mark(U31(X)) -> active(U31(mark(X))) active(U71(tt, L, N)) -> mark(U72(isNat(N), L)) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) active(U72(tt, L)) -> mark(s(length(L))) mark(U42(X)) -> active(U42(mark(X))) active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N)) mark(isNatIList(X)) -> active(isNatIList(X)) active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N)) mark(U51(X1, X2)) -> active(U51(mark(X1), X2)) active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) mark(U52(X)) -> active(U52(mark(X))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) mark(isNatList(X)) -> active(isNatList(X)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) active(isNatIList(V)) -> mark(U31(isNatList(V))) mark(U62(X)) -> active(U62(mark(X))) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3)) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2)) mark(isNat(X)) -> active(isNat(X)) active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N)) mark(s(X)) -> active(s(mark(X))) active(take(0, IL)) -> mark(U81(isNatIList(IL))) mark(length(X)) -> active(length(mark(X))) active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N)) mark(U81(X)) -> active(U81(mark(X))) mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4)) mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4)) mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4)) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) mark(0) -> active(0) mark(tt) -> active(tt) mark(nil) -> active(nil) U41(X1, mark(X2)) -> U41(X1, X2) U41(mark(X1), X2) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) length(active(X)) -> length(X) length(mark(X)) -> length(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4) U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4) U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4) U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4) U51(X1, mark(X2)) -> U51(X1, X2) U51(mark(X1), X2) -> U51(X1, X2) U51(active(X1), X2) -> U51(X1, X2) U51(X1, active(X2)) -> U51(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(mark(X1), X2) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) cons(X1, mark(X2)) -> cons(X1, X2) cons(mark(X1), X2) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U61(X1, mark(X2)) -> U61(X1, X2) U61(mark(X1), X2) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U81(active(X)) -> U81(X) U81(mark(X)) -> U81(X) U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4) U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U42(tt)) -> mark(tt) active(U52(tt)) -> mark(tt) active(U62(tt)) -> mark(tt) active(U81(tt)) -> mark(nil) active(isNat(0)) -> mark(tt) active(isNatIList(zeros)) -> mark(tt) active(isNatList(nil)) -> mark(tt) active(length(nil)) -> mark(0) ---------------------------------------- (119) Obligation: Q DP problem: The TRS P consists of the following rules: ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2))) MARK(cons(X1, X2)) -> MARK(X1) MARK(zeros) -> ACTIVE(zeros) ACTIVE(zeros) -> MARK(cons(0, zeros)) ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2))) MARK(U11(X)) -> MARK(X) ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIList(V2))) MARK(U21(X)) -> MARK(X) ACTIVE(U71(tt, L, N)) -> MARK(U72(isNat(N), L)) MARK(U31(X)) -> MARK(X) MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) ACTIVE(U72(tt, L)) -> MARK(s(length(L))) MARK(U41(X1, X2)) -> MARK(X1) ACTIVE(U91(tt, IL, M, N)) -> MARK(U92(isNat(M), IL, M, N)) MARK(U42(X)) -> MARK(X) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) ACTIVE(U92(tt, IL, M, N)) -> MARK(U93(isNat(N), IL, M, N)) MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2)) ACTIVE(U93(tt, IL, M, N)) -> MARK(cons(N, take(M, IL))) MARK(U51(X1, X2)) -> MARK(X1) ACTIVE(isNat(length(V1))) -> MARK(U11(isNatList(V1))) MARK(U52(X)) -> MARK(X) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2)) ACTIVE(isNatIList(V)) -> MARK(U31(isNatList(V))) MARK(U61(X1, X2)) -> MARK(X1) ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNat(V1), V2)) MARK(U62(X)) -> MARK(X) MARK(U71(X1, X2, X3)) -> ACTIVE(U71(mark(X1), X2, X3)) ACTIVE(isNatList(cons(V1, V2))) -> MARK(U51(isNat(V1), V2)) MARK(U72(X1, X2)) -> ACTIVE(U72(mark(X1), X2)) ACTIVE(isNatList(take(V1, V2))) -> MARK(U61(isNat(V1), V2)) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(length(cons(N, L))) -> MARK(U71(isNatList(L), L, N)) ACTIVE(take(0, IL)) -> MARK(U81(isNatIList(IL))) MARK(s(X)) -> MARK(X) MARK(length(X)) -> ACTIVE(length(mark(X))) ACTIVE(take(s(M), cons(N, IL))) -> MARK(U91(isNatIList(IL), IL, M, N)) MARK(U81(X)) -> MARK(X) MARK(U91(X1, X2, X3, X4)) -> ACTIVE(U91(mark(X1), X2, X3, X4)) MARK(U91(X1, X2, X3, X4)) -> MARK(X1) MARK(U92(X1, X2, X3, X4)) -> ACTIVE(U92(mark(X1), X2, X3, X4)) MARK(U92(X1, X2, X3, X4)) -> MARK(X1) MARK(U93(X1, X2, X3, X4)) -> ACTIVE(U93(mark(X1), X2, X3, X4)) MARK(U93(X1, X2, X3, X4)) -> MARK(X1) MARK(take(X1, X2)) -> ACTIVE(take(mark(X1), mark(X2))) MARK(take(X1, X2)) -> MARK(X1) MARK(take(X1, X2)) -> MARK(X2) The TRS R consists of the following rules: active(zeros) -> mark(cons(0, zeros)) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatIList(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, L, N)) -> mark(U72(isNat(N), L)) active(U72(tt, L)) -> mark(s(length(L))) active(U81(tt)) -> mark(nil) active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N)) active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N)) active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(isNat(0)) -> mark(tt) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2)) active(length(nil)) -> mark(0) active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N)) active(take(0, IL)) -> mark(U81(isNatIList(IL))) active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N)) mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) mark(0) -> active(0) mark(U11(X)) -> active(U11(mark(X))) mark(tt) -> active(tt) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1, X2)) -> active(U51(mark(X1), X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil) -> active(nil) mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4)) mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4)) mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4)) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) cons(mark(X1), X2) -> cons(X1, X2) cons(X1, mark(X2)) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1), X2) -> U41(X1, X2) U41(X1, mark(X2)) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1), X2) -> U51(X1, X2) U51(X1, mark(X2)) -> U51(X1, X2) U51(active(X1), X2) -> U51(X1, X2) U51(X1, active(X2)) -> U51(X1, X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4) U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4) U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4) U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4) U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4) U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4) take(mark(X1), X2) -> take(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (120) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MARK(cons(X1, X2)) -> MARK(X1) ACTIVE(take(0, IL)) -> MARK(U81(isNatIList(IL))) ACTIVE(take(s(M), cons(N, IL))) -> MARK(U91(isNatIList(IL), IL, M, N)) MARK(U91(X1, X2, X3, X4)) -> MARK(X1) MARK(U92(X1, X2, X3, X4)) -> MARK(X1) MARK(U93(X1, X2, X3, X4)) -> MARK(X1) MARK(take(X1, X2)) -> MARK(X1) MARK(take(X1, X2)) -> MARK(X2) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( ACTIVE_1(x_1) ) = 2x_1 POL( MARK_1(x_1) ) = 2x_1 POL( U42_1(x_1) ) = 2x_1 POL( U62_1(x_1) ) = 2x_1 POL( U81_1(x_1) ) = x_1 POL( U91_4(x_1, ..., x_4) ) = 2x_1 + x_3 + 2x_4 + 2 POL( isNatIList_1(x_1) ) = 0 POL( active_1(x_1) ) = x_1 POL( mark_1(x_1) ) = x_1 POL( U41_2(x_1, x_2) ) = x_1 POL( U51_2(x_1, x_2) ) = 2x_1 POL( U61_2(x_1, x_2) ) = 2x_1 POL( U71_3(x_1, ..., x_3) ) = 2 POL( U72_2(x_1, x_2) ) = 2 POL( U92_4(x_1, ..., x_4) ) = 2x_1 + 2x_4 + 2 POL( U93_4(x_1, ..., x_4) ) = x_1 + 2x_4 + 2 POL( length_1(x_1) ) = 2 POL( take_2(x_1, x_2) ) = 2x_1 + x_2 + 1 POL( U11_1(x_1) ) = x_1 POL( U31_1(x_1) ) = 2x_1 POL( U52_1(x_1) ) = x_1 POL( isNatList_1(x_1) ) = 0 POL( U21_1(x_1) ) = 2x_1 POL( isNat_1(x_1) ) = 0 POL( cons_2(x_1, x_2) ) = 2x_1 + 2 POL( zeros ) = 2 POL( 0 ) = 0 POL( tt ) = 0 POL( s_1(x_1) ) = x_1 POL( nil ) = 0 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U62(active(X)) -> U62(X) U62(mark(X)) -> U62(X) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) U72(X1, mark(X2)) -> U72(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) active(zeros) -> mark(cons(0, zeros)) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) mark(zeros) -> active(zeros) mark(U11(X)) -> active(U11(mark(X))) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) mark(U21(X)) -> active(U21(mark(X))) active(U61(tt, V2)) -> mark(U62(isNatIList(V2))) mark(U31(X)) -> active(U31(mark(X))) active(U71(tt, L, N)) -> mark(U72(isNat(N), L)) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) active(U72(tt, L)) -> mark(s(length(L))) mark(U42(X)) -> active(U42(mark(X))) active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N)) mark(isNatIList(X)) -> active(isNatIList(X)) active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N)) mark(U51(X1, X2)) -> active(U51(mark(X1), X2)) active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) mark(U52(X)) -> active(U52(mark(X))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) mark(isNatList(X)) -> active(isNatList(X)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) active(isNatIList(V)) -> mark(U31(isNatList(V))) mark(U62(X)) -> active(U62(mark(X))) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3)) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2)) mark(isNat(X)) -> active(isNat(X)) active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N)) mark(s(X)) -> active(s(mark(X))) active(take(0, IL)) -> mark(U81(isNatIList(IL))) mark(length(X)) -> active(length(mark(X))) active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N)) mark(U81(X)) -> active(U81(mark(X))) mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4)) mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4)) mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4)) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) mark(0) -> active(0) mark(tt) -> active(tt) mark(nil) -> active(nil) U41(X1, mark(X2)) -> U41(X1, X2) U41(mark(X1), X2) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) length(active(X)) -> length(X) length(mark(X)) -> length(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4) U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4) U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4) U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4) U51(X1, mark(X2)) -> U51(X1, X2) U51(mark(X1), X2) -> U51(X1, X2) U51(active(X1), X2) -> U51(X1, X2) U51(X1, active(X2)) -> U51(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(mark(X1), X2) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) cons(X1, mark(X2)) -> cons(X1, X2) cons(mark(X1), X2) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U61(X1, mark(X2)) -> U61(X1, X2) U61(mark(X1), X2) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U81(active(X)) -> U81(X) U81(mark(X)) -> U81(X) U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4) U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U42(tt)) -> mark(tt) active(U52(tt)) -> mark(tt) active(U62(tt)) -> mark(tt) active(U81(tt)) -> mark(nil) active(isNat(0)) -> mark(tt) active(isNatIList(zeros)) -> mark(tt) active(isNatList(nil)) -> mark(tt) active(length(nil)) -> mark(0) ---------------------------------------- (121) Obligation: Q DP problem: The TRS P consists of the following rules: ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2))) MARK(zeros) -> ACTIVE(zeros) ACTIVE(zeros) -> MARK(cons(0, zeros)) ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2))) MARK(U11(X)) -> MARK(X) ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIList(V2))) MARK(U21(X)) -> MARK(X) ACTIVE(U71(tt, L, N)) -> MARK(U72(isNat(N), L)) MARK(U31(X)) -> MARK(X) MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) ACTIVE(U72(tt, L)) -> MARK(s(length(L))) MARK(U41(X1, X2)) -> MARK(X1) ACTIVE(U91(tt, IL, M, N)) -> MARK(U92(isNat(M), IL, M, N)) MARK(U42(X)) -> MARK(X) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) ACTIVE(U92(tt, IL, M, N)) -> MARK(U93(isNat(N), IL, M, N)) MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2)) ACTIVE(U93(tt, IL, M, N)) -> MARK(cons(N, take(M, IL))) MARK(U51(X1, X2)) -> MARK(X1) ACTIVE(isNat(length(V1))) -> MARK(U11(isNatList(V1))) MARK(U52(X)) -> MARK(X) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2)) ACTIVE(isNatIList(V)) -> MARK(U31(isNatList(V))) MARK(U61(X1, X2)) -> MARK(X1) ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNat(V1), V2)) MARK(U62(X)) -> MARK(X) MARK(U71(X1, X2, X3)) -> ACTIVE(U71(mark(X1), X2, X3)) ACTIVE(isNatList(cons(V1, V2))) -> MARK(U51(isNat(V1), V2)) MARK(U72(X1, X2)) -> ACTIVE(U72(mark(X1), X2)) ACTIVE(isNatList(take(V1, V2))) -> MARK(U61(isNat(V1), V2)) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(length(cons(N, L))) -> MARK(U71(isNatList(L), L, N)) MARK(s(X)) -> MARK(X) MARK(length(X)) -> ACTIVE(length(mark(X))) MARK(U81(X)) -> MARK(X) MARK(U91(X1, X2, X3, X4)) -> ACTIVE(U91(mark(X1), X2, X3, X4)) MARK(U92(X1, X2, X3, X4)) -> ACTIVE(U92(mark(X1), X2, X3, X4)) MARK(U93(X1, X2, X3, X4)) -> ACTIVE(U93(mark(X1), X2, X3, X4)) MARK(take(X1, X2)) -> ACTIVE(take(mark(X1), mark(X2))) The TRS R consists of the following rules: active(zeros) -> mark(cons(0, zeros)) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatIList(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, L, N)) -> mark(U72(isNat(N), L)) active(U72(tt, L)) -> mark(s(length(L))) active(U81(tt)) -> mark(nil) active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N)) active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N)) active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(isNat(0)) -> mark(tt) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2)) active(length(nil)) -> mark(0) active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N)) active(take(0, IL)) -> mark(U81(isNatIList(IL))) active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N)) mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) mark(0) -> active(0) mark(U11(X)) -> active(U11(mark(X))) mark(tt) -> active(tt) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1, X2)) -> active(U51(mark(X1), X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil) -> active(nil) mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4)) mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4)) mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4)) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) cons(mark(X1), X2) -> cons(X1, X2) cons(X1, mark(X2)) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1), X2) -> U41(X1, X2) U41(X1, mark(X2)) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1), X2) -> U51(X1, X2) U51(X1, mark(X2)) -> U51(X1, X2) U51(active(X1), X2) -> U51(X1, X2) U51(X1, active(X2)) -> U51(X1, X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4) U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4) U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4) U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4) U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4) U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4) take(mark(X1), X2) -> take(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (122) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes. ---------------------------------------- (123) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(U11(X)) -> MARK(X) MARK(U21(X)) -> MARK(X) MARK(U31(X)) -> MARK(X) MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2))) MARK(U41(X1, X2)) -> MARK(X1) MARK(U42(X)) -> MARK(X) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2))) MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2)) ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIList(V2))) MARK(U51(X1, X2)) -> MARK(X1) MARK(U52(X)) -> MARK(X) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) ACTIVE(U71(tt, L, N)) -> MARK(U72(isNat(N), L)) MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2)) ACTIVE(U72(tt, L)) -> MARK(s(length(L))) MARK(U61(X1, X2)) -> MARK(X1) MARK(U62(X)) -> MARK(X) MARK(U71(X1, X2, X3)) -> ACTIVE(U71(mark(X1), X2, X3)) ACTIVE(U91(tt, IL, M, N)) -> MARK(U92(isNat(M), IL, M, N)) MARK(U72(X1, X2)) -> ACTIVE(U72(mark(X1), X2)) ACTIVE(U92(tt, IL, M, N)) -> MARK(U93(isNat(N), IL, M, N)) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(U93(tt, IL, M, N)) -> MARK(cons(N, take(M, IL))) MARK(s(X)) -> MARK(X) MARK(length(X)) -> ACTIVE(length(mark(X))) ACTIVE(isNat(length(V1))) -> MARK(U11(isNatList(V1))) MARK(U81(X)) -> MARK(X) MARK(U91(X1, X2, X3, X4)) -> ACTIVE(U91(mark(X1), X2, X3, X4)) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) MARK(U92(X1, X2, X3, X4)) -> ACTIVE(U92(mark(X1), X2, X3, X4)) ACTIVE(isNatIList(V)) -> MARK(U31(isNatList(V))) MARK(U93(X1, X2, X3, X4)) -> ACTIVE(U93(mark(X1), X2, X3, X4)) ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNat(V1), V2)) MARK(take(X1, X2)) -> ACTIVE(take(mark(X1), mark(X2))) ACTIVE(isNatList(cons(V1, V2))) -> MARK(U51(isNat(V1), V2)) ACTIVE(isNatList(take(V1, V2))) -> MARK(U61(isNat(V1), V2)) ACTIVE(length(cons(N, L))) -> MARK(U71(isNatList(L), L, N)) The TRS R consists of the following rules: active(zeros) -> mark(cons(0, zeros)) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatIList(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, L, N)) -> mark(U72(isNat(N), L)) active(U72(tt, L)) -> mark(s(length(L))) active(U81(tt)) -> mark(nil) active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N)) active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N)) active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(isNat(0)) -> mark(tt) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2)) active(length(nil)) -> mark(0) active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N)) active(take(0, IL)) -> mark(U81(isNatIList(IL))) active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N)) mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) mark(0) -> active(0) mark(U11(X)) -> active(U11(mark(X))) mark(tt) -> active(tt) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1, X2)) -> active(U51(mark(X1), X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil) -> active(nil) mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4)) mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4)) mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4)) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) cons(mark(X1), X2) -> cons(X1, X2) cons(X1, mark(X2)) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1), X2) -> U41(X1, X2) U41(X1, mark(X2)) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1), X2) -> U51(X1, X2) U51(X1, mark(X2)) -> U51(X1, X2) U51(active(X1), X2) -> U51(X1, X2) U51(X1, active(X2)) -> U51(X1, X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4) U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4) U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4) U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4) U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4) U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4) take(mark(X1), X2) -> take(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (124) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MARK(U81(X)) -> MARK(X) MARK(U91(X1, X2, X3, X4)) -> ACTIVE(U91(mark(X1), X2, X3, X4)) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( ACTIVE_1(x_1) ) = max{0, -2} POL( U41_2(x_1, x_2) ) = 2x_1 + 1 POL( U51_2(x_1, x_2) ) = x_1 + 1 POL( U61_2(x_1, x_2) ) = 2x_1 + 1 POL( U71_3(x_1, ..., x_3) ) = max{0, -2} POL( U72_2(x_1, x_2) ) = max{0, -2} POL( U91_4(x_1, ..., x_4) ) = x_1 + x_2 + 2x_3 + 2x_4 + 2 POL( U92_4(x_1, ..., x_4) ) = max{0, -2} POL( U93_4(x_1, ..., x_4) ) = 0 POL( length_1(x_1) ) = 0 POL( take_2(x_1, x_2) ) = 0 POL( mark_1(x_1) ) = x_1 + 1 POL( cons_2(x_1, x_2) ) = 0 POL( active_1(x_1) ) = 2x_1 + 1 POL( zeros ) = 1 POL( 0 ) = 0 POL( tt ) = 0 POL( U42_1(x_1) ) = 2x_1 + 1 POL( isNatIList_1(x_1) ) = 0 POL( U11_1(x_1) ) = x_1 + 1 POL( U52_1(x_1) ) = 2x_1 + 1 POL( isNatList_1(x_1) ) = 0 POL( U21_1(x_1) ) = x_1 + 1 POL( U62_1(x_1) ) = 2x_1 + 1 POL( U31_1(x_1) ) = x_1 + 1 POL( isNat_1(x_1) ) = 0 POL( s_1(x_1) ) = 2x_1 + 1 POL( U81_1(x_1) ) = x_1 + 2 POL( nil ) = 1 POL( MARK_1(x_1) ) = max{0, x_1 - 1} The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: U41(X1, mark(X2)) -> U41(X1, X2) U41(mark(X1), X2) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U51(X1, mark(X2)) -> U51(X1, X2) U51(mark(X1), X2) -> U51(X1, X2) U51(active(X1), X2) -> U51(X1, X2) U51(X1, active(X2)) -> U51(X1, X2) U62(active(X)) -> U62(X) U62(mark(X)) -> U62(X) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) U72(X1, mark(X2)) -> U72(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(mark(X1), X2) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) length(active(X)) -> length(X) length(mark(X)) -> length(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4) U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4) U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4) U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4) cons(X1, mark(X2)) -> cons(X1, X2) cons(mark(X1), X2) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) ---------------------------------------- (125) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(U11(X)) -> MARK(X) MARK(U21(X)) -> MARK(X) MARK(U31(X)) -> MARK(X) MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2))) MARK(U41(X1, X2)) -> MARK(X1) MARK(U42(X)) -> MARK(X) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2))) MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2)) ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIList(V2))) MARK(U51(X1, X2)) -> MARK(X1) MARK(U52(X)) -> MARK(X) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) ACTIVE(U71(tt, L, N)) -> MARK(U72(isNat(N), L)) MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2)) ACTIVE(U72(tt, L)) -> MARK(s(length(L))) MARK(U61(X1, X2)) -> MARK(X1) MARK(U62(X)) -> MARK(X) MARK(U71(X1, X2, X3)) -> ACTIVE(U71(mark(X1), X2, X3)) ACTIVE(U91(tt, IL, M, N)) -> MARK(U92(isNat(M), IL, M, N)) MARK(U72(X1, X2)) -> ACTIVE(U72(mark(X1), X2)) ACTIVE(U92(tt, IL, M, N)) -> MARK(U93(isNat(N), IL, M, N)) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(U93(tt, IL, M, N)) -> MARK(cons(N, take(M, IL))) MARK(s(X)) -> MARK(X) MARK(length(X)) -> ACTIVE(length(mark(X))) ACTIVE(isNat(length(V1))) -> MARK(U11(isNatList(V1))) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) MARK(U92(X1, X2, X3, X4)) -> ACTIVE(U92(mark(X1), X2, X3, X4)) ACTIVE(isNatIList(V)) -> MARK(U31(isNatList(V))) MARK(U93(X1, X2, X3, X4)) -> ACTIVE(U93(mark(X1), X2, X3, X4)) ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNat(V1), V2)) MARK(take(X1, X2)) -> ACTIVE(take(mark(X1), mark(X2))) ACTIVE(isNatList(cons(V1, V2))) -> MARK(U51(isNat(V1), V2)) ACTIVE(isNatList(take(V1, V2))) -> MARK(U61(isNat(V1), V2)) ACTIVE(length(cons(N, L))) -> MARK(U71(isNatList(L), L, N)) The TRS R consists of the following rules: active(zeros) -> mark(cons(0, zeros)) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatIList(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, L, N)) -> mark(U72(isNat(N), L)) active(U72(tt, L)) -> mark(s(length(L))) active(U81(tt)) -> mark(nil) active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N)) active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N)) active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(isNat(0)) -> mark(tt) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2)) active(length(nil)) -> mark(0) active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N)) active(take(0, IL)) -> mark(U81(isNatIList(IL))) active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N)) mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) mark(0) -> active(0) mark(U11(X)) -> active(U11(mark(X))) mark(tt) -> active(tt) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1, X2)) -> active(U51(mark(X1), X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil) -> active(nil) mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4)) mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4)) mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4)) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) cons(mark(X1), X2) -> cons(X1, X2) cons(X1, mark(X2)) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1), X2) -> U41(X1, X2) U41(X1, mark(X2)) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1), X2) -> U51(X1, X2) U51(X1, mark(X2)) -> U51(X1, X2) U51(active(X1), X2) -> U51(X1, X2) U51(X1, active(X2)) -> U51(X1, X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4) U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4) U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4) U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4) U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4) U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4) take(mark(X1), X2) -> take(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (126) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. ACTIVE(U91(tt, IL, M, N)) -> MARK(U92(isNat(M), IL, M, N)) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( ACTIVE_1(x_1) ) = max{0, x_1 - 2} POL( U41_2(x_1, x_2) ) = max{0, -2} POL( U51_2(x_1, x_2) ) = 2 POL( U61_2(x_1, x_2) ) = 2 POL( U71_3(x_1, ..., x_3) ) = max{0, -2} POL( U72_2(x_1, x_2) ) = 2 POL( U92_4(x_1, ..., x_4) ) = 2 POL( U93_4(x_1, ..., x_4) ) = 2 POL( length_1(x_1) ) = 2 POL( take_2(x_1, x_2) ) = max{0, -2} POL( mark_1(x_1) ) = max{0, x_1 - 2} POL( cons_2(x_1, x_2) ) = max{0, 2x_2 - 2} POL( active_1(x_1) ) = 2x_1 + 1 POL( zeros ) = 0 POL( 0 ) = 0 POL( tt ) = 2 POL( U42_1(x_1) ) = max{0, x_1 - 2} POL( isNatIList_1(x_1) ) = 0 POL( U11_1(x_1) ) = max{0, x_1 - 1} POL( U52_1(x_1) ) = max{0, -2} POL( isNatList_1(x_1) ) = 0 POL( U21_1(x_1) ) = max{0, x_1 - 1} POL( U62_1(x_1) ) = max{0, x_1 - 2} POL( U31_1(x_1) ) = max{0, x_1 - 2} POL( isNat_1(x_1) ) = 2 POL( s_1(x_1) ) = max{0, x_1 - 2} POL( U91_4(x_1, ..., x_4) ) = 2x_1 + 2x_2 + 2x_3 + x_4 POL( U81_1(x_1) ) = max{0, x_1 - 2} POL( nil ) = 0 POL( MARK_1(x_1) ) = max{0, -2} The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: U41(X1, mark(X2)) -> U41(X1, X2) U41(mark(X1), X2) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) U51(X1, mark(X2)) -> U51(X1, X2) U51(mark(X1), X2) -> U51(X1, X2) U51(active(X1), X2) -> U51(X1, X2) U51(X1, active(X2)) -> U51(X1, X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) U72(X1, mark(X2)) -> U72(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(mark(X1), X2) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) length(active(X)) -> length(X) length(mark(X)) -> length(X) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4) U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4) U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4) U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4) take(X1, mark(X2)) -> take(X1, X2) take(mark(X1), X2) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) ---------------------------------------- (127) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(U11(X)) -> MARK(X) MARK(U21(X)) -> MARK(X) MARK(U31(X)) -> MARK(X) MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2))) MARK(U41(X1, X2)) -> MARK(X1) MARK(U42(X)) -> MARK(X) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2))) MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2)) ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIList(V2))) MARK(U51(X1, X2)) -> MARK(X1) MARK(U52(X)) -> MARK(X) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) ACTIVE(U71(tt, L, N)) -> MARK(U72(isNat(N), L)) MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2)) ACTIVE(U72(tt, L)) -> MARK(s(length(L))) MARK(U61(X1, X2)) -> MARK(X1) MARK(U62(X)) -> MARK(X) MARK(U71(X1, X2, X3)) -> ACTIVE(U71(mark(X1), X2, X3)) MARK(U72(X1, X2)) -> ACTIVE(U72(mark(X1), X2)) ACTIVE(U92(tt, IL, M, N)) -> MARK(U93(isNat(N), IL, M, N)) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(U93(tt, IL, M, N)) -> MARK(cons(N, take(M, IL))) MARK(s(X)) -> MARK(X) MARK(length(X)) -> ACTIVE(length(mark(X))) ACTIVE(isNat(length(V1))) -> MARK(U11(isNatList(V1))) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) MARK(U92(X1, X2, X3, X4)) -> ACTIVE(U92(mark(X1), X2, X3, X4)) ACTIVE(isNatIList(V)) -> MARK(U31(isNatList(V))) MARK(U93(X1, X2, X3, X4)) -> ACTIVE(U93(mark(X1), X2, X3, X4)) ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNat(V1), V2)) MARK(take(X1, X2)) -> ACTIVE(take(mark(X1), mark(X2))) ACTIVE(isNatList(cons(V1, V2))) -> MARK(U51(isNat(V1), V2)) ACTIVE(isNatList(take(V1, V2))) -> MARK(U61(isNat(V1), V2)) ACTIVE(length(cons(N, L))) -> MARK(U71(isNatList(L), L, N)) The TRS R consists of the following rules: active(zeros) -> mark(cons(0, zeros)) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatIList(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, L, N)) -> mark(U72(isNat(N), L)) active(U72(tt, L)) -> mark(s(length(L))) active(U81(tt)) -> mark(nil) active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N)) active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N)) active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(isNat(0)) -> mark(tt) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2)) active(length(nil)) -> mark(0) active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N)) active(take(0, IL)) -> mark(U81(isNatIList(IL))) active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N)) mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) mark(0) -> active(0) mark(U11(X)) -> active(U11(mark(X))) mark(tt) -> active(tt) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1, X2)) -> active(U51(mark(X1), X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil) -> active(nil) mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4)) mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4)) mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4)) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) cons(mark(X1), X2) -> cons(X1, X2) cons(X1, mark(X2)) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1), X2) -> U41(X1, X2) U41(X1, mark(X2)) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1), X2) -> U51(X1, X2) U51(X1, mark(X2)) -> U51(X1, X2) U51(active(X1), X2) -> U51(X1, X2) U51(X1, active(X2)) -> U51(X1, X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4) U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4) U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4) U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4) U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4) U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4) take(mark(X1), X2) -> take(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (128) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MARK(take(X1, X2)) -> ACTIVE(take(mark(X1), mark(X2))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( ACTIVE_1(x_1) ) = max{0, 2x_1 - 2} POL( U41_2(x_1, x_2) ) = 2 POL( U51_2(x_1, x_2) ) = 2 POL( U61_2(x_1, x_2) ) = 2 POL( U71_3(x_1, ..., x_3) ) = 2 POL( U72_2(x_1, x_2) ) = 2 POL( U92_4(x_1, ..., x_4) ) = 2 POL( U93_4(x_1, ..., x_4) ) = 2 POL( length_1(x_1) ) = 2 POL( take_2(x_1, x_2) ) = 0 POL( mark_1(x_1) ) = max{0, -2} POL( cons_2(x_1, x_2) ) = 2 POL( active_1(x_1) ) = max{0, -2} POL( zeros ) = 0 POL( 0 ) = 0 POL( tt ) = 0 POL( U42_1(x_1) ) = max{0, -2} POL( isNatIList_1(x_1) ) = 2 POL( U11_1(x_1) ) = max{0, 2x_1 - 2} POL( U52_1(x_1) ) = max{0, x_1 - 2} POL( isNatList_1(x_1) ) = 2 POL( U21_1(x_1) ) = max{0, x_1 - 2} POL( U62_1(x_1) ) = max{0, -2} POL( U31_1(x_1) ) = max{0, x_1 - 2} POL( isNat_1(x_1) ) = 2 POL( s_1(x_1) ) = 0 POL( U91_4(x_1, ..., x_4) ) = max{0, x_1 + x_3 - 2} POL( U81_1(x_1) ) = max{0, -2} POL( nil ) = 0 POL( MARK_1(x_1) ) = 2 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: U41(X1, mark(X2)) -> U41(X1, X2) U41(mark(X1), X2) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) U51(X1, mark(X2)) -> U51(X1, X2) U51(mark(X1), X2) -> U51(X1, X2) U51(active(X1), X2) -> U51(X1, X2) U51(X1, active(X2)) -> U51(X1, X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) U72(X1, mark(X2)) -> U72(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(mark(X1), X2) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) length(active(X)) -> length(X) length(mark(X)) -> length(X) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4) U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4) take(X1, mark(X2)) -> take(X1, X2) take(mark(X1), X2) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4) U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4) ---------------------------------------- (129) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(U11(X)) -> MARK(X) MARK(U21(X)) -> MARK(X) MARK(U31(X)) -> MARK(X) MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2))) MARK(U41(X1, X2)) -> MARK(X1) MARK(U42(X)) -> MARK(X) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2))) MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2)) ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIList(V2))) MARK(U51(X1, X2)) -> MARK(X1) MARK(U52(X)) -> MARK(X) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) ACTIVE(U71(tt, L, N)) -> MARK(U72(isNat(N), L)) MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2)) ACTIVE(U72(tt, L)) -> MARK(s(length(L))) MARK(U61(X1, X2)) -> MARK(X1) MARK(U62(X)) -> MARK(X) MARK(U71(X1, X2, X3)) -> ACTIVE(U71(mark(X1), X2, X3)) MARK(U72(X1, X2)) -> ACTIVE(U72(mark(X1), X2)) ACTIVE(U92(tt, IL, M, N)) -> MARK(U93(isNat(N), IL, M, N)) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(U93(tt, IL, M, N)) -> MARK(cons(N, take(M, IL))) MARK(s(X)) -> MARK(X) MARK(length(X)) -> ACTIVE(length(mark(X))) ACTIVE(isNat(length(V1))) -> MARK(U11(isNatList(V1))) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) MARK(U92(X1, X2, X3, X4)) -> ACTIVE(U92(mark(X1), X2, X3, X4)) ACTIVE(isNatIList(V)) -> MARK(U31(isNatList(V))) MARK(U93(X1, X2, X3, X4)) -> ACTIVE(U93(mark(X1), X2, X3, X4)) ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNat(V1), V2)) ACTIVE(isNatList(cons(V1, V2))) -> MARK(U51(isNat(V1), V2)) ACTIVE(isNatList(take(V1, V2))) -> MARK(U61(isNat(V1), V2)) ACTIVE(length(cons(N, L))) -> MARK(U71(isNatList(L), L, N)) The TRS R consists of the following rules: active(zeros) -> mark(cons(0, zeros)) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatIList(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, L, N)) -> mark(U72(isNat(N), L)) active(U72(tt, L)) -> mark(s(length(L))) active(U81(tt)) -> mark(nil) active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N)) active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N)) active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(isNat(0)) -> mark(tt) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2)) active(length(nil)) -> mark(0) active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N)) active(take(0, IL)) -> mark(U81(isNatIList(IL))) active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N)) mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) mark(0) -> active(0) mark(U11(X)) -> active(U11(mark(X))) mark(tt) -> active(tt) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1, X2)) -> active(U51(mark(X1), X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil) -> active(nil) mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4)) mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4)) mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4)) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) cons(mark(X1), X2) -> cons(X1, X2) cons(X1, mark(X2)) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1), X2) -> U41(X1, X2) U41(X1, mark(X2)) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1), X2) -> U51(X1, X2) U51(X1, mark(X2)) -> U51(X1, X2) U51(active(X1), X2) -> U51(X1, X2) U51(X1, active(X2)) -> U51(X1, X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4) U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4) U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4) U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4) U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4) U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4) take(mark(X1), X2) -> take(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (130) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MARK(U92(X1, X2, X3, X4)) -> ACTIVE(U92(mark(X1), X2, X3, X4)) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( ACTIVE_1(x_1) ) = 2 POL( U41_2(x_1, x_2) ) = 2x_1 POL( U51_2(x_1, x_2) ) = 2x_1 POL( U61_2(x_1, x_2) ) = x_1 POL( U71_3(x_1, ..., x_3) ) = max{0, -2} POL( U72_2(x_1, x_2) ) = max{0, -2} POL( U92_4(x_1, ..., x_4) ) = 1 POL( U93_4(x_1, ..., x_4) ) = max{0, -2} POL( length_1(x_1) ) = max{0, -2} POL( mark_1(x_1) ) = 2x_1 POL( cons_2(x_1, x_2) ) = x_2 POL( active_1(x_1) ) = x_1 + 2 POL( zeros ) = 0 POL( 0 ) = 0 POL( tt ) = 1 POL( U42_1(x_1) ) = 2x_1 POL( isNatIList_1(x_1) ) = 0 POL( U11_1(x_1) ) = x_1 POL( U52_1(x_1) ) = x_1 POL( isNatList_1(x_1) ) = 0 POL( U21_1(x_1) ) = x_1 POL( U62_1(x_1) ) = 2x_1 POL( U31_1(x_1) ) = x_1 POL( isNat_1(x_1) ) = 0 POL( s_1(x_1) ) = 2x_1 POL( U91_4(x_1, ..., x_4) ) = x_3 + x_4 POL( take_2(x_1, x_2) ) = max{0, -2} POL( U81_1(x_1) ) = 1 POL( nil ) = 0 POL( MARK_1(x_1) ) = x_1 + 2 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: U41(X1, mark(X2)) -> U41(X1, X2) U41(mark(X1), X2) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U51(X1, mark(X2)) -> U51(X1, X2) U51(mark(X1), X2) -> U51(X1, X2) U51(active(X1), X2) -> U51(X1, X2) U51(X1, active(X2)) -> U51(X1, X2) U62(active(X)) -> U62(X) U62(mark(X)) -> U62(X) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) U72(X1, mark(X2)) -> U72(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(mark(X1), X2) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) length(active(X)) -> length(X) length(mark(X)) -> length(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4) U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4) take(X1, mark(X2)) -> take(X1, X2) take(mark(X1), X2) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) cons(X1, mark(X2)) -> cons(X1, X2) cons(mark(X1), X2) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) ---------------------------------------- (131) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(U11(X)) -> MARK(X) MARK(U21(X)) -> MARK(X) MARK(U31(X)) -> MARK(X) MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2))) MARK(U41(X1, X2)) -> MARK(X1) MARK(U42(X)) -> MARK(X) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2))) MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2)) ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIList(V2))) MARK(U51(X1, X2)) -> MARK(X1) MARK(U52(X)) -> MARK(X) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) ACTIVE(U71(tt, L, N)) -> MARK(U72(isNat(N), L)) MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2)) ACTIVE(U72(tt, L)) -> MARK(s(length(L))) MARK(U61(X1, X2)) -> MARK(X1) MARK(U62(X)) -> MARK(X) MARK(U71(X1, X2, X3)) -> ACTIVE(U71(mark(X1), X2, X3)) MARK(U72(X1, X2)) -> ACTIVE(U72(mark(X1), X2)) ACTIVE(U92(tt, IL, M, N)) -> MARK(U93(isNat(N), IL, M, N)) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(U93(tt, IL, M, N)) -> MARK(cons(N, take(M, IL))) MARK(s(X)) -> MARK(X) MARK(length(X)) -> ACTIVE(length(mark(X))) ACTIVE(isNat(length(V1))) -> MARK(U11(isNatList(V1))) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) ACTIVE(isNatIList(V)) -> MARK(U31(isNatList(V))) MARK(U93(X1, X2, X3, X4)) -> ACTIVE(U93(mark(X1), X2, X3, X4)) ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNat(V1), V2)) ACTIVE(isNatList(cons(V1, V2))) -> MARK(U51(isNat(V1), V2)) ACTIVE(isNatList(take(V1, V2))) -> MARK(U61(isNat(V1), V2)) ACTIVE(length(cons(N, L))) -> MARK(U71(isNatList(L), L, N)) The TRS R consists of the following rules: active(zeros) -> mark(cons(0, zeros)) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatIList(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, L, N)) -> mark(U72(isNat(N), L)) active(U72(tt, L)) -> mark(s(length(L))) active(U81(tt)) -> mark(nil) active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N)) active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N)) active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(isNat(0)) -> mark(tt) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2)) active(length(nil)) -> mark(0) active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N)) active(take(0, IL)) -> mark(U81(isNatIList(IL))) active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N)) mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) mark(0) -> active(0) mark(U11(X)) -> active(U11(mark(X))) mark(tt) -> active(tt) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1, X2)) -> active(U51(mark(X1), X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil) -> active(nil) mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4)) mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4)) mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4)) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) cons(mark(X1), X2) -> cons(X1, X2) cons(X1, mark(X2)) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1), X2) -> U41(X1, X2) U41(X1, mark(X2)) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1), X2) -> U51(X1, X2) U51(X1, mark(X2)) -> U51(X1, X2) U51(active(X1), X2) -> U51(X1, X2) U51(X1, active(X2)) -> U51(X1, X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4) U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4) U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4) U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4) U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4) U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4) take(mark(X1), X2) -> take(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (132) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. ACTIVE(U92(tt, IL, M, N)) -> MARK(U93(isNat(N), IL, M, N)) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( ACTIVE_1(x_1) ) = max{0, 2x_1 - 2} POL( U41_2(x_1, x_2) ) = 2 POL( U51_2(x_1, x_2) ) = 2 POL( U61_2(x_1, x_2) ) = 2 POL( U71_3(x_1, ..., x_3) ) = 2 POL( U72_2(x_1, x_2) ) = 2 POL( U93_4(x_1, ..., x_4) ) = 2 POL( length_1(x_1) ) = 2 POL( mark_1(x_1) ) = 2 POL( cons_2(x_1, x_2) ) = max{0, x_1 - 2} POL( active_1(x_1) ) = max{0, -2} POL( zeros ) = 0 POL( 0 ) = 2 POL( tt ) = 2 POL( U42_1(x_1) ) = max{0, x_1 - 2} POL( isNatIList_1(x_1) ) = 2 POL( U11_1(x_1) ) = max{0, 2x_1 - 2} POL( U52_1(x_1) ) = max{0, -2} POL( isNatList_1(x_1) ) = 2 POL( U21_1(x_1) ) = 2 POL( U62_1(x_1) ) = max{0, -2} POL( U31_1(x_1) ) = max{0, -2} POL( isNat_1(x_1) ) = 2 POL( s_1(x_1) ) = 2 POL( U91_4(x_1, ..., x_4) ) = max{0, 2x_2 + 2x_4 - 2} POL( U92_4(x_1, ..., x_4) ) = 2x_1 + 2x_2 + x_3 + x_4 + 2 POL( take_2(x_1, x_2) ) = max{0, x_1 - 2} POL( U81_1(x_1) ) = 2 POL( nil ) = 0 POL( MARK_1(x_1) ) = 2 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: U41(X1, mark(X2)) -> U41(X1, X2) U41(mark(X1), X2) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) U51(X1, mark(X2)) -> U51(X1, X2) U51(mark(X1), X2) -> U51(X1, X2) U51(active(X1), X2) -> U51(X1, X2) U51(X1, active(X2)) -> U51(X1, X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) U72(X1, mark(X2)) -> U72(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(mark(X1), X2) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) length(active(X)) -> length(X) length(mark(X)) -> length(X) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4) U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4) ---------------------------------------- (133) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(U11(X)) -> MARK(X) MARK(U21(X)) -> MARK(X) MARK(U31(X)) -> MARK(X) MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2))) MARK(U41(X1, X2)) -> MARK(X1) MARK(U42(X)) -> MARK(X) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2))) MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2)) ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIList(V2))) MARK(U51(X1, X2)) -> MARK(X1) MARK(U52(X)) -> MARK(X) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) ACTIVE(U71(tt, L, N)) -> MARK(U72(isNat(N), L)) MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2)) ACTIVE(U72(tt, L)) -> MARK(s(length(L))) MARK(U61(X1, X2)) -> MARK(X1) MARK(U62(X)) -> MARK(X) MARK(U71(X1, X2, X3)) -> ACTIVE(U71(mark(X1), X2, X3)) MARK(U72(X1, X2)) -> ACTIVE(U72(mark(X1), X2)) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(U93(tt, IL, M, N)) -> MARK(cons(N, take(M, IL))) MARK(s(X)) -> MARK(X) MARK(length(X)) -> ACTIVE(length(mark(X))) ACTIVE(isNat(length(V1))) -> MARK(U11(isNatList(V1))) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) ACTIVE(isNatIList(V)) -> MARK(U31(isNatList(V))) MARK(U93(X1, X2, X3, X4)) -> ACTIVE(U93(mark(X1), X2, X3, X4)) ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNat(V1), V2)) ACTIVE(isNatList(cons(V1, V2))) -> MARK(U51(isNat(V1), V2)) ACTIVE(isNatList(take(V1, V2))) -> MARK(U61(isNat(V1), V2)) ACTIVE(length(cons(N, L))) -> MARK(U71(isNatList(L), L, N)) The TRS R consists of the following rules: active(zeros) -> mark(cons(0, zeros)) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatIList(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, L, N)) -> mark(U72(isNat(N), L)) active(U72(tt, L)) -> mark(s(length(L))) active(U81(tt)) -> mark(nil) active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N)) active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N)) active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(isNat(0)) -> mark(tt) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2)) active(length(nil)) -> mark(0) active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N)) active(take(0, IL)) -> mark(U81(isNatIList(IL))) active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N)) mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) mark(0) -> active(0) mark(U11(X)) -> active(U11(mark(X))) mark(tt) -> active(tt) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1, X2)) -> active(U51(mark(X1), X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil) -> active(nil) mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4)) mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4)) mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4)) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) cons(mark(X1), X2) -> cons(X1, X2) cons(X1, mark(X2)) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1), X2) -> U41(X1, X2) U41(X1, mark(X2)) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1), X2) -> U51(X1, X2) U51(X1, mark(X2)) -> U51(X1, X2) U51(active(X1), X2) -> U51(X1, X2) U51(X1, active(X2)) -> U51(X1, X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4) U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4) U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4) U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4) U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4) U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4) take(mark(X1), X2) -> take(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (134) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MARK(U93(X1, X2, X3, X4)) -> ACTIVE(U93(mark(X1), X2, X3, X4)) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( ACTIVE_1(x_1) ) = max{0, -2} POL( U41_2(x_1, x_2) ) = x_1 + 1 POL( U51_2(x_1, x_2) ) = x_1 + 1 POL( U61_2(x_1, x_2) ) = x_1 + 1 POL( U71_3(x_1, ..., x_3) ) = max{0, -2} POL( U72_2(x_1, x_2) ) = max{0, -2} POL( U93_4(x_1, ..., x_4) ) = x_2 + 2x_3 + 2x_4 + 2 POL( length_1(x_1) ) = 0 POL( mark_1(x_1) ) = x_1 + 2 POL( cons_2(x_1, x_2) ) = max{0, -2} POL( active_1(x_1) ) = 2x_1 + 2 POL( zeros ) = 2 POL( 0 ) = 2 POL( tt ) = 2 POL( U42_1(x_1) ) = 2x_1 + 1 POL( isNatIList_1(x_1) ) = 0 POL( U11_1(x_1) ) = 2x_1 + 1 POL( U52_1(x_1) ) = x_1 + 1 POL( isNatList_1(x_1) ) = 0 POL( U21_1(x_1) ) = x_1 + 1 POL( U62_1(x_1) ) = x_1 + 1 POL( U31_1(x_1) ) = x_1 + 1 POL( isNat_1(x_1) ) = 0 POL( s_1(x_1) ) = x_1 + 1 POL( U91_4(x_1, ..., x_4) ) = 2x_2 + x_4 + 2 POL( U92_4(x_1, ..., x_4) ) = max{0, -2} POL( take_2(x_1, x_2) ) = max{0, x_1 - 2} POL( U81_1(x_1) ) = 2x_1 + 2 POL( nil ) = 2 POL( MARK_1(x_1) ) = max{0, x_1 - 1} The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: U41(X1, mark(X2)) -> U41(X1, X2) U41(mark(X1), X2) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U51(X1, mark(X2)) -> U51(X1, X2) U51(mark(X1), X2) -> U51(X1, X2) U51(active(X1), X2) -> U51(X1, X2) U51(X1, active(X2)) -> U51(X1, X2) U62(active(X)) -> U62(X) U62(mark(X)) -> U62(X) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) U72(X1, mark(X2)) -> U72(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(mark(X1), X2) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) length(active(X)) -> length(X) length(mark(X)) -> length(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) cons(X1, mark(X2)) -> cons(X1, X2) cons(mark(X1), X2) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) ---------------------------------------- (135) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(U11(X)) -> MARK(X) MARK(U21(X)) -> MARK(X) MARK(U31(X)) -> MARK(X) MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2))) MARK(U41(X1, X2)) -> MARK(X1) MARK(U42(X)) -> MARK(X) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2))) MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2)) ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIList(V2))) MARK(U51(X1, X2)) -> MARK(X1) MARK(U52(X)) -> MARK(X) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) ACTIVE(U71(tt, L, N)) -> MARK(U72(isNat(N), L)) MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2)) ACTIVE(U72(tt, L)) -> MARK(s(length(L))) MARK(U61(X1, X2)) -> MARK(X1) MARK(U62(X)) -> MARK(X) MARK(U71(X1, X2, X3)) -> ACTIVE(U71(mark(X1), X2, X3)) MARK(U72(X1, X2)) -> ACTIVE(U72(mark(X1), X2)) MARK(isNat(X)) -> ACTIVE(isNat(X)) ACTIVE(U93(tt, IL, M, N)) -> MARK(cons(N, take(M, IL))) MARK(s(X)) -> MARK(X) MARK(length(X)) -> ACTIVE(length(mark(X))) ACTIVE(isNat(length(V1))) -> MARK(U11(isNatList(V1))) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) ACTIVE(isNatIList(V)) -> MARK(U31(isNatList(V))) ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNat(V1), V2)) ACTIVE(isNatList(cons(V1, V2))) -> MARK(U51(isNat(V1), V2)) ACTIVE(isNatList(take(V1, V2))) -> MARK(U61(isNat(V1), V2)) ACTIVE(length(cons(N, L))) -> MARK(U71(isNatList(L), L, N)) The TRS R consists of the following rules: active(zeros) -> mark(cons(0, zeros)) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatIList(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, L, N)) -> mark(U72(isNat(N), L)) active(U72(tt, L)) -> mark(s(length(L))) active(U81(tt)) -> mark(nil) active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N)) active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N)) active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(isNat(0)) -> mark(tt) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2)) active(length(nil)) -> mark(0) active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N)) active(take(0, IL)) -> mark(U81(isNatIList(IL))) active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N)) mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) mark(0) -> active(0) mark(U11(X)) -> active(U11(mark(X))) mark(tt) -> active(tt) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1, X2)) -> active(U51(mark(X1), X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil) -> active(nil) mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4)) mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4)) mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4)) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) cons(mark(X1), X2) -> cons(X1, X2) cons(X1, mark(X2)) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1), X2) -> U41(X1, X2) U41(X1, mark(X2)) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1), X2) -> U51(X1, X2) U51(X1, mark(X2)) -> U51(X1, X2) U51(active(X1), X2) -> U51(X1, X2) U51(X1, active(X2)) -> U51(X1, X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4) U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4) U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4) U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4) U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4) U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4) take(mark(X1), X2) -> take(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (136) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. ACTIVE(U93(tt, IL, M, N)) -> MARK(cons(N, take(M, IL))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( ACTIVE_1(x_1) ) = x_1 POL( U41_2(x_1, x_2) ) = 2 POL( U51_2(x_1, x_2) ) = 2 POL( U61_2(x_1, x_2) ) = 2 POL( U71_3(x_1, ..., x_3) ) = 2 POL( U72_2(x_1, x_2) ) = 2 POL( length_1(x_1) ) = 2 POL( mark_1(x_1) ) = 2x_1 + 2 POL( cons_2(x_1, x_2) ) = max{0, x_1 - 2} POL( active_1(x_1) ) = max{0, -2} POL( zeros ) = 0 POL( 0 ) = 0 POL( tt ) = 2 POL( U42_1(x_1) ) = 2 POL( isNatIList_1(x_1) ) = 2 POL( U11_1(x_1) ) = max{0, -2} POL( U52_1(x_1) ) = max{0, -2} POL( isNatList_1(x_1) ) = 2 POL( U21_1(x_1) ) = max{0, -2} POL( U62_1(x_1) ) = max{0, x_1 - 1} POL( U31_1(x_1) ) = 2 POL( isNat_1(x_1) ) = 2 POL( s_1(x_1) ) = max{0, x_1 - 2} POL( U91_4(x_1, ..., x_4) ) = max{0, 2x_4 - 2} POL( U92_4(x_1, ..., x_4) ) = 2 POL( U93_4(x_1, ..., x_4) ) = max{0, 2x_1 - 1} POL( take_2(x_1, x_2) ) = max{0, x_1 + x_2 - 2} POL( U81_1(x_1) ) = x_1 + 1 POL( nil ) = 2 POL( MARK_1(x_1) ) = 2 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: U41(X1, mark(X2)) -> U41(X1, X2) U41(mark(X1), X2) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) U51(X1, mark(X2)) -> U51(X1, X2) U51(mark(X1), X2) -> U51(X1, X2) U51(active(X1), X2) -> U51(X1, X2) U51(X1, active(X2)) -> U51(X1, X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) U72(X1, mark(X2)) -> U72(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(mark(X1), X2) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) length(active(X)) -> length(X) length(mark(X)) -> length(X) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) ---------------------------------------- (137) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(U11(X)) -> MARK(X) MARK(U21(X)) -> MARK(X) MARK(U31(X)) -> MARK(X) MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2))) MARK(U41(X1, X2)) -> MARK(X1) MARK(U42(X)) -> MARK(X) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2))) MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2)) ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIList(V2))) MARK(U51(X1, X2)) -> MARK(X1) MARK(U52(X)) -> MARK(X) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) ACTIVE(U71(tt, L, N)) -> MARK(U72(isNat(N), L)) MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2)) ACTIVE(U72(tt, L)) -> MARK(s(length(L))) MARK(U61(X1, X2)) -> MARK(X1) MARK(U62(X)) -> MARK(X) MARK(U71(X1, X2, X3)) -> ACTIVE(U71(mark(X1), X2, X3)) MARK(U72(X1, X2)) -> ACTIVE(U72(mark(X1), X2)) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(s(X)) -> MARK(X) MARK(length(X)) -> ACTIVE(length(mark(X))) ACTIVE(isNat(length(V1))) -> MARK(U11(isNatList(V1))) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) ACTIVE(isNatIList(V)) -> MARK(U31(isNatList(V))) ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNat(V1), V2)) ACTIVE(isNatList(cons(V1, V2))) -> MARK(U51(isNat(V1), V2)) ACTIVE(isNatList(take(V1, V2))) -> MARK(U61(isNat(V1), V2)) ACTIVE(length(cons(N, L))) -> MARK(U71(isNatList(L), L, N)) The TRS R consists of the following rules: active(zeros) -> mark(cons(0, zeros)) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatIList(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, L, N)) -> mark(U72(isNat(N), L)) active(U72(tt, L)) -> mark(s(length(L))) active(U81(tt)) -> mark(nil) active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N)) active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N)) active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(isNat(0)) -> mark(tt) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2)) active(length(nil)) -> mark(0) active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N)) active(take(0, IL)) -> mark(U81(isNatIList(IL))) active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N)) mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) mark(0) -> active(0) mark(U11(X)) -> active(U11(mark(X))) mark(tt) -> active(tt) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1, X2)) -> active(U51(mark(X1), X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil) -> active(nil) mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4)) mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4)) mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4)) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) cons(mark(X1), X2) -> cons(X1, X2) cons(X1, mark(X2)) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1), X2) -> U41(X1, X2) U41(X1, mark(X2)) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1), X2) -> U51(X1, X2) U51(X1, mark(X2)) -> U51(X1, X2) U51(active(X1), X2) -> U51(X1, X2) U51(X1, active(X2)) -> U51(X1, X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4) U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4) U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4) U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4) U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4) U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4) take(mark(X1), X2) -> take(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (138) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MARK(U41(X1, X2)) -> MARK(X1) ACTIVE(U61(tt, V2)) -> MARK(U62(isNatIList(V2))) MARK(U61(X1, X2)) -> MARK(X1) ACTIVE(isNatIList(V)) -> MARK(U31(isNatList(V))) ACTIVE(isNatList(take(V1, V2))) -> MARK(U61(isNat(V1), V2)) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( ACTIVE_1(x_1) ) = x_1 POL( U41_2(x_1, x_2) ) = 2x_1 + 2x_2 + 1 POL( U51_2(x_1, x_2) ) = 2x_1 + 2x_2 POL( U61_2(x_1, x_2) ) = 2x_1 + 2x_2 + 2 POL( U71_3(x_1, ..., x_3) ) = x_2 + 2x_3 POL( U72_2(x_1, x_2) ) = x_2 POL( length_1(x_1) ) = x_1 POL( mark_1(x_1) ) = x_1 POL( cons_2(x_1, x_2) ) = 2x_1 + x_2 POL( active_1(x_1) ) = x_1 POL( zeros ) = 0 POL( 0 ) = 0 POL( tt ) = 0 POL( U42_1(x_1) ) = x_1 POL( isNatIList_1(x_1) ) = 2x_1 + 1 POL( U11_1(x_1) ) = x_1 POL( U52_1(x_1) ) = x_1 POL( isNatList_1(x_1) ) = 2x_1 POL( U21_1(x_1) ) = x_1 POL( U62_1(x_1) ) = x_1 POL( U31_1(x_1) ) = x_1 POL( isNat_1(x_1) ) = 2x_1 POL( s_1(x_1) ) = x_1 POL( U91_4(x_1, ..., x_4) ) = x_2 + 2x_3 + 2x_4 + 2 POL( U92_4(x_1, ..., x_4) ) = x_2 + 2x_3 + 2x_4 + 2 POL( U93_4(x_1, ..., x_4) ) = x_2 + 2x_3 + 2x_4 + 2 POL( take_2(x_1, x_2) ) = 2x_1 + x_2 + 2 POL( U81_1(x_1) ) = 2 POL( nil ) = 2 POL( MARK_1(x_1) ) = x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) active(zeros) -> mark(cons(0, zeros)) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) mark(zeros) -> active(zeros) mark(U11(X)) -> active(U11(mark(X))) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) mark(U21(X)) -> active(U21(mark(X))) active(U61(tt, V2)) -> mark(U62(isNatIList(V2))) mark(U31(X)) -> active(U31(mark(X))) active(U71(tt, L, N)) -> mark(U72(isNat(N), L)) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) active(U72(tt, L)) -> mark(s(length(L))) mark(U42(X)) -> active(U42(mark(X))) active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N)) mark(isNatIList(X)) -> active(isNatIList(X)) active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N)) mark(U51(X1, X2)) -> active(U51(mark(X1), X2)) active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) mark(U52(X)) -> active(U52(mark(X))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) mark(isNatList(X)) -> active(isNatList(X)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) active(isNatIList(V)) -> mark(U31(isNatList(V))) mark(U62(X)) -> active(U62(mark(X))) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3)) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2)) mark(isNat(X)) -> active(isNat(X)) active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N)) mark(s(X)) -> active(s(mark(X))) active(take(0, IL)) -> mark(U81(isNatIList(IL))) mark(length(X)) -> active(length(mark(X))) active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N)) mark(U81(X)) -> active(U81(mark(X))) mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4)) mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4)) mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4)) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) mark(0) -> active(0) mark(tt) -> active(tt) mark(nil) -> active(nil) U41(X1, mark(X2)) -> U41(X1, X2) U41(mark(X1), X2) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U51(X1, mark(X2)) -> U51(X1, X2) U51(mark(X1), X2) -> U51(X1, X2) U51(active(X1), X2) -> U51(X1, X2) U51(X1, active(X2)) -> U51(X1, X2) U62(active(X)) -> U62(X) U62(mark(X)) -> U62(X) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) U72(X1, mark(X2)) -> U72(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(mark(X1), X2) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) length(active(X)) -> length(X) length(mark(X)) -> length(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) cons(X1, mark(X2)) -> cons(X1, X2) cons(mark(X1), X2) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U42(tt)) -> mark(tt) active(U52(tt)) -> mark(tt) active(U62(tt)) -> mark(tt) active(U81(tt)) -> mark(nil) active(isNat(0)) -> mark(tt) active(isNatIList(zeros)) -> mark(tt) active(isNatList(nil)) -> mark(tt) active(length(nil)) -> mark(0) U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4) U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4) U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4) U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4) take(X1, mark(X2)) -> take(X1, X2) take(mark(X1), X2) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) U81(active(X)) -> U81(X) U81(mark(X)) -> U81(X) U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4) U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4) ---------------------------------------- (139) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(U11(X)) -> MARK(X) MARK(U21(X)) -> MARK(X) MARK(U31(X)) -> MARK(X) MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2))) MARK(U42(X)) -> MARK(X) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2))) MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2)) MARK(U51(X1, X2)) -> MARK(X1) MARK(U52(X)) -> MARK(X) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) ACTIVE(U71(tt, L, N)) -> MARK(U72(isNat(N), L)) MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2)) ACTIVE(U72(tt, L)) -> MARK(s(length(L))) MARK(U62(X)) -> MARK(X) MARK(U71(X1, X2, X3)) -> ACTIVE(U71(mark(X1), X2, X3)) MARK(U72(X1, X2)) -> ACTIVE(U72(mark(X1), X2)) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(s(X)) -> MARK(X) MARK(length(X)) -> ACTIVE(length(mark(X))) ACTIVE(isNat(length(V1))) -> MARK(U11(isNatList(V1))) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNat(V1), V2)) ACTIVE(isNatList(cons(V1, V2))) -> MARK(U51(isNat(V1), V2)) ACTIVE(length(cons(N, L))) -> MARK(U71(isNatList(L), L, N)) The TRS R consists of the following rules: active(zeros) -> mark(cons(0, zeros)) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatIList(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, L, N)) -> mark(U72(isNat(N), L)) active(U72(tt, L)) -> mark(s(length(L))) active(U81(tt)) -> mark(nil) active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N)) active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N)) active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(isNat(0)) -> mark(tt) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2)) active(length(nil)) -> mark(0) active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N)) active(take(0, IL)) -> mark(U81(isNatIList(IL))) active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N)) mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) mark(0) -> active(0) mark(U11(X)) -> active(U11(mark(X))) mark(tt) -> active(tt) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1, X2)) -> active(U51(mark(X1), X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil) -> active(nil) mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4)) mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4)) mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4)) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) cons(mark(X1), X2) -> cons(X1, X2) cons(X1, mark(X2)) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1), X2) -> U41(X1, X2) U41(X1, mark(X2)) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1), X2) -> U51(X1, X2) U51(X1, mark(X2)) -> U51(X1, X2) U51(active(X1), X2) -> U51(X1, X2) U51(X1, active(X2)) -> U51(X1, X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4) U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4) U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4) U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4) U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4) U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4) take(mark(X1), X2) -> take(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (140) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MARK(U31(X)) -> MARK(X) MARK(U61(X1, X2)) -> ACTIVE(U61(mark(X1), X2)) MARK(U62(X)) -> MARK(X) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( ACTIVE_1(x_1) ) = 0 POL( U41_2(x_1, x_2) ) = 0 POL( U51_2(x_1, x_2) ) = 2x_1 + 1 POL( U61_2(x_1, x_2) ) = 2 POL( U71_3(x_1, ..., x_3) ) = max{0, -2} POL( U72_2(x_1, x_2) ) = 0 POL( length_1(x_1) ) = 0 POL( mark_1(x_1) ) = 2x_1 POL( cons_2(x_1, x_2) ) = x_1 + 2x_2 + 2 POL( active_1(x_1) ) = 2x_1 + 2 POL( zeros ) = 0 POL( 0 ) = 0 POL( tt ) = 0 POL( U42_1(x_1) ) = 2x_1 + 1 POL( isNatIList_1(x_1) ) = 0 POL( U11_1(x_1) ) = 2x_1 + 1 POL( U52_1(x_1) ) = 2x_1 + 1 POL( isNatList_1(x_1) ) = 0 POL( U21_1(x_1) ) = x_1 + 1 POL( U62_1(x_1) ) = x_1 + 2 POL( U31_1(x_1) ) = x_1 + 2 POL( isNat_1(x_1) ) = 0 POL( s_1(x_1) ) = 2x_1 + 1 POL( U91_4(x_1, ..., x_4) ) = max{0, x_1 + x_2 + x_4 - 2} POL( U92_4(x_1, ..., x_4) ) = x_1 + 2x_2 + 2x_3 + 2 POL( U93_4(x_1, ..., x_4) ) = 2x_2 + x_3 + 2 POL( take_2(x_1, x_2) ) = max{0, 2x_1 + x_2 - 2} POL( U81_1(x_1) ) = x_1 + 2 POL( nil ) = 0 POL( MARK_1(x_1) ) = max{0, x_1 - 1} The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: U41(X1, mark(X2)) -> U41(X1, X2) U41(mark(X1), X2) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U51(X1, mark(X2)) -> U51(X1, X2) U51(mark(X1), X2) -> U51(X1, X2) U51(active(X1), X2) -> U51(X1, X2) U51(X1, active(X2)) -> U51(X1, X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) U72(X1, mark(X2)) -> U72(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) length(active(X)) -> length(X) length(mark(X)) -> length(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) ---------------------------------------- (141) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(U11(X)) -> MARK(X) MARK(U21(X)) -> MARK(X) MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2))) MARK(U42(X)) -> MARK(X) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2))) MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2)) MARK(U51(X1, X2)) -> MARK(X1) MARK(U52(X)) -> MARK(X) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) ACTIVE(U71(tt, L, N)) -> MARK(U72(isNat(N), L)) ACTIVE(U72(tt, L)) -> MARK(s(length(L))) MARK(U71(X1, X2, X3)) -> ACTIVE(U71(mark(X1), X2, X3)) MARK(U72(X1, X2)) -> ACTIVE(U72(mark(X1), X2)) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(s(X)) -> MARK(X) MARK(length(X)) -> ACTIVE(length(mark(X))) ACTIVE(isNat(length(V1))) -> MARK(U11(isNatList(V1))) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNat(V1), V2)) ACTIVE(isNatList(cons(V1, V2))) -> MARK(U51(isNat(V1), V2)) ACTIVE(length(cons(N, L))) -> MARK(U71(isNatList(L), L, N)) The TRS R consists of the following rules: active(zeros) -> mark(cons(0, zeros)) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatIList(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, L, N)) -> mark(U72(isNat(N), L)) active(U72(tt, L)) -> mark(s(length(L))) active(U81(tt)) -> mark(nil) active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N)) active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N)) active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(isNat(0)) -> mark(tt) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2)) active(length(nil)) -> mark(0) active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N)) active(take(0, IL)) -> mark(U81(isNatIList(IL))) active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N)) mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) mark(0) -> active(0) mark(U11(X)) -> active(U11(mark(X))) mark(tt) -> active(tt) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1, X2)) -> active(U51(mark(X1), X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil) -> active(nil) mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4)) mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4)) mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4)) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) cons(mark(X1), X2) -> cons(X1, X2) cons(X1, mark(X2)) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1), X2) -> U41(X1, X2) U41(X1, mark(X2)) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1), X2) -> U51(X1, X2) U51(X1, mark(X2)) -> U51(X1, X2) U51(active(X1), X2) -> U51(X1, X2) U51(X1, active(X2)) -> U51(X1, X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4) U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4) U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4) U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4) U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4) U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4) take(mark(X1), X2) -> take(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (142) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MARK(U11(X)) -> MARK(X) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( ACTIVE_1(x_1) ) = x_1 POL( U41_2(x_1, x_2) ) = max{0, -2} POL( U51_2(x_1, x_2) ) = 2x_1 + 2x_2 POL( U71_3(x_1, ..., x_3) ) = 2x_2 + 2x_3 + 1 POL( U72_2(x_1, x_2) ) = 2x_2 + 1 POL( length_1(x_1) ) = 2x_1 + 1 POL( mark_1(x_1) ) = x_1 POL( cons_2(x_1, x_2) ) = 2x_1 + x_2 POL( active_1(x_1) ) = x_1 POL( zeros ) = 0 POL( 0 ) = 0 POL( tt ) = 0 POL( U42_1(x_1) ) = x_1 POL( isNatIList_1(x_1) ) = 0 POL( U11_1(x_1) ) = x_1 + 2 POL( U52_1(x_1) ) = x_1 POL( isNatList_1(x_1) ) = 2x_1 POL( U21_1(x_1) ) = x_1 POL( U61_2(x_1, x_2) ) = 2 POL( U62_1(x_1) ) = 0 POL( U31_1(x_1) ) = max{0, -2} POL( isNat_1(x_1) ) = 2x_1 POL( s_1(x_1) ) = x_1 POL( U91_4(x_1, ..., x_4) ) = x_2 + 2x_4 + 2 POL( U92_4(x_1, ..., x_4) ) = x_2 + 2x_4 + 2 POL( U93_4(x_1, ..., x_4) ) = x_2 + 2x_4 + 2 POL( take_2(x_1, x_2) ) = x_2 + 2 POL( U81_1(x_1) ) = max{0, -2} POL( nil ) = 0 POL( MARK_1(x_1) ) = x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) active(zeros) -> mark(cons(0, zeros)) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) mark(zeros) -> active(zeros) mark(U11(X)) -> active(U11(mark(X))) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) mark(U21(X)) -> active(U21(mark(X))) active(U61(tt, V2)) -> mark(U62(isNatIList(V2))) mark(U31(X)) -> active(U31(mark(X))) active(U71(tt, L, N)) -> mark(U72(isNat(N), L)) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) active(U72(tt, L)) -> mark(s(length(L))) mark(U42(X)) -> active(U42(mark(X))) active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N)) mark(isNatIList(X)) -> active(isNatIList(X)) active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N)) mark(U51(X1, X2)) -> active(U51(mark(X1), X2)) active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) mark(U52(X)) -> active(U52(mark(X))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) mark(isNatList(X)) -> active(isNatList(X)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) active(isNatIList(V)) -> mark(U31(isNatList(V))) mark(U62(X)) -> active(U62(mark(X))) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3)) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2)) mark(isNat(X)) -> active(isNat(X)) active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N)) mark(s(X)) -> active(s(mark(X))) active(take(0, IL)) -> mark(U81(isNatIList(IL))) mark(length(X)) -> active(length(mark(X))) active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N)) mark(U81(X)) -> active(U81(mark(X))) mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4)) mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4)) mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4)) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) mark(0) -> active(0) mark(tt) -> active(tt) mark(nil) -> active(nil) U41(X1, mark(X2)) -> U41(X1, X2) U41(mark(X1), X2) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U51(X1, mark(X2)) -> U51(X1, X2) U51(mark(X1), X2) -> U51(X1, X2) U51(active(X1), X2) -> U51(X1, X2) U51(X1, active(X2)) -> U51(X1, X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) U72(X1, mark(X2)) -> U72(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) length(active(X)) -> length(X) length(mark(X)) -> length(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) cons(X1, mark(X2)) -> cons(X1, X2) cons(mark(X1), X2) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U42(tt)) -> mark(tt) active(U52(tt)) -> mark(tt) active(U62(tt)) -> mark(tt) active(U81(tt)) -> mark(nil) active(isNat(0)) -> mark(tt) active(isNatIList(zeros)) -> mark(tt) active(isNatList(nil)) -> mark(tt) active(length(nil)) -> mark(0) U62(active(X)) -> U62(X) U62(mark(X)) -> U62(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4) U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4) U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4) U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4) take(X1, mark(X2)) -> take(X1, X2) take(mark(X1), X2) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(mark(X1), X2) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U81(active(X)) -> U81(X) U81(mark(X)) -> U81(X) U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4) U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4) ---------------------------------------- (143) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(U21(X)) -> MARK(X) MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2))) MARK(U42(X)) -> MARK(X) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2))) MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2)) MARK(U51(X1, X2)) -> MARK(X1) MARK(U52(X)) -> MARK(X) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) ACTIVE(U71(tt, L, N)) -> MARK(U72(isNat(N), L)) ACTIVE(U72(tt, L)) -> MARK(s(length(L))) MARK(U71(X1, X2, X3)) -> ACTIVE(U71(mark(X1), X2, X3)) MARK(U72(X1, X2)) -> ACTIVE(U72(mark(X1), X2)) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(s(X)) -> MARK(X) MARK(length(X)) -> ACTIVE(length(mark(X))) ACTIVE(isNat(length(V1))) -> MARK(U11(isNatList(V1))) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNat(V1), V2)) ACTIVE(isNatList(cons(V1, V2))) -> MARK(U51(isNat(V1), V2)) ACTIVE(length(cons(N, L))) -> MARK(U71(isNatList(L), L, N)) The TRS R consists of the following rules: active(zeros) -> mark(cons(0, zeros)) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatIList(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, L, N)) -> mark(U72(isNat(N), L)) active(U72(tt, L)) -> mark(s(length(L))) active(U81(tt)) -> mark(nil) active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N)) active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N)) active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(isNat(0)) -> mark(tt) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2)) active(length(nil)) -> mark(0) active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N)) active(take(0, IL)) -> mark(U81(isNatIList(IL))) active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N)) mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) mark(0) -> active(0) mark(U11(X)) -> active(U11(mark(X))) mark(tt) -> active(tt) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1, X2)) -> active(U51(mark(X1), X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil) -> active(nil) mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4)) mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4)) mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4)) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) cons(mark(X1), X2) -> cons(X1, X2) cons(X1, mark(X2)) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1), X2) -> U41(X1, X2) U41(X1, mark(X2)) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1), X2) -> U51(X1, X2) U51(X1, mark(X2)) -> U51(X1, X2) U51(active(X1), X2) -> U51(X1, X2) U51(X1, active(X2)) -> U51(X1, X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4) U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4) U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4) U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4) U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4) U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4) take(mark(X1), X2) -> take(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (144) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MARK(U51(X1, X2)) -> MARK(X1) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( ACTIVE_1(x_1) ) = 2x_1 + 2 POL( U41_2(x_1, x_2) ) = max{0, -2} POL( U51_2(x_1, x_2) ) = 2x_1 + 1 POL( U71_3(x_1, ..., x_3) ) = max{0, -2} POL( U72_2(x_1, x_2) ) = max{0, -2} POL( length_1(x_1) ) = 0 POL( mark_1(x_1) ) = x_1 POL( cons_2(x_1, x_2) ) = max{0, -2} POL( active_1(x_1) ) = x_1 POL( zeros ) = 0 POL( 0 ) = 0 POL( tt ) = 0 POL( U42_1(x_1) ) = 2x_1 POL( isNatIList_1(x_1) ) = 0 POL( U11_1(x_1) ) = max{0, -2} POL( U52_1(x_1) ) = x_1 POL( isNatList_1(x_1) ) = 1 POL( U21_1(x_1) ) = x_1 POL( U61_2(x_1, x_2) ) = max{0, -2} POL( U62_1(x_1) ) = max{0, -2} POL( U31_1(x_1) ) = max{0, -2} POL( isNat_1(x_1) ) = 0 POL( s_1(x_1) ) = x_1 POL( U91_4(x_1, ..., x_4) ) = max{0, -2} POL( U92_4(x_1, ..., x_4) ) = 0 POL( U93_4(x_1, ..., x_4) ) = max{0, -2} POL( take_2(x_1, x_2) ) = max{0, -2} POL( U81_1(x_1) ) = max{0, -2} POL( nil ) = 0 POL( MARK_1(x_1) ) = 2x_1 + 2 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) active(zeros) -> mark(cons(0, zeros)) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) mark(zeros) -> active(zeros) mark(U11(X)) -> active(U11(mark(X))) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) mark(U21(X)) -> active(U21(mark(X))) active(U61(tt, V2)) -> mark(U62(isNatIList(V2))) mark(U31(X)) -> active(U31(mark(X))) active(U71(tt, L, N)) -> mark(U72(isNat(N), L)) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) active(U72(tt, L)) -> mark(s(length(L))) mark(U42(X)) -> active(U42(mark(X))) active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N)) mark(isNatIList(X)) -> active(isNatIList(X)) active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N)) mark(U51(X1, X2)) -> active(U51(mark(X1), X2)) active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) mark(U52(X)) -> active(U52(mark(X))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) mark(isNatList(X)) -> active(isNatList(X)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) active(isNatIList(V)) -> mark(U31(isNatList(V))) mark(U62(X)) -> active(U62(mark(X))) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3)) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2)) mark(isNat(X)) -> active(isNat(X)) active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N)) mark(s(X)) -> active(s(mark(X))) active(take(0, IL)) -> mark(U81(isNatIList(IL))) mark(length(X)) -> active(length(mark(X))) active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N)) mark(U81(X)) -> active(U81(mark(X))) mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4)) mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4)) mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4)) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) mark(0) -> active(0) mark(tt) -> active(tt) mark(nil) -> active(nil) U41(X1, mark(X2)) -> U41(X1, X2) U41(mark(X1), X2) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U51(X1, mark(X2)) -> U51(X1, X2) U51(mark(X1), X2) -> U51(X1, X2) U51(active(X1), X2) -> U51(X1, X2) U51(X1, active(X2)) -> U51(X1, X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) U72(X1, mark(X2)) -> U72(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) length(active(X)) -> length(X) length(mark(X)) -> length(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) cons(X1, mark(X2)) -> cons(X1, X2) cons(mark(X1), X2) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U42(tt)) -> mark(tt) active(U52(tt)) -> mark(tt) active(U62(tt)) -> mark(tt) active(U81(tt)) -> mark(nil) active(isNat(0)) -> mark(tt) active(isNatIList(zeros)) -> mark(tt) active(isNatList(nil)) -> mark(tt) active(length(nil)) -> mark(0) U62(active(X)) -> U62(X) U62(mark(X)) -> U62(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4) U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4) U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4) U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4) take(X1, mark(X2)) -> take(X1, X2) take(mark(X1), X2) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(mark(X1), X2) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U81(active(X)) -> U81(X) U81(mark(X)) -> U81(X) U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4) U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4) ---------------------------------------- (145) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(U21(X)) -> MARK(X) MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2))) MARK(U42(X)) -> MARK(X) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2))) MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2)) MARK(U52(X)) -> MARK(X) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) ACTIVE(U71(tt, L, N)) -> MARK(U72(isNat(N), L)) ACTIVE(U72(tt, L)) -> MARK(s(length(L))) MARK(U71(X1, X2, X3)) -> ACTIVE(U71(mark(X1), X2, X3)) MARK(U72(X1, X2)) -> ACTIVE(U72(mark(X1), X2)) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(s(X)) -> MARK(X) MARK(length(X)) -> ACTIVE(length(mark(X))) ACTIVE(isNat(length(V1))) -> MARK(U11(isNatList(V1))) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNat(V1), V2)) ACTIVE(isNatList(cons(V1, V2))) -> MARK(U51(isNat(V1), V2)) ACTIVE(length(cons(N, L))) -> MARK(U71(isNatList(L), L, N)) The TRS R consists of the following rules: active(zeros) -> mark(cons(0, zeros)) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatIList(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, L, N)) -> mark(U72(isNat(N), L)) active(U72(tt, L)) -> mark(s(length(L))) active(U81(tt)) -> mark(nil) active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N)) active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N)) active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(isNat(0)) -> mark(tt) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2)) active(length(nil)) -> mark(0) active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N)) active(take(0, IL)) -> mark(U81(isNatIList(IL))) active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N)) mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) mark(0) -> active(0) mark(U11(X)) -> active(U11(mark(X))) mark(tt) -> active(tt) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1, X2)) -> active(U51(mark(X1), X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil) -> active(nil) mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4)) mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4)) mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4)) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) cons(mark(X1), X2) -> cons(X1, X2) cons(X1, mark(X2)) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1), X2) -> U41(X1, X2) U41(X1, mark(X2)) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1), X2) -> U51(X1, X2) U51(X1, mark(X2)) -> U51(X1, X2) U51(active(X1), X2) -> U51(X1, X2) U51(X1, active(X2)) -> U51(X1, X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4) U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4) U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4) U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4) U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4) U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4) take(mark(X1), X2) -> take(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (146) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. ACTIVE(isNat(length(V1))) -> MARK(U11(isNatList(V1))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( ACTIVE_1(x_1) ) = max{0, 2x_1 - 2} POL( U41_2(x_1, x_2) ) = max{0, -2} POL( U51_2(x_1, x_2) ) = 2 POL( U71_3(x_1, ..., x_3) ) = max{0, -2} POL( U72_2(x_1, x_2) ) = max{0, -2} POL( length_1(x_1) ) = max{0, -2} POL( mark_1(x_1) ) = 2x_1 POL( cons_2(x_1, x_2) ) = max{0, x_1 - 2} POL( active_1(x_1) ) = 2x_1 + 2 POL( zeros ) = 0 POL( 0 ) = 0 POL( tt ) = 0 POL( U42_1(x_1) ) = 2x_1 POL( isNatIList_1(x_1) ) = 0 POL( U11_1(x_1) ) = 1 POL( U52_1(x_1) ) = x_1 POL( isNatList_1(x_1) ) = 2 POL( U21_1(x_1) ) = x_1 POL( U61_2(x_1, x_2) ) = max{0, 2x_1 - 2} POL( U62_1(x_1) ) = 2 POL( U31_1(x_1) ) = max{0, -2} POL( isNat_1(x_1) ) = 2 POL( s_1(x_1) ) = x_1 POL( U91_4(x_1, ..., x_4) ) = x_2 + x_3 + 2 POL( U92_4(x_1, ..., x_4) ) = max{0, 2x_1 + 2x_3 + x_4 - 2} POL( U93_4(x_1, ..., x_4) ) = max{0, 2x_2 - 2} POL( take_2(x_1, x_2) ) = max{0, x_1 - 2} POL( U81_1(x_1) ) = max{0, -2} POL( nil ) = 0 POL( MARK_1(x_1) ) = x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: U41(X1, mark(X2)) -> U41(X1, X2) U41(mark(X1), X2) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U51(X1, mark(X2)) -> U51(X1, X2) U51(mark(X1), X2) -> U51(X1, X2) U51(active(X1), X2) -> U51(X1, X2) U51(X1, active(X2)) -> U51(X1, X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) U72(X1, mark(X2)) -> U72(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) length(active(X)) -> length(X) length(mark(X)) -> length(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) ---------------------------------------- (147) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(U21(X)) -> MARK(X) MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2))) MARK(U42(X)) -> MARK(X) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2))) MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2)) MARK(U52(X)) -> MARK(X) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) ACTIVE(U71(tt, L, N)) -> MARK(U72(isNat(N), L)) ACTIVE(U72(tt, L)) -> MARK(s(length(L))) MARK(U71(X1, X2, X3)) -> ACTIVE(U71(mark(X1), X2, X3)) MARK(U72(X1, X2)) -> ACTIVE(U72(mark(X1), X2)) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(s(X)) -> MARK(X) MARK(length(X)) -> ACTIVE(length(mark(X))) ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNat(V1), V2)) ACTIVE(isNatList(cons(V1, V2))) -> MARK(U51(isNat(V1), V2)) ACTIVE(length(cons(N, L))) -> MARK(U71(isNatList(L), L, N)) The TRS R consists of the following rules: active(zeros) -> mark(cons(0, zeros)) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatIList(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, L, N)) -> mark(U72(isNat(N), L)) active(U72(tt, L)) -> mark(s(length(L))) active(U81(tt)) -> mark(nil) active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N)) active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N)) active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(isNat(0)) -> mark(tt) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2)) active(length(nil)) -> mark(0) active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N)) active(take(0, IL)) -> mark(U81(isNatIList(IL))) active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N)) mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) mark(0) -> active(0) mark(U11(X)) -> active(U11(mark(X))) mark(tt) -> active(tt) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1, X2)) -> active(U51(mark(X1), X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil) -> active(nil) mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4)) mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4)) mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4)) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) cons(mark(X1), X2) -> cons(X1, X2) cons(X1, mark(X2)) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1), X2) -> U41(X1, X2) U41(X1, mark(X2)) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1), X2) -> U51(X1, X2) U51(X1, mark(X2)) -> U51(X1, X2) U51(active(X1), X2) -> U51(X1, X2) U51(X1, active(X2)) -> U51(X1, X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4) U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4) U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4) U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4) U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4) U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4) take(mark(X1), X2) -> take(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (148) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. ACTIVE(isNat(s(V1))) -> MARK(U21(isNat(V1))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( ACTIVE_1(x_1) ) = max{0, x_1 - 2} POL( U41_2(x_1, x_2) ) = 0 POL( U51_2(x_1, x_2) ) = max{0, -2} POL( U71_3(x_1, ..., x_3) ) = 2 POL( U72_2(x_1, x_2) ) = 2 POL( length_1(x_1) ) = max{0, -2} POL( mark_1(x_1) ) = x_1 + 2 POL( cons_2(x_1, x_2) ) = max{0, 2x_1 - 2} POL( active_1(x_1) ) = 2x_1 POL( zeros ) = 0 POL( 0 ) = 0 POL( tt ) = 2 POL( U42_1(x_1) ) = x_1 + 2 POL( isNatIList_1(x_1) ) = 0 POL( U11_1(x_1) ) = max{0, x_1 - 2} POL( U52_1(x_1) ) = 2x_1 + 2 POL( isNatList_1(x_1) ) = 0 POL( U21_1(x_1) ) = x_1 + 2 POL( U61_2(x_1, x_2) ) = x_1 + 2 POL( U62_1(x_1) ) = 2x_1 + 2 POL( U31_1(x_1) ) = 2 POL( isNat_1(x_1) ) = 2x_1 + 2 POL( s_1(x_1) ) = 2x_1 + 2 POL( U91_4(x_1, ..., x_4) ) = max{0, 2x_3 - 2} POL( U92_4(x_1, ..., x_4) ) = max{0, 2x_4 - 2} POL( U93_4(x_1, ..., x_4) ) = 2 POL( take_2(x_1, x_2) ) = max{0, -2} POL( U81_1(x_1) ) = max{0, -2} POL( nil ) = 0 POL( MARK_1(x_1) ) = max{0, x_1 - 2} The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: U41(X1, mark(X2)) -> U41(X1, X2) U41(mark(X1), X2) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U51(X1, mark(X2)) -> U51(X1, X2) U51(mark(X1), X2) -> U51(X1, X2) U51(active(X1), X2) -> U51(X1, X2) U51(X1, active(X2)) -> U51(X1, X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) U72(X1, mark(X2)) -> U72(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) length(active(X)) -> length(X) length(mark(X)) -> length(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) ---------------------------------------- (149) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(U21(X)) -> MARK(X) MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2))) MARK(U42(X)) -> MARK(X) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2))) MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2)) MARK(U52(X)) -> MARK(X) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) ACTIVE(U71(tt, L, N)) -> MARK(U72(isNat(N), L)) ACTIVE(U72(tt, L)) -> MARK(s(length(L))) MARK(U71(X1, X2, X3)) -> ACTIVE(U71(mark(X1), X2, X3)) MARK(U72(X1, X2)) -> ACTIVE(U72(mark(X1), X2)) MARK(isNat(X)) -> ACTIVE(isNat(X)) MARK(s(X)) -> MARK(X) MARK(length(X)) -> ACTIVE(length(mark(X))) ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNat(V1), V2)) ACTIVE(isNatList(cons(V1, V2))) -> MARK(U51(isNat(V1), V2)) ACTIVE(length(cons(N, L))) -> MARK(U71(isNatList(L), L, N)) The TRS R consists of the following rules: active(zeros) -> mark(cons(0, zeros)) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatIList(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, L, N)) -> mark(U72(isNat(N), L)) active(U72(tt, L)) -> mark(s(length(L))) active(U81(tt)) -> mark(nil) active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N)) active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N)) active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(isNat(0)) -> mark(tt) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2)) active(length(nil)) -> mark(0) active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N)) active(take(0, IL)) -> mark(U81(isNatIList(IL))) active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N)) mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) mark(0) -> active(0) mark(U11(X)) -> active(U11(mark(X))) mark(tt) -> active(tt) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1, X2)) -> active(U51(mark(X1), X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil) -> active(nil) mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4)) mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4)) mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4)) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) cons(mark(X1), X2) -> cons(X1, X2) cons(X1, mark(X2)) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1), X2) -> U41(X1, X2) U41(X1, mark(X2)) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1), X2) -> U51(X1, X2) U51(X1, mark(X2)) -> U51(X1, X2) U51(active(X1), X2) -> U51(X1, X2) U51(X1, active(X2)) -> U51(X1, X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4) U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4) U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4) U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4) U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4) U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4) take(mark(X1), X2) -> take(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (150) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MARK(isNat(X)) -> ACTIVE(isNat(X)) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( ACTIVE_1(x_1) ) = max{0, 2x_1 - 2} POL( U41_2(x_1, x_2) ) = 2 POL( U51_2(x_1, x_2) ) = 2 POL( U71_3(x_1, ..., x_3) ) = 2 POL( U72_2(x_1, x_2) ) = 2 POL( length_1(x_1) ) = 2 POL( mark_1(x_1) ) = 2 POL( cons_2(x_1, x_2) ) = max{0, 2x_2 - 2} POL( active_1(x_1) ) = max{0, x_1 - 2} POL( zeros ) = 0 POL( 0 ) = 0 POL( tt ) = 2 POL( U42_1(x_1) ) = max{0, 2x_1 - 2} POL( isNatIList_1(x_1) ) = 2 POL( U11_1(x_1) ) = 0 POL( U52_1(x_1) ) = max{0, -2} POL( isNatList_1(x_1) ) = 2 POL( U21_1(x_1) ) = max{0, -2} POL( U61_2(x_1, x_2) ) = x_1 + x_2 + 2 POL( U62_1(x_1) ) = 2 POL( U31_1(x_1) ) = x_1 + 1 POL( isNat_1(x_1) ) = 0 POL( s_1(x_1) ) = max{0, -2} POL( U91_4(x_1, ..., x_4) ) = max{0, 2x_1 + 2x_4 - 2} POL( U92_4(x_1, ..., x_4) ) = 2x_1 + x_4 + 2 POL( U93_4(x_1, ..., x_4) ) = max{0, x_2 + x_3 + 2x_4 - 2} POL( take_2(x_1, x_2) ) = 2 POL( U81_1(x_1) ) = 1 POL( nil ) = 0 POL( MARK_1(x_1) ) = 2 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: U41(X1, mark(X2)) -> U41(X1, X2) U41(mark(X1), X2) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) U51(X1, mark(X2)) -> U51(X1, X2) U51(mark(X1), X2) -> U51(X1, X2) U51(active(X1), X2) -> U51(X1, X2) U51(X1, active(X2)) -> U51(X1, X2) isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) U72(X1, mark(X2)) -> U72(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) length(active(X)) -> length(X) length(mark(X)) -> length(X) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) ---------------------------------------- (151) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(U21(X)) -> MARK(X) MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2))) MARK(U42(X)) -> MARK(X) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2))) MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2)) MARK(U52(X)) -> MARK(X) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) ACTIVE(U71(tt, L, N)) -> MARK(U72(isNat(N), L)) ACTIVE(U72(tt, L)) -> MARK(s(length(L))) MARK(U71(X1, X2, X3)) -> ACTIVE(U71(mark(X1), X2, X3)) MARK(U72(X1, X2)) -> ACTIVE(U72(mark(X1), X2)) MARK(s(X)) -> MARK(X) MARK(length(X)) -> ACTIVE(length(mark(X))) ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNat(V1), V2)) ACTIVE(isNatList(cons(V1, V2))) -> MARK(U51(isNat(V1), V2)) ACTIVE(length(cons(N, L))) -> MARK(U71(isNatList(L), L, N)) The TRS R consists of the following rules: active(zeros) -> mark(cons(0, zeros)) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatIList(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, L, N)) -> mark(U72(isNat(N), L)) active(U72(tt, L)) -> mark(s(length(L))) active(U81(tt)) -> mark(nil) active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N)) active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N)) active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(isNat(0)) -> mark(tt) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2)) active(length(nil)) -> mark(0) active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N)) active(take(0, IL)) -> mark(U81(isNatIList(IL))) active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N)) mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) mark(0) -> active(0) mark(U11(X)) -> active(U11(mark(X))) mark(tt) -> active(tt) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1, X2)) -> active(U51(mark(X1), X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil) -> active(nil) mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4)) mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4)) mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4)) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) cons(mark(X1), X2) -> cons(X1, X2) cons(X1, mark(X2)) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1), X2) -> U41(X1, X2) U41(X1, mark(X2)) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1), X2) -> U51(X1, X2) U51(X1, mark(X2)) -> U51(X1, X2) U51(active(X1), X2) -> U51(X1, X2) U51(X1, active(X2)) -> U51(X1, X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4) U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4) U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4) U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4) U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4) U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4) take(mark(X1), X2) -> take(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (152) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MARK(U21(X)) -> MARK(X) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( ACTIVE_1(x_1) ) = max{0, -2} POL( U41_2(x_1, x_2) ) = max{0, -2} POL( U51_2(x_1, x_2) ) = max{0, -2} POL( U71_3(x_1, ..., x_3) ) = max{0, -2} POL( U72_2(x_1, x_2) ) = max{0, -2} POL( length_1(x_1) ) = 0 POL( mark_1(x_1) ) = 2x_1 + 2 POL( cons_2(x_1, x_2) ) = max{0, 2x_1 - 2} POL( active_1(x_1) ) = 2x_1 POL( zeros ) = 0 POL( 0 ) = 0 POL( tt ) = 0 POL( U42_1(x_1) ) = x_1 + 1 POL( isNatIList_1(x_1) ) = 0 POL( U11_1(x_1) ) = 0 POL( U52_1(x_1) ) = 2x_1 + 1 POL( isNatList_1(x_1) ) = 0 POL( U21_1(x_1) ) = 2x_1 + 2 POL( U61_2(x_1, x_2) ) = max{0, x_2 - 2} POL( U62_1(x_1) ) = x_1 + 2 POL( U31_1(x_1) ) = 2 POL( isNat_1(x_1) ) = 0 POL( s_1(x_1) ) = 2x_1 + 1 POL( U91_4(x_1, ..., x_4) ) = 2 POL( U92_4(x_1, ..., x_4) ) = max{0, x_2 + 2x_3 - 2} POL( U93_4(x_1, ..., x_4) ) = 2 POL( take_2(x_1, x_2) ) = max{0, x_1 - 2} POL( U81_1(x_1) ) = 2 POL( nil ) = 1 POL( MARK_1(x_1) ) = max{0, x_1 - 1} The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: U41(X1, mark(X2)) -> U41(X1, X2) U41(mark(X1), X2) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U51(X1, mark(X2)) -> U51(X1, X2) U51(mark(X1), X2) -> U51(X1, X2) U51(active(X1), X2) -> U51(X1, X2) U51(X1, active(X2)) -> U51(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) length(active(X)) -> length(X) length(mark(X)) -> length(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) ---------------------------------------- (153) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2))) MARK(U42(X)) -> MARK(X) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2))) MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2)) MARK(U52(X)) -> MARK(X) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) ACTIVE(U71(tt, L, N)) -> MARK(U72(isNat(N), L)) ACTIVE(U72(tt, L)) -> MARK(s(length(L))) MARK(U71(X1, X2, X3)) -> ACTIVE(U71(mark(X1), X2, X3)) MARK(U72(X1, X2)) -> ACTIVE(U72(mark(X1), X2)) MARK(s(X)) -> MARK(X) MARK(length(X)) -> ACTIVE(length(mark(X))) ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNat(V1), V2)) ACTIVE(isNatList(cons(V1, V2))) -> MARK(U51(isNat(V1), V2)) ACTIVE(length(cons(N, L))) -> MARK(U71(isNatList(L), L, N)) The TRS R consists of the following rules: active(zeros) -> mark(cons(0, zeros)) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatIList(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, L, N)) -> mark(U72(isNat(N), L)) active(U72(tt, L)) -> mark(s(length(L))) active(U81(tt)) -> mark(nil) active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N)) active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N)) active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(isNat(0)) -> mark(tt) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2)) active(length(nil)) -> mark(0) active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N)) active(take(0, IL)) -> mark(U81(isNatIList(IL))) active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N)) mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) mark(0) -> active(0) mark(U11(X)) -> active(U11(mark(X))) mark(tt) -> active(tt) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1, X2)) -> active(U51(mark(X1), X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil) -> active(nil) mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4)) mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4)) mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4)) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) cons(mark(X1), X2) -> cons(X1, X2) cons(X1, mark(X2)) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1), X2) -> U41(X1, X2) U41(X1, mark(X2)) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1), X2) -> U51(X1, X2) U51(X1, mark(X2)) -> U51(X1, X2) U51(active(X1), X2) -> U51(X1, X2) U51(X1, active(X2)) -> U51(X1, X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4) U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4) U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4) U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4) U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4) U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4) take(mark(X1), X2) -> take(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (154) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. ACTIVE(isNatList(cons(V1, V2))) -> MARK(U51(isNat(V1), V2)) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( ACTIVE_1(x_1) ) = x_1 + 2 POL( U41_2(x_1, x_2) ) = max{0, -2} POL( U51_2(x_1, x_2) ) = 2x_2 POL( U71_3(x_1, ..., x_3) ) = max{0, -2} POL( U72_2(x_1, x_2) ) = max{0, -2} POL( length_1(x_1) ) = 0 POL( mark_1(x_1) ) = 2x_1 + 2 POL( cons_2(x_1, x_2) ) = x_2 + 1 POL( active_1(x_1) ) = 2x_1 + 2 POL( zeros ) = 0 POL( 0 ) = 0 POL( tt ) = 0 POL( U42_1(x_1) ) = 2x_1 POL( isNatIList_1(x_1) ) = 0 POL( U11_1(x_1) ) = 2x_1 + 2 POL( U52_1(x_1) ) = x_1 POL( isNatList_1(x_1) ) = 2x_1 POL( U21_1(x_1) ) = max{0, -2} POL( U61_2(x_1, x_2) ) = max{0, 2x_1 - 2} POL( U62_1(x_1) ) = max{0, -2} POL( U31_1(x_1) ) = max{0, -2} POL( isNat_1(x_1) ) = 0 POL( s_1(x_1) ) = x_1 POL( U91_4(x_1, ..., x_4) ) = max{0, 2x_2 + 2x_3 + 2x_4 - 2} POL( U92_4(x_1, ..., x_4) ) = max{0, 2x_2 - 2} POL( U93_4(x_1, ..., x_4) ) = x_2 + 2 POL( take_2(x_1, x_2) ) = max{0, 2x_2 - 2} POL( U81_1(x_1) ) = 0 POL( nil ) = 1 POL( MARK_1(x_1) ) = x_1 + 2 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: U41(X1, mark(X2)) -> U41(X1, X2) U41(mark(X1), X2) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U51(X1, mark(X2)) -> U51(X1, X2) U51(mark(X1), X2) -> U51(X1, X2) U51(active(X1), X2) -> U51(X1, X2) U51(X1, active(X2)) -> U51(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) length(active(X)) -> length(X) length(mark(X)) -> length(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) ---------------------------------------- (155) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2))) MARK(U42(X)) -> MARK(X) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2))) MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2)) MARK(U52(X)) -> MARK(X) MARK(isNatList(X)) -> ACTIVE(isNatList(X)) ACTIVE(U71(tt, L, N)) -> MARK(U72(isNat(N), L)) ACTIVE(U72(tt, L)) -> MARK(s(length(L))) MARK(U71(X1, X2, X3)) -> ACTIVE(U71(mark(X1), X2, X3)) MARK(U72(X1, X2)) -> ACTIVE(U72(mark(X1), X2)) MARK(s(X)) -> MARK(X) MARK(length(X)) -> ACTIVE(length(mark(X))) ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNat(V1), V2)) ACTIVE(length(cons(N, L))) -> MARK(U71(isNatList(L), L, N)) The TRS R consists of the following rules: active(zeros) -> mark(cons(0, zeros)) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatIList(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, L, N)) -> mark(U72(isNat(N), L)) active(U72(tt, L)) -> mark(s(length(L))) active(U81(tt)) -> mark(nil) active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N)) active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N)) active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(isNat(0)) -> mark(tt) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2)) active(length(nil)) -> mark(0) active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N)) active(take(0, IL)) -> mark(U81(isNatIList(IL))) active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N)) mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) mark(0) -> active(0) mark(U11(X)) -> active(U11(mark(X))) mark(tt) -> active(tt) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1, X2)) -> active(U51(mark(X1), X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil) -> active(nil) mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4)) mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4)) mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4)) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) cons(mark(X1), X2) -> cons(X1, X2) cons(X1, mark(X2)) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1), X2) -> U41(X1, X2) U41(X1, mark(X2)) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1), X2) -> U51(X1, X2) U51(X1, mark(X2)) -> U51(X1, X2) U51(active(X1), X2) -> U51(X1, X2) U51(X1, active(X2)) -> U51(X1, X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4) U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4) U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4) U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4) U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4) U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4) take(mark(X1), X2) -> take(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (156) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MARK(isNatList(X)) -> ACTIVE(isNatList(X)) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( ACTIVE_1(x_1) ) = max{0, 2x_1 - 2} POL( U41_2(x_1, x_2) ) = 2 POL( U51_2(x_1, x_2) ) = 2 POL( U71_3(x_1, ..., x_3) ) = 2 POL( U72_2(x_1, x_2) ) = 2 POL( length_1(x_1) ) = 2 POL( mark_1(x_1) ) = 2 POL( cons_2(x_1, x_2) ) = max{0, 2x_1 - 2} POL( active_1(x_1) ) = max{0, 2x_1 - 2} POL( zeros ) = 2 POL( 0 ) = 0 POL( tt ) = 0 POL( U42_1(x_1) ) = max{0, -2} POL( isNatIList_1(x_1) ) = 2 POL( U11_1(x_1) ) = x_1 + 1 POL( U52_1(x_1) ) = max{0, x_1 - 2} POL( isNatList_1(x_1) ) = 0 POL( U21_1(x_1) ) = 2 POL( U61_2(x_1, x_2) ) = x_2 + 2 POL( U62_1(x_1) ) = 2 POL( U31_1(x_1) ) = 2 POL( isNat_1(x_1) ) = 0 POL( s_1(x_1) ) = 2 POL( U91_4(x_1, ..., x_4) ) = max{0, x_3 - 1} POL( U92_4(x_1, ..., x_4) ) = 2 POL( U93_4(x_1, ..., x_4) ) = 2x_1 + x_2 + 2x_3 + 2 POL( take_2(x_1, x_2) ) = max{0, 2x_1 - 2} POL( U81_1(x_1) ) = max{0, -2} POL( nil ) = 0 POL( MARK_1(x_1) ) = 2 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: U41(X1, mark(X2)) -> U41(X1, X2) U41(mark(X1), X2) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) U51(X1, mark(X2)) -> U51(X1, X2) U51(mark(X1), X2) -> U51(X1, X2) U51(active(X1), X2) -> U51(X1, X2) U51(X1, active(X2)) -> U51(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) length(active(X)) -> length(X) length(mark(X)) -> length(X) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) ---------------------------------------- (157) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2))) MARK(U42(X)) -> MARK(X) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2))) MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2)) MARK(U52(X)) -> MARK(X) ACTIVE(U71(tt, L, N)) -> MARK(U72(isNat(N), L)) ACTIVE(U72(tt, L)) -> MARK(s(length(L))) MARK(U71(X1, X2, X3)) -> ACTIVE(U71(mark(X1), X2, X3)) MARK(U72(X1, X2)) -> ACTIVE(U72(mark(X1), X2)) MARK(s(X)) -> MARK(X) MARK(length(X)) -> ACTIVE(length(mark(X))) ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNat(V1), V2)) ACTIVE(length(cons(N, L))) -> MARK(U71(isNatList(L), L, N)) The TRS R consists of the following rules: active(zeros) -> mark(cons(0, zeros)) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatIList(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, L, N)) -> mark(U72(isNat(N), L)) active(U72(tt, L)) -> mark(s(length(L))) active(U81(tt)) -> mark(nil) active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N)) active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N)) active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(isNat(0)) -> mark(tt) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2)) active(length(nil)) -> mark(0) active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N)) active(take(0, IL)) -> mark(U81(isNatIList(IL))) active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N)) mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) mark(0) -> active(0) mark(U11(X)) -> active(U11(mark(X))) mark(tt) -> active(tt) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1, X2)) -> active(U51(mark(X1), X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil) -> active(nil) mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4)) mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4)) mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4)) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) cons(mark(X1), X2) -> cons(X1, X2) cons(X1, mark(X2)) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1), X2) -> U41(X1, X2) U41(X1, mark(X2)) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1), X2) -> U51(X1, X2) U51(X1, mark(X2)) -> U51(X1, X2) U51(active(X1), X2) -> U51(X1, X2) U51(X1, active(X2)) -> U51(X1, X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4) U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4) U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4) U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4) U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4) U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4) take(mark(X1), X2) -> take(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (158) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MARK(U51(X1, X2)) -> ACTIVE(U51(mark(X1), X2)) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( ACTIVE_1(x_1) ) = max{0, -2} POL( U41_2(x_1, x_2) ) = max{0, -2} POL( U51_2(x_1, x_2) ) = 2 POL( U71_3(x_1, ..., x_3) ) = 0 POL( U72_2(x_1, x_2) ) = max{0, -2} POL( length_1(x_1) ) = max{0, -2} POL( mark_1(x_1) ) = 2x_1 + 2 POL( cons_2(x_1, x_2) ) = max{0, -2} POL( active_1(x_1) ) = x_1 + 2 POL( zeros ) = 0 POL( 0 ) = 0 POL( tt ) = 2 POL( U42_1(x_1) ) = 2x_1 + 1 POL( isNatIList_1(x_1) ) = 0 POL( U11_1(x_1) ) = max{0, -2} POL( U52_1(x_1) ) = 2x_1 + 1 POL( isNatList_1(x_1) ) = 0 POL( U21_1(x_1) ) = 2 POL( U61_2(x_1, x_2) ) = max{0, 2x_2 - 2} POL( U62_1(x_1) ) = 2 POL( U31_1(x_1) ) = max{0, -2} POL( isNat_1(x_1) ) = 0 POL( s_1(x_1) ) = x_1 + 1 POL( U91_4(x_1, ..., x_4) ) = max{0, x_2 + x_3 - 2} POL( U92_4(x_1, ..., x_4) ) = x_1 + x_2 + x_3 + x_4 + 2 POL( U93_4(x_1, ..., x_4) ) = x_1 + x_3 + 2x_4 + 2 POL( take_2(x_1, x_2) ) = 0 POL( U81_1(x_1) ) = max{0, -2} POL( nil ) = 2 POL( MARK_1(x_1) ) = max{0, x_1 - 1} The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: U41(X1, mark(X2)) -> U41(X1, X2) U41(mark(X1), X2) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U72(X1, mark(X2)) -> U72(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) length(active(X)) -> length(X) length(mark(X)) -> length(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) ---------------------------------------- (159) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2))) MARK(U42(X)) -> MARK(X) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2))) MARK(U52(X)) -> MARK(X) ACTIVE(U71(tt, L, N)) -> MARK(U72(isNat(N), L)) ACTIVE(U72(tt, L)) -> MARK(s(length(L))) MARK(U71(X1, X2, X3)) -> ACTIVE(U71(mark(X1), X2, X3)) MARK(U72(X1, X2)) -> ACTIVE(U72(mark(X1), X2)) MARK(s(X)) -> MARK(X) MARK(length(X)) -> ACTIVE(length(mark(X))) ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNat(V1), V2)) ACTIVE(length(cons(N, L))) -> MARK(U71(isNatList(L), L, N)) The TRS R consists of the following rules: active(zeros) -> mark(cons(0, zeros)) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatIList(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, L, N)) -> mark(U72(isNat(N), L)) active(U72(tt, L)) -> mark(s(length(L))) active(U81(tt)) -> mark(nil) active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N)) active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N)) active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(isNat(0)) -> mark(tt) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2)) active(length(nil)) -> mark(0) active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N)) active(take(0, IL)) -> mark(U81(isNatIList(IL))) active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N)) mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) mark(0) -> active(0) mark(U11(X)) -> active(U11(mark(X))) mark(tt) -> active(tt) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1, X2)) -> active(U51(mark(X1), X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil) -> active(nil) mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4)) mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4)) mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4)) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) cons(mark(X1), X2) -> cons(X1, X2) cons(X1, mark(X2)) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1), X2) -> U41(X1, X2) U41(X1, mark(X2)) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1), X2) -> U51(X1, X2) U51(X1, mark(X2)) -> U51(X1, X2) U51(active(X1), X2) -> U51(X1, X2) U51(X1, active(X2)) -> U51(X1, X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4) U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4) U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4) U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4) U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4) U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4) take(mark(X1), X2) -> take(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (160) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. ACTIVE(U51(tt, V2)) -> MARK(U52(isNatList(V2))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( ACTIVE_1(x_1) ) = max{0, x_1 - 2} POL( U41_2(x_1, x_2) ) = max{0, -2} POL( U71_3(x_1, ..., x_3) ) = 2 POL( U72_2(x_1, x_2) ) = max{0, -2} POL( length_1(x_1) ) = 2 POL( mark_1(x_1) ) = max{0, 2x_1 - 2} POL( cons_2(x_1, x_2) ) = max{0, 2x_2 - 2} POL( active_1(x_1) ) = 2 POL( zeros ) = 2 POL( 0 ) = 0 POL( tt ) = 2 POL( U42_1(x_1) ) = max{0, 2x_1 - 2} POL( isNatIList_1(x_1) ) = 0 POL( U11_1(x_1) ) = x_1 + 2 POL( U51_2(x_1, x_2) ) = 2x_1 + 2 POL( U52_1(x_1) ) = max{0, 2x_1 - 2} POL( isNatList_1(x_1) ) = 2x_1 + 2 POL( U21_1(x_1) ) = 2 POL( U61_2(x_1, x_2) ) = max{0, 2x_1 - 2} POL( U62_1(x_1) ) = max{0, 2x_1 - 2} POL( U31_1(x_1) ) = max{0, x_1 - 2} POL( isNat_1(x_1) ) = 2x_1 POL( s_1(x_1) ) = 0 POL( U91_4(x_1, ..., x_4) ) = max{0, 2x_2 + x_3 + 2x_4 - 1} POL( U92_4(x_1, ..., x_4) ) = max{0, x_2 + 2x_3 - 2} POL( U93_4(x_1, ..., x_4) ) = 2x_1 + x_3 + 2 POL( take_2(x_1, x_2) ) = 2x_1 + 2 POL( U81_1(x_1) ) = x_1 POL( nil ) = 2 POL( MARK_1(x_1) ) = max{0, -2} The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: U41(X1, mark(X2)) -> U41(X1, X2) U41(mark(X1), X2) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) U72(X1, mark(X2)) -> U72(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) length(active(X)) -> length(X) length(mark(X)) -> length(X) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) ---------------------------------------- (161) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2))) MARK(U42(X)) -> MARK(X) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) MARK(U52(X)) -> MARK(X) ACTIVE(U71(tt, L, N)) -> MARK(U72(isNat(N), L)) ACTIVE(U72(tt, L)) -> MARK(s(length(L))) MARK(U71(X1, X2, X3)) -> ACTIVE(U71(mark(X1), X2, X3)) MARK(U72(X1, X2)) -> ACTIVE(U72(mark(X1), X2)) MARK(s(X)) -> MARK(X) MARK(length(X)) -> ACTIVE(length(mark(X))) ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNat(V1), V2)) ACTIVE(length(cons(N, L))) -> MARK(U71(isNatList(L), L, N)) The TRS R consists of the following rules: active(zeros) -> mark(cons(0, zeros)) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatIList(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, L, N)) -> mark(U72(isNat(N), L)) active(U72(tt, L)) -> mark(s(length(L))) active(U81(tt)) -> mark(nil) active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N)) active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N)) active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(isNat(0)) -> mark(tt) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2)) active(length(nil)) -> mark(0) active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N)) active(take(0, IL)) -> mark(U81(isNatIList(IL))) active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N)) mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) mark(0) -> active(0) mark(U11(X)) -> active(U11(mark(X))) mark(tt) -> active(tt) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1, X2)) -> active(U51(mark(X1), X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil) -> active(nil) mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4)) mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4)) mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4)) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) cons(mark(X1), X2) -> cons(X1, X2) cons(X1, mark(X2)) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1), X2) -> U41(X1, X2) U41(X1, mark(X2)) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1), X2) -> U51(X1, X2) U51(X1, mark(X2)) -> U51(X1, X2) U51(active(X1), X2) -> U51(X1, X2) U51(X1, active(X2)) -> U51(X1, X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4) U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4) U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4) U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4) U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4) U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4) take(mark(X1), X2) -> take(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (162) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MARK(U52(X)) -> MARK(X) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( ACTIVE_1(x_1) ) = max{0, -2} POL( U41_2(x_1, x_2) ) = max{0, -2} POL( U71_3(x_1, ..., x_3) ) = 1 POL( U72_2(x_1, x_2) ) = max{0, -2} POL( length_1(x_1) ) = max{0, -2} POL( mark_1(x_1) ) = 2x_1 POL( cons_2(x_1, x_2) ) = max{0, 2x_2 - 2} POL( active_1(x_1) ) = x_1 POL( zeros ) = 2 POL( 0 ) = 0 POL( tt ) = 0 POL( U42_1(x_1) ) = x_1 + 1 POL( isNatIList_1(x_1) ) = 0 POL( U11_1(x_1) ) = 1 POL( U51_2(x_1, x_2) ) = max{0, x_1 - 2} POL( U52_1(x_1) ) = 2x_1 + 2 POL( isNatList_1(x_1) ) = 2 POL( U21_1(x_1) ) = max{0, 2x_1 - 2} POL( U61_2(x_1, x_2) ) = 2x_1 + 2 POL( U62_1(x_1) ) = max{0, x_1 - 2} POL( U31_1(x_1) ) = 1 POL( isNat_1(x_1) ) = 0 POL( s_1(x_1) ) = x_1 + 1 POL( U91_4(x_1, ..., x_4) ) = 2 POL( U92_4(x_1, ..., x_4) ) = 2 POL( U93_4(x_1, ..., x_4) ) = 2x_3 + 2x_4 + 2 POL( take_2(x_1, x_2) ) = max{0, 2x_1 - 2} POL( U81_1(x_1) ) = x_1 POL( nil ) = 0 POL( MARK_1(x_1) ) = max{0, 2x_1 - 2} The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: U41(X1, mark(X2)) -> U41(X1, X2) U41(mark(X1), X2) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U72(X1, mark(X2)) -> U72(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) length(active(X)) -> length(X) length(mark(X)) -> length(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) ---------------------------------------- (163) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2))) MARK(U42(X)) -> MARK(X) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) ACTIVE(U71(tt, L, N)) -> MARK(U72(isNat(N), L)) ACTIVE(U72(tt, L)) -> MARK(s(length(L))) MARK(U71(X1, X2, X3)) -> ACTIVE(U71(mark(X1), X2, X3)) MARK(U72(X1, X2)) -> ACTIVE(U72(mark(X1), X2)) MARK(s(X)) -> MARK(X) MARK(length(X)) -> ACTIVE(length(mark(X))) ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNat(V1), V2)) ACTIVE(length(cons(N, L))) -> MARK(U71(isNatList(L), L, N)) The TRS R consists of the following rules: active(zeros) -> mark(cons(0, zeros)) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatIList(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, L, N)) -> mark(U72(isNat(N), L)) active(U72(tt, L)) -> mark(s(length(L))) active(U81(tt)) -> mark(nil) active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N)) active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N)) active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(isNat(0)) -> mark(tt) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2)) active(length(nil)) -> mark(0) active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N)) active(take(0, IL)) -> mark(U81(isNatIList(IL))) active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N)) mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) mark(0) -> active(0) mark(U11(X)) -> active(U11(mark(X))) mark(tt) -> active(tt) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1, X2)) -> active(U51(mark(X1), X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil) -> active(nil) mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4)) mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4)) mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4)) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) cons(mark(X1), X2) -> cons(X1, X2) cons(X1, mark(X2)) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1), X2) -> U41(X1, X2) U41(X1, mark(X2)) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1), X2) -> U51(X1, X2) U51(X1, mark(X2)) -> U51(X1, X2) U51(active(X1), X2) -> U51(X1, X2) U51(X1, active(X2)) -> U51(X1, X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4) U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4) U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4) U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4) U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4) U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4) take(mark(X1), X2) -> take(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (164) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. ACTIVE(isNatIList(cons(V1, V2))) -> MARK(U41(isNat(V1), V2)) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( ACTIVE_1(x_1) ) = max{0, x_1 - 1} POL( U41_2(x_1, x_2) ) = 2x_2 + 2 POL( U71_3(x_1, ..., x_3) ) = max{0, -2} POL( U72_2(x_1, x_2) ) = max{0, -2} POL( length_1(x_1) ) = max{0, -2} POL( mark_1(x_1) ) = x_1 POL( cons_2(x_1, x_2) ) = 2x_1 + 2x_2 + 1 POL( active_1(x_1) ) = x_1 + 2 POL( zeros ) = 0 POL( 0 ) = 0 POL( tt ) = 0 POL( U42_1(x_1) ) = x_1 + 1 POL( isNatIList_1(x_1) ) = 2x_1 + 1 POL( U11_1(x_1) ) = max{0, x_1 - 2} POL( U51_2(x_1, x_2) ) = max{0, x_2 - 2} POL( U52_1(x_1) ) = max{0, -2} POL( isNatList_1(x_1) ) = 0 POL( U21_1(x_1) ) = max{0, x_1 - 2} POL( U61_2(x_1, x_2) ) = max{0, x_1 - 1} POL( U62_1(x_1) ) = max{0, 2x_1 - 2} POL( U31_1(x_1) ) = x_1 POL( isNat_1(x_1) ) = 2x_1 POL( s_1(x_1) ) = 2x_1 + 1 POL( U91_4(x_1, ..., x_4) ) = max{0, 2x_4 - 1} POL( U92_4(x_1, ..., x_4) ) = max{0, x_1 + x_2 + 2x_4 - 2} POL( U93_4(x_1, ..., x_4) ) = 2x_2 + 2x_3 + 1 POL( take_2(x_1, x_2) ) = 0 POL( U81_1(x_1) ) = max{0, -2} POL( nil ) = 0 POL( MARK_1(x_1) ) = max{0, x_1 - 1} The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: U41(X1, mark(X2)) -> U41(X1, X2) U41(mark(X1), X2) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U72(X1, mark(X2)) -> U72(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) length(active(X)) -> length(X) length(mark(X)) -> length(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) ---------------------------------------- (165) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2))) MARK(U42(X)) -> MARK(X) MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) ACTIVE(U71(tt, L, N)) -> MARK(U72(isNat(N), L)) ACTIVE(U72(tt, L)) -> MARK(s(length(L))) MARK(U71(X1, X2, X3)) -> ACTIVE(U71(mark(X1), X2, X3)) MARK(U72(X1, X2)) -> ACTIVE(U72(mark(X1), X2)) MARK(s(X)) -> MARK(X) MARK(length(X)) -> ACTIVE(length(mark(X))) ACTIVE(length(cons(N, L))) -> MARK(U71(isNatList(L), L, N)) The TRS R consists of the following rules: active(zeros) -> mark(cons(0, zeros)) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatIList(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, L, N)) -> mark(U72(isNat(N), L)) active(U72(tt, L)) -> mark(s(length(L))) active(U81(tt)) -> mark(nil) active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N)) active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N)) active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(isNat(0)) -> mark(tt) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2)) active(length(nil)) -> mark(0) active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N)) active(take(0, IL)) -> mark(U81(isNatIList(IL))) active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N)) mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) mark(0) -> active(0) mark(U11(X)) -> active(U11(mark(X))) mark(tt) -> active(tt) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1, X2)) -> active(U51(mark(X1), X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil) -> active(nil) mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4)) mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4)) mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4)) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) cons(mark(X1), X2) -> cons(X1, X2) cons(X1, mark(X2)) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1), X2) -> U41(X1, X2) U41(X1, mark(X2)) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1), X2) -> U51(X1, X2) U51(X1, mark(X2)) -> U51(X1, X2) U51(active(X1), X2) -> U51(X1, X2) U51(X1, active(X2)) -> U51(X1, X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4) U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4) U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4) U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4) U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4) U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4) take(mark(X1), X2) -> take(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (166) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MARK(isNatIList(X)) -> ACTIVE(isNatIList(X)) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( ACTIVE_1(x_1) ) = max{0, x_1 - 1} POL( U41_2(x_1, x_2) ) = 2 POL( U71_3(x_1, ..., x_3) ) = 2 POL( U72_2(x_1, x_2) ) = 2 POL( length_1(x_1) ) = 2 POL( mark_1(x_1) ) = max{0, x_1 - 2} POL( cons_2(x_1, x_2) ) = max{0, 2x_1 - 2} POL( active_1(x_1) ) = max{0, 2x_1 - 2} POL( zeros ) = 2 POL( 0 ) = 0 POL( tt ) = 0 POL( U42_1(x_1) ) = 0 POL( isNatIList_1(x_1) ) = 0 POL( U11_1(x_1) ) = max{0, x_1 - 2} POL( U51_2(x_1, x_2) ) = max{0, x_2 - 2} POL( U52_1(x_1) ) = max{0, 2x_1 - 2} POL( isNatList_1(x_1) ) = 0 POL( U21_1(x_1) ) = max{0, x_1 - 2} POL( U61_2(x_1, x_2) ) = x_1 + 2x_2 + 2 POL( U62_1(x_1) ) = 2 POL( U31_1(x_1) ) = max{0, x_1 - 2} POL( isNat_1(x_1) ) = 0 POL( s_1(x_1) ) = max{0, -2} POL( U91_4(x_1, ..., x_4) ) = max{0, 2x_1 + 2x_2 + x_3 - 2} POL( U92_4(x_1, ..., x_4) ) = max{0, x_1 + 2x_4 - 2} POL( U93_4(x_1, ..., x_4) ) = max{0, x_1 + x_2 - 2} POL( take_2(x_1, x_2) ) = max{0, -2} POL( U81_1(x_1) ) = max{0, 2x_1 - 2} POL( nil ) = 0 POL( MARK_1(x_1) ) = 1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: U41(X1, mark(X2)) -> U41(X1, X2) U41(mark(X1), X2) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) U72(X1, mark(X2)) -> U72(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) length(active(X)) -> length(X) length(mark(X)) -> length(X) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) ---------------------------------------- (167) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2))) MARK(U42(X)) -> MARK(X) ACTIVE(U71(tt, L, N)) -> MARK(U72(isNat(N), L)) ACTIVE(U72(tt, L)) -> MARK(s(length(L))) MARK(U71(X1, X2, X3)) -> ACTIVE(U71(mark(X1), X2, X3)) MARK(U72(X1, X2)) -> ACTIVE(U72(mark(X1), X2)) MARK(s(X)) -> MARK(X) MARK(length(X)) -> ACTIVE(length(mark(X))) ACTIVE(length(cons(N, L))) -> MARK(U71(isNatList(L), L, N)) The TRS R consists of the following rules: active(zeros) -> mark(cons(0, zeros)) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatIList(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, L, N)) -> mark(U72(isNat(N), L)) active(U72(tt, L)) -> mark(s(length(L))) active(U81(tt)) -> mark(nil) active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N)) active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N)) active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(isNat(0)) -> mark(tt) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2)) active(length(nil)) -> mark(0) active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N)) active(take(0, IL)) -> mark(U81(isNatIList(IL))) active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N)) mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) mark(0) -> active(0) mark(U11(X)) -> active(U11(mark(X))) mark(tt) -> active(tt) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1, X2)) -> active(U51(mark(X1), X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil) -> active(nil) mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4)) mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4)) mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4)) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) cons(mark(X1), X2) -> cons(X1, X2) cons(X1, mark(X2)) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1), X2) -> U41(X1, X2) U41(X1, mark(X2)) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1), X2) -> U51(X1, X2) U51(X1, mark(X2)) -> U51(X1, X2) U51(active(X1), X2) -> U51(X1, X2) U51(X1, active(X2)) -> U51(X1, X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4) U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4) U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4) U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4) U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4) U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4) take(mark(X1), X2) -> take(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (168) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MARK(U41(X1, X2)) -> ACTIVE(U41(mark(X1), X2)) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( ACTIVE_1(x_1) ) = max{0, -2} POL( U41_2(x_1, x_2) ) = x_2 + 2 POL( U71_3(x_1, ..., x_3) ) = max{0, -2} POL( U72_2(x_1, x_2) ) = max{0, -2} POL( length_1(x_1) ) = 0 POL( mark_1(x_1) ) = x_1 + 2 POL( cons_2(x_1, x_2) ) = max{0, x_2 - 2} POL( active_1(x_1) ) = x_1 POL( zeros ) = 1 POL( 0 ) = 0 POL( tt ) = 0 POL( U42_1(x_1) ) = 2x_1 POL( isNatIList_1(x_1) ) = 0 POL( U11_1(x_1) ) = 2 POL( U51_2(x_1, x_2) ) = max{0, x_1 - 2} POL( U52_1(x_1) ) = max{0, x_1 - 2} POL( isNatList_1(x_1) ) = 0 POL( U21_1(x_1) ) = max{0, 2x_1 - 2} POL( U61_2(x_1, x_2) ) = 2 POL( U62_1(x_1) ) = 2 POL( U31_1(x_1) ) = max{0, 2x_1 - 2} POL( isNat_1(x_1) ) = 0 POL( s_1(x_1) ) = x_1 POL( U91_4(x_1, ..., x_4) ) = x_3 + 2x_4 + 2 POL( U92_4(x_1, ..., x_4) ) = 2x_3 + 2 POL( U93_4(x_1, ..., x_4) ) = 2 POL( take_2(x_1, x_2) ) = 2 POL( U81_1(x_1) ) = max{0, -2} POL( nil ) = 0 POL( MARK_1(x_1) ) = x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U72(X1, mark(X2)) -> U72(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) length(active(X)) -> length(X) length(mark(X)) -> length(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) ---------------------------------------- (169) Obligation: Q DP problem: The TRS P consists of the following rules: ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2))) MARK(U42(X)) -> MARK(X) ACTIVE(U71(tt, L, N)) -> MARK(U72(isNat(N), L)) ACTIVE(U72(tt, L)) -> MARK(s(length(L))) MARK(U71(X1, X2, X3)) -> ACTIVE(U71(mark(X1), X2, X3)) MARK(U72(X1, X2)) -> ACTIVE(U72(mark(X1), X2)) MARK(s(X)) -> MARK(X) MARK(length(X)) -> ACTIVE(length(mark(X))) ACTIVE(length(cons(N, L))) -> MARK(U71(isNatList(L), L, N)) The TRS R consists of the following rules: active(zeros) -> mark(cons(0, zeros)) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatIList(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, L, N)) -> mark(U72(isNat(N), L)) active(U72(tt, L)) -> mark(s(length(L))) active(U81(tt)) -> mark(nil) active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N)) active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N)) active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(isNat(0)) -> mark(tt) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2)) active(length(nil)) -> mark(0) active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N)) active(take(0, IL)) -> mark(U81(isNatIList(IL))) active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N)) mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) mark(0) -> active(0) mark(U11(X)) -> active(U11(mark(X))) mark(tt) -> active(tt) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1, X2)) -> active(U51(mark(X1), X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil) -> active(nil) mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4)) mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4)) mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4)) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) cons(mark(X1), X2) -> cons(X1, X2) cons(X1, mark(X2)) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1), X2) -> U41(X1, X2) U41(X1, mark(X2)) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1), X2) -> U51(X1, X2) U51(X1, mark(X2)) -> U51(X1, X2) U51(active(X1), X2) -> U51(X1, X2) U51(X1, active(X2)) -> U51(X1, X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4) U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4) U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4) U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4) U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4) U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4) take(mark(X1), X2) -> take(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (170) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. ACTIVE(U41(tt, V2)) -> MARK(U42(isNatIList(V2))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( MARK_1(x_1) ) = max{0, -2} POL( U42_1(x_1) ) = max{0, x_1 - 2} POL( ACTIVE_1(x_1) ) = max{0, x_1 - 2} POL( U71_3(x_1, ..., x_3) ) = max{0, -2} POL( active_1(x_1) ) = 2 POL( U72_2(x_1, x_2) ) = max{0, -2} POL( length_1(x_1) ) = 2 POL( mark_1(x_1) ) = max{0, -2} POL( U62_1(x_1) ) = max{0, -2} POL( U81_1(x_1) ) = 2 POL( U91_4(x_1, ..., x_4) ) = max{0, x_2 - 2} POL( isNatIList_1(x_1) ) = 0 POL( U21_1(x_1) ) = 2x_1 + 2 POL( U41_2(x_1, x_2) ) = 2x_1 + 1 POL( U51_2(x_1, x_2) ) = 2x_2 + 2 POL( U61_2(x_1, x_2) ) = 2x_2 + 2 POL( U92_4(x_1, ..., x_4) ) = max{0, 2x_4 - 2} POL( U93_4(x_1, ..., x_4) ) = max{0, 2x_2 + x_3 - 2} POL( isNat_1(x_1) ) = 0 POL( s_1(x_1) ) = 0 POL( cons_2(x_1, x_2) ) = max{0, x_1 + 2x_2 - 1} POL( zeros ) = 0 POL( 0 ) = 0 POL( tt ) = 2 POL( U11_1(x_1) ) = 1 POL( U52_1(x_1) ) = max{0, 2x_1 - 2} POL( isNatList_1(x_1) ) = 0 POL( U31_1(x_1) ) = 2 POL( take_2(x_1, x_2) ) = 0 POL( nil ) = 0 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: U72(X1, mark(X2)) -> U72(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) length(active(X)) -> length(X) length(mark(X)) -> length(X) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) ---------------------------------------- (171) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(U42(X)) -> MARK(X) ACTIVE(U71(tt, L, N)) -> MARK(U72(isNat(N), L)) ACTIVE(U72(tt, L)) -> MARK(s(length(L))) MARK(U71(X1, X2, X3)) -> ACTIVE(U71(mark(X1), X2, X3)) MARK(U72(X1, X2)) -> ACTIVE(U72(mark(X1), X2)) MARK(s(X)) -> MARK(X) MARK(length(X)) -> ACTIVE(length(mark(X))) ACTIVE(length(cons(N, L))) -> MARK(U71(isNatList(L), L, N)) The TRS R consists of the following rules: active(zeros) -> mark(cons(0, zeros)) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatIList(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, L, N)) -> mark(U72(isNat(N), L)) active(U72(tt, L)) -> mark(s(length(L))) active(U81(tt)) -> mark(nil) active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N)) active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N)) active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(isNat(0)) -> mark(tt) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2)) active(length(nil)) -> mark(0) active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N)) active(take(0, IL)) -> mark(U81(isNatIList(IL))) active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N)) mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) mark(0) -> active(0) mark(U11(X)) -> active(U11(mark(X))) mark(tt) -> active(tt) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1, X2)) -> active(U51(mark(X1), X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil) -> active(nil) mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4)) mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4)) mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4)) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) cons(mark(X1), X2) -> cons(X1, X2) cons(X1, mark(X2)) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1), X2) -> U41(X1, X2) U41(X1, mark(X2)) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1), X2) -> U51(X1, X2) U51(X1, mark(X2)) -> U51(X1, X2) U51(active(X1), X2) -> U51(X1, X2) U51(X1, active(X2)) -> U51(X1, X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4) U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4) U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4) U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4) U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4) U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4) take(mark(X1), X2) -> take(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (172) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MARK(U42(X)) -> MARK(X) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( MARK_1(x_1) ) = max{0, 2x_1 - 2} POL( U72_2(x_1, x_2) ) = max{0, -2} POL( ACTIVE_1(x_1) ) = 0 POL( U71_3(x_1, ..., x_3) ) = 1 POL( active_1(x_1) ) = 2x_1 POL( mark_1(x_1) ) = 2x_1 POL( length_1(x_1) ) = 0 POL( U21_1(x_1) ) = 2 POL( U41_2(x_1, x_2) ) = 2 POL( U51_2(x_1, x_2) ) = max{0, x_1 + 2x_2 - 2} POL( U61_2(x_1, x_2) ) = x_2 + 2 POL( U92_4(x_1, ..., x_4) ) = 2 POL( U93_4(x_1, ..., x_4) ) = 2x_2 + x_3 + 2 POL( isNat_1(x_1) ) = 0 POL( s_1(x_1) ) = x_1 + 1 POL( cons_2(x_1, x_2) ) = max{0, 2x_1 - 2} POL( zeros ) = 0 POL( 0 ) = 0 POL( tt ) = 0 POL( U42_1(x_1) ) = 2x_1 + 2 POL( isNatIList_1(x_1) ) = 2x_1 + 1 POL( U11_1(x_1) ) = 2 POL( U52_1(x_1) ) = max{0, x_1 - 2} POL( isNatList_1(x_1) ) = x_1 + 1 POL( U62_1(x_1) ) = 2 POL( U31_1(x_1) ) = 2 POL( U91_4(x_1, ..., x_4) ) = max{0, x_1 + x_2 + 2x_4 - 1} POL( take_2(x_1, x_2) ) = 2 POL( U81_1(x_1) ) = 1 POL( nil ) = 0 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: U72(X1, mark(X2)) -> U72(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) length(active(X)) -> length(X) length(mark(X)) -> length(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) ---------------------------------------- (173) Obligation: Q DP problem: The TRS P consists of the following rules: ACTIVE(U71(tt, L, N)) -> MARK(U72(isNat(N), L)) ACTIVE(U72(tt, L)) -> MARK(s(length(L))) MARK(U71(X1, X2, X3)) -> ACTIVE(U71(mark(X1), X2, X3)) MARK(U72(X1, X2)) -> ACTIVE(U72(mark(X1), X2)) MARK(s(X)) -> MARK(X) MARK(length(X)) -> ACTIVE(length(mark(X))) ACTIVE(length(cons(N, L))) -> MARK(U71(isNatList(L), L, N)) The TRS R consists of the following rules: active(zeros) -> mark(cons(0, zeros)) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatIList(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, L, N)) -> mark(U72(isNat(N), L)) active(U72(tt, L)) -> mark(s(length(L))) active(U81(tt)) -> mark(nil) active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N)) active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N)) active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(isNat(0)) -> mark(tt) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2)) active(length(nil)) -> mark(0) active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N)) active(take(0, IL)) -> mark(U81(isNatIList(IL))) active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N)) mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) mark(0) -> active(0) mark(U11(X)) -> active(U11(mark(X))) mark(tt) -> active(tt) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1, X2)) -> active(U51(mark(X1), X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil) -> active(nil) mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4)) mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4)) mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4)) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) cons(mark(X1), X2) -> cons(X1, X2) cons(X1, mark(X2)) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1), X2) -> U41(X1, X2) U41(X1, mark(X2)) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1), X2) -> U51(X1, X2) U51(X1, mark(X2)) -> U51(X1, X2) U51(active(X1), X2) -> U51(X1, X2) U51(X1, active(X2)) -> U51(X1, X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4) U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4) U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4) U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4) U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4) U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4) take(mark(X1), X2) -> take(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (174) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MARK(s(X)) -> MARK(X) The remaining pairs can at least be oriented weakly. Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]: <<< POL(ACTIVE(x_1)) = [[1A]] + [[0A]] * x_1 >>> <<< POL(U71(x_1, x_2, x_3)) = [[-I]] + [[0A]] * x_1 + [[1A]] * x_2 + [[0A]] * x_3 >>> <<< POL(tt) = [[2A]] >>> <<< POL(MARK(x_1)) = [[1A]] + [[0A]] * x_1 >>> <<< POL(U72(x_1, x_2)) = [[0A]] + [[0A]] * x_1 + [[1A]] * x_2 >>> <<< POL(isNat(x_1)) = [[2A]] + [[0A]] * x_1 >>> <<< POL(s(x_1)) = [[2A]] + [[1A]] * x_1 >>> <<< POL(length(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(mark(x_1)) = [[1A]] + [[0A]] * x_1 >>> <<< POL(cons(x_1, x_2)) = [[-I]] + [[0A]] * x_1 + [[1A]] * x_2 >>> <<< POL(isNatList(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(active(x_1)) = [[1A]] + [[0A]] * x_1 >>> <<< POL(zeros) = [[0A]] >>> <<< POL(0) = [[0A]] >>> <<< POL(U41(x_1, x_2)) = [[2A]] + [[-I]] * x_1 + [[-I]] * x_2 >>> <<< POL(U42(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(isNatIList(x_1)) = [[2A]] + [[-I]] * x_1 >>> <<< POL(U11(x_1)) = [[2A]] + [[-I]] * x_1 >>> <<< POL(U51(x_1, x_2)) = [[0A]] + [[-I]] * x_1 + [[1A]] * x_2 >>> <<< POL(U52(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(U21(x_1)) = [[2A]] + [[-I]] * x_1 >>> <<< POL(U61(x_1, x_2)) = [[3A]] + [[0A]] * x_1 + [[-I]] * x_2 >>> <<< POL(U62(x_1)) = [[-I]] + [[0A]] * x_1 >>> <<< POL(U31(x_1)) = [[2A]] + [[-I]] * x_1 >>> <<< POL(U91(x_1, x_2, x_3, x_4)) = [[3A]] + [[2A]] * x_1 + [[3A]] * x_2 + [[3A]] * x_3 + [[0A]] * x_4 >>> <<< POL(U92(x_1, x_2, x_3, x_4)) = [[4A]] + [[-I]] * x_1 + [[3A]] * x_2 + [[3A]] * x_3 + [[0A]] * x_4 >>> <<< POL(U93(x_1, x_2, x_3, x_4)) = [[4A]] + [[0A]] * x_1 + [[3A]] * x_2 + [[3A]] * x_3 + [[0A]] * x_4 >>> <<< POL(take(x_1, x_2)) = [[3A]] + [[2A]] * x_1 + [[2A]] * x_2 >>> <<< POL(U81(x_1)) = [[0A]] + [[0A]] * x_1 >>> <<< POL(nil) = [[2A]] >>> The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: isNat(active(X)) -> isNat(X) isNat(mark(X)) -> isNat(X) U72(X1, mark(X2)) -> U72(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) length(active(X)) -> length(X) length(mark(X)) -> length(X) s(active(X)) -> s(X) s(mark(X)) -> s(X) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) active(zeros) -> mark(cons(0, zeros)) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) mark(zeros) -> active(zeros) mark(U11(X)) -> active(U11(mark(X))) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) mark(U21(X)) -> active(U21(mark(X))) active(U61(tt, V2)) -> mark(U62(isNatIList(V2))) mark(U31(X)) -> active(U31(mark(X))) active(U71(tt, L, N)) -> mark(U72(isNat(N), L)) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) active(U72(tt, L)) -> mark(s(length(L))) mark(U42(X)) -> active(U42(mark(X))) active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N)) mark(isNatIList(X)) -> active(isNatIList(X)) active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N)) mark(U51(X1, X2)) -> active(U51(mark(X1), X2)) active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) mark(U52(X)) -> active(U52(mark(X))) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) mark(isNatList(X)) -> active(isNatList(X)) active(isNat(s(V1))) -> mark(U21(isNat(V1))) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) active(isNatIList(V)) -> mark(U31(isNatList(V))) mark(U62(X)) -> active(U62(mark(X))) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3)) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2)) mark(isNat(X)) -> active(isNat(X)) active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N)) mark(s(X)) -> active(s(mark(X))) active(take(0, IL)) -> mark(U81(isNatIList(IL))) mark(length(X)) -> active(length(mark(X))) active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N)) mark(U81(X)) -> active(U81(mark(X))) mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4)) mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4)) mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4)) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) mark(0) -> active(0) mark(tt) -> active(tt) mark(nil) -> active(nil) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) isNatList(active(X)) -> isNatList(X) isNatList(mark(X)) -> isNatList(X) cons(X1, mark(X2)) -> cons(X1, X2) cons(mark(X1), X2) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U42(tt)) -> mark(tt) active(U52(tt)) -> mark(tt) active(U62(tt)) -> mark(tt) active(U81(tt)) -> mark(nil) active(isNat(0)) -> mark(tt) active(isNatIList(zeros)) -> mark(tt) active(isNatList(nil)) -> mark(tt) active(length(nil)) -> mark(0) isNatIList(active(X)) -> isNatIList(X) isNatIList(mark(X)) -> isNatIList(X) U42(active(X)) -> U42(X) U42(mark(X)) -> U42(X) U11(active(X)) -> U11(X) U11(mark(X)) -> U11(X) U52(active(X)) -> U52(X) U52(mark(X)) -> U52(X) U21(active(X)) -> U21(X) U21(mark(X)) -> U21(X) U62(active(X)) -> U62(X) U62(mark(X)) -> U62(X) U31(active(X)) -> U31(X) U31(mark(X)) -> U31(X) U41(X1, mark(X2)) -> U41(X1, X2) U41(mark(X1), X2) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4) U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4) U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4) U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4) U51(X1, mark(X2)) -> U51(X1, X2) U51(mark(X1), X2) -> U51(X1, X2) U51(active(X1), X2) -> U51(X1, X2) U51(X1, active(X2)) -> U51(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(mark(X1), X2) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(mark(X1), X2) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U81(active(X)) -> U81(X) U81(mark(X)) -> U81(X) U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4) U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4) ---------------------------------------- (175) Obligation: Q DP problem: The TRS P consists of the following rules: ACTIVE(U71(tt, L, N)) -> MARK(U72(isNat(N), L)) ACTIVE(U72(tt, L)) -> MARK(s(length(L))) MARK(U71(X1, X2, X3)) -> ACTIVE(U71(mark(X1), X2, X3)) MARK(U72(X1, X2)) -> ACTIVE(U72(mark(X1), X2)) MARK(length(X)) -> ACTIVE(length(mark(X))) ACTIVE(length(cons(N, L))) -> MARK(U71(isNatList(L), L, N)) The TRS R consists of the following rules: active(zeros) -> mark(cons(0, zeros)) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatIList(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, L, N)) -> mark(U72(isNat(N), L)) active(U72(tt, L)) -> mark(s(length(L))) active(U81(tt)) -> mark(nil) active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N)) active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N)) active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(isNat(0)) -> mark(tt) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2)) active(length(nil)) -> mark(0) active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N)) active(take(0, IL)) -> mark(U81(isNatIList(IL))) active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N)) mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) mark(0) -> active(0) mark(U11(X)) -> active(U11(mark(X))) mark(tt) -> active(tt) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1, X2)) -> active(U51(mark(X1), X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil) -> active(nil) mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4)) mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4)) mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4)) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) cons(mark(X1), X2) -> cons(X1, X2) cons(X1, mark(X2)) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1), X2) -> U41(X1, X2) U41(X1, mark(X2)) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1), X2) -> U51(X1, X2) U51(X1, mark(X2)) -> U51(X1, X2) U51(active(X1), X2) -> U51(X1, X2) U51(X1, active(X2)) -> U51(X1, X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4) U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4) U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4) U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4) U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4) U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4) take(mark(X1), X2) -> take(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (176) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. MARK(length(X)) -> ACTIVE(length(mark(X))) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial Order [NEGPOLO,POLO] with Interpretation: POL( MARK_1(x_1) ) = x_1 + 2 POL( U72_2(x_1, x_2) ) = 0 POL( ACTIVE_1(x_1) ) = 2 POL( U71_3(x_1, ..., x_3) ) = max{0, -2} POL( active_1(x_1) ) = 2 POL( mark_1(x_1) ) = max{0, 2x_1 - 2} POL( length_1(x_1) ) = 2 POL( U21_1(x_1) ) = 2x_1 + 2 POL( U41_2(x_1, x_2) ) = max{0, 2x_1 - 2} POL( U51_2(x_1, x_2) ) = max{0, x_1 - 2} POL( U61_2(x_1, x_2) ) = max{0, 2x_2 - 2} POL( U92_4(x_1, ..., x_4) ) = max{0, 2x_3 + 2x_4 - 2} POL( U93_4(x_1, ..., x_4) ) = max{0, x_1 + x_2 - 2} POL( isNat_1(x_1) ) = x_1 POL( s_1(x_1) ) = max{0, -2} POL( cons_2(x_1, x_2) ) = max{0, 2x_1 - 2} POL( zeros ) = 0 POL( 0 ) = 0 POL( tt ) = 0 POL( U42_1(x_1) ) = max{0, 2x_1 - 2} POL( isNatIList_1(x_1) ) = 2x_1 POL( U11_1(x_1) ) = 2x_1 + 1 POL( U52_1(x_1) ) = 2x_1 + 2 POL( isNatList_1(x_1) ) = 2 POL( U62_1(x_1) ) = max{0, 2x_1 - 2} POL( U31_1(x_1) ) = max{0, 2x_1 - 2} POL( U91_4(x_1, ..., x_4) ) = max{0, 2x_3 + 2x_4 - 2} POL( take_2(x_1, x_2) ) = max{0, x_1 - 2} POL( U81_1(x_1) ) = 2 POL( nil ) = 0 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: U72(X1, mark(X2)) -> U72(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) ---------------------------------------- (177) Obligation: Q DP problem: The TRS P consists of the following rules: ACTIVE(U71(tt, L, N)) -> MARK(U72(isNat(N), L)) ACTIVE(U72(tt, L)) -> MARK(s(length(L))) MARK(U71(X1, X2, X3)) -> ACTIVE(U71(mark(X1), X2, X3)) MARK(U72(X1, X2)) -> ACTIVE(U72(mark(X1), X2)) ACTIVE(length(cons(N, L))) -> MARK(U71(isNatList(L), L, N)) The TRS R consists of the following rules: active(zeros) -> mark(cons(0, zeros)) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatIList(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, L, N)) -> mark(U72(isNat(N), L)) active(U72(tt, L)) -> mark(s(length(L))) active(U81(tt)) -> mark(nil) active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N)) active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N)) active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(isNat(0)) -> mark(tt) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2)) active(length(nil)) -> mark(0) active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N)) active(take(0, IL)) -> mark(U81(isNatIList(IL))) active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N)) mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) mark(0) -> active(0) mark(U11(X)) -> active(U11(mark(X))) mark(tt) -> active(tt) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1, X2)) -> active(U51(mark(X1), X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil) -> active(nil) mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4)) mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4)) mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4)) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) cons(mark(X1), X2) -> cons(X1, X2) cons(X1, mark(X2)) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1), X2) -> U41(X1, X2) U41(X1, mark(X2)) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1), X2) -> U51(X1, X2) U51(X1, mark(X2)) -> U51(X1, X2) U51(active(X1), X2) -> U51(X1, X2) U51(X1, active(X2)) -> U51(X1, X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4) U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4) U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4) U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4) U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4) U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4) take(mark(X1), X2) -> take(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (178) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. ACTIVE(U71(tt, L, N)) -> MARK(U72(isNat(N), L)) ACTIVE(U72(tt, L)) -> MARK(s(length(L))) ACTIVE(length(cons(N, L))) -> MARK(U71(isNatList(L), L, N)) The remaining pairs can at least be oriented weakly. Used ordering: Combined order from the following AFS and order. ACTIVE(x1) = x1 U71(x1, x2, x3) = U71 MARK(x1) = x1 U72(x1, x2) = U72 s(x1) = s length(x1) = length Knuth-Bendix order [KBO] with precedence:trivial and weight map: s=1 U72=2 length=4 U71=3 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: U72(X1, mark(X2)) -> U72(X1, X2) U72(mark(X1), X2) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) s(active(X)) -> s(X) s(mark(X)) -> s(X) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) ---------------------------------------- (179) Obligation: Q DP problem: The TRS P consists of the following rules: MARK(U71(X1, X2, X3)) -> ACTIVE(U71(mark(X1), X2, X3)) MARK(U72(X1, X2)) -> ACTIVE(U72(mark(X1), X2)) The TRS R consists of the following rules: active(zeros) -> mark(cons(0, zeros)) active(U11(tt)) -> mark(tt) active(U21(tt)) -> mark(tt) active(U31(tt)) -> mark(tt) active(U41(tt, V2)) -> mark(U42(isNatIList(V2))) active(U42(tt)) -> mark(tt) active(U51(tt, V2)) -> mark(U52(isNatList(V2))) active(U52(tt)) -> mark(tt) active(U61(tt, V2)) -> mark(U62(isNatIList(V2))) active(U62(tt)) -> mark(tt) active(U71(tt, L, N)) -> mark(U72(isNat(N), L)) active(U72(tt, L)) -> mark(s(length(L))) active(U81(tt)) -> mark(nil) active(U91(tt, IL, M, N)) -> mark(U92(isNat(M), IL, M, N)) active(U92(tt, IL, M, N)) -> mark(U93(isNat(N), IL, M, N)) active(U93(tt, IL, M, N)) -> mark(cons(N, take(M, IL))) active(isNat(0)) -> mark(tt) active(isNat(length(V1))) -> mark(U11(isNatList(V1))) active(isNat(s(V1))) -> mark(U21(isNat(V1))) active(isNatIList(V)) -> mark(U31(isNatList(V))) active(isNatIList(zeros)) -> mark(tt) active(isNatIList(cons(V1, V2))) -> mark(U41(isNat(V1), V2)) active(isNatList(nil)) -> mark(tt) active(isNatList(cons(V1, V2))) -> mark(U51(isNat(V1), V2)) active(isNatList(take(V1, V2))) -> mark(U61(isNat(V1), V2)) active(length(nil)) -> mark(0) active(length(cons(N, L))) -> mark(U71(isNatList(L), L, N)) active(take(0, IL)) -> mark(U81(isNatIList(IL))) active(take(s(M), cons(N, IL))) -> mark(U91(isNatIList(IL), IL, M, N)) mark(zeros) -> active(zeros) mark(cons(X1, X2)) -> active(cons(mark(X1), X2)) mark(0) -> active(0) mark(U11(X)) -> active(U11(mark(X))) mark(tt) -> active(tt) mark(U21(X)) -> active(U21(mark(X))) mark(U31(X)) -> active(U31(mark(X))) mark(U41(X1, X2)) -> active(U41(mark(X1), X2)) mark(U42(X)) -> active(U42(mark(X))) mark(isNatIList(X)) -> active(isNatIList(X)) mark(U51(X1, X2)) -> active(U51(mark(X1), X2)) mark(U52(X)) -> active(U52(mark(X))) mark(isNatList(X)) -> active(isNatList(X)) mark(U61(X1, X2)) -> active(U61(mark(X1), X2)) mark(U62(X)) -> active(U62(mark(X))) mark(U71(X1, X2, X3)) -> active(U71(mark(X1), X2, X3)) mark(U72(X1, X2)) -> active(U72(mark(X1), X2)) mark(isNat(X)) -> active(isNat(X)) mark(s(X)) -> active(s(mark(X))) mark(length(X)) -> active(length(mark(X))) mark(U81(X)) -> active(U81(mark(X))) mark(nil) -> active(nil) mark(U91(X1, X2, X3, X4)) -> active(U91(mark(X1), X2, X3, X4)) mark(U92(X1, X2, X3, X4)) -> active(U92(mark(X1), X2, X3, X4)) mark(U93(X1, X2, X3, X4)) -> active(U93(mark(X1), X2, X3, X4)) mark(take(X1, X2)) -> active(take(mark(X1), mark(X2))) cons(mark(X1), X2) -> cons(X1, X2) cons(X1, mark(X2)) -> cons(X1, X2) cons(active(X1), X2) -> cons(X1, X2) cons(X1, active(X2)) -> cons(X1, X2) U11(mark(X)) -> U11(X) U11(active(X)) -> U11(X) U21(mark(X)) -> U21(X) U21(active(X)) -> U21(X) U31(mark(X)) -> U31(X) U31(active(X)) -> U31(X) U41(mark(X1), X2) -> U41(X1, X2) U41(X1, mark(X2)) -> U41(X1, X2) U41(active(X1), X2) -> U41(X1, X2) U41(X1, active(X2)) -> U41(X1, X2) U42(mark(X)) -> U42(X) U42(active(X)) -> U42(X) isNatIList(mark(X)) -> isNatIList(X) isNatIList(active(X)) -> isNatIList(X) U51(mark(X1), X2) -> U51(X1, X2) U51(X1, mark(X2)) -> U51(X1, X2) U51(active(X1), X2) -> U51(X1, X2) U51(X1, active(X2)) -> U51(X1, X2) U52(mark(X)) -> U52(X) U52(active(X)) -> U52(X) isNatList(mark(X)) -> isNatList(X) isNatList(active(X)) -> isNatList(X) U61(mark(X1), X2) -> U61(X1, X2) U61(X1, mark(X2)) -> U61(X1, X2) U61(active(X1), X2) -> U61(X1, X2) U61(X1, active(X2)) -> U61(X1, X2) U62(mark(X)) -> U62(X) U62(active(X)) -> U62(X) U71(mark(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, mark(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, mark(X3)) -> U71(X1, X2, X3) U71(active(X1), X2, X3) -> U71(X1, X2, X3) U71(X1, active(X2), X3) -> U71(X1, X2, X3) U71(X1, X2, active(X3)) -> U71(X1, X2, X3) U72(mark(X1), X2) -> U72(X1, X2) U72(X1, mark(X2)) -> U72(X1, X2) U72(active(X1), X2) -> U72(X1, X2) U72(X1, active(X2)) -> U72(X1, X2) isNat(mark(X)) -> isNat(X) isNat(active(X)) -> isNat(X) s(mark(X)) -> s(X) s(active(X)) -> s(X) length(mark(X)) -> length(X) length(active(X)) -> length(X) U81(mark(X)) -> U81(X) U81(active(X)) -> U81(X) U91(mark(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, mark(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, mark(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, mark(X4)) -> U91(X1, X2, X3, X4) U91(active(X1), X2, X3, X4) -> U91(X1, X2, X3, X4) U91(X1, active(X2), X3, X4) -> U91(X1, X2, X3, X4) U91(X1, X2, active(X3), X4) -> U91(X1, X2, X3, X4) U91(X1, X2, X3, active(X4)) -> U91(X1, X2, X3, X4) U92(mark(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, mark(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, mark(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, mark(X4)) -> U92(X1, X2, X3, X4) U92(active(X1), X2, X3, X4) -> U92(X1, X2, X3, X4) U92(X1, active(X2), X3, X4) -> U92(X1, X2, X3, X4) U92(X1, X2, active(X3), X4) -> U92(X1, X2, X3, X4) U92(X1, X2, X3, active(X4)) -> U92(X1, X2, X3, X4) U93(mark(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, mark(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, mark(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, mark(X4)) -> U93(X1, X2, X3, X4) U93(active(X1), X2, X3, X4) -> U93(X1, X2, X3, X4) U93(X1, active(X2), X3, X4) -> U93(X1, X2, X3, X4) U93(X1, X2, active(X3), X4) -> U93(X1, X2, X3, X4) U93(X1, X2, X3, active(X4)) -> U93(X1, X2, X3, X4) take(mark(X1), X2) -> take(X1, X2) take(X1, mark(X2)) -> take(X1, X2) take(active(X1), X2) -> take(X1, X2) take(X1, active(X2)) -> take(X1, X2) Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (180) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes. ---------------------------------------- (181) TRUE